week1_tutorial.ipynb

{
 "metadata": {
  "name": "week1_tutorial"
 },
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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Tutorial #1: Probability distributions and error bars"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Set matplotlib to plot in the notebook\n",
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note: unlike MATLAB, Python has the concepts of namespaces and explicit imports.\n",
      "Objects from the ``numpy`` and ``matplotlib`` namespaces are availible because we are using ``pylab``,\n",
      "although this is fine in an interactive or notebook session it's considered proper to explicitly name them\n",
      "(i.e. ``import numpy as np; imoprt matplotlib.pyplot as plt``) when writing persistant code.\n",
      "\n",
      "Let's also import our plotting backend manipulator for attractive plots and some utilities to go along with the course"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import utils\n",
      "import seaborn\n",
      "seaborn.set()\n",
      "colors = seaborn.color_palette()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "(1) Exploring a simple data set; one variable, one condition"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Simulating data is incredibly useful.\n",
      "\n",
      "The function ``randn()`` generates data\n",
      "containing values that are randomly drawn from a gaussian distribution\n",
      "with a zero mean and unit variance (a standard normal).\n",
      "\n",
      "You can also generate random gaussian values with the ``normal()`` function,\n",
      "which allows you to specify the mean and variance of the distribution you will sample from.\n",
      "\n",
      "Take care not to confuse ``randn()`` with ``rand()``, which draws from a uniform distribution between 0 and 1.\n",
      "\n",
      "We will be using this function regularly throughout the course.\n",
      "\n",
      "Note that if you are not using ``pylab``, both these functions are found in the ``numpy.random`` subpackage of numpy."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Start by generating a dataset and calculating some statistics\n",
      "d = randn(100)\n",
      "m = d.mean()\n",
      "s = d.std()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Now plot a histogram of the data with 15 bins\n",
      "hist(d, 15)\n",
      "\n",
      "m_y = 2\n",
      "\n",
      "# Add a plot of the mean and standard deviation range\n",
      "plot(m, m_y, \"ko\")\n",
      "plot([m - s, m + s], [m_y] * 2, \"k--\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Now let's find the medium and interquartile range\n",
      "med = median(d)\n",
      "iqr = utils.percentiles(d, [25, 75])\n",
      "\n",
      "# Plot the data with this summary\n",
      "hist(d, 15)\n",
      "plot(med, m_y, \"ko\")\n",
      "plot(iqr, [m_y] * 2, \"k--\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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wjDAGAMAywhgAAMsIYwAALCOMAQCwjDAGAMAywhgAAMsIYwAALCOMAQCwjDAGAMAywhgA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\nDH5vS4mg0AAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that the mean and median are very similar for these data. This is an important feature of the gaussian distribution.\n",
      "\n",
      "If data is gaussian then (1) the mean and the median will on-average approximate\n",
      "each other (more so, the more data you have) and (2) mean +/- 1 standard\n",
      "deviation spans from the 16th to the 84th percentile (i.e. 50% +/- 68%/2).\n",
      "\n",
      "Lets check this assertion by plotting these percentiles around the mean\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Let's define a function to streamline the central tendency and error bar plots from above\n",
      "def ctend_plot(point, ci, y, color, label):\n",
      "    plot(ci, [y, y], \"-\", color=color, linewidth=4, label=label)\n",
      "    plot(point, y, \"o\", color=color, markersize=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hist(d, 15)\n",
      "ctend_plot(m, [m - s, m + s], m_y, colors[1], \"std dev\")\n",
      "\n",
      "ci = utils.percentiles(d, [16, 84])\n",
      "ctend_plot(med, ci,  m_y - 1, colors[2], \"68% CI\")\n",
      "legend();"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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O9UI9+z46fEiKiWh0nYtjIxS25Y8a2CG+ZZpw8v70Y62He07yz46A7yGMgRC0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\njDAGAMCw/wNyudKUGlHObAAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now let's look at the effect of an outlier. Notice that when we add\n",
      "outliers to the data the mean does not accurately describe the\n",
      "central tendency of the data and the standard deviation does not\n",
      "accurately reflect the spread in the data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "d1 = concatenate([d, [40, 41, 45]])\n",
      "m1 = d1.mean()\n",
      "s1 = d1.std()\n",
      "med1 = median(d1)\n",
      "ci1 = percentile(d1, [16, 84])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hist(d1, 15)\n",
      "# Find some space near the top of the plot\n",
      "m_y = histogram(d1, 15)[0].max() + 5\n",
      "ctend_plot(m1, [m1 - s1, m1 + s1], m_y, colors[1], \"std dev\")\n",
      "ctend_plot(med1, ci1, m_y - 2.5, colors[2], \"68% CI\")\n",
      "legend();"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that in this plot the mean is much higher than the median and the\n",
      "standard deviation is massive.  \n",
      "\n",
      "Neither the median nor the percentiles are drastically affected by the outliers.\n",
      "\n",
      "Notice that the mean is higher than the 84th percentile.  This means that the mean is higher than 84% of\n",
      "the values in the data.\n",
      "\n",
      "Given this distribution the mean and standard deviation are poor representations of the central tendency and spread of the data."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "(2) Probability distributions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The equation for a gaussian is:\n",
      "\n",
      "$$p(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}}\\exp[-\\frac{1}{2}(\\frac{x - \\mu}{\\sigma})^2]$$\n",
      "\n",
      "where $\\mu$ is the mean and $\\sigma$ is the standard deviation.\n",
      "\n",
      "We can plot the gaussian function with the approriate parameters on both histograms and see how well they fit the data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# First create a vector of x values spanning the support of the histogram\n",
      "x = linspace(d.min(), d.max(), 1000)\n",
      "\n",
      "# Let's define a function to calculate the gaussian function for a given x, mean, and std\n",
      "gauss = lambda x, m, s: (1 / (s * sqrt(2 * pi)) * exp(-0.5 * ((x - m) / s) ** 2))\n",
      "\n",
      "# We could iterate through the x values and calculate a y value for each using this equation\n",
      "y0 = zeros_like(x)\n",
      "for i, x_i in enumerate(x):\n",
      "    y0[i] = gauss(x_i, m, s)\n",
      "\n",
      "# You can also do it in one step with a vectorized computation.\n",
      "# In general, avoid for loops, as vectorized expressions are much faster.\n",
      "y = gauss(x, m, s)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "NOTE: MATLAB and Python (specifically numpy) differ here.\n",
      "Regular arithmetic is elementwise in numpy\n",
      "i.e.: x * y in python is equivalent to x *. y in MATLAB\n",
      "Use the dot() function or np.matrix objects for matrix operations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now plot the gaussian curve over the histogram"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Use a normed histogram to match the range of the probaiblity density function\n",
      "hist(d, 15, normed=True)\n",
      "plot(x, y);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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rUo/ccUghWMzkUp92HEWDqRn35myGnneOIprWhtQ1CAsw4tXa3bwDFQFgMZML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\nRK5CYII=\n"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Calculate the height of the gaussian for the outlier distribution and plot\n",
      "x1 = linspace(d1.min(), d1.max(), 10000)\n",
      "y1 = gauss(x1, m1, s1)\n",
      "hist(d1, 15, normed=True)\n",
      "plot(x1, y1);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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TseExio8aKXfkSEVb0RoZHquY8BhFu6IUHRalKFdUvz+HO8IIeAyZro42LV97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\nYII=\n"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that the Gaussian is a good description of the data when the data\n",
      "does in fact come from a Gaussian distribution but is a poor\n",
      "description when the data is non-Gaussian because of the outliers."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In case you were wondering, there are easier ways to generate gaussian distributions in Python.\n",
      "\n",
      "The ``scipy.stats`` package offers many distribution objects that let you evaluate pdfs, cdfs, and draw random values"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy import stats\n",
      "x = linspace(-4, 4, 10001)\n",
      "hist(stats.norm(0, 1).rvs(1000), 30, normed=True, alpha=.6)\n",
      "plot(x, stats.norm(0, 1).pdf(x), linewidth=6)\n",
      "plot(x, gauss(x, 0, 1), linewidth=3);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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7cUdMyh8vLJfYUe7GZ5XjWH04xpaXa1bxqMDJrDs+iUpm2KQkoMcXHRFCrDtJ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\nAAAAAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note how the distributions lie directly on top of each other"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "(3) Error bars"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First lets simulate a population of 10,000 data points.\n",
      "Assume that for whatever we are  measuring this represents the full\n",
      "population of measurements."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s = 10000\n",
      "N = randn(s)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To randomly sample from the population, we will use the ``randint`` function, which draws from a \"discrete uniform\" distibution.\n",
      "\n",
      "We will use the returned values as indices into the population array.\n",
      "\n",
      "Note that this will sample *with* replacement."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sample_size = 100\n",
      "n = N[randint(0, s, sample_size)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Get the standard error of the mean and 95% CI for the sample\n",
      "sem1 = n.std() / sqrt(sample_size)\n",
      "m1 = n.mean()\n",
      "ci1 = m1 - 2 * sem1, m1 + 2 * sem1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can show that the 95% confidence interval obtained from the standard\n",
      "error of the mean aproximates the variability in the mean that would be\n",
      "obtained when many independent random samples are taken from the\n",
      "population."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n_samples = 10000\n",
      "mu = array([N[randint(0, s, sample_size)].mean() for i in xrange(n_samples)])\n",
      "sem2 = mu.std()\n",
      "ci2 = utils.percentiles(mu, [2.5, 97.5])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we'll visualize these results."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hist(n)\n",
      "ctend_plot(N.mean(), ci2, 3, colors[4], \"poplation mean\")\n",
      "ctend_plot(n.mean(), ci1, 2, colors[2], \"sample mean\")\n",
      "legend(loc=\"best\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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20cnKSn1S9J4+/uotfds2UBckZ7g6JoBaULyAB6mqKFevih81KzykxvXtfH2V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\np7A0cZsAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "(4) Nonparametric approaches to error bars"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As we showed above when the data is gaussian then the standard error of\n",
      "the mean, can be calculated with the equation sd ./ sqrt(n).  \n",
      "\n",
      "In the above simulation roughly 95% of the sample means fell within the 95%\n",
      "confidence interval derived from the standard error. This will not be the\n",
      "case if the data is not gaussian.\n",
      "\n",
      "If we do not want to make the gaussian\n",
      "assumption we can calculate central tendency (median) and error by\n",
      "randomly sampling from our data with replacement."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sample_size = 20\n",
      "s = 10000\n",
      "n = N[randint(0, s, sample_size)]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n_boots = 10000\n",
      "\n",
      "# We can bootstrap the two different statistics\n",
      "me = zeros(n_boots)\n",
      "mn = zeros(n_boots)\n",
      "for i in xrange(n_boots):\n",
      "    sample = n[randint(0, sample_size, sample_size)]\n",
      "    me[i] = median(sample)\n",
      "    mn[i] = mean(sample)\n",
      "\n",
      "# Compute the median and its confidence interval\n",
      "med = median(me)\n",
      "ci95med = utils.percentiles(me, [2.5, 97.5])\n",
      "\n",
      "# Do the same for the mean\n",
      "xbar = mean(mn)\n",
      "ci95mean = utils.percentiles(mn, [2.5, 97.5])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The MATLAB tutorial asks us to think of a good way to visualize the data.\n",
      "\n",
      "This might be one?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Define a function to streamline the central tendency and error bar plots from above\n",
      "def ctend_plot(point, ci, y, color, label):\n",
      "    plot(ci, [y, y], \"-\", color=color, linewidth=4, label=label)\n",
      "    plot(point, y, \"o\", color=color, markersize=10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hist(me, 20, alpha=0.5, normed=True)\n",
      "hist(mn, 20, alpha=0.5, normed=True)\n",
      "plot_height = ylim()[1]\n",
      "ctend_plot(med, ci95med, plot_height * .75, colors[0], \"median\")\n",
      "ctend_plot(xbar, ci95mean, plot_height * .70, colors[1], \"mean\")\n",
      "legend(loc=\"best\");"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 22
    }
   ],
   "metadata": {}
  }
 ]
}