S08b_Excercises_solutions.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Execute before using this notebook if using google colab\n",
    "kernel = str(get_ipython())\n",
    "if 'google.colab' in kernel:    \n",
    "    !wget https://raw.githubusercontent.com/fredzett/rmqa/master/utils.py -P local_modules -nc \n",
    "    !npx degit fredzett/rmqa/data data\n",
    "    import sys\n",
    "    sys.path.append('local_modules')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from sklearn.cluster import KMeans\n",
    "import matplotlib.pyplot as plt\n",
    "from utils import plot_scatter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement the formula for the __Euclidian distance__ in python. \n",
    "\n",
    "\n",
    "$$d(p,q) = \\sqrt{\\sum_{n=1}^N (p_i - q_i)^2}$$\n",
    "\n",
    "Use the following assumptions:\n",
    "\n",
    "- solution should be a function that takes two parameters $p$ and $q$\n",
    "\n",
    "- the parameters should be of the type `np.array`\n",
    "\n",
    "Calculate the euclidean distance for the following points:\n",
    "\n",
    "- $p = (2, 3, 1)$ and $q = (4, 1, 2)$\n",
    "\n",
    "- $p = (2, 3, 1)$ and $q = (2, 3, 1)$\n",
    "\n",
    "- $p = (1, 2, 3)$ and $q = (-1, -2, -3)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def euclidean(p,q):\n",
    "    return np.sqrt(np.sum((p-q)**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = np.array((2,3,1))\n",
    "q = np.array((4,1,2))\n",
    "euclidean(p,q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = np.array((2,3,1))\n",
    "q = p\n",
    "euclidean(p,q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.483314773547883"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = np.array((1, 2, 3))\n",
    "q = np.array((-1, -2, -3))\n",
    "euclidean(p,q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Your are given the below data set containing $2$ explanatory variables and the result of a segmentation exercise conducted by a marketing company. \n",
    "\n",
    "You are asked to conduct a clustering analysis using the __K-Means algorithm__ to analyse if the segmentation can be validated by thorough data analysis. \n",
    "\n",
    "Answer the following questions:\n",
    "\n",
    "- what $K$ did you choose? Why?\n",
    "\n",
    "- how many iterations did the algorithm need? (note: use `n_iter_` to find out)\n",
    "\n",
    "- how do the segmentation result and the clustering results compare?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "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></th>\n",
       "      <th>X1</th>\n",
       "      <th>X2</th>\n",
       "      <th>Segmentation</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>8.032108</td>\n",
       "      <td>-4.157312</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-8.769474</td>\n",
       "      <td>7.120189</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>9.906003</td>\n",
       "      <td>9.891514</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-3.237460</td>\n",
       "      <td>9.197322</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-9.101472</td>\n",
       "      <td>6.525046</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         X1        X2  Segmentation\n",
       "0  8.032108 -4.157312             4\n",
       "1 -8.769474  7.120189             3\n",
       "2  9.906003  9.891514             0\n",
       "3 -3.237460  9.197322             1\n",
       "4 -9.101472  6.525046             3"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_csv(\"./data/cluster2.csv\")\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solution Exercise 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = df.loc[:,[\"X1\",\"X2\"]].values # Convert dataframe to nummpy (not needed but easier to slice data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is no way to determine what the correct number of K is. However, given our data set is 2-dimensional we can plot the data and visually inspect if clusters or groups seem to be appear."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can visually identify 5 clusters. It seems to be a sensible approach to start with 5 clusters, i.e. set $K = 5$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "KMeans(n_clusters=5)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = 5\n",
    "model = KMeans(n_clusters=K)\n",
    "model.fit(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The algorithm needed only two iterations to come up with a result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.n_iter_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How does our model compare with the marketing segmentation?\n",
    "\n",
    "Problem: we cannot compare the labels directly as the assigning of numbers to a specific cluster of observations is arbitrarily. We can, however, for example compare if the mean for each variable is similar. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "df[\"Cluster\"] = model.labels_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Segmentation\n",
       "0    9.352633\n",
       "1    2.403563\n",
       "2    5.495775\n",
       "3   -0.980329\n",
       "4    1.879628\n",
       "dtype: float64"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.groupby(\"Segmentation\")[[\"X1\",\"X2\"]].mean().mean(axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Cluster\n",
       "0    1.879628\n",
       "1    2.403563\n",
       "2    9.352633\n",
       "3    5.495775\n",
       "4   -0.980329\n",
       "dtype: float64"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.groupby(\"Cluster\")[[\"X1\",\"X2\"]].mean().mean(axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we could also plot the clusters to see if we identified the visually apparent groups. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1], c=df[\"Cluster\"]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The algorithm seems to have identified the five segments. "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "mlbasic",
   "language": "python",
   "name": "mlbasic"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}