doubly_sim.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "london_flows=pd.read_csv(\"data/london_flows.csv\")\n",
    "london_flows=london_flows[london_flows['station_origin']!=london_flows[\"station_destination\"]]\n",
    "london_flows=london_flows[london_flows['distance'] != 0]# drop intra station\n",
    "london_flows=london_flows[london_flows['station_origin'] != 'Battersea Park']\n",
    "london_flows=london_flows[london_flows['station_destination'] != 'Battersea Park']\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "# change column name\n",
    "london_flows=london_flows.rename(columns={\"station_origin\":\"Orig\",\"station_destination\":'Dest','flows':'Total','distance':'Dist','population':'Oi_origpop','jobs':'Dj_destjob'})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a pivot table to monitor flows\n",
    "ldf_submat=pd.pivot_table(london_flows,values =\"Total\", index=\"Orig\", columns = \"Dest\",\n",
    "                            aggfunc=np.sum, margins=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    .dataframe tbody tr th:only-of-type {\n",
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       "    }\n",
       "\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Dest</th>\n",
       "      <th>Abbey Road</th>\n",
       "      <th>Acton Central</th>\n",
       "      <th>Acton Town</th>\n",
       "      <th>Aldgate</th>\n",
       "      <th>Aldgate East</th>\n",
       "      <th>All Saints</th>\n",
       "      <th>Alperton</th>\n",
       "      <th>Amersham</th>\n",
       "      <th>Anerley</th>\n",
       "      <th>Angel</th>\n",
       "      <th>...</th>\n",
       "      <th>Wimbledon</th>\n",
       "      <th>Wimbledon Park</th>\n",
       "      <th>Wood Green</th>\n",
       "      <th>Wood Lane</th>\n",
       "      <th>Wood Street</th>\n",
       "      <th>Woodford</th>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <th>Woodside Park</th>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <th>All</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Orig</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Abbey Road</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>32.0</td>\n",
       "      <td>599</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Central</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1224</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Town</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>35.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>11.0</td>\n",
       "      <td>...</td>\n",
       "      <td>77.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3745</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>17.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>4.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2886</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate East</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>20.0</td>\n",
       "      <td>...</td>\n",
       "      <td>24.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3172</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodford</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>47.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>22.0</td>\n",
       "      <td>...</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>4868</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>530</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodside Park</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>26.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>59.0</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3093</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <td>20.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>7.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>7892</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>All</th>\n",
       "      <td>345.0</td>\n",
       "      <td>750.0</td>\n",
       "      <td>2202.0</td>\n",
       "      <td>7782.0</td>\n",
       "      <td>7932.0</td>\n",
       "      <td>444.0</td>\n",
       "      <td>741.0</td>\n",
       "      <td>256.0</td>\n",
       "      <td>173.0</td>\n",
       "      <td>8103.0</td>\n",
       "      <td>...</td>\n",
       "      <td>6295.0</td>\n",
       "      <td>596.0</td>\n",
       "      <td>2146.0</td>\n",
       "      <td>1578.0</td>\n",
       "      <td>404.0</td>\n",
       "      <td>706.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>745.0</td>\n",
       "      <td>4428.0</td>\n",
       "      <td>1542283</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>399 rows × 399 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "Dest              Abbey Road  Acton Central  Acton Town  Aldgate  \\\n",
       "Orig                                                               \n",
       "Abbey Road               NaN            NaN         NaN      NaN   \n",
       "Acton Central            NaN            NaN         NaN      NaN   \n",
       "Acton Town               NaN            NaN         NaN      3.0   \n",
       "Aldgate                  NaN            NaN         0.0      NaN   \n",
       "Aldgate East             NaN            NaN         2.0      0.0   \n",
       "...                      ...            ...         ...      ...   \n",
       "Woodford                 NaN            NaN         2.0      5.0   \n",
       "Woodgrange Park          NaN            0.0         NaN      NaN   \n",
       "Woodside Park            NaN            NaN         1.0     26.0   \n",
       "Woolwich Arsenal        20.0            NaN         NaN      NaN   \n",
       "All                    345.0          750.0      2202.0   7782.0   \n",
       "\n",
       "Dest              Aldgate East  All Saints  Alperton  Amersham  Anerley  \\\n",
       "Orig                                                                      \n",
       "Abbey Road                 NaN         NaN       NaN       NaN      NaN   \n",
       "Acton Central              NaN         NaN       NaN       NaN      NaN   \n",
       "Acton Town                17.0         NaN      35.0       0.0      NaN   \n",
       "Aldgate                    0.0         NaN       NaN       0.0      NaN   \n",
       "Aldgate East               NaN         NaN       0.0       0.0      NaN   \n",
       "...                        ...         ...       ...       ...      ...   \n",
       "Woodford                  47.0         NaN       NaN       NaN      NaN   \n",
       "Woodgrange Park            NaN         NaN       NaN       NaN      NaN   \n",
       "Woodside Park             11.0         NaN       0.0       NaN      NaN   \n",
       "Woolwich Arsenal           NaN         7.0       NaN       NaN      NaN   \n",
       "All                     7932.0       444.0     741.0     256.0    173.0   \n",
       "\n",
       "Dest               Angel  ...  Wimbledon  Wimbledon Park  Wood Green  \\\n",
       "Orig                      ...                                          \n",
       "Abbey Road           NaN  ...        NaN             NaN         NaN   \n",
       "Acton Central        NaN  ...        NaN             NaN         NaN   \n",
       "Acton Town          11.0  ...       77.0             3.0         6.0   \n",
       "Aldgate             17.0  ...        0.0             NaN         4.0   \n",
       "Aldgate East        20.0  ...       24.0             0.0         0.0   \n",
       "...                  ...  ...        ...             ...         ...   \n",
       "Woodford            22.0  ...        2.0             NaN         1.0   \n",
       "Woodgrange Park      NaN  ...        NaN             NaN         NaN   \n",
       "Woodside Park       59.0  ...        0.0             NaN         0.0   \n",
       "Woolwich Arsenal     NaN  ...        NaN             NaN         NaN   \n",
       "All               8103.0  ...     6295.0           596.0      2146.0   \n",
       "\n",
       "Dest              Wood Lane  Wood Street  Woodford  Woodgrange Park  \\\n",
       "Orig                                                                  \n",
       "Abbey Road              NaN          NaN       NaN              NaN   \n",
       "Acton Central           NaN          NaN       NaN              0.0   \n",
       "Acton Town              9.0          NaN       0.0              NaN   \n",
       "Aldgate                 8.0          NaN       0.0              NaN   \n",
       "Aldgate East           12.0          NaN       1.0              NaN   \n",
       "...                     ...          ...       ...              ...   \n",
       "Woodford                NaN          NaN       NaN              NaN   \n",
       "Woodgrange Park         NaN          NaN       NaN              NaN   \n",
       "Woodside Park           NaN          NaN       NaN              NaN   \n",
       "Woolwich Arsenal        NaN          NaN       NaN              NaN   \n",
       "All                  1578.0        404.0     706.0            242.0   \n",
       "\n",
       "Dest              Woodside Park  Woolwich Arsenal      All  \n",
       "Orig                                                        \n",
       "Abbey Road                  NaN              32.0      599  \n",
       "Acton Central               NaN               NaN     1224  \n",
       "Acton Town                  0.0               NaN     3745  \n",
       "Aldgate                     0.0               NaN     2886  \n",
       "Aldgate East                1.0               NaN     3172  \n",
       "...                         ...               ...      ...  \n",
       "Woodford                    NaN               NaN     4868  \n",
       "Woodgrange Park             NaN               NaN      530  \n",
       "Woodside Park               NaN               NaN     3093  \n",
       "Woolwich Arsenal            NaN               NaN     7892  \n",
       "All                       745.0            4428.0  1542283  \n",
       "\n",
       "[399 rows x 399 columns]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ldf_submat"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Evalutation of model\n",
    "Goodness of fit\n",
    "*coefficient of determination ($r^2$)\n",
    " $r^2$ is popular as it is quite intuitive and can be compared across models.\n",
    "*the Square Root of Mean Squared Error (RMSE).\n",
    "Less intuitive but can be used to comparing changes to the same model.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.stats\n",
    "\n",
    "def CalcRSqaured(observed, estimated):\n",
    "    \"\"\"Calculate the r^2 from a series of observed and estimated target values\n",
    "    inputs:\n",
    "    Observed: Series of actual observed values\n",
    "    estimated: Series of predicted values\"\"\"\n",
    "    \n",
    "    r, p = scipy.stats.pearsonr(observed, estimated)\n",
    "    R2 = r **2\n",
    "    \n",
    "    return R2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "\n",
    "def CalcRMSE(observed, estimated):\n",
    "    \"\"\"Calculate Root Mean Square Error between a series of observed and estimated values\n",
    "    inputs:\n",
    "    Observed: Series of actual observed values\n",
    "    estimated: Series of predicted values\"\"\"\n",
    "    \n",
    "    res = (observed -estimated)**2\n",
    "    RMSE = round(sqrt(res.mean()), 3)\n",
    "    \n",
    "    return RMSE\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "# take log of data\n",
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf\n",
    "\n",
    "#take the variables and produce logarithms of them\n",
    "x_variables = [\"Oi_origpop\", \"Dj_destjob\", \"Dist\"]\n",
    "log_x_vars = []\n",
    "for x in x_variables:\n",
    "    london_flows[f\"log_{x}\"] = np.log(london_flows[x])\n",
    "    log_x_vars.append(f\"log_{x}\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "london_flows.head()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Doubly Constrained Model\n",
    "\n",
    "For the doubly constrained model:\n",
    "\n",
    "- We might be interested in understanding the changing patterns of commuting or migration over time. Data from the Census allows us to know an accurate snap-shot of migrating and commuting patterns every 10 years. In these full data matrices, we know both the numbers of commuters/migrants leaving origins and arriving at destinations as well as the interactions between them. If we constrain our model estimates to this known information at origin and destination, we can examine various things, including:\n",
    "    - The ways that the patterns of commuting/migration differ from the model predictions - where we might get more migrant/commuter flows than we would expect\n",
    "    - How the model parameters vary over time - for example how does distance / cost of travel affect flows over time? Are people prepared to travel further or less far than before?\n",
    "\n",
    "\n",
    "Which we can now introduce.\n",
    "\n",
    "Let us begin with the formula:\n",
    "\n",
    "\\begin{equation} \\tag{9}\n",
    "T_{ij} = A_i B_j O_i D_j d_{ij}^{-\\beta}\n",
    "\\end{equation}\n",
    "\n",
    "Where\n",
    "\n",
    "\\begin{equation} \\tag{10}\n",
    "O_i = \\sum_j T_{ij}\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation} \\tag{11}\n",
    "D_j = \\sum_i T_{ij} \n",
    "\\end{equation}\n",
    "\n",
    "and\n",
    "\n",
    "\\begin{equation} \\tag{12}\n",
    "A_i = \\frac{1}{\\sum_j B_j D_j d_{ij}^{-\\beta}}\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation} \\tag{13}\n",
    "B_j = \\frac{1}{\\sum_i A_i O_i d_{ij}^{-\\beta}}\n",
    "\\end{equation}\n",
    "\n",
    "Now, the astute will have noticed that the calculation of $A_i$ relies on knowing $B_j$ and the calculation of $B_j$ relies on knowing $A_i$. A conundrum!! If I don’t know $A_i$ how can I calcuate $B_j$ and then in turn $A_i$ and then $B_j$ ad infinitum???!!\n",
    "\n",
    "Well, I wrestled with that for a while until I came across [this paper by Martyn Senior](http://journals.sagepub.com/doi/abs/10.1177/030913257900300218) where he sketches out a very useful algorithm for iteratively arriving at values for $A_i$ and $B_j$ by setting each to equal to 1 initially and then continuing to calculate each in turn until the difference between each value is small enough not to matter.\n",
    "\n",
    "We will return to this later, but for now, we will once again used the awesome power of Python to deal with all this difficulty for us!\n",
    "\n",
    "We can run the doubly constrained model in exactly the same way as we ran the singly constrained models:\n",
    "\n",
    "\\begin{equation} \\tag{14}\n",
    "\\lambda_{ij} = \\exp (\\alpha_i + \\gamma_j -\\beta \\ln d_{ij})\n",
    "\\end{equation}\n",
    "\n",
    "now in python:"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model calibration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                 Generalized Linear Model Regression Results                  \n",
      "==============================================================================\n",
      "Dep. Variable:                  Total   No. Observations:                61413\n",
      "Model:                            GLM   Df Residuals:                    60617\n",
      "Model Family:                 Poisson   Df Model:                          795\n",
      "Link Function:                    log   Scale:                          1.0000\n",
      "Method:                          IRLS   Log-Likelihood:            -9.7074e+05\n",
      "Date:                Mon, 01 May 2023   Deviance:                   1.7693e+06\n",
      "Time:                        13:32:04   Pearson chi2:                 2.47e+06\n",
      "No. Iterations:                     8                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "=======================================================================================================\n",
      "                                          coef    std err          z      P>|z|      [0.025      0.975]\n",
      "-------------------------------------------------------------------------------------------------------\n",
      "Dest[Abbey Road]                        7.7407      0.068    113.624      0.000       7.607       7.874\n",
      "Dest[Acton Central]                     9.3149      0.056    166.710      0.000       9.205       9.424\n",
      "Dest[Acton Town]                        9.3644      0.047    198.490      0.000       9.272       9.457\n",
      "Dest[Aldgate]                          10.1768      0.044    233.904      0.000      10.091      10.262\n",
      "Dest[Aldgate East]                     10.1538      0.043    233.519      0.000      10.069      10.239\n",
      "Dest[All Saints]                        8.1657      0.063    128.940      0.000       8.042       8.290\n",
      "Dest[Alperton]                          8.7481      0.056    156.412      0.000       8.638       8.858\n",
      "Dest[Amersham]                          8.5157      0.076    112.733      0.000       8.368       8.664\n",
      "Dest[Anerley]                           7.7917      0.087     89.341      0.000       7.621       7.963\n",
      "Dest[Angel]                            10.2056      0.043    234.713      0.000      10.120      10.291\n",
      "Dest[Archway]                           9.2953      0.047    196.840      0.000       9.203       9.388\n",
      "Dest[Arnos Grove]                       8.5423      0.056    153.095      0.000       8.433       8.652\n",
      "Dest[Arsenal]                           7.0747      0.070    100.697      0.000       6.937       7.212\n",
      "Dest[Baker Street]                     10.7941      0.043    252.154      0.000      10.710      10.878\n",
      "Dest[Balham]                            9.8393      0.045    218.323      0.000       9.751       9.928\n",
      "Dest[Bank and Monument]                12.1944      0.042    289.697      0.000      12.112      12.277\n",
      "Dest[Barbican]                         10.2420      0.043    236.580      0.000      10.157      10.327\n",
      "Dest[Barking]                          10.4204      0.044    237.787      0.000      10.334      10.506\n",
      "Dest[Barkingside]                       7.6979      0.075    102.963      0.000       7.551       7.844\n",
      "Dest[Barons Court]                      9.3979      0.046    204.822      0.000       9.308       9.488\n",
      "Dest[Bayswater]                         8.0400      0.058    138.003      0.000       7.926       8.154\n",
      "Dest[Beckton]                           8.9551      0.063    141.119      0.000       8.831       9.079\n",
      "Dest[Beckton Park]                      7.1675      0.115     62.532      0.000       6.943       7.392\n",
      "Dest[Becontree]                         8.1398      0.063    129.202      0.000       8.016       8.263\n",
      "Dest[Belsize Park]                      9.3472      0.047    198.523      0.000       9.255       9.440\n",
      "Dest[Bermondsey]                        9.3458      0.046    204.826      0.000       9.256       9.435\n",
      "Dest[Bethnal Green]                     9.6059      0.045    215.670      0.000       9.519       9.693\n",
      "Dest[Blackfriars]                      10.4557      0.043    242.526      0.000      10.371      10.540\n",
      "Dest[Blackhorse Road]                   9.4647      0.046    206.043      0.000       9.375       9.555\n",
      "Dest[Blackwall]                         8.4846      0.057    148.924      0.000       8.373       8.596\n",
      "Dest[Bond Street]                      11.0996      0.043    260.619      0.000      11.016      11.183\n",
      "Dest[Borough]                           9.1746      0.045    201.742      0.000       9.086       9.264\n",
      "Dest[Boston Manor]                      8.7798      0.054    163.118      0.000       8.674       8.885\n",
      "Dest[Bounds Green]                      8.4682      0.055    153.088      0.000       8.360       8.577\n",
      "Dest[Bow Church]                        9.0428      0.053    169.720      0.000       8.938       9.147\n",
      "Dest[Bow Road]                          8.4614      0.051    165.843      0.000       8.361       8.561\n",
      "Dest[Brent Cross]                       8.1057      0.063    128.194      0.000       7.982       8.230\n",
      "Dest[Brentwood]                         8.8093      0.056    156.509      0.000       8.699       8.920\n",
      "Dest[Brixton]                          10.0643      0.044    226.725      0.000       9.977      10.151\n",
      "Dest[Brockley]                          8.3210      0.063    132.726      0.000       8.198       8.444\n",
      "Dest[Bromley-by-Bow]                    8.7824      0.049    179.850      0.000       8.687       8.878\n",
      "Dest[Brondesbury]                       8.7723      0.056    155.525      0.000       8.662       8.883\n",
      "Dest[Brondesbury Park]                  9.0775      0.054    168.581      0.000       8.972       9.183\n",
      "Dest[Bruce Grove]                       8.1855      0.068    120.223      0.000       8.052       8.319\n",
      "Dest[Buckhurst Hill]                    7.6461      0.079     97.402      0.000       7.492       7.800\n",
      "Dest[Burnt Oak]                         8.6074      0.058    149.396      0.000       8.494       8.720\n",
      "Dest[Bush Hill Park]                    7.6789      0.087     88.678      0.000       7.509       7.849\n",
      "Dest[Bushey]                            9.2858      0.064    145.561      0.000       9.161       9.411\n",
      "Dest[Caledonian Road]                   8.8058      0.048    182.745      0.000       8.711       8.900\n",
      "Dest[Caledonian Road & Barnsbury]       9.3630      0.049    192.424      0.000       9.268       9.458\n",
      "Dest[Cambridge Heath]                   8.5325      0.063    135.434      0.000       8.409       8.656\n",
      "Dest[Camden Road]                      10.2129      0.045    225.869      0.000      10.124      10.302\n",
      "Dest[Camden Town]                      10.0510      0.044    229.018      0.000       9.965      10.137\n",
      "Dest[Canada Water]                     10.8897      0.043    255.397      0.000      10.806      10.973\n",
      "Dest[Canary Wharf]                     12.1008      0.042    287.040      0.000      12.018      12.183\n",
      "Dest[Canning Town]                     10.8283      0.043    253.891      0.000      10.745      10.912\n",
      "Dest[Cannon Street]                     9.4690      0.044    213.253      0.000       9.382       9.556\n",
      "Dest[Canonbury]                         8.5516      0.054    157.438      0.000       8.445       8.658\n",
      "Dest[Canons Park]                       8.4701      0.061    139.608      0.000       8.351       8.589\n",
      "Dest[Carpenders Park]                   9.1192      0.069    131.808      0.000       8.984       9.255\n",
      "Dest[Chadwell Heath]                    7.7277      0.060    129.184      0.000       7.610       7.845\n",
      "Dest[Chalfont & Latimer]                9.3882      0.060    156.469      0.000       9.271       9.506\n",
      "Dest[Chalk Farm]                        8.7071      0.050    173.327      0.000       8.609       8.806\n",
      "Dest[Chancery Lane]                    10.8010      0.043    252.361      0.000      10.717      10.885\n",
      "Dest[Charing Cross]                     9.8984      0.044    226.780      0.000       9.813       9.984\n",
      "Dest[Chesham]                           8.6965      0.077    113.051      0.000       8.546       8.847\n",
      "Dest[Cheshunt]                          7.9942      0.103     77.942      0.000       7.793       8.195\n",
      "Dest[Chigwell]                          7.5496      0.092     81.734      0.000       7.369       7.731\n",
      "Dest[Chingford]                         8.4020      0.078    108.310      0.000       8.250       8.554\n",
      "Dest[Chiswick Park]                     9.0365      0.049    183.317      0.000       8.940       9.133\n",
      "Dest[Chorleywood]                       9.5788      0.057    168.879      0.000       9.468       9.690\n",
      "Dest[Clapham Common]                    9.0412      0.048    190.067      0.000       8.948       9.134\n",
      "Dest[Clapham High Street]               9.7569      0.051    192.056      0.000       9.657       9.856\n",
      "Dest[Clapham Junction]                 11.0923      0.045    248.191      0.000      11.005      11.180\n",
      "Dest[Clapham North]                     8.5025      0.051    167.622      0.000       8.403       8.602\n",
      "Dest[Clapham South]                     8.8501      0.049    179.551      0.000       8.753       8.947\n",
      "Dest[Clapton]                           8.5749      0.064    133.531      0.000       8.449       8.701\n",
      "Dest[Cockfosters]                       7.9285      0.075    105.928      0.000       7.782       8.075\n",
      "Dest[Colindale]                         9.6894      0.048    203.766      0.000       9.596       9.783\n",
      "Dest[Colliers Wood]                     8.5231      0.054    157.565      0.000       8.417       8.629\n",
      "Dest[Covent Garden]                     9.4837      0.045    212.336      0.000       9.396       9.571\n",
      "Dest[Crossharbour]                      9.2392      0.051    181.832      0.000       9.140       9.339\n",
      "Dest[Crouch Hill]                       8.8347      0.065    135.085      0.000       8.706       8.963\n",
      "Dest[Croxley]                           8.6108      0.069    125.214      0.000       8.476       8.746\n",
      "Dest[Crystal Palace]                    9.1422      0.062    147.673      0.000       9.021       9.264\n",
      "Dest[Custom House]                      8.8922      0.054    164.326      0.000       8.786       8.998\n",
      "Dest[Cutty Sark]                        9.7691      0.048    202.241      0.000       9.674       9.864\n",
      "Dest[Cyprus]                            9.3200      0.054    171.950      0.000       9.214       9.426\n",
      "Dest[Dagenham East]                     8.6239      0.058    149.703      0.000       8.511       8.737\n",
      "Dest[Dagenham Heathway]                 9.0400      0.052    174.593      0.000       8.939       9.141\n",
      "Dest[Dalston Junction]                  9.3372      0.051    184.719      0.000       9.238       9.436\n",
      "Dest[Dalston Kingsland]                 9.6843      0.047    204.197      0.000       9.591       9.777\n",
      "Dest[Debden]                            9.1276      0.054    168.372      0.000       9.021       9.234\n",
      "Dest[Denmark Hill]                     10.1989      0.047    218.492      0.000      10.107      10.290\n",
      "Dest[Deptford Bridge]                   9.0028      0.054    165.354      0.000       8.896       9.110\n",
      "Dest[Devons Road]                       8.5796      0.058    147.558      0.000       8.466       8.694\n",
      "Dest[Dollis Hill]                       8.4710      0.054    155.899      0.000       8.365       8.578\n",
      "Dest[Ealing Broadway]                  10.3179      0.045    231.627      0.000      10.231      10.405\n",
      "Dest[Ealing Common]                     8.6003      0.053    160.760      0.000       8.495       8.705\n",
      "Dest[Earl's Court]                     10.0944      0.044    230.514      0.000      10.009      10.180\n",
      "Dest[East Acton]                        9.0449      0.049    183.121      0.000       8.948       9.142\n",
      "Dest[East Finchley]                     8.9753      0.051    176.851      0.000       8.876       9.075\n",
      "Dest[East Ham]                          9.2898      0.048    195.572      0.000       9.197       9.383\n",
      "Dest[East India]                        9.3126      0.049    190.220      0.000       9.217       9.409\n",
      "Dest[East Putney]                       9.2722      0.048    193.018      0.000       9.178       9.366\n",
      "Dest[Eastcote]                          8.0381      0.067    119.526      0.000       7.906       8.170\n",
      "Dest[Edgware]                           8.8979      0.055    160.331      0.000       8.789       9.007\n",
      "Dest[Edgware Road]                     10.0230      0.044    229.366      0.000       9.937      10.109\n",
      "Dest[Edmonton Green]                    8.8759      0.055    160.076      0.000       8.767       8.985\n",
      "Dest[Elephant & Castle]                 9.9084      0.044    226.099      0.000       9.823       9.994\n",
      "Dest[Elm Park]                          7.9462      0.074    107.150      0.000       7.801       8.092\n",
      "Dest[Elverson Road]                     7.3829      0.079     93.725      0.000       7.229       7.537\n",
      "Dest[Embankment]                       10.2016      0.043    236.678      0.000      10.117      10.286\n",
      "Dest[Emerson Park]                      6.5646      0.161     40.887      0.000       6.250       6.879\n",
      "Dest[Enfield Town]                      9.2567      0.056    164.914      0.000       9.147       9.367\n",
      "Dest[Epping]                            8.8542      0.062    143.604      0.000       8.733       8.975\n",
      "Dest[Euston]                           10.8771      0.043    254.639      0.000      10.793      10.961\n",
      "Dest[Euston Square]                    10.3494      0.043    238.097      0.000      10.264      10.435\n",
      "Dest[Fairlop]                           7.4289      0.085     87.146      0.000       7.262       7.596\n",
      "Dest[Farringdon]                       11.3264      0.042    266.534      0.000      11.243      11.410\n",
      "Dest[Finchley Central]                  9.5209      0.048    198.563      0.000       9.427       9.615\n",
      "Dest[Finchley Road]                     9.5967      0.045    211.402      0.000       9.508       9.686\n",
      "Dest[Finchley Road & Frognal]           9.4419      0.049    191.685      0.000       9.345       9.538\n",
      "Dest[Finsbury Park]                    10.2044      0.044    233.145      0.000      10.119      10.290\n",
      "Dest[Forest Gate]                       8.3562      0.055    152.421      0.000       8.249       8.464\n",
      "Dest[Forest Hill]                       8.6304      0.059    146.854      0.000       8.515       8.746\n",
      "Dest[Fulham Broadway]                   9.7670      0.045    217.996      0.000       9.679       9.855\n",
      "Dest[Gallions Reach]                    9.0340      0.060    151.104      0.000       8.917       9.151\n",
      "Dest[Gants Hill]                        8.3276      0.059    141.983      0.000       8.213       8.443\n",
      "Dest[Gidea Park]                        7.1011      0.072     98.917      0.000       6.960       7.242\n",
      "Dest[Gloucester Road]                   9.8258      0.044    221.983      0.000       9.739       9.913\n",
      "Dest[Golders Green]                     9.4217      0.048    195.891      0.000       9.327       9.516\n",
      "Dest[Goldhawk Road]                     7.6038      0.067    113.717      0.000       7.473       7.735\n",
      "Dest[Goodge Street]                     9.9759      0.044    227.625      0.000       9.890      10.062\n",
      "Dest[Goodmayes]                         7.6995      0.057    134.873      0.000       7.588       7.811\n",
      "Dest[Gospel Oak]                        9.6901      0.048    203.152      0.000       9.597       9.784\n",
      "Dest[Grange Hill]                       6.7852      0.122     55.528      0.000       6.546       7.025\n",
      "Dest[Great Portland Street]            10.1041      0.044    231.199      0.000      10.018      10.190\n",
      "Dest[Green Park]                       11.2897      0.042    265.958      0.000      11.207      11.373\n",
      "Dest[Greenford]                         9.1819      0.051    178.476      0.000       9.081       9.283\n",
      "Dest[Greenwich]                         9.3164      0.051    182.294      0.000       9.216       9.417\n",
      "Dest[Gunnersbury]                       9.9713      0.045    223.692      0.000       9.884      10.059\n",
      "Dest[Hackney Central]                   9.8639      0.047    211.084      0.000       9.772       9.956\n",
      "Dest[Hackney Downs]                     9.0888      0.055    166.254      0.000       8.982       9.196\n",
      "Dest[Hackney Wick]                      8.9651      0.051    177.316      0.000       8.866       9.064\n",
      "Dest[Haggerston]                        9.0797      0.051    177.251      0.000       8.979       9.180\n",
      "Dest[Hainault]                          8.0430      0.068    118.739      0.000       7.910       8.176\n",
      "Dest[Hammersmith]                      11.2739      0.043    263.897      0.000      11.190      11.358\n",
      "Dest[Hampstead]                         9.3349      0.048    195.806      0.000       9.241       9.428\n",
      "Dest[Hampstead Heath]                   9.9726      0.046    215.594      0.000       9.882      10.063\n",
      "Dest[Hanger Lane]                       9.3652      0.050    189.175      0.000       9.268       9.462\n",
      "Dest[Harlesden]                         9.0784      0.049    186.206      0.000       8.983       9.174\n",
      "Dest[Harold Wood]                       7.9740      0.061    131.115      0.000       7.855       8.093\n",
      "Dest[Harringay Green Lanes]             9.0381      0.060    149.852      0.000       8.920       9.156\n",
      "Dest[Harrow & Wealdstone]               9.3005      0.050    184.532      0.000       9.202       9.399\n",
      "Dest[Harrow-on-the-Hill]                9.8918      0.046    215.575      0.000       9.802       9.982\n",
      "Dest[Hatch End]                         7.7421      0.112     68.927      0.000       7.522       7.962\n",
      "Dest[Hatton Cross]                      9.4599      0.051    185.277      0.000       9.360       9.560\n",
      "Dest[Headstone Lane]                    9.0813      0.071    127.943      0.000       8.942       9.220\n",
      "Dest[Heathrow Terminal 4]               8.9510      0.057    157.437      0.000       8.840       9.062\n",
      "Dest[Heathrow Terminal 5]               9.3868      0.053    176.788      0.000       9.283       9.491\n",
      "Dest[Heathrow Terminals 2 & 3]         10.1229      0.047    215.504      0.000      10.031      10.215\n",
      "Dest[Hendon Central]                    9.5052      0.048    197.700      0.000       9.411       9.599\n",
      "Dest[Heron Quays]                      10.6049      0.044    242.108      0.000      10.519      10.691\n",
      "Dest[High Barnet]                       9.1605      0.053    171.955      0.000       9.056       9.265\n",
      "Dest[High Street Kensington]            9.9057      0.044    224.207      0.000       9.819       9.992\n",
      "Dest[Highams Park]                      8.5013      0.066    128.830      0.000       8.372       8.631\n",
      "Dest[Highbury & Islington]             11.1441      0.043    262.075      0.000      11.061      11.227\n",
      "Dest[Highgate]                          8.4938      0.054    156.556      0.000       8.388       8.600\n",
      "Dest[Hillingdon]                        8.1933      0.070    116.737      0.000       8.056       8.331\n",
      "Dest[Holborn]                          11.1229      0.043    261.421      0.000      11.039      11.206\n",
      "Dest[Holland Park]                      9.0562      0.047    190.802      0.000       8.963       9.149\n",
      "Dest[Holloway Road]                     8.7322      0.049    179.913      0.000       8.637       8.827\n",
      "Dest[Homerton]                          9.5043      0.048    199.353      0.000       9.411       9.598\n",
      "Dest[Honor Oak Park]                    8.0283      0.069    116.647      0.000       7.893       8.163\n",
      "Dest[Hornchurch]                        8.6000      0.063    135.971      0.000       8.476       8.724\n",
      "Dest[Hounslow Central]                  8.6327      0.056    152.807      0.000       8.522       8.743\n",
      "Dest[Hounslow East]                     8.7719      0.054    161.106      0.000       8.665       8.879\n",
      "Dest[Hounslow West]                     8.2279      0.065    125.806      0.000       8.100       8.356\n",
      "Dest[Hoxton]                            9.7034      0.047    205.160      0.000       9.611       9.796\n",
      "Dest[Hyde Park Corner]                  9.0086      0.047    192.835      0.000       8.917       9.100\n",
      "Dest[Ickenham]                          8.1289      0.073    111.403      0.000       7.986       8.272\n",
      "Dest[Ilford]                            9.0227      0.049    184.413      0.000       8.927       9.119\n",
      "Dest[Imperial Wharf]                    9.9443      0.048    208.489      0.000       9.851      10.038\n",
      "Dest[Island Gardens]                    8.7730      0.057    154.974      0.000       8.662       8.884\n",
      "Dest[Kennington]                        8.2660      0.051    161.291      0.000       8.166       8.366\n",
      "Dest[Kensal Green]                      7.8853      0.062    126.364      0.000       7.763       8.008\n",
      "Dest[Kensal Rise]                       9.4053      0.050    187.799      0.000       9.307       9.503\n",
      "Dest[Kensington]                        9.3029      0.047    197.541      0.000       9.211       9.395\n",
      "Dest[Kentish Town]                      9.5705      0.045    210.622      0.000       9.481       9.660\n",
      "Dest[Kentish Town West]                 9.0907      0.052    176.026      0.000       8.989       9.192\n",
      "Dest[Kenton]                            9.0182      0.056    162.202      0.000       8.909       9.127\n",
      "Dest[Kew Gardens]                       8.3405      0.055    151.203      0.000       8.232       8.449\n",
      "Dest[Kilburn]                           9.1626      0.048    191.799      0.000       9.069       9.256\n",
      "Dest[Kilburn High Road]                 8.8126      0.075    118.004      0.000       8.666       8.959\n",
      "Dest[Kilburn Park]                      8.4238      0.055    154.136      0.000       8.317       8.531\n",
      "Dest[King George V]                     9.1057      0.055    166.035      0.000       8.998       9.213\n",
      "Dest[King's Cross St. Pancras]         11.5616      0.042    272.893      0.000      11.479      11.645\n",
      "Dest[Kingsbury]                         8.9362      0.053    170.114      0.000       8.833       9.039\n",
      "Dest[Knightsbridge]                    10.0872      0.044    230.479      0.000      10.001      10.173\n",
      "Dest[Ladbroke Grove]                    9.1718      0.047    193.741      0.000       9.079       9.265\n",
      "Dest[Lambeth North]                     8.1948      0.050    163.337      0.000       8.096       8.293\n",
      "Dest[Lancaster Gate]                    8.9334      0.048    185.493      0.000       8.839       9.028\n",
      "Dest[Langdon Park]                      9.7395      0.047    205.340      0.000       9.647       9.832\n",
      "Dest[Latimer Road]                      8.5999      0.052    166.346      0.000       8.499       8.701\n",
      "Dest[Leicester Square]                 10.1449      0.043    234.184      0.000      10.060      10.230\n",
      "Dest[Lewisham]                         10.5471      0.047    225.075      0.000      10.455      10.639\n",
      "Dest[Leyton]                            8.8109      0.050    177.760      0.000       8.714       8.908\n",
      "Dest[Leyton Midland Road]               8.7230      0.064    136.429      0.000       8.598       8.848\n",
      "Dest[Leytonstone]                       9.1805      0.048    190.966      0.000       9.086       9.275\n",
      "Dest[Leytonstone High Road]             8.4207      0.071    118.768      0.000       8.282       8.560\n",
      "Dest[Limehouse]                        10.2421      0.046    223.905      0.000      10.152      10.332\n",
      "Dest[Liverpool Street]                 11.8532      0.042    280.764      0.000      11.770      11.936\n",
      "Dest[London Bridge]                    11.4039      0.042    269.027      0.000      11.321      11.487\n",
      "Dest[London City Airport]               9.9703      0.048    209.316      0.000       9.877      10.064\n",
      "Dest[London Fields]                     8.9357      0.055    161.817      0.000       8.827       9.044\n",
      "Dest[Loughton]                          8.7504      0.058    152.154      0.000       8.638       8.863\n",
      "Dest[Maida Vale]                        8.4119      0.053    158.286      0.000       8.308       8.516\n",
      "Dest[Manor House]                       8.5955      0.050    170.915      0.000       8.497       8.694\n",
      "Dest[Manor Park]                        7.5785      0.064    118.655      0.000       7.453       7.704\n",
      "Dest[Mansion House]                    10.0754      0.043    231.783      0.000       9.990      10.161\n",
      "Dest[Marble Arch]                       9.9688      0.044    226.856      0.000       9.883      10.055\n",
      "Dest[Maryland]                          7.3512      0.068    108.168      0.000       7.218       7.484\n",
      "Dest[Marylebone]                        9.5728      0.045    214.010      0.000       9.485       9.660\n",
      "Dest[Mile End]                          9.5286      0.045    211.423      0.000       9.440       9.617\n",
      "Dest[Mill Hill East]                    8.7137      0.057    152.902      0.000       8.602       8.825\n",
      "Dest[Moor Park]                         9.0522      0.060    150.867      0.000       8.935       9.170\n",
      "Dest[Moorgate]                         11.0987      0.042    261.573      0.000      11.016      11.182\n",
      "Dest[Morden]                            9.2755      0.050    186.583      0.000       9.178       9.373\n",
      "Dest[Mornington Crescent]               9.2532      0.046    201.565      0.000       9.163       9.343\n",
      "Dest[Mudchute]                          7.9008      0.071    110.890      0.000       7.761       8.040\n",
      "Dest[Neasden]                           8.3887      0.056    149.730      0.000       8.279       8.499\n",
      "Dest[New Cross]                         9.6559      0.053    183.669      0.000       9.553       9.759\n",
      "Dest[New Cross Gate]                    9.3137      0.050    184.974      0.000       9.215       9.412\n",
      "Dest[Newbury Park]                      8.3769      0.060    140.272      0.000       8.260       8.494\n",
      "Dest[North Acton]                       9.8312      0.046    214.975      0.000       9.742       9.921\n",
      "Dest[North Ealing]                      8.1395      0.069    117.640      0.000       8.004       8.275\n",
      "Dest[North Greenwich]                  10.2631      0.044    233.627      0.000      10.177      10.349\n",
      "Dest[North Harrow]                      7.4304      0.084     88.575      0.000       7.266       7.595\n",
      "Dest[North Wembley]                     7.9858      0.067    118.734      0.000       7.854       8.118\n",
      "Dest[Northfields]                       8.6651      0.054    160.030      0.000       8.559       8.771\n",
      "Dest[Northolt]                          8.7313      0.057    152.139      0.000       8.619       8.844\n",
      "Dest[Northwick Park]                    9.3030      0.049    191.680      0.000       9.208       9.398\n",
      "Dest[Northwood]                         9.2917      0.053    176.694      0.000       9.189       9.395\n",
      "Dest[Northwood Hills]                   8.5266      0.062    137.807      0.000       8.405       8.648\n",
      "Dest[Norwood Junction]                  8.7268      0.066    133.066      0.000       8.598       8.855\n",
      "Dest[Notting Hill Gate]                 9.9187      0.044    224.644      0.000       9.832      10.005\n",
      "Dest[Oakwood]                           8.2711      0.064    128.442      0.000       8.145       8.397\n",
      "Dest[Old Street]                       10.8587      0.043    253.854      0.000      10.775      10.943\n",
      "Dest[Osterley]                          8.3714      0.061    137.662      0.000       8.252       8.491\n",
      "Dest[Oval]                              9.1287      0.046    196.422      0.000       9.038       9.220\n",
      "Dest[Oxford Circus]                    11.8001      0.042    279.093      0.000      11.717      11.883\n",
      "Dest[Paddington]                       11.2441      0.043    263.674      0.000      11.160      11.328\n",
      "Dest[Park Royal]                        9.3842      0.049    190.770      0.000       9.288       9.481\n",
      "Dest[Parsons Green]                     9.4296      0.046    203.846      0.000       9.339       9.520\n",
      "Dest[Peckham Rye]                       9.2654      0.052    177.293      0.000       9.163       9.368\n",
      "Dest[Penge West]                        7.2630      0.092     78.752      0.000       7.082       7.444\n",
      "Dest[Perivale]                          8.4244      0.061    139.233      0.000       8.306       8.543\n",
      "Dest[Piccadilly Circus]                10.5880      0.043    247.012      0.000      10.504      10.672\n",
      "Dest[Pimlico]                           9.8404      0.044    223.173      0.000       9.754       9.927\n",
      "Dest[Pinner]                            7.9202      0.072    109.798      0.000       7.779       8.062\n",
      "Dest[Plaistow]                          8.4097      0.052    161.357      0.000       8.308       8.512\n",
      "Dest[Pontoon Dock]                      9.3179      0.051    181.392      0.000       9.217       9.419\n",
      "Dest[Poplar]                            9.0590      0.050    180.785      0.000       8.961       9.157\n",
      "Dest[Preston Road]                      7.9098      0.065    121.974      0.000       7.783       8.037\n",
      "Dest[Prince Regent]                     8.9252      0.055    160.984      0.000       8.817       9.034\n",
      "Dest[Pudding Mill Lane]                 8.0571      0.064    126.339      0.000       7.932       8.182\n",
      "Dest[Putney Bridge]                     9.4216      0.047    202.117      0.000       9.330       9.513\n",
      "Dest[Queen's Park]                      9.1464      0.047    193.412      0.000       9.054       9.239\n",
      "Dest[Queens Road Peckham]               8.9521      0.056    159.975      0.000       8.842       9.062\n",
      "Dest[Queensbury]                        8.1737      0.064    128.340      0.000       8.049       8.298\n",
      "Dest[Queensway]                         8.9782      0.048    187.207      0.000       8.884       9.072\n",
      "Dest[Ravenscourt Park]                  9.0629      0.048    189.181      0.000       8.969       9.157\n",
      "Dest[Rayners Lane]                      8.0569      0.064    125.504      0.000       7.931       8.183\n",
      "Dest[Rectory Road]                      7.7036      0.080     96.630      0.000       7.547       7.860\n",
      "Dest[Redbridge]                         7.2072      0.083     87.265      0.000       7.045       7.369\n",
      "Dest[Regent's Park]                     9.3973      0.045    206.841      0.000       9.308       9.486\n",
      "Dest[Richmond]                         10.3595      0.044    233.357      0.000      10.272      10.446\n",
      "Dest[Rickmansworth]                     9.4826      0.053    177.416      0.000       9.378       9.587\n",
      "Dest[Roding Valley]                     6.9739      0.117     59.637      0.000       6.745       7.203\n",
      "Dest[Romford]                           9.1466      0.048    191.487      0.000       9.053       9.240\n",
      "Dest[Rotherhithe]                       7.4780      0.070    107.092      0.000       7.341       7.615\n",
      "Dest[Royal Albert]                      8.9904      0.057    158.580      0.000       8.879       9.102\n",
      "Dest[Royal Oak]                         8.1985      0.054    151.310      0.000       8.092       8.305\n",
      "Dest[Royal Victoria]                    8.7393      0.055    157.840      0.000       8.631       8.848\n",
      "Dest[Ruislip]                           7.8457      0.075    104.224      0.000       7.698       7.993\n",
      "Dest[Ruislip Gardens]                   7.7801      0.088     88.236      0.000       7.607       7.953\n",
      "Dest[Ruislip Manor]                     8.2224      0.065    126.498      0.000       8.095       8.350\n",
      "Dest[Russell Square]                    9.6716      0.044    218.553      0.000       9.585       9.758\n",
      "Dest[Seven Kings]                       7.5634      0.060    126.366      0.000       7.446       7.681\n",
      "Dest[Seven Sisters]                    10.5261      0.043    243.562      0.000      10.441      10.611\n",
      "Dest[Shadwell]                         10.4161      0.044    239.330      0.000      10.331      10.501\n",
      "Dest[Shenfield]                         8.9879      0.054    166.718      0.000       8.882       9.094\n",
      "Dest[Shepherd's Bush]                  10.4819      0.043    242.147      0.000      10.397      10.567\n",
      "Dest[Shepherd's Bush Market]            7.6954      0.065    119.118      0.000       7.569       7.822\n",
      "Dest[Shoreditch High Street]           10.9660      0.044    251.029      0.000      10.880      11.052\n",
      "Dest[Silver Street]                     9.1656      0.053    171.744      0.000       9.061       9.270\n",
      "Dest[Sloane Square]                    10.2325      0.043    235.477      0.000      10.147      10.318\n",
      "Dest[Snaresbrook]                       8.0898      0.061    132.603      0.000       7.970       8.209\n",
      "Dest[South Acton]                       9.6252      0.053    181.364      0.000       9.521       9.729\n",
      "Dest[South Ealing]                      8.3588      0.057    146.282      0.000       8.247       8.471\n",
      "Dest[South Hampstead]                   8.8908      0.078    114.152      0.000       8.738       9.043\n",
      "Dest[South Harrow]                      8.1810      0.066    123.886      0.000       8.052       8.310\n",
      "Dest[South Kensington]                 10.9022      0.043    254.644      0.000      10.818      10.986\n",
      "Dest[South Kenton]                      7.9494      0.075    106.068      0.000       7.803       8.096\n",
      "Dest[South Quay]                       10.5191      0.044    237.107      0.000      10.432      10.606\n",
      "Dest[South Ruislip]                     8.1505      0.072    112.528      0.000       8.009       8.292\n",
      "Dest[South Tottenham]                   9.2642      0.060    153.947      0.000       9.146       9.382\n",
      "Dest[South Wimbledon]                   9.0902      0.050    181.618      0.000       8.992       9.188\n",
      "Dest[South Woodford]                    8.5002      0.055    153.208      0.000       8.391       8.609\n",
      "Dest[Southbury]                         9.1864      0.061    151.505      0.000       9.068       9.305\n",
      "Dest[Southfields]                       9.0676      0.050    180.719      0.000       8.969       9.166\n",
      "Dest[Southgate]                         8.7863      0.054    162.446      0.000       8.680       8.892\n",
      "Dest[Southwark]                        10.2246      0.043    237.306      0.000      10.140      10.309\n",
      "Dest[St James Street]                   7.4935      0.085     88.318      0.000       7.327       7.660\n",
      "Dest[St. James's Park]                 10.5603      0.043    246.366      0.000      10.476      10.644\n",
      "Dest[St. John's Wood]                   9.7110      0.045    215.972      0.000       9.623       9.799\n",
      "Dest[St. Paul's]                       10.6262      0.043    247.758      0.000      10.542      10.710\n",
      "Dest[Stamford Brook]                    8.3166      0.055    150.736      0.000       8.208       8.425\n",
      "Dest[Stamford Hill]                     8.1983      0.068    120.689      0.000       8.065       8.331\n",
      "Dest[Stanmore]                          9.1403      0.053    172.688      0.000       9.037       9.244\n",
      "Dest[Star Lane]                         8.9130      0.052    172.626      0.000       8.812       9.014\n",
      "Dest[Stepney Green]                     8.8272      0.048    184.132      0.000       8.733       8.921\n",
      "Dest[Stockwell]                         8.6856      0.049    177.980      0.000       8.590       8.781\n",
      "Dest[Stoke Newington]                   8.4187      0.062    135.315      0.000       8.297       8.541\n",
      "Dest[Stonebridge Park]                  8.3826      0.056    149.287      0.000       8.273       8.493\n",
      "Dest[Stratford]                        11.6810      0.042    277.366      0.000      11.598      11.764\n",
      "Dest[Stratford High Street]             8.2815      0.054    152.362      0.000       8.175       8.388\n",
      "Dest[Stratford International]           9.5151      0.049    195.335      0.000       9.420       9.611\n",
      "Dest[Sudbury Hill]                      7.8074      0.073    106.364      0.000       7.664       7.951\n",
      "Dest[Sudbury Town]                      7.6838      0.080     96.300      0.000       7.527       7.840\n",
      "Dest[Surrey Quays]                      8.6560      0.053    164.760      0.000       8.553       8.759\n",
      "Dest[Swiss Cottage]                     9.3322      0.046    201.064      0.000       9.241       9.423\n",
      "Dest[Sydenham]                          8.4406      0.061    137.671      0.000       8.320       8.561\n",
      "Dest[Temple]                            9.9550      0.044    227.634      0.000       9.869      10.041\n",
      "Dest[Theobalds Grove]                   8.3717      0.086     97.580      0.000       8.204       8.540\n",
      "Dest[Theydon Bois]                      7.3979      0.109     67.565      0.000       7.183       7.613\n",
      "Dest[Tooting Bec]                       8.7042      0.051    171.578      0.000       8.605       8.804\n",
      "Dest[Tooting Broadway]                 10.0715      0.045    225.627      0.000       9.984      10.159\n",
      "Dest[Tottenham Court Road]             10.9600      0.043    257.078      0.000      10.876      11.044\n",
      "Dest[Tottenham Hale]                    9.6450      0.046    211.734      0.000       9.556       9.734\n",
      "Dest[Totteridge & Whetstone]            7.8605      0.073    108.324      0.000       7.718       8.003\n",
      "Dest[Tower Gateway]                    10.6434      0.045    234.015      0.000      10.554      10.733\n",
      "Dest[Tower Hill]                       10.1496      0.043    233.772      0.000      10.064      10.235\n",
      "Dest[Tufnell Park]                      7.8874      0.060    132.069      0.000       7.770       8.004\n",
      "Dest[Turkey Street]                     9.1665      0.063    144.603      0.000       9.042       9.291\n",
      "Dest[Turnham Green]                     8.9181      0.049    180.891      0.000       8.821       9.015\n",
      "Dest[Turnpike Lane]                     8.5172      0.053    161.565      0.000       8.414       8.620\n",
      "Dest[Upminster]                         9.0504      0.052    172.766      0.000       8.948       9.153\n",
      "Dest[Upminster Bridge]                  9.0788      0.060    152.492      0.000       8.962       9.196\n",
      "Dest[Upney]                             8.1506      0.062    131.643      0.000       8.029       8.272\n",
      "Dest[Upper Holloway]                    9.2946      0.054    172.553      0.000       9.189       9.400\n",
      "Dest[Upton Park]                        8.9537      0.049    183.655      0.000       8.858       9.049\n",
      "Dest[Uxbridge]                         10.6866      0.045    237.727      0.000      10.598      10.775\n",
      "Dest[Vauxhall]                         11.0281      0.043    257.774      0.000      10.944      11.112\n",
      "Dest[Victoria]                         11.6119      0.042    273.957      0.000      11.529      11.695\n",
      "Dest[Walthamstow Central]              10.1067      0.044    228.071      0.000      10.020      10.194\n",
      "Dest[Walthamstow Queens Road]          10.2577      0.051    200.140      0.000      10.157      10.358\n",
      "Dest[Wandsworth Road]                   8.6255      0.062    139.181      0.000       8.504       8.747\n",
      "Dest[Wanstead]                          7.9158      0.064    122.789      0.000       7.789       8.042\n",
      "Dest[Wanstead Park]                     8.8805      0.065    136.245      0.000       8.753       9.008\n",
      "Dest[Wapping]                           8.8411      0.052    169.887      0.000       8.739       8.943\n",
      "Dest[Warren Street]                    10.8597      0.043    254.429      0.000      10.776      10.943\n",
      "Dest[Warwick Avenue]                    8.5065      0.051    166.218      0.000       8.406       8.607\n",
      "Dest[Waterloo]                         11.2222      0.043    263.947      0.000      11.139      11.305\n",
      "Dest[Watford]                           9.6303      0.053    180.953      0.000       9.526       9.735\n",
      "Dest[Watford High Street]               9.9609      0.056    177.331      0.000       9.851      10.071\n",
      "Dest[Watford Junction]                 10.6629      0.051    209.326      0.000      10.563      10.763\n",
      "Dest[Wembley Central]                   9.0075      0.050    178.850      0.000       8.909       9.106\n",
      "Dest[Wembley Park]                     10.4588      0.044    236.890      0.000      10.372      10.545\n",
      "Dest[West Acton]                        8.1932      0.062    132.513      0.000       8.072       8.314\n",
      "Dest[West Brompton]                     9.7896      0.044    222.348      0.000       9.703       9.876\n",
      "Dest[West Croydon]                     10.6705      0.048    221.791      0.000      10.576      10.765\n",
      "Dest[West Finchley]                     6.6352      0.113     58.734      0.000       6.414       6.857\n",
      "Dest[West Ham]                          9.7295      0.044    222.758      0.000       9.644       9.815\n",
      "Dest[West Hampstead]                    9.8862      0.044    224.445      0.000       9.800       9.973\n",
      "Dest[West Harrow]                       6.8885      0.100     69.006      0.000       6.693       7.084\n",
      "Dest[West India Quay]                   7.7585      0.065    119.019      0.000       7.631       7.886\n",
      "Dest[West Kensington]                   8.6051      0.050    171.755      0.000       8.507       8.703\n",
      "Dest[West Ruislip]                      8.0931      0.083     97.094      0.000       7.930       8.256\n",
      "Dest[West Silvertown]                   9.0087      0.054    168.359      0.000       8.904       9.114\n",
      "Dest[Westbourne Park]                   8.7024      0.050    173.578      0.000       8.604       8.801\n",
      "Dest[Westferry]                         9.1319      0.050    180.959      0.000       9.033       9.231\n",
      "Dest[Westminster]                      10.6534      0.043    249.324      0.000      10.570      10.737\n",
      "Dest[White City]                       10.3094      0.044    234.029      0.000      10.223      10.396\n",
      "Dest[White Hart Lane]                   8.5858      0.058    148.375      0.000       8.472       8.699\n",
      "Dest[Whitechapel]                      10.7288      0.043    251.184      0.000      10.645      10.812\n",
      "Dest[Willesden Green]                   8.3740      0.055    153.468      0.000       8.267       8.481\n",
      "Dest[Willesden Junction]                9.7160      0.045    216.360      0.000       9.628       9.804\n",
      "Dest[Wimbledon]                        10.7770      0.044    244.491      0.000      10.691      10.863\n",
      "Dest[Wimbledon Park]                    8.4544      0.059    143.882      0.000       8.339       8.570\n",
      "Dest[Wood Green]                        9.3119      0.047    196.843      0.000       9.219       9.405\n",
      "Dest[Wood Lane]                         9.0417      0.049    184.413      0.000       8.946       9.138\n",
      "Dest[Wood Street]                       8.2855      0.065    127.144      0.000       8.158       8.413\n",
      "Dest[Woodford]                          8.5368      0.056    151.109      0.000       8.426       8.648\n",
      "Dest[Woodgrange Park]                   8.3353      0.077    107.773      0.000       8.184       8.487\n",
      "Dest[Woodside Park]                     8.7161      0.056    156.025      0.000       8.607       8.826\n",
      "Dest[Woolwich Arsenal]                 11.5667      0.045    259.447      0.000      11.479      11.654\n",
      "Orig[T.Acton Central]                   1.6537      0.050     32.953      0.000       1.555       1.752\n",
      "Orig[T.Acton Town]                      1.4186      0.044     32.133      0.000       1.332       1.505\n",
      "Orig[T.Aldgate]                         0.1860      0.045      4.132      0.000       0.098       0.274\n",
      "Orig[T.Aldgate East]                    0.3274      0.045      7.333      0.000       0.240       0.415\n",
      "Orig[T.All Saints]                     -0.0749      0.055     -1.361      0.174      -0.183       0.033\n",
      "Orig[T.Alperton]                        1.0326      0.048     21.535      0.000       0.939       1.127\n",
      "Orig[T.Amersham]                        1.4704      0.050     29.235      0.000       1.372       1.569\n",
      "Orig[T.Anerley]                         1.7327      0.057     30.199      0.000       1.620       1.845\n",
      "Orig[T.Angel]                           0.6948      0.044     15.867      0.000       0.609       0.781\n",
      "Orig[T.Archway]                         1.5856      0.043     36.784      0.000       1.501       1.670\n",
      "Orig[T.Arnos Grove]                     1.4625      0.045     32.614      0.000       1.375       1.550\n",
      "Orig[T.Arsenal]                         0.4333      0.046      9.359      0.000       0.343       0.524\n",
      "Orig[T.Baker Street]                    1.2401      0.043     29.115      0.000       1.157       1.324\n",
      "Orig[T.Balham]                          2.3182      0.042     54.863      0.000       2.235       2.401\n",
      "Orig[T.Bank and Monument]               2.3056      0.041     55.715      0.000       2.224       2.387\n",
      "Orig[T.Barbican]                       -0.7520      0.050    -15.080      0.000      -0.850      -0.654\n",
      "Orig[T.Barking]                         2.3347      0.042     54.969      0.000       2.251       2.418\n",
      "Orig[T.Barkingside]                     0.9644      0.050     19.474      0.000       0.867       1.061\n",
      "Orig[T.Barons Court]                    1.0648      0.044     24.253      0.000       0.979       1.151\n",
      "Orig[T.Bayswater]                       0.0731      0.048      1.518      0.129      -0.021       0.167\n",
      "Orig[T.Beckton]                         1.4588      0.050     29.004      0.000       1.360       1.557\n",
      "Orig[T.Beckton Park]                   -0.2218      0.071     -3.106      0.002      -0.362      -0.082\n",
      "Orig[T.Becontree]                       1.3477      0.046     29.327      0.000       1.258       1.438\n",
      "Orig[T.Belsize Park]                    0.7816      0.045     17.283      0.000       0.693       0.870\n",
      "Orig[T.Bermondsey]                      1.1004      0.043     25.412      0.000       1.016       1.185\n",
      "Orig[T.Bethnal Green]                   1.4202      0.043     33.232      0.000       1.336       1.504\n",
      "Orig[T.Blackfriars]                     1.0182      0.043     23.824      0.000       0.934       1.102\n",
      "Orig[T.Blackhorse Road]                 2.1958      0.042     51.736      0.000       2.113       2.279\n",
      "Orig[T.Blackwall]                       0.1779      0.052      3.406      0.001       0.076       0.280\n",
      "Orig[T.Bond Street]                    -0.2649      0.047     -5.686      0.000      -0.356      -0.174\n",
      "Orig[T.Borough]                        -0.5603      0.049    -11.383      0.000      -0.657      -0.464\n",
      "Orig[T.Boston Manor]                    0.5766      0.050     11.464      0.000       0.478       0.675\n",
      "Orig[T.Bounds Green]                    1.7418      0.044     39.884      0.000       1.656       1.827\n",
      "Orig[T.Bow Church]                      1.1285      0.048     23.653      0.000       1.035       1.222\n",
      "Orig[T.Bow Road]                        1.1098      0.044     25.271      0.000       1.024       1.196\n",
      "Orig[T.Brent Cross]                     0.6557      0.048     13.522      0.000       0.561       0.751\n",
      "Orig[T.Brentwood]                       4.0802      0.047     87.458      0.000       3.989       4.172\n",
      "Orig[T.Brixton]                         2.9717      0.042     71.580      0.000       2.890       3.053\n",
      "Orig[T.Brockley]                        2.3918      0.046     51.778      0.000       2.301       2.482\n",
      "Orig[T.Bromley-by-Bow]                  0.7335      0.045     16.123      0.000       0.644       0.823\n",
      "Orig[T.Brondesbury]                     1.7944      0.047     38.225      0.000       1.702       1.886\n",
      "Orig[T.Brondesbury Park]                0.7621      0.056     13.594      0.000       0.652       0.872\n",
      "Orig[T.Bruce Grove]                     1.6775      0.052     32.027      0.000       1.575       1.780\n",
      "Orig[T.Buckhurst Hill]                  1.1557      0.048     24.088      0.000       1.062       1.250\n",
      "Orig[T.Burnt Oak]                       1.6174      0.045     36.033      0.000       1.529       1.705\n",
      "Orig[T.Bush Hill Park]                  2.3761      0.049     48.164      0.000       2.279       2.473\n",
      "Orig[T.Bushey]                          0.4629      0.077      5.974      0.000       0.311       0.615\n",
      "Orig[T.Caledonian Road]                 0.5647      0.045     12.527      0.000       0.476       0.653\n",
      "Orig[T.Caledonian Road & Barnsbury]     0.1843      0.059      3.102      0.002       0.068       0.301\n",
      "Orig[T.Cambridge Heath]                 0.1788      0.069      2.597      0.009       0.044       0.314\n",
      "Orig[T.Camden Road]                     1.1682      0.050     23.275      0.000       1.070       1.267\n",
      "Orig[T.Camden Town]                     0.6157      0.045     13.835      0.000       0.528       0.703\n",
      "Orig[T.Canada Water]                    2.7498      0.041     66.374      0.000       2.669       2.831\n",
      "Orig[T.Canary Wharf]                    2.1221      0.042     50.779      0.000       2.040       2.204\n",
      "Orig[T.Canning Town]                    2.6710      0.042     64.348      0.000       2.590       2.752\n",
      "Orig[T.Cannon Street]                   0.1154      0.044      2.622      0.009       0.029       0.202\n",
      "Orig[T.Canonbury]                       1.5841      0.046     34.537      0.000       1.494       1.674\n",
      "Orig[T.Canons Park]                     1.4761      0.046     32.087      0.000       1.386       1.566\n",
      "Orig[T.Carpenders Park]                 2.2767      0.050     45.543      0.000       2.179       2.375\n",
      "Orig[T.Chadwell Heath]                  4.0494      0.044     91.618      0.000       3.963       4.136\n",
      "Orig[T.Chalfont & Latimer]              1.2726      0.052     24.253      0.000       1.170       1.375\n",
      "Orig[T.Chalk Farm]                      0.4811      0.046     10.502      0.000       0.391       0.571\n",
      "Orig[T.Chancery Lane]                  -0.7902      0.051    -15.646      0.000      -0.889      -0.691\n",
      "Orig[T.Charing Cross]                   0.7871      0.043     18.380      0.000       0.703       0.871\n",
      "Orig[T.Chesham]                         1.3800      0.053     26.257      0.000       1.277       1.483\n",
      "Orig[T.Cheshunt]                        0.9150      0.087     10.497      0.000       0.744       1.086\n",
      "Orig[T.Chigwell]                        0.1346      0.062      2.181      0.029       0.014       0.255\n",
      "Orig[T.Chingford]                       2.8871      0.047     61.792      0.000       2.795       2.979\n",
      "Orig[T.Chiswick Park]                   0.3316      0.050      6.684      0.000       0.234       0.429\n",
      "Orig[T.Chorleywood]                     1.1675      0.053     22.064      0.000       1.064       1.271\n",
      "Orig[T.Clapham Common]                  1.4020      0.043     32.298      0.000       1.317       1.487\n",
      "Orig[T.Clapham High Street]             2.1683      0.049     44.319      0.000       2.072       2.264\n",
      "Orig[T.Clapham Junction]                3.4865      0.043     81.265      0.000       3.402       3.571\n",
      "Orig[T.Clapham North]                   0.7820      0.045     17.348      0.000       0.694       0.870\n",
      "Orig[T.Clapham South]                   1.8231      0.043     42.598      0.000       1.739       1.907\n",
      "Orig[T.Clapton]                         1.9056      0.049     39.264      0.000       1.811       2.001\n",
      "Orig[T.Cockfosters]                     0.6879      0.052     13.278      0.000       0.586       0.789\n",
      "Orig[T.Colindale]                       2.1099      0.043     48.808      0.000       2.025       2.195\n",
      "Orig[T.Colliers Wood]                   1.9654      0.043     45.539      0.000       1.881       2.050\n",
      "Orig[T.Covent Garden]                  -1.9129      0.064    -30.101      0.000      -2.037      -1.788\n",
      "Orig[T.Crossharbour]                    0.8297      0.048     17.449      0.000       0.736       0.923\n",
      "Orig[T.Crouch Hill]                     0.6845      0.065     10.471      0.000       0.556       0.813\n",
      "Orig[T.Croxley]                         1.1043      0.052     21.436      0.000       1.003       1.205\n",
      "Orig[T.Crystal Palace]                  2.9580      0.047     62.653      0.000       2.866       3.051\n",
      "Orig[T.Custom House]                    0.4143      0.054      7.625      0.000       0.308       0.521\n",
      "Orig[T.Cutty Sark]                      1.4608      0.046     31.477      0.000       1.370       1.552\n",
      "Orig[T.Cyprus]                          0.3936      0.061      6.485      0.000       0.275       0.513\n",
      "Orig[T.Dagenham East]                   1.3116      0.047     27.942      0.000       1.220       1.404\n",
      "Orig[T.Dagenham Heathway]               1.7749      0.044     39.890      0.000       1.688       1.862\n",
      "Orig[T.Dalston Junction]                2.3540      0.044     53.346      0.000       2.268       2.441\n",
      "Orig[T.Dalston Kingsland]               2.0292      0.045     44.994      0.000       1.941       2.118\n",
      "Orig[T.Debden]                          1.3342      0.048     27.847      0.000       1.240       1.428\n",
      "Orig[T.Denmark Hill]                    2.2318      0.047     47.081      0.000       2.139       2.325\n",
      "Orig[T.Deptford Bridge]                 1.7973      0.046     39.303      0.000       1.708       1.887\n",
      "Orig[T.Devons Road]                     0.8388      0.049     17.129      0.000       0.743       0.935\n",
      "Orig[T.Dollis Hill]                     1.0643      0.046     23.362      0.000       0.975       1.154\n",
      "Orig[T.Ealing Broadway]                 2.7969      0.042     66.659      0.000       2.715       2.879\n",
      "Orig[T.Ealing Common]                   0.8536      0.047     18.275      0.000       0.762       0.945\n",
      "Orig[T.Earl's Court]                    1.6265      0.042     38.345      0.000       1.543       1.710\n",
      "Orig[T.East Acton]                      0.8906      0.046     19.407      0.000       0.801       0.981\n",
      "Orig[T.East Finchley]                   1.9229      0.043     44.669      0.000       1.839       2.007\n",
      "Orig[T.East Ham]                        2.2119      0.043     52.036      0.000       2.129       2.295\n",
      "Orig[T.East India]                      0.8334      0.047     17.595      0.000       0.741       0.926\n",
      "Orig[T.East Putney]                     1.7163      0.043     39.741      0.000       1.632       1.801\n",
      "Orig[T.Eastcote]                        1.6269      0.046     35.664      0.000       1.537       1.716\n",
      "Orig[T.Edgware]                         1.8188      0.044     40.901      0.000       1.732       1.906\n",
      "Orig[T.Edgware Road]                    0.3790      0.045      8.407      0.000       0.291       0.467\n",
      "Orig[T.Edmonton Green]                  2.4802      0.047     53.033      0.000       2.389       2.572\n",
      "Orig[T.Elephant & Castle]               1.4913      0.042     35.281      0.000       1.408       1.574\n",
      "Orig[T.Elm Park]                        1.7635      0.045     38.952      0.000       1.675       1.852\n",
      "Orig[T.Elverson Road]                   1.0089      0.051     19.849      0.000       0.909       1.108\n",
      "Orig[T.Embankment]                      0.3526      0.044      8.036      0.000       0.267       0.439\n",
      "Orig[T.Emerson Park]                    3.2328      0.071     45.429      0.000       3.093       3.372\n",
      "Orig[T.Enfield Town]                    2.7359      0.047     57.624      0.000       2.643       2.829\n",
      "Orig[T.Epping]                          2.0742      0.045     46.027      0.000       1.986       2.162\n",
      "Orig[T.Euston]                          1.9599      0.042     47.037      0.000       1.878       2.042\n",
      "Orig[T.Euston Square]                   0.8786      0.043     20.247      0.000       0.794       0.964\n",
      "Orig[T.Fairlop]                         0.7685      0.052     14.820      0.000       0.667       0.870\n",
      "Orig[T.Farringdon]                      0.8064      0.043     18.612      0.000       0.722       0.891\n",
      "Orig[T.Finchley Central]                2.1585      0.043     50.290      0.000       2.074       2.243\n",
      "Orig[T.Finchley Road]                   1.3016      0.043     29.925      0.000       1.216       1.387\n",
      "Orig[T.Finchley Road & Frognal]         1.2087      0.051     23.813      0.000       1.109       1.308\n",
      "Orig[T.Finsbury Park]                   2.9347      0.041     70.747      0.000       2.853       3.016\n",
      "Orig[T.Forest Gate]                     2.7269      0.046     59.519      0.000       2.637       2.817\n",
      "Orig[T.Forest Hill]                     2.7307      0.046     59.288      0.000       2.640       2.821\n",
      "Orig[T.Fulham Broadway]                 0.9005      0.045     20.211      0.000       0.813       0.988\n",
      "Orig[T.Gallions Reach]                  0.7219      0.057     12.645      0.000       0.610       0.834\n",
      "Orig[T.Gants Hill]                      2.1120      0.043     48.799      0.000       2.027       2.197\n",
      "Orig[T.Gidea Park]                      4.0985      0.045     91.411      0.000       4.011       4.186\n",
      "Orig[T.Gloucester Road]                 0.9811      0.043     22.570      0.000       0.896       1.066\n",
      "Orig[T.Golders Green]                   1.6881      0.043     38.827      0.000       1.603       1.773\n",
      "Orig[T.Goldhawk Road]                   0.1982      0.049      4.027      0.000       0.102       0.295\n",
      "Orig[T.Goodge Street]                  -1.9406      0.066    -29.215      0.000      -2.071      -1.810\n",
      "Orig[T.Goodmayes]                       3.7140      0.045     82.625      0.000       3.626       3.802\n",
      "Orig[T.Gospel Oak]                      0.9749      0.051     19.066      0.000       0.875       1.075\n",
      "Orig[T.Grange Hill]                     0.4682      0.057      8.259      0.000       0.357       0.579\n",
      "Orig[T.Great Portland Street]          -0.5486      0.051    -10.843      0.000      -0.648      -0.449\n",
      "Orig[T.Green Park]                     -0.2419      0.047     -5.200      0.000      -0.333      -0.151\n",
      "Orig[T.Greenford]                       1.5300      0.045     33.770      0.000       1.441       1.619\n",
      "Orig[T.Greenwich]                       1.1089      0.049     22.564      0.000       1.013       1.205\n",
      "Orig[T.Gunnersbury]                     0.8141      0.046     17.671      0.000       0.724       0.904\n",
      "Orig[T.Hackney Central]                 2.2622      0.045     50.775      0.000       2.175       2.349\n",
      "Orig[T.Hackney Downs]                   1.6357      0.054     30.376      0.000       1.530       1.741\n",
      "Orig[T.Hackney Wick]                    1.1915      0.049     24.537      0.000       1.096       1.287\n",
      "Orig[T.Haggerston]                      1.6130      0.047     34.523      0.000       1.521       1.705\n",
      "Orig[T.Hainault]                        1.8667      0.045     41.686      0.000       1.779       1.954\n",
      "Orig[T.Hammersmith]                     2.2721      0.042     54.071      0.000       2.190       2.354\n",
      "Orig[T.Hampstead]                       0.5733      0.047     12.212      0.000       0.481       0.665\n",
      "Orig[T.Hampstead Heath]                 0.8635      0.053     16.361      0.000       0.760       0.967\n",
      "Orig[T.Hanger Lane]                     1.0756      0.047     23.018      0.000       0.984       1.167\n",
      "Orig[T.Harlesden]                       0.4498      0.048      9.282      0.000       0.355       0.545\n",
      "Orig[T.Harold Wood]                     4.1763      0.045     92.439      0.000       4.088       4.265\n",
      "Orig[T.Harringay Green Lanes]           1.1040      0.061     18.180      0.000       0.985       1.223\n",
      "Orig[T.Harrow & Wealdstone]             1.0702      0.048     22.513      0.000       0.977       1.163\n",
      "Orig[T.Harrow-on-the-Hill]              2.3054      0.043     53.718      0.000       2.221       2.390\n",
      "Orig[T.Hatch End]                       1.9856      0.054     36.806      0.000       1.880       2.091\n",
      "Orig[T.Hatton Cross]                    1.0480      0.049     21.294      0.000       0.952       1.144\n",
      "Orig[T.Headstone Lane]                  1.2986      0.062     20.903      0.000       1.177       1.420\n",
      "Orig[T.Heathrow Terminal 4]             0.5564      0.054     10.299      0.000       0.451       0.662\n",
      "Orig[T.Heathrow Terminal 5]             0.8265      0.052     15.754      0.000       0.724       0.929\n",
      "Orig[T.Heathrow Terminals 2 & 3]        1.5446      0.046     33.383      0.000       1.454       1.635\n",
      "Orig[T.Hendon Central]                  1.6919      0.044     38.523      0.000       1.606       1.778\n",
      "Orig[T.Heron Quays]                     0.2474      0.047      5.227      0.000       0.155       0.340\n",
      "Orig[T.High Barnet]                     1.8393      0.045     41.231      0.000       1.752       1.927\n",
      "Orig[T.High Street Kensington]          0.2863      0.046      6.183      0.000       0.196       0.377\n",
      "Orig[T.Highams Park]                    2.8801      0.045     63.897      0.000       2.792       2.968\n",
      "Orig[T.Highbury & Islington]            2.6051      0.041     62.773      0.000       2.524       2.686\n",
      "Orig[T.Highgate]                        1.5472      0.044     35.502      0.000       1.462       1.633\n",
      "Orig[T.Hillingdon]                      1.1593      0.049     23.495      0.000       1.063       1.256\n",
      "Orig[T.Holborn]                        -0.4947      0.048    -10.358      0.000      -0.588      -0.401\n",
      "Orig[T.Holland Park]                    0.0620      0.048      1.285      0.199      -0.033       0.157\n",
      "Orig[T.Holloway Road]                   0.7045      0.045     15.720      0.000       0.617       0.792\n",
      "Orig[T.Homerton]                        2.1341      0.044     48.070      0.000       2.047       2.221\n",
      "Orig[T.Honor Oak Park]                  2.2780      0.048     47.734      0.000       2.184       2.371\n",
      "Orig[T.Hornchurch]                      1.4016      0.048     29.410      0.000       1.308       1.495\n",
      "Orig[T.Hounslow Central]                1.3437      0.046     29.071      0.000       1.253       1.434\n",
      "Orig[T.Hounslow East]                   1.3223      0.046     28.552      0.000       1.232       1.413\n",
      "Orig[T.Hounslow West]                   1.3681      0.047     29.390      0.000       1.277       1.459\n",
      "Orig[T.Hoxton]                          0.8902      0.052     17.274      0.000       0.789       0.991\n",
      "Orig[T.Hyde Park Corner]               -2.1774      0.074    -29.252      0.000      -2.323      -2.032\n",
      "Orig[T.Ickenham]                        0.6438      0.055     11.671      0.000       0.536       0.752\n",
      "Orig[T.Ilford]                          4.0990      0.043     94.622      0.000       4.014       4.184\n",
      "Orig[T.Imperial Wharf]                  1.1154      0.053     20.899      0.000       1.011       1.220\n",
      "Orig[T.Island Gardens]                  0.9143      0.049     18.706      0.000       0.819       1.010\n",
      "Orig[T.Kennington]                      0.6446      0.044     14.565      0.000       0.558       0.731\n",
      "Orig[T.Kensal Green]                    0.6216      0.047     13.185      0.000       0.529       0.714\n",
      "Orig[T.Kensal Rise]                     1.7649      0.047     37.566      0.000       1.673       1.857\n",
      "Orig[T.Kensington]                     -1.1388      0.063    -18.016      0.000      -1.263      -1.015\n",
      "Orig[T.Kentish Town]                    0.5097      0.046     11.137      0.000       0.420       0.599\n",
      "Orig[T.Kentish Town West]               0.5307      0.056      9.535      0.000       0.422       0.640\n",
      "Orig[T.Kenton]                          0.8247      0.051     16.055      0.000       0.724       0.925\n",
      "Orig[T.Kew Gardens]                     0.9171      0.046     19.877      0.000       0.827       1.007\n",
      "Orig[T.Kilburn]                         1.4678      0.044     33.698      0.000       1.382       1.553\n",
      "Orig[T.Kilburn High Road]               0.7910      0.068     11.625      0.000       0.658       0.924\n",
      "Orig[T.Kilburn Park]                    0.5102      0.047     10.849      0.000       0.418       0.602\n",
      "Orig[T.King George V]                   0.8882      0.052     17.038      0.000       0.786       0.990\n",
      "Orig[T.King's Cross St. Pancras]        2.4875      0.041     60.066      0.000       2.406       2.569\n",
      "Orig[T.Kingsbury]                       1.4708      0.045     32.483      0.000       1.382       1.560\n",
      "Orig[T.Knightsbridge]                  -0.5866      0.051    -11.573      0.000      -0.686      -0.487\n",
      "Orig[T.Ladbroke Grove]                  0.6549      0.046     14.283      0.000       0.565       0.745\n",
      "Orig[T.Lambeth North]                  -0.7085      0.051    -13.955      0.000      -0.808      -0.609\n",
      "Orig[T.Lancaster Gate]                  0.7318      0.044     16.511      0.000       0.645       0.819\n",
      "Orig[T.Langdon Park]                    0.7978      0.049     16.432      0.000       0.703       0.893\n",
      "Orig[T.Latimer Road]                   -0.2955      0.053     -5.542      0.000      -0.400      -0.191\n",
      "Orig[T.Leicester Square]               -1.1353      0.052    -21.867      0.000      -1.237      -1.034\n",
      "Orig[T.Lewisham]                        3.2716      0.043     76.907      0.000       3.188       3.355\n",
      "Orig[T.Leyton]                          2.0550      0.043     48.308      0.000       1.972       2.138\n",
      "Orig[T.Leyton Midland Road]             1.8772      0.054     34.597      0.000       1.771       1.984\n",
      "Orig[T.Leytonstone]                     2.0772      0.043     48.560      0.000       1.993       2.161\n",
      "Orig[T.Leytonstone High Road]           1.7170      0.058     29.602      0.000       1.603       1.831\n",
      "Orig[T.Limehouse]                       2.0278      0.043     46.963      0.000       1.943       2.112\n",
      "Orig[T.Liverpool Street]                2.4412      0.041     59.014      0.000       2.360       2.522\n",
      "Orig[T.London Bridge]                   2.5622      0.041     61.965      0.000       2.481       2.643\n",
      "Orig[T.London City Airport]             1.6072      0.047     34.473      0.000       1.516       1.699\n",
      "Orig[T.London Fields]                   1.6294      0.051     31.855      0.000       1.529       1.730\n",
      "Orig[T.Loughton]                        1.7264      0.045     38.130      0.000       1.638       1.815\n",
      "Orig[T.Maida Vale]                      0.5022      0.046     10.836      0.000       0.411       0.593\n",
      "Orig[T.Manor House]                     1.5427      0.043     35.732      0.000       1.458       1.627\n",
      "Orig[T.Manor Park]                      2.6933      0.047     56.704      0.000       2.600       2.786\n",
      "Orig[T.Mansion House]                  -1.6861      0.060    -28.034      0.000      -1.804      -1.568\n",
      "Orig[T.Marble Arch]                    -0.1974      0.047     -4.176      0.000      -0.290      -0.105\n",
      "Orig[T.Maryland]                        1.1096      0.053     20.886      0.000       1.005       1.214\n",
      "Orig[T.Marylebone]                      1.0786      0.043     25.121      0.000       0.994       1.163\n",
      "Orig[T.Mile End]                        1.6078      0.043     37.702      0.000       1.524       1.691\n",
      "Orig[T.Mill Hill East]                  0.7581      0.050     15.295      0.000       0.661       0.855\n",
      "Orig[T.Moor Park]                       0.5987      0.057     10.535      0.000       0.487       0.710\n",
      "Orig[T.Moorgate]                        0.3620      0.044      8.292      0.000       0.276       0.448\n",
      "Orig[T.Morden]                          2.5202      0.043     59.274      0.000       2.437       2.603\n",
      "Orig[T.Mornington Crescent]            -1.0731      0.056    -19.189      0.000      -1.183      -0.963\n",
      "Orig[T.Mudchute]                        0.7266      0.049     14.771      0.000       0.630       0.823\n",
      "Orig[T.Neasden]                         1.0224      0.046     22.253      0.000       0.932       1.112\n",
      "Orig[T.New Cross]                       2.0002      0.049     40.910      0.000       1.904       2.096\n",
      "Orig[T.New Cross Gate]                  2.0043      0.048     41.930      0.000       1.911       2.098\n",
      "Orig[T.Newbury Park]                    2.0112      0.044     45.891      0.000       1.925       2.097\n",
      "Orig[T.North Acton]                     1.2534      0.045     27.980      0.000       1.166       1.341\n",
      "Orig[T.North Ealing]                   -0.2019      0.061     -3.336      0.001      -0.321      -0.083\n",
      "Orig[T.North Greenwich]                 2.1981      0.042     52.250      0.000       2.116       2.281\n",
      "Orig[T.North Harrow]                    1.3265      0.047     28.233      0.000       1.234       1.419\n",
      "Orig[T.North Wembley]                   0.5009      0.051      9.817      0.000       0.401       0.601\n",
      "Orig[T.Northfields]                     1.4075      0.045     31.301      0.000       1.319       1.496\n",
      "Orig[T.Northolt]                        1.9085      0.044     43.074      0.000       1.822       1.995\n",
      "Orig[T.Northwick Park]                  1.1891      0.046     25.605      0.000       1.098       1.280\n",
      "Orig[T.Northwood]                       1.5549      0.047     33.307      0.000       1.463       1.646\n",
      "Orig[T.Northwood Hills]                 1.1738      0.049     24.095      0.000       1.078       1.269\n",
      "Orig[T.Norwood Junction]                1.7376      0.057     30.654      0.000       1.627       1.849\n",
      "Orig[T.Notting Hill Gate]               1.0961      0.043     25.262      0.000       1.011       1.181\n",
      "Orig[T.Oakwood]                         1.1686      0.047     24.646      0.000       1.076       1.261\n",
      "Orig[T.Old Street]                      0.8774      0.043     20.313      0.000       0.793       0.962\n",
      "Orig[T.Osterley]                        0.8840      0.049     18.091      0.000       0.788       0.980\n",
      "Orig[T.Oval]                            0.8163      0.044     18.458      0.000       0.730       0.903\n",
      "Orig[T.Oxford Circus]                   0.0270      0.045      0.599      0.549      -0.061       0.116\n",
      "Orig[T.Paddington]                      2.6547      0.042     63.953      0.000       2.573       2.736\n",
      "Orig[T.Park Royal]                     -0.1185      0.059     -2.011      0.044      -0.234      -0.003\n",
      "Orig[T.Parsons Green]                   1.2640      0.044     28.918      0.000       1.178       1.350\n",
      "Orig[T.Peckham Rye]                     2.3070      0.046     50.055      0.000       2.217       2.397\n",
      "Orig[T.Penge West]                      1.2406      0.067     18.505      0.000       1.109       1.372\n",
      "Orig[T.Perivale]                        0.9800      0.048     20.439      0.000       0.886       1.074\n",
      "Orig[T.Piccadilly Circus]              -1.0476      0.051    -20.371      0.000      -1.148      -0.947\n",
      "Orig[T.Pimlico]                         0.7623      0.044     17.329      0.000       0.676       0.849\n",
      "Orig[T.Pinner]                          1.7713      0.045     39.290      0.000       1.683       1.860\n",
      "Orig[T.Plaistow]                        1.2013      0.044     27.182      0.000       1.115       1.288\n",
      "Orig[T.Pontoon Dock]                    1.4486      0.047     31.060      0.000       1.357       1.540\n",
      "Orig[T.Poplar]                          0.6538      0.046     14.204      0.000       0.564       0.744\n",
      "Orig[T.Preston Road]                    1.2226      0.046     26.634      0.000       1.133       1.313\n",
      "Orig[T.Prince Regent]                   0.6736      0.052     12.852      0.000       0.571       0.776\n",
      "Orig[T.Pudding Mill Lane]              -0.2973      0.063     -4.708      0.000      -0.421      -0.174\n",
      "Orig[T.Putney Bridge]                   0.9834      0.045     21.879      0.000       0.895       1.071\n",
      "Orig[T.Queen's Park]                    1.4528      0.043     33.578      0.000       1.368       1.538\n",
      "Orig[T.Queens Road Peckham]             2.1081      0.047     44.734      0.000       2.016       2.200\n",
      "Orig[T.Queensbury]                      1.6007      0.045     35.542      0.000       1.512       1.689\n",
      "Orig[T.Queensway]                       0.5319      0.045     11.752      0.000       0.443       0.621\n",
      "Orig[T.Ravenscourt Park]                0.2502      0.048      5.188      0.000       0.156       0.345\n",
      "Orig[T.Rayners Lane]                    1.8065      0.044     40.662      0.000       1.719       1.894\n",
      "Orig[T.Rectory Road]                    1.4515      0.052     27.698      0.000       1.349       1.554\n",
      "Orig[T.Redbridge]                       1.1347      0.047     24.376      0.000       1.043       1.226\n",
      "Orig[T.Regent's Park]                  -1.7932      0.066    -27.110      0.000      -1.923      -1.664\n",
      "Orig[T.Richmond]                        1.9025      0.043     43.911      0.000       1.818       1.987\n",
      "Orig[T.Rickmansworth]                   1.2581      0.050     25.251      0.000       1.160       1.356\n",
      "Orig[T.Roding Valley]                   0.0670      0.064      1.042      0.298      -0.059       0.193\n",
      "Orig[T.Romford]                         4.4073      0.044    100.379      0.000       4.321       4.493\n",
      "Orig[T.Rotherhithe]                     1.1325      0.048     23.468      0.000       1.038       1.227\n",
      "Orig[T.Royal Albert]                    0.4738      0.057      8.354      0.000       0.363       0.585\n",
      "Orig[T.Royal Oak]                      -0.0036      0.049     -0.073      0.941      -0.099       0.092\n",
      "Orig[T.Royal Victoria]                  1.2625      0.046     27.157      0.000       1.171       1.354\n",
      "Orig[T.Ruislip]                         1.0845      0.050     21.900      0.000       0.987       1.182\n",
      "Orig[T.Ruislip Gardens]                 0.9292      0.052     17.809      0.000       0.827       1.031\n",
      "Orig[T.Ruislip Manor]                   1.1119      0.049     22.693      0.000       1.016       1.208\n",
      "Orig[T.Russell Square]                 -0.5275      0.048    -10.898      0.000      -0.622      -0.433\n",
      "Orig[T.Seven Kings]                     3.6887      0.045     82.447      0.000       3.601       3.776\n",
      "Orig[T.Seven Sisters]                   2.8947      0.042     69.499      0.000       2.813       2.976\n",
      "Orig[T.Shadwell]                        1.8893      0.043     44.211      0.000       1.806       1.973\n",
      "Orig[T.Shenfield]                       2.0552      0.076     27.102      0.000       1.907       2.204\n",
      "Orig[T.Shepherd's Bush]                 1.9546      0.042     46.383      0.000       1.872       2.037\n",
      "Orig[T.Shepherd's Bush Market]          0.4808      0.047     10.164      0.000       0.388       0.574\n",
      "Orig[T.Shoreditch High Street]          0.8915      0.053     16.955      0.000       0.788       0.995\n",
      "Orig[T.Silver Street]                   2.3022      0.048     47.785      0.000       2.208       2.397\n",
      "Orig[T.Sloane Square]                   0.9309      0.043     21.461      0.000       0.846       1.016\n",
      "Orig[T.Snaresbrook]                     0.9423      0.047     19.898      0.000       0.850       1.035\n",
      "Orig[T.South Acton]                     1.0562      0.057     18.512      0.000       0.944       1.168\n",
      "Orig[T.South Ealing]                    1.0652      0.046     23.011      0.000       0.974       1.156\n",
      "Orig[T.South Hampstead]                 0.7299      0.076      9.606      0.000       0.581       0.879\n",
      "Orig[T.South Harrow]                    1.0887      0.048     22.488      0.000       0.994       1.184\n",
      "Orig[T.South Kensington]                1.0306      0.043     23.776      0.000       0.946       1.116\n",
      "Orig[T.South Kenton]                    0.6037      0.051     11.803      0.000       0.503       0.704\n",
      "Orig[T.South Quay]                      0.6739      0.048     14.126      0.000       0.580       0.767\n",
      "Orig[T.South Ruislip]                   1.1397      0.049     23.260      0.000       1.044       1.236\n",
      "Orig[T.South Tottenham]                 1.1046      0.064     17.254      0.000       0.979       1.230\n",
      "Orig[T.South Wimbledon]                 1.7098      0.044     38.747      0.000       1.623       1.796\n",
      "Orig[T.South Woodford]                  1.8066      0.044     41.194      0.000       1.721       1.893\n",
      "Orig[T.Southbury]                       1.6869      0.059     28.369      0.000       1.570       1.803\n",
      "Orig[T.Southfields]                     1.7983      0.043     41.544      0.000       1.713       1.883\n",
      "Orig[T.Southgate]                       1.7499      0.044     39.561      0.000       1.663       1.837\n",
      "Orig[T.Southwark]                       0.6832      0.043     15.739      0.000       0.598       0.768\n",
      "Orig[T.St James Street]                 1.7345      0.052     33.678      0.000       1.634       1.835\n",
      "Orig[T.St. James's Park]               -0.2818      0.047     -5.964      0.000      -0.374      -0.189\n",
      "Orig[T.St. John's Wood]                 0.7942      0.045     17.807      0.000       0.707       0.882\n",
      "Orig[T.St. Paul's]                     -1.0771      0.053    -20.290      0.000      -1.181      -0.973\n",
      "Orig[T.Stamford Brook]                  0.5168      0.047     10.955      0.000       0.424       0.609\n",
      "Orig[T.Stamford Hill]                   1.0275      0.061     16.801      0.000       0.908       1.147\n",
      "Orig[T.Stanmore]                        1.6765      0.045     36.906      0.000       1.587       1.765\n",
      "Orig[T.Star Lane]                      -0.3313      0.063     -5.268      0.000      -0.455      -0.208\n",
      "Orig[T.Stepney Green]                   0.5390      0.045     11.918      0.000       0.450       0.628\n",
      "Orig[T.Stockwell]                       1.9607      0.042     46.552      0.000       1.878       2.043\n",
      "Orig[T.Stoke Newington]                 1.6295      0.051     31.913      0.000       1.529       1.730\n",
      "Orig[T.Stonebridge Park]                0.5538      0.049     11.407      0.000       0.459       0.649\n",
      "Orig[T.Stratford]                       3.6991      0.041     89.871      0.000       3.618       3.780\n",
      "Orig[T.Stratford High Street]          -0.4826      0.066     -7.348      0.000      -0.611      -0.354\n",
      "Orig[T.Stratford International]         1.6721      0.046     36.409      0.000       1.582       1.762\n",
      "Orig[T.Sudbury Hill]                    0.9217      0.049     18.689      0.000       0.825       1.018\n",
      "Orig[T.Sudbury Town]                    1.0314      0.049     21.259      0.000       0.936       1.126\n",
      "Orig[T.Surrey Quays]                    2.0367      0.045     45.693      0.000       1.949       2.124\n",
      "Orig[T.Swiss Cottage]                   0.9689      0.044     21.864      0.000       0.882       1.056\n",
      "Orig[T.Sydenham]                        2.5180      0.048     52.873      0.000       2.425       2.611\n",
      "Orig[T.Temple]                         -1.9301      0.067    -28.630      0.000      -2.062      -1.798\n",
      "Orig[T.Theobalds Grove]                 1.9118      0.060     31.609      0.000       1.793       2.030\n",
      "Orig[T.Theydon Bois]                    0.6817      0.056     12.122      0.000       0.571       0.792\n",
      "Orig[T.Tooting Bec]                     1.9793      0.043     46.265      0.000       1.895       2.063\n",
      "Orig[T.Tooting Broadway]                2.4358      0.042     57.623      0.000       2.353       2.519\n",
      "Orig[T.Tottenham Court Road]           -0.4794      0.047    -10.121      0.000      -0.572      -0.387\n",
      "Orig[T.Tottenham Hale]                  2.2453      0.042     53.008      0.000       2.162       2.328\n",
      "Orig[T.Totteridge & Whetstone]          1.3872      0.046     30.140      0.000       1.297       1.477\n",
      "Orig[T.Tower Gateway]                   0.5789      0.053     10.847      0.000       0.474       0.683\n",
      "Orig[T.Tower Hill]                      1.2058      0.042     28.446      0.000       1.123       1.289\n",
      "Orig[T.Tufnell Park]                    0.7472      0.045     16.523      0.000       0.659       0.836\n",
      "Orig[T.Turkey Street]                   2.1766      0.054     39.956      0.000       2.070       2.283\n",
      "Orig[T.Turnham Green]                   1.2370      0.044     27.974      0.000       1.150       1.324\n",
      "Orig[T.Turnpike Lane]                   1.9507      0.043     45.588      0.000       1.867       2.035\n",
      "Orig[T.Upminster]                       0.6050      0.055     10.934      0.000       0.497       0.713\n",
      "Orig[T.Upminster Bridge]                0.7480      0.054     13.730      0.000       0.641       0.855\n",
      "Orig[T.Upney]                           0.9643      0.047     20.302      0.000       0.871       1.057\n",
      "Orig[T.Upper Holloway]                  0.4722      0.066      7.127      0.000       0.342       0.602\n",
      "Orig[T.Upton Park]                      1.8282      0.043     42.537      0.000       1.744       1.912\n",
      "Orig[T.Uxbridge]                        1.9144      0.045     42.301      0.000       1.826       2.003\n",
      "Orig[T.Vauxhall]                        2.3565      0.042     56.461      0.000       2.275       2.438\n",
      "Orig[T.Victoria]                        2.8352      0.041     68.632      0.000       2.754       2.916\n",
      "Orig[T.Walthamstow Central]             3.0439      0.042     72.983      0.000       2.962       3.126\n",
      "Orig[T.Walthamstow Queens Road]         2.3987      0.052     45.811      0.000       2.296       2.501\n",
      "Orig[T.Wandsworth Road]                 1.0263      0.058     17.693      0.000       0.913       1.140\n",
      "Orig[T.Wanstead]                        1.0259      0.047     21.948      0.000       0.934       1.118\n",
      "Orig[T.Wanstead Park]                   2.1033      0.055     37.999      0.000       1.995       2.212\n",
      "Orig[T.Wapping]                         1.0747      0.050     21.516      0.000       0.977       1.173\n",
      "Orig[T.Warren Street]                  -0.3049      0.047     -6.497      0.000      -0.397      -0.213\n",
      "Orig[T.Warwick Avenue]                  0.4759      0.046     10.387      0.000       0.386       0.566\n",
      "Orig[T.Waterloo]                        3.3025      0.041     80.229      0.000       3.222       3.383\n",
      "Orig[T.Watford]                         1.4600      0.049     29.791      0.000       1.364       1.556\n",
      "Orig[T.Watford High Street]             1.1329      0.063     17.933      0.000       1.009       1.257\n",
      "Orig[T.Watford Junction]                1.5039      0.062     24.245      0.000       1.382       1.625\n",
      "Orig[T.Wembley Central]                 1.0469      0.046     22.694      0.000       0.957       1.137\n",
      "Orig[T.Wembley Park]                    2.2496      0.043     52.670      0.000       2.166       2.333\n",
      "Orig[T.West Acton]                      0.4521      0.050      8.978      0.000       0.353       0.551\n",
      "Orig[T.West Brompton]                   0.8139      0.044     18.470      0.000       0.728       0.900\n",
      "Orig[T.West Croydon]                    2.9102      0.050     58.632      0.000       2.813       3.007\n",
      "Orig[T.West Finchley]                   0.8875      0.048     18.393      0.000       0.793       0.982\n",
      "Orig[T.West Ham]                        1.8148      0.042     42.917      0.000       1.732       1.898\n",
      "Orig[T.West Hampstead]                  1.9969      0.042     47.361      0.000       1.914       2.080\n",
      "Orig[T.West Harrow]                     0.5995      0.052     11.615      0.000       0.498       0.701\n",
      "Orig[T.West India Quay]                -2.3001      0.089    -25.953      0.000      -2.474      -2.126\n",
      "Orig[T.West Kensington]                 0.6124      0.045     13.528      0.000       0.524       0.701\n",
      "Orig[T.West Ruislip]                    1.0828      0.051     21.199      0.000       0.983       1.183\n",
      "Orig[T.West Silvertown]                 0.5577      0.053     10.560      0.000       0.454       0.661\n",
      "Orig[T.Westbourne Park]                 0.6346      0.046     13.866      0.000       0.545       0.724\n",
      "Orig[T.Westferry]                       0.8404      0.046     18.206      0.000       0.750       0.931\n",
      "Orig[T.Westminster]                    -0.5319      0.048    -10.999      0.000      -0.627      -0.437\n",
      "Orig[T.White City]                      0.7784      0.046     17.028      0.000       0.689       0.868\n",
      "Orig[T.White Hart Lane]                 1.6550      0.052     32.026      0.000       1.554       1.756\n",
      "Orig[T.Whitechapel]                     1.7522      0.042     41.762      0.000       1.670       1.834\n",
      "Orig[T.Willesden Green]                 1.7551      0.043     40.721      0.000       1.671       1.840\n",
      "Orig[T.Willesden Junction]              1.5299      0.043     35.363      0.000       1.445       1.615\n",
      "Orig[T.Wimbledon]                       2.3354      0.043     54.676      0.000       2.252       2.419\n",
      "Orig[T.Wimbledon Park]                  0.8045      0.048     16.836      0.000       0.711       0.898\n",
      "Orig[T.Wood Green]                      2.0298      0.043     47.426      0.000       1.946       2.114\n",
      "Orig[T.Wood Lane]                       0.0123      0.051      0.241      0.810      -0.088       0.112\n",
      "Orig[T.Wood Street]                     1.9917      0.049     40.444      0.000       1.895       2.088\n",
      "Orig[T.Woodford]                        2.0625      0.043     47.470      0.000       1.977       2.148\n",
      "Orig[T.Woodgrange Park]                 1.6669      0.060     27.654      0.000       1.549       1.785\n",
      "Orig[T.Woodside Park]                   1.5896      0.045     35.485      0.000       1.502       1.677\n",
      "Orig[T.Woolwich Arsenal]                3.5490      0.042     83.526      0.000       3.466       3.632\n",
      "log_Dist                               -0.9097      0.001   -776.770      0.000      -0.912      -0.907\n",
      "=======================================================================================================\n"
     ]
    }
   ],
   "source": [
    "#create the formula (the \"-1\" indicates no intercept in the regression model).\n",
    "#create the formula (the \"-1\" indicates no intercept in the regression model).\n",
    "dbl_form = 'Total ~ Dest + Orig + log_Dist-1'\n",
    "#run a doubly constrained sim\n",
    "doubSim = smf.glm(formula = dbl_form, data=london_flows, family=sm.families.Poisson()).fit()\n",
    "#let's have a look at it's summary\n",
    "#print(doubSim.summary())"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\beta$=0.0907"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9096556995428264"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta_pow=-doubSim.params[-1]\n",
    "beta_pow"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "#get the estimates\n",
    "london_flows[\"doubsimfitted_pow\"] = np.round(doubSim.mu)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Dest</th>\n",
       "      <th>Abbey Road</th>\n",
       "      <th>Acton Central</th>\n",
       "      <th>Acton Town</th>\n",
       "      <th>Aldgate</th>\n",
       "      <th>Aldgate East</th>\n",
       "      <th>All Saints</th>\n",
       "      <th>Alperton</th>\n",
       "      <th>Amersham</th>\n",
       "      <th>Anerley</th>\n",
       "      <th>Angel</th>\n",
       "      <th>...</th>\n",
       "      <th>Wimbledon</th>\n",
       "      <th>Wimbledon Park</th>\n",
       "      <th>Wood Green</th>\n",
       "      <th>Wood Lane</th>\n",
       "      <th>Wood Street</th>\n",
       "      <th>Woodford</th>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <th>Woodside Park</th>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <th>All</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Orig</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Abbey Road</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>26.0</td>\n",
       "      <td>600.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Central</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1224.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Town</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>15.0</td>\n",
       "      <td>15.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>11.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>17.0</td>\n",
       "      <td>...</td>\n",
       "      <td>30.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3747.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>42.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>19.0</td>\n",
       "      <td>...</td>\n",
       "      <td>7.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2873.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate East</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>49.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>21.0</td>\n",
       "      <td>...</td>\n",
       "      <td>8.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3172.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodford</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>7.0</td>\n",
       "      <td>31.0</td>\n",
       "      <td>33.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>29.0</td>\n",
       "      <td>...</td>\n",
       "      <td>29.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>10.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>4862.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>530.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodside Park</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>5.0</td>\n",
       "      <td>18.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>22.0</td>\n",
       "      <td>...</td>\n",
       "      <td>21.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3093.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <td>20.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>28.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>7893.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>All</th>\n",
       "      <td>346.0</td>\n",
       "      <td>751.0</td>\n",
       "      <td>2198.0</td>\n",
       "      <td>7788.0</td>\n",
       "      <td>7934.0</td>\n",
       "      <td>446.0</td>\n",
       "      <td>741.0</td>\n",
       "      <td>256.0</td>\n",
       "      <td>174.0</td>\n",
       "      <td>8108.0</td>\n",
       "      <td>...</td>\n",
       "      <td>6278.0</td>\n",
       "      <td>599.0</td>\n",
       "      <td>2141.0</td>\n",
       "      <td>1576.0</td>\n",
       "      <td>402.0</td>\n",
       "      <td>700.0</td>\n",
       "      <td>240.0</td>\n",
       "      <td>740.0</td>\n",
       "      <td>4427.0</td>\n",
       "      <td>1541959.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>399 rows × 399 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "Dest              Abbey Road  Acton Central  Acton Town  Aldgate  \\\n",
       "Orig                                                               \n",
       "Abbey Road               NaN            NaN         NaN      NaN   \n",
       "Acton Central            NaN            NaN         NaN      NaN   \n",
       "Acton Town               NaN            NaN         NaN     15.0   \n",
       "Aldgate                  NaN            NaN         2.0      NaN   \n",
       "Aldgate East             NaN            NaN         2.0     49.0   \n",
       "...                      ...            ...         ...      ...   \n",
       "Woodford                 NaN            NaN         7.0     31.0   \n",
       "Woodgrange Park          NaN            6.0         NaN      NaN   \n",
       "Woodside Park            NaN            NaN         5.0     18.0   \n",
       "Woolwich Arsenal        20.0            NaN         NaN      NaN   \n",
       "All                    346.0          751.0      2198.0   7788.0   \n",
       "\n",
       "Dest              Aldgate East  All Saints  Alperton  Amersham  Anerley  \\\n",
       "Orig                                                                      \n",
       "Abbey Road                 NaN         NaN       NaN       NaN      NaN   \n",
       "Acton Central              NaN         NaN       NaN       NaN      NaN   \n",
       "Acton Town                15.0         NaN      11.0       1.0      NaN   \n",
       "Aldgate                   42.0         NaN       NaN       0.0      NaN   \n",
       "Aldgate East               NaN         NaN       1.0       0.0      NaN   \n",
       "...                        ...         ...       ...       ...      ...   \n",
       "Woodford                  33.0         NaN       NaN       NaN      NaN   \n",
       "Woodgrange Park            NaN         NaN       NaN       NaN      NaN   \n",
       "Woodside Park             17.0         NaN       3.0       NaN      NaN   \n",
       "Woolwich Arsenal           NaN        28.0       NaN       NaN      NaN   \n",
       "All                     7934.0       446.0     741.0     256.0    174.0   \n",
       "\n",
       "Dest               Angel  ...  Wimbledon  Wimbledon Park  Wood Green  \\\n",
       "Orig                      ...                                          \n",
       "Abbey Road           NaN  ...        NaN             NaN         NaN   \n",
       "Acton Central        NaN  ...        NaN             NaN         NaN   \n",
       "Acton Town          17.0  ...       30.0             3.0         5.0   \n",
       "Aldgate             19.0  ...        7.0             NaN         2.0   \n",
       "Aldgate East        21.0  ...        8.0             1.0         3.0   \n",
       "...                  ...  ...        ...             ...         ...   \n",
       "Woodford            29.0  ...       29.0             NaN        10.0   \n",
       "Woodgrange Park      NaN  ...        NaN             NaN         NaN   \n",
       "Woodside Park       22.0  ...       21.0             NaN         6.0   \n",
       "Woolwich Arsenal     NaN  ...        NaN             NaN         NaN   \n",
       "All               8108.0  ...     6278.0           599.0      2141.0   \n",
       "\n",
       "Dest              Wood Lane  Wood Street  Woodford  Woodgrange Park  \\\n",
       "Orig                                                                  \n",
       "Abbey Road              NaN          NaN       NaN              NaN   \n",
       "Acton Central           NaN          NaN       NaN              2.0   \n",
       "Acton Town             12.0          NaN       2.0              NaN   \n",
       "Aldgate                 2.0          NaN       1.0              NaN   \n",
       "Aldgate East            2.0          NaN       1.0              NaN   \n",
       "...                     ...          ...       ...              ...   \n",
       "Woodford                NaN          NaN       NaN              NaN   \n",
       "Woodgrange Park         NaN          NaN       NaN              NaN   \n",
       "Woodside Park           NaN          NaN       NaN              NaN   \n",
       "Woolwich Arsenal        NaN          NaN       NaN              NaN   \n",
       "All                  1576.0        402.0     700.0            240.0   \n",
       "\n",
       "Dest              Woodside Park  Woolwich Arsenal        All  \n",
       "Orig                                                          \n",
       "Abbey Road                  NaN              26.0      600.0  \n",
       "Acton Central               NaN               NaN     1224.0  \n",
       "Acton Town                  2.0               NaN     3747.0  \n",
       "Aldgate                     1.0               NaN     2873.0  \n",
       "Aldgate East                1.0               NaN     3172.0  \n",
       "...                         ...               ...        ...  \n",
       "Woodford                    NaN               NaN     4862.0  \n",
       "Woodgrange Park             NaN               NaN      530.0  \n",
       "Woodside Park               NaN               NaN     3093.0  \n",
       "Woolwich Arsenal            NaN               NaN     7893.0  \n",
       "All                       740.0            4427.0  1541959.0  \n",
       "\n",
       "[399 rows x 399 columns]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.pivot_table(london_flows,values =\"doubsimfitted_pow\", index=\"Orig\", columns = \"Dest\",\n",
    "                            aggfunc=np.sum, margins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.40772748658809344"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check goodness of fit\n",
    "CalcRSqaured(london_flows['Total'],london_flows['doubsimfitted_pow'])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With these parameters, the inverse power function has a far more rapid distance decay effect than the negative exponential function. In real life, what this means is that if the observed interactions drop off very rapidly with distance, then they might be more likely to follow an inverse power law. This might be the case when looking at trips to the local convenience store by walking, for example. On the other hand, if the effect of distance is less severe (for example migration across the country for a new job) then the negative exponential funtion might be more appropriate.\n",
    "\n",
    "There is no hard and fast rule as to which function to pick, it will just come down to which fits the data better…\n",
    "\n",
    "As [Taylor Oshan points out in his excellent Primer](http://openjournals.wu.ac.at/region/paper_175/175.html) what this means in our Poisson regression model is that we simply substitute $-\\beta \\ln d_{ij}$ for $-\\beta d_{ij}$ in our model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Run a doubly constrained SIM with a negative exponential cost function.\n",
    "doubsim_form = \"Total ~ Orig + Dest + Dist -1\"\n",
    "doubsim_exp = smf.glm(formula=doubsim_form, data = london_flows, family = sm.families.Poisson()).fit()\n",
    "#print(doubsim_exp.summary())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00015436845765522456"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta_exp=-doubsim_exp.params[-1]\n",
    "beta_exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "# fit to make predictions\n",
    "london_flows['doubsimfitted_exp']=np.round(doubsim_exp.mu,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Dest</th>\n",
       "      <th>Abbey Road</th>\n",
       "      <th>Acton Central</th>\n",
       "      <th>Acton Town</th>\n",
       "      <th>Aldgate</th>\n",
       "      <th>Aldgate East</th>\n",
       "      <th>All Saints</th>\n",
       "      <th>Alperton</th>\n",
       "      <th>Amersham</th>\n",
       "      <th>Anerley</th>\n",
       "      <th>Angel</th>\n",
       "      <th>...</th>\n",
       "      <th>Wimbledon</th>\n",
       "      <th>Wimbledon Park</th>\n",
       "      <th>Wood Green</th>\n",
       "      <th>Wood Lane</th>\n",
       "      <th>Wood Street</th>\n",
       "      <th>Woodford</th>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <th>Woodside Park</th>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <th>All</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Orig</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Abbey Road</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>31.0</td>\n",
       "      <td>603.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Central</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1221.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Town</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>11.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>17.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>12.0</td>\n",
       "      <td>...</td>\n",
       "      <td>40.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>19.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3752.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>32.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>23.0</td>\n",
       "      <td>...</td>\n",
       "      <td>7.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2883.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate East</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>38.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>24.0</td>\n",
       "      <td>...</td>\n",
       "      <td>7.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3167.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodford</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>38.0</td>\n",
       "      <td>44.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>25.0</td>\n",
       "      <td>...</td>\n",
       "      <td>7.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>7.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>4862.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>528.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodside Park</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>2.0</td>\n",
       "      <td>17.0</td>\n",
       "      <td>15.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>25.0</td>\n",
       "      <td>...</td>\n",
       "      <td>10.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>4.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>3093.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <td>28.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>29.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>...</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>7896.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>All</th>\n",
       "      <td>344.0</td>\n",
       "      <td>750.0</td>\n",
       "      <td>2204.0</td>\n",
       "      <td>7776.0</td>\n",
       "      <td>7933.0</td>\n",
       "      <td>442.0</td>\n",
       "      <td>739.0</td>\n",
       "      <td>249.0</td>\n",
       "      <td>175.0</td>\n",
       "      <td>8097.0</td>\n",
       "      <td>...</td>\n",
       "      <td>6296.0</td>\n",
       "      <td>594.0</td>\n",
       "      <td>2140.0</td>\n",
       "      <td>1578.0</td>\n",
       "      <td>402.0</td>\n",
       "      <td>692.0</td>\n",
       "      <td>242.0</td>\n",
       "      <td>747.0</td>\n",
       "      <td>4426.0</td>\n",
       "      <td>1541437.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>399 rows × 399 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "Dest              Abbey Road  Acton Central  Acton Town  Aldgate  \\\n",
       "Orig                                                               \n",
       "Abbey Road               NaN            NaN         NaN      NaN   \n",
       "Acton Central            NaN            NaN         NaN      NaN   \n",
       "Acton Town               NaN            NaN         NaN     11.0   \n",
       "Aldgate                  NaN            NaN         1.0      NaN   \n",
       "Aldgate East             NaN            NaN         2.0     38.0   \n",
       "...                      ...            ...         ...      ...   \n",
       "Woodford                 NaN            NaN         2.0     38.0   \n",
       "Woodgrange Park          NaN            2.0         NaN      NaN   \n",
       "Woodside Park            NaN            NaN         2.0     17.0   \n",
       "Woolwich Arsenal        28.0            NaN         NaN      NaN   \n",
       "All                    344.0          750.0      2204.0   7776.0   \n",
       "\n",
       "Dest              Aldgate East  All Saints  Alperton  Amersham  Anerley  \\\n",
       "Orig                                                                      \n",
       "Abbey Road                 NaN         NaN       NaN       NaN      NaN   \n",
       "Acton Central              NaN         NaN       NaN       NaN      NaN   \n",
       "Acton Town                10.0         NaN      17.0       0.0      NaN   \n",
       "Aldgate                   32.0         NaN       NaN       0.0      NaN   \n",
       "Aldgate East               NaN         NaN       0.0       0.0      NaN   \n",
       "...                        ...         ...       ...       ...      ...   \n",
       "Woodford                  44.0         NaN       NaN       NaN      NaN   \n",
       "Woodgrange Park            NaN         NaN       NaN       NaN      NaN   \n",
       "Woodside Park             15.0         NaN       0.0       NaN      NaN   \n",
       "Woolwich Arsenal           NaN        29.0       NaN       NaN      NaN   \n",
       "All                     7933.0       442.0     739.0     249.0    175.0   \n",
       "\n",
       "Dest               Angel  ...  Wimbledon  Wimbledon Park  Wood Green  \\\n",
       "Orig                      ...                                          \n",
       "Abbey Road           NaN  ...        NaN             NaN         NaN   \n",
       "Acton Central        NaN  ...        NaN             NaN         NaN   \n",
       "Acton Town          12.0  ...       40.0             4.0         2.0   \n",
       "Aldgate             23.0  ...        7.0             NaN         3.0   \n",
       "Aldgate East        24.0  ...        7.0             1.0         3.0   \n",
       "...                  ...  ...        ...             ...         ...   \n",
       "Woodford            25.0  ...        7.0             NaN         7.0   \n",
       "Woodgrange Park      NaN  ...        NaN             NaN         NaN   \n",
       "Woodside Park       25.0  ...       10.0             NaN         4.0   \n",
       "Woolwich Arsenal     NaN  ...        NaN             NaN         NaN   \n",
       "All               8097.0  ...     6296.0           594.0      2140.0   \n",
       "\n",
       "Dest              Wood Lane  Wood Street  Woodford  Woodgrange Park  \\\n",
       "Orig                                                                  \n",
       "Abbey Road              NaN          NaN       NaN              NaN   \n",
       "Acton Central           NaN          NaN       NaN              0.0   \n",
       "Acton Town             19.0          NaN       0.0              NaN   \n",
       "Aldgate                 2.0          NaN       1.0              NaN   \n",
       "Aldgate East            2.0          NaN       1.0              NaN   \n",
       "...                     ...          ...       ...              ...   \n",
       "Woodford                NaN          NaN       NaN              NaN   \n",
       "Woodgrange Park         NaN          NaN       NaN              NaN   \n",
       "Woodside Park           NaN          NaN       NaN              NaN   \n",
       "Woolwich Arsenal        NaN          NaN       NaN              NaN   \n",
       "All                  1578.0        402.0     692.0            242.0   \n",
       "\n",
       "Dest              Woodside Park  Woolwich Arsenal        All  \n",
       "Orig                                                          \n",
       "Abbey Road                  NaN              31.0      603.0  \n",
       "Acton Central               NaN               NaN     1221.0  \n",
       "Acton Town                  1.0               NaN     3752.0  \n",
       "Aldgate                     1.0               NaN     2883.0  \n",
       "Aldgate East                1.0               NaN     3167.0  \n",
       "...                         ...               ...        ...  \n",
       "Woodford                    NaN               NaN     4862.0  \n",
       "Woodgrange Park             NaN               NaN      528.0  \n",
       "Woodside Park               NaN               NaN     3093.0  \n",
       "Woolwich Arsenal            NaN               NaN     7896.0  \n",
       "All                       747.0            4426.0  1541437.0  \n",
       "\n",
       "[399 rows x 399 columns]"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.pivot_table(london_flows,values =\"doubsimfitted_exp\", index=\"Orig\", columns = \"Dest\",\n",
    "                            aggfunc=np.sum, margins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4978623570468335"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check the goodness-of-fit\n",
    "CalcRSqaured(london_flows['Total'],london_flows['doubsimfitted_exp'])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plug the parameter estimates back into Wilson's entropy maximising multiplicative models \n",
    "(To generate estimates)\n",
    "\n",
    "the key to the doubly constrained models is the $A_i$ and $B_j$ balancing factors and as they rely on each other, they need to be calculated iteratively. We can do this using [Senior’s algorthim](http://journals.sagepub.com/doi/abs/10.1177/030913257900300218) also mentioned earlier.\n",
    "\n",
    "Here is the code as provided by [Dan Lewis](https://github.com/danlewis85/UCL_CASA_Urban_Simulation/blob/master/Constrained%20SIM.ipynb) who in a departure from Dennet rewrites the algorithm as a function, which can then be called subject to the required parameters. In order for it to work it requires:\n",
    "\n",
    "- pd - a pandas dataframe of origin-destination pairwise flows and associated data.\n",
    "- orig_field - the name of the dataframe field in pd that uniquely labels origin zones.\n",
    "- dest_field - the name of the dataframe field in pd that uniquely labels destination zones.\n",
    "- Oi_field - the name of the dataframe field that stores total flows from a given origin $i$\n",
    "- Dj_field - the name of the dataframe field that stores total flows to a given destination $j$\n",
    "- cij_field - the name of the dataframe field that stores the pairwise cost (e.g. distance) between $i$ and $j$\n",
    "- beta - a constant for the beta parameter you wish to use in the model\n",
    "- cost_function - a string representing the cost function, either 'power' or 'exponential'\n",
    "- Ainame - What you want to call the new field in pd that will hold $A_{i}$ values, defaults to \"Ai_new\"\n",
    "- Bjname - What you want to call the new field in pd that will hold $B_{j}$ values, defaults to \"Bj_new\"\n",
    "- converge - A threshold value at which a model can be said to have converged, the default of 0.001 seems to work fine."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "#create some Oi and Dj columns in the dataframe and store row and column totals in them:\n",
    "#to create O_i, take london_flows ...then... group by origcodenew ...then... summarise by calculating the sum of Total\n",
    "O_i = pd.DataFrame(london_flows.groupby([\"Orig\"])[\"Total\"].agg(np.sum))\n",
    "O_i.rename(columns={\"Total\":\"O_i\"}, inplace = True)\n",
    "london_flows = london_flows.merge(O_i, on = \"Orig\", how = \"left\" )\n",
    "\n",
    "D_j = pd.DataFrame(london_flows.groupby([\"Dest\"])[\"Total\"].agg(np.sum))\n",
    "D_j.rename(columns={\"Total\":\"D_j\"}, inplace = True)\n",
    "london_flows = london_flows.merge(D_j, on = \"Dest\", how = \"left\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# Here is the entropy maximising approach for a known beta.\n",
    "# Plug in the required values in this function to solve.\n",
    "\n",
    "def balance_doubly_constrained(pd, orig_field, dest_field, Oi_field, Dj_field, cij_field, beta, \n",
    "                               cost_function, Ai_name = \"Ai_new\", Bj_name = \"Bj_new\", converge=0.001):\n",
    "    # Define some variables\n",
    "    Oi = pd[[orig_field, Oi_field]]\n",
    "    Dj = pd[[dest_field,Dj_field]]    \n",
    "    if cost_function.lower() in ['power','pow']:\n",
    "        beta_cij = np.exp(beta * np.log(pd[cij_field]))\n",
    "    elif cost_function.lower() in ['exponential','exp']:\n",
    "        beta_cij = np.exp(beta * pd[cij_field])\n",
    "    else:\n",
    "        return \"Cost function not specified properly, use 'exp' or 'pow'\"\n",
    "    \n",
    "    # Create some helper variables\n",
    "    cnvg = 1\n",
    "    iteration = 0\n",
    "    # Now iteratively rebalance the Ai and Bj terms until convergence\n",
    "    while cnvg > converge:\n",
    "        if iteration == 0:\n",
    "            # This first condition sets starting values for Ai and Bj\n",
    "            # NB sets starting value of Ai assuming Bj is a vector of 1s.\n",
    "            # We've already established beta_cij with the appropriate cost function, so...\n",
    "            Oi = Oi.assign(Ai = Dj[Dj_field] * beta_cij)\n",
    "            # Aggregate Ai and take inverse\n",
    "            Ai = 1.0/Oi.groupby(orig_field)['Ai'].sum().to_frame()\n",
    "            # Merge new Ais \n",
    "            Oi = Oi.merge(Ai,left_on = orig_field, right_index = True, suffixes = ('','_old'))\n",
    "            # Drop the temporary Ai field we created, leaving Ai_old\n",
    "            Oi.drop('Ai', axis=1, inplace=True)\n",
    "            \n",
    "            # Now set up Bjs using starting values of Ai\n",
    "            Dj = Dj.assign(Bj = Oi['Ai_old'] * Oi[Oi_field] * beta_cij)\n",
    "            # Aggregate Bj and take inverse\n",
    "            Bj = 1.0/Dj.groupby(dest_field)['Bj'].sum().to_frame()\n",
    "            # Merge new Bjs\n",
    "            Dj = Dj.merge(Bj,left_on = dest_field, right_index = True, suffixes = ('','_old'))\n",
    "            # Drop the temporary Bj field we created, leaving Bj_old\n",
    "            Dj.drop('Bj', axis=1, inplace=True)\n",
    "            \n",
    "            # Increment loop\n",
    "            iteration += 1\n",
    "        else:\n",
    "            # This bit is the iterated bit of the loop which refines the values of Ai and Bj\n",
    "            # First Ai\n",
    "            Oi['Ai'] = Dj['Bj_old'] * Dj[Dj_field] * beta_cij\n",
    "            # Aggregate Ai and take inverse\n",
    "            Ai = 1.0/Oi.groupby(orig_field)['Ai'].sum().to_frame()\n",
    "            # Drop temporary Ai\n",
    "            Oi.drop('Ai', axis=1, inplace=True)\n",
    "            # Merge new Ais \n",
    "            Oi = Oi.merge(Ai,left_on = orig_field, right_index = True)\n",
    "            # Calculate the difference between old and new Ais\n",
    "            Oi['diff'] = np.absolute((Oi['Ai_old'] - Oi['Ai'])/Oi['Ai_old'])\n",
    "            # Set new Ais to Ai_old\n",
    "            Oi['Ai_old'] = Oi['Ai']\n",
    "            # Drop the temporary Ai field we created, leaving Ai_old\n",
    "            Oi.drop('Ai', axis=1, inplace=True)\n",
    "            \n",
    "            # Then Bj\n",
    "            Dj['Bj'] = Oi['Ai_old'] * Oi[Oi_field] * beta_cij\n",
    "            # Aggregate Bj and take inverse\n",
    "            Bj = 1.0/Dj.groupby(dest_field)['Bj'].sum().to_frame()\n",
    "            # Drop temporary Bj\n",
    "            Dj.drop('Bj', axis=1, inplace=True)\n",
    "            # Merge new Bjs\n",
    "            Dj = Dj.merge(Bj,left_on = dest_field, right_index = True)\n",
    "            # Calculate the difference between old and new Bjs\n",
    "            Dj['diff'] = np.absolute((Dj['Bj_old'] - Dj['Bj'])/Dj['Bj_old'])\n",
    "            # Set new Bjs to Bj_old\n",
    "            Dj['Bj_old'] = Dj['Bj']\n",
    "            # Drop the temporary Bj field we created, leaving Bj_old\n",
    "            Dj.drop('Bj', axis=1, inplace=True)\n",
    "            \n",
    "            # Assign higher sum difference from Ai or Bj to cnvg\n",
    "            cnvg = np.maximum(Oi['diff'].sum(),Dj['diff'].sum())\n",
    "            \n",
    "            # Print and increment loop\n",
    "            print(\"Iteration:\", iteration)\n",
    "            iteration += 1\n",
    "\n",
    "    # When the while loop finishes add the computed Ai_old and Bj_old to the dataframe and return\n",
    "    pd[Ai_name] = Oi['Ai_old']\n",
    "    pd[Bj_name] = Dj['Bj_old']\n",
    "    return pd"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the maximise entropy function to calculate $A_1$ and $B_j$ for the Poisson model \n",
    " by plugging the beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration: 1\n",
      "Iteration: 2\n",
      "Iteration: 3\n",
      "Iteration: 4\n",
      "Iteration: 5\n",
      "Iteration: 6\n",
      "Iteration: 7\n",
      "Iteration: 8\n",
      "Iteration: 9\n",
      "Iteration: 10\n",
      "Iteration: 11\n",
      "Iteration: 12\n",
      "Iteration: 13\n",
      "Iteration: 14\n",
      "Iteration: 15\n",
      "Iteration: 16\n",
      "Iteration: 17\n",
      "Iteration: 18\n",
      "Iteration: 19\n",
      "Iteration: 20\n",
      "Iteration: 21\n",
      "Iteration: 22\n"
     ]
    }
   ],
   "source": [
    "# Inver power cost function\n",
    "# Get the balancing factors.\n",
    "london_flows = balance_doubly_constrained(london_flows,'Orig','Dest','O_i','D_j','Dist',-beta_pow,'power')\n",
    "\n",
    "# Now predict the model again using the new Ai and Dj fields.\n",
    "london_flows['SIM_est_pow'] = np.round(london_flows['O_i'] * london_flows['Ai_new'] * london_flows['D_j'] * london_flows['Bj_new'] * \n",
    "                                   np.exp(np.log(london_flows['Dist'])*-beta_pow))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Dest</th>\n",
       "      <th>Abbey Road</th>\n",
       "      <th>Acton Central</th>\n",
       "      <th>Acton Town</th>\n",
       "      <th>Aldgate</th>\n",
       "      <th>Aldgate East</th>\n",
       "      <th>All Saints</th>\n",
       "      <th>Alperton</th>\n",
       "      <th>Amersham</th>\n",
       "      <th>Anerley</th>\n",
       "      <th>Angel</th>\n",
       "      <th>...</th>\n",
       "      <th>Wimbledon</th>\n",
       "      <th>Wimbledon Park</th>\n",
       "      <th>Wood Green</th>\n",
       "      <th>Wood Lane</th>\n",
       "      <th>Wood Street</th>\n",
       "      <th>Woodford</th>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <th>Woodside Park</th>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <th>All</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Orig</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Abbey Road</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>26</td>\n",
       "      <td>600.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Central</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1224.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Acton Town</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>15</td>\n",
       "      <td>15</td>\n",
       "      <td>0</td>\n",
       "      <td>11</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>17</td>\n",
       "      <td>...</td>\n",
       "      <td>30</td>\n",
       "      <td>3</td>\n",
       "      <td>5</td>\n",
       "      <td>12</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>3747.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>42</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>19</td>\n",
       "      <td>...</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>2873.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Aldgate East</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>49</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>21</td>\n",
       "      <td>...</td>\n",
       "      <td>8</td>\n",
       "      <td>1</td>\n",
       "      <td>3</td>\n",
       "      <td>2</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>3172.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodford</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>7</td>\n",
       "      <td>31</td>\n",
       "      <td>33</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>29</td>\n",
       "      <td>...</td>\n",
       "      <td>29</td>\n",
       "      <td>0</td>\n",
       "      <td>10</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>4862.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodgrange Park</th>\n",
       "      <td>0</td>\n",
       "      <td>6</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>530.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woodside Park</th>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "      <td>18</td>\n",
       "      <td>17</td>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>22</td>\n",
       "      <td>...</td>\n",
       "      <td>21</td>\n",
       "      <td>0</td>\n",
       "      <td>6</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>3093.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Woolwich Arsenal</th>\n",
       "      <td>20</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>28</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>...</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>7893.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>All</th>\n",
       "      <td>346</td>\n",
       "      <td>751</td>\n",
       "      <td>2198</td>\n",
       "      <td>7788</td>\n",
       "      <td>7934</td>\n",
       "      <td>446</td>\n",
       "      <td>741</td>\n",
       "      <td>256</td>\n",
       "      <td>174</td>\n",
       "      <td>8108</td>\n",
       "      <td>...</td>\n",
       "      <td>6278</td>\n",
       "      <td>599</td>\n",
       "      <td>2141</td>\n",
       "      <td>1576</td>\n",
       "      <td>402</td>\n",
       "      <td>700</td>\n",
       "      <td>240</td>\n",
       "      <td>740</td>\n",
       "      <td>4427</td>\n",
       "      <td>1541959.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>399 rows × 399 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "Dest              Abbey Road  Acton Central  Acton Town  Aldgate  \\\n",
       "Orig                                                               \n",
       "Abbey Road                 0              0           0        0   \n",
       "Acton Central              0              0           0        0   \n",
       "Acton Town                 0              0           0       15   \n",
       "Aldgate                    0              0           2        0   \n",
       "Aldgate East               0              0           2       49   \n",
       "...                      ...            ...         ...      ...   \n",
       "Woodford                   0              0           7       31   \n",
       "Woodgrange Park            0              6           0        0   \n",
       "Woodside Park              0              0           5       18   \n",
       "Woolwich Arsenal          20              0           0        0   \n",
       "All                      346            751        2198     7788   \n",
       "\n",
       "Dest              Aldgate East  All Saints  Alperton  Amersham  Anerley  \\\n",
       "Orig                                                                      \n",
       "Abbey Road                   0           0         0         0        0   \n",
       "Acton Central                0           0         0         0        0   \n",
       "Acton Town                  15           0        11         1        0   \n",
       "Aldgate                     42           0         0         0        0   \n",
       "Aldgate East                 0           0         1         0        0   \n",
       "...                        ...         ...       ...       ...      ...   \n",
       "Woodford                    33           0         0         0        0   \n",
       "Woodgrange Park              0           0         0         0        0   \n",
       "Woodside Park               17           0         3         0        0   \n",
       "Woolwich Arsenal             0          28         0         0        0   \n",
       "All                       7934         446       741       256      174   \n",
       "\n",
       "Dest              Angel  ...  Wimbledon  Wimbledon Park  Wood Green  \\\n",
       "Orig                     ...                                          \n",
       "Abbey Road            0  ...          0               0           0   \n",
       "Acton Central         0  ...          0               0           0   \n",
       "Acton Town           17  ...         30               3           5   \n",
       "Aldgate              19  ...          7               0           2   \n",
       "Aldgate East         21  ...          8               1           3   \n",
       "...                 ...  ...        ...             ...         ...   \n",
       "Woodford             29  ...         29               0          10   \n",
       "Woodgrange Park       0  ...          0               0           0   \n",
       "Woodside Park        22  ...         21               0           6   \n",
       "Woolwich Arsenal      0  ...          0               0           0   \n",
       "All                8108  ...       6278             599        2141   \n",
       "\n",
       "Dest              Wood Lane  Wood Street  Woodford  Woodgrange Park  \\\n",
       "Orig                                                                  \n",
       "Abbey Road                0            0         0                0   \n",
       "Acton Central             0            0         0                2   \n",
       "Acton Town               12            0         2                0   \n",
       "Aldgate                   2            0         1                0   \n",
       "Aldgate East              2            0         1                0   \n",
       "...                     ...          ...       ...              ...   \n",
       "Woodford                  0            0         0                0   \n",
       "Woodgrange Park           0            0         0                0   \n",
       "Woodside Park             0            0         0                0   \n",
       "Woolwich Arsenal          0            0         0                0   \n",
       "All                    1576          402       700              240   \n",
       "\n",
       "Dest              Woodside Park  Woolwich Arsenal        All  \n",
       "Orig                                                          \n",
       "Abbey Road                    0                26      600.0  \n",
       "Acton Central                 0                 0     1224.0  \n",
       "Acton Town                    2                 0     3747.0  \n",
       "Aldgate                       1                 0     2873.0  \n",
       "Aldgate East                  1                 0     3172.0  \n",
       "...                         ...               ...        ...  \n",
       "Woodford                      0                 0     4862.0  \n",
       "Woodgrange Park               0                 0      530.0  \n",
       "Woodside Park                 0                 0     3093.0  \n",
       "Woolwich Arsenal              0                 0     7893.0  \n",
       "All                         740              4427  1541959.0  \n",
       "\n",
       "[399 rows x 399 columns]"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "# Check out the matrix\n",
    "pd.pivot_table(london_flows,values='SIM_est_pow',index ='Orig',columns='Dest',fill_value=0,aggfunc=sum,margins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# check the goodness-of-fit\n",
    "CalcRSqaured (london_flows['Total'],london_flows['SIM_est_pow'])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exponential cost function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get the balancing factors. NB Setting of new field names for Ai and Bj.\n",
    "london_flows = balance_doubly_constrained(london_flows,'Orig','Dest','O_i','D_j','Dist',-beta_exp,'exponential','Ai_exp','Bj_exp')\n",
    "\n",
    "# Now predict the model again using the new Ai and Dj fields.\n",
    "london_flows['SIM_est_exp'] = np.round(london_flows['O_i'] * london_flows['Ai_exp'] * london_flows['D_j'] * london_flows['Bj_exp'] * \n",
    "                                   np.exp(london_flows['Dist']*-beta_exp))\n",
    "# Check out the matrix\n",
    "pd.pivot_table(london_flows,values='SIM_est_exp',index ='Orig',columns='Dest',fill_value=0,aggfunc=sum,margins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# check the goodness-of-fit\n",
    "CalcRSqaured (london_flows['Total'],london_flows['SIM_est_exp'])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scenario"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scenario A:\n",
    "\n",
    " assume that **Canary Wharf** has a 50% decrease in jobs after Brexit. Using the \n",
    "calibrated parameter $\\beta$, compute the new flows for scenario A. Make sure the number of commuters is conserved, and explain how you ensured this.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a new variable with the altered jobs\n",
    "def new_job(row):\n",
    "    if row[\"Dest\"] == \"Canary Wharf\":\n",
    "        val = 0.5*row[\"Dj_destjob\"]\n",
    "    else:\n",
    "        val = row[\"Dj_destjob\"]\n",
    "    return val\n",
    "        \n",
    "london_flows[\"Dj_destjob_a\"] = london_flows.apply(new_job, axis =1)\n",
    "#london_flows.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get the balancing factors. NB Setting of new field names for Ai and Bj.\n",
    "london_flows = balance_doubly_constrained(london_flows,'Orig','Dest','O_i',\"Dj_destjob_a\",'Dist',-beta_exp,'exponential','Ai_exp_a','Bj_exp_a')\n",
    "\n",
    "# Now predict the model again using the new Ai and Dj fields.\n",
    "london_flows['SIM_est_exp_a'] = np.round(london_flows['O_i'] * london_flows['Ai_exp_a'] * london_flows['Dj_destjob_a'] * london_flows['Bj_exp_a'] * \n",
    "                                   np.exp(london_flows['Dist']*-beta_exp))\n",
    "# Check out the matrix\n",
    "pd.pivot_table(london_flows,values='SIM_est_exp',index ='Orig',columns='Dest',fill_value=0,aggfunc=sum,margins=True)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scenario B\n",
    "\n",
    "set up the adjusted $\\beta$, we assume that $k$ is the increase in cost of transport, meaning the original cost is $c$ and new cost is $kc$\n",
    "\n",
    "According to research (nature of transport cost and pricing), the  transport costs are proportional to distance\n",
    "$f(d)=kd_{i j}$\n",
    "Given a cost function that aligns with power law\n",
    "$f(d)=d_{i j}^{-\\beta}$\n",
    "To adjust the parameter $\\beta$ in the cost function reflecting the increase in cost of transport\n",
    "\n",
    "$(kd_{i j})^{-\\beta}=d_{i j}^{-\\beta_{adjusted}}$\n",
    "\n",
    "After taking logarithmic of both sides, the equation will look like the following \n",
    "\n",
    "$ \\beta(log(k)+ log(d_{i j})=\\beta_{adjusted}log(d_{ij})$\n",
    "\n",
    "\n",
    "So $\\beta_{adjusted}=\\beta \\frac{ log(d_{i j})+log(k)}{log(d_{i j})}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def beta_b_pow(beta,k):\n",
    "    #beta: the calibrated beta\n",
    "    beta_adj=beta*(london_flows['log_Dist']+np.log(k))/london_flows['log_Dist']\n",
    "    return beta_adj.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# assume the cost of transport has increased 25%\n",
    "beta_pow_25=beta_b(beta_pow,1.25)\n",
    "beta_pow_25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Inver power cost function\n",
    "# Get the balancing factors.\n",
    "london_flows_b = balance_doubly_constrained(london_flows.copy(),'Orig','Dest','O_i','D_j','Dist',-beta_pow_25,'power')\n",
    "\n",
    "# Now predict the model again using the new Ai and Dj fields.\n",
    "london_flows_b['SIM_est_pow_25'] = np.round(london_flows['O_i'] * london_flows['Ai_new'] * london_flows['D_j'] * london_flows['Bj_new'] * \n",
    "                                   np.exp(np.log(london_flows['Dist'])*-beta_pow_25))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# Check out the matrix\n",
    "pd.pivot_table(london_flows_b,values='SIM_est_pow_25',index ='Orig',columns='Dest',fill_value=0,aggfunc=sum,margins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# assume the cost of transport has increased 50%\n",
    "beta_exp_50=beta_b(beta_exp,1.5)\n",
    "beta_exp_50"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Get the balancing factors. NB Setting of new field names for Ai and Bj.\n",
    "london_flows_b = balance_doubly_constrained(london_flows,'Orig','Dest','O_i','D_j','Dist',-beta_exp_50,'exponential','Ai_exp','Bj_exp')\n",
    "\n",
    "# Now predict the model again using the new Ai and Dj fields.\n",
    "london_flows_b['SIM_est_exp_b2'] = np.round(london_flows['O_i'] * london_flows['Ai_exp'] * london_flows['D_j'] * london_flows['Bj_exp'] * \n",
    "                                   np.exp(london_flows['Dist']*-beta_exp_50))\n",
    "# Check out the matrix\n",
    "pd.pivot_table(london_flows,values='SIM_est_exp_b2',index ='Orig',columns='Dest',fill_value=0,aggfunc=sum,margins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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