01-Regression/LRinference.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Inference statistics for linear regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have seen how we can [fit a model](https://github.com/Mashimo/datascience/blob/master/01-Regression/moneyball.ipynb) to existing data using [linear regression](https://github.com/Mashimo/datascience/blob/master/01-Regression/LinearRegression.ipynb). Now we want to assess how well the model describes those data points (every Outcome = Model + Error) and will use some statistics for it. \n",
    "\n",
    "As an example we access some available diamond data: prices in Singapore dollars and weights in carats (the standard measure of diamond mass, equal to 0.2 g).  \n",
    "The diamond data can be downloaded from [the Journal of Statistics Education](http://ww2.amstat.org/publications/jse/jse_data_archive.htm)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "diamondData = pd.read_csv(\"../datasets/diamond.dat.txt\", delim_whitespace=True, header=None, names=[\"carats\",\"price\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>carats</th>\n",
       "      <th>price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.17</td>\n",
       "      <td>355</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.16</td>\n",
       "      <td>328</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.17</td>\n",
       "      <td>350</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.18</td>\n",
       "      <td>325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.25</td>\n",
       "      <td>642</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   carats  price\n",
       "0    0.17    355\n",
       "1    0.16    328\n",
       "2    0.17    350\n",
       "3    0.18    325\n",
       "4    0.25    642"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diamondData.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fit a model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Is there a relationship between the diamond price and its weight?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our first goal should be to determine whether the data provide evidence of an association between price and carats. If the evidence is weak, then one might argue that bigger diamonds are not better!\n",
    "\n",
    "To evaluate the model we will use a special Python package, **statsmodel**, which has nice available functions for this.  \n",
    "Statsmodel is a package based on the original - later removed - statistics module of SciPy (Scientific Python) by Jonathan Taylor, corrected, improved, tested and released as a new package during the Google Summer of Code 2009.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![statsmodels](../images/statsmodels_hybi_banner.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since statsmodels offers also functions to fit a linear regression model, we do not need to import and use *sklearn* to fit the model but we can do everything with *statsmodels*.  \n",
    "We will use its function *OLS()* that fits a linear regression based on the Ordinary Least Squares algorithm.  \n",
    "\n",
    "The model we want to get is :  \n",
    "y_hat = beta0 + beta1 * X\n",
    "\n",
    "where y_hat is the estimated Diamond Price (the dependent variable) and x is the diamond Weight (the independent variable).  \n",
    "\n",
    "An intercept is not included by default and should be added by the user:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = sm.add_constant(diamondData.carats) # this append a column of ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "simpleModel = sm.OLS(diamondData.price, X).fit() # fit the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "const     -259.625907\n",
       "carats    3721.024852\n",
       "dtype: float64"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.params  # here are the beta coefficients (intercept and slope of the linear regression line)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The intercept (beta0) is -259.6 and the slope (beta1) is 3721  \n",
    "Therefore our simple (one variable only) model looks like:  \n",
    "\n",
    "Diamond Price = -259.6 + 3721 * Diamond Weight"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot the obtained regression line together with the input data X.\n",
    "We can do it by drawing a line using the beta parameters just calculated or also plotting the fitted values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mPBsTlTz9YD0yBoCqqqqgQ/CF5Rk+Uck1LHl+8AEMGuSKl+uvh7FjISfnx/Vh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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fb57c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1,1,1)\n",
    "\n",
    "ax.scatter(diamondData.carats, diamondData.price)\n",
    "   # draw linear regression line\n",
    "x = [0.1,0.4]\n",
    "y = [-259.6 + 3721 * i for i in x] \n",
    "ax.plot(x, y)\n",
    "   # alternatively, plot the fitted values\n",
    "#y_hat = simpleModel.fittedvalues\n",
    "#ax.plot(diamondData.carats, y_hat)\n",
    "   # pretty-up the plot\n",
    "fig.suptitle(\"Relation between diamonds' price and weight\")\n",
    "ax.set_ylabel('Price [SIN $]')\n",
    "ax.set_xlabel('Weight [carat]')\n",
    "ax.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This answers our first question.  \n",
    "There is a relationship between price and weight of diamond and we can model it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analyse the model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Which kind of relation is between weight and price?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next question would be to find out if the relation is linear and how it looks like."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Coefficients interpretation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Beta1 (the slope of the regression line) is the **expected change in response for a 1 unit change in the predictor.**  \n",
    "In our case, we expect 3721 Singapore dollars increase in price for every carat increase in mass of diamond.  \n",
    "This is within the restricted range considered; extrapolation of the regression line for bigger diamond stones would not be advisable as these stones are rarer and command a different price range.\n",
    "\n",
    "Beta0 (the intercept of the regression line) is the **expected price when the weight is zero.**\n",
    "This does not always make sense and in our case the negative intercept is even more puzzling because it suggests that a zero-carat diamond ring has a negative economic value!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting a more interpretable intercept"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The intercept -259.63 is the expected price of a 0 carat diamond.  \n",
    "Which does not make much sense (unless you consider it the cost of bothering the diamond expert when you ask the price of a non-existing diamond :) )  \n",
    "\n",
    "It can be ignored (the model applies only to a restricted range of the data, say starting from 0.1 Carats) or can be an indication that a different model could be more precise, for example a non-linear regression.  \n",
    "\n",
    "We can also use instead the expected price for a more suitable weight, for example the average diamond weight."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2041666666666667"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diamondData.carats.mean() # this is the weight mean of our dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of X as input for the model, we take X centered around the mean, i.e. we shift X of a value equal to the sample mean:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "XmeanCentered = diamondData.carats - diamondData.carats.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "const      500.083333\n",
       "carats    3721.024852\n",
       "dtype: float64"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "XmeanCentered = sm.add_constant(XmeanCentered) # this append a column of ones\n",
    "meanCenteredModel = sm.OLS(diamondData.price, XmeanCentered).fit() # fit a new model\n",
    "meanCenteredModel.params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, **the slope is the same as the previous model, only the intercept shifted**.  \n",
    "This is always valid when you shift your X values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus $500.1 is the **expected price for the average sized diamond** of the initial dataset (=0.2042 carats).  \n",
    "This is an intercept making much more sense."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can shift the X input by a certain value but you can also re-scale them.  \n",
    "This can be useful when one unit is quite large and we would prefer a finer unit.  \n",
    "For example, in our case, one carat is worth of almost 4K SIN$ and could make sense to talk about tenth of carats (= 1/10). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Xtenth = diamondData.carats *10 # rescale the X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "const    -259.625907\n",
       "carats    372.102485\n",
       "dtype: float64"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Xtenth = sm.add_constant(Xtenth)\n",
    "tenthModel = sm.OLS(diamondData.price, Xtenth).fit() # again fit the model\n",
    "tenthModel.params"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The intercept is the same as in the original model, only the slope coefficient is divided by 10**.  \n",
    "This is always valid when you re-scale the X values.  \n",
    "\n",
    "We expect a **372.102 (SIN) dollar change in price for every 1/10th of a carat increase** in mass of diamond.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Predicting the price of a diamond"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Once we have a model of the relation, we can use it for predictions.  \n",
    "The statsmodel package has a method *predict()* associated to each model, that takes a new set of input and will output the predicted values, according to the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say that I want to buy a 0.2 carats diamond. How much should I expect it to cost?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I can use the beta parameters estimated by the model and just putting them into the linear regression formula:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "484.57906311853975"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.params[0] + 0.2*simpleModel.params[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I expect to pay around 485 SIN $.  \n",
    "\n",
    "Or I can use the *predict()* function available in the *statsmodel* package:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "newDiamond = [1, 0.2] # remember to add always the intercept!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 484.57906312])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.predict(newDiamond)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's also possible to pass a list of values to predict:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "newDiamonds = sm.add_constant([0.16, 0.27, 0.34]) # add the intecept"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  335.73806906,   745.05080273,  1005.52254234])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.predict(newDiamonds)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Result: for 0.16, 0.27, and 0.34 carats, we predict the prices to be 335.74, 745.05, 1005.52 (SIN) dollars"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Model fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How strong is the relationship?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know that there is a relationship between diamonds carats and prices, we would like to know the strength of this relationship. In other words, given a certain diamond weight, can we predict the price with a high level of accuracy? This would be a strong relationship. Or is a prediction of prices based on weight  only slightly better than a random guess? This would be a weak relationship."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Residuals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we have seen previously, the residuals are the difference between the observed (y) and the predicted outcome (y_hat):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = diamondData.price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_hat = simpleModel.fittedvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "85.158566087530232"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max (abs (y - y_hat))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Conveniently, residuals are also stored in the results attribute *resid*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "residuals = simpleModel.resid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "85.158566087530232"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max(abs(residuals))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "85 SIN$ (per defect or excess) is the biggest difference done by the model.\n",
    "\n",
    "Don't confuse errors and residuals.  \n",
    "The error is the deviation of the observed value from the (unobservable) true value of a quantity of interest (for example, a population mean), and the residual is the difference between the observed value and the estimated value of the quantity of interest (for example, a sample mean).  \n",
    " \n",
    "We can learn many things from the residuals.\n",
    "One is that their distribution and properties can give us an indication about the model fit."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Residuals should not show any pattern"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The residuals and their plot can highlight a poor model fit.  \n",
    "Here we plot the residuals versus the fitted values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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lrDTKW3TKWfKpt1SR1FtKRESkNOotJSIiIjIEKm5EREQkVVTcSEVt3bo17hBqkvIWnXJW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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11248c4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1,1,1)\n",
    "ax.plot(simpleModel.fittedvalues, residuals, 'o') # as round marks\n",
    "\n",
    "   # pretty-up the plot\n",
    "ax.plot ((0, 1200), (0,0)) # draw also a line at y=0\n",
    "fig.suptitle(\"Residuals versus fitted prices\")\n",
    "ax.set_ylabel('Residuals [SIN $]')\n",
    "ax.set_xlabel('Price [SIN $]')\n",
    "ax.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The residuals should be **distributed uniformly without showing any pattern** and having a constant variance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we see from the residuals vs. fitted plot that **the variance of the residuals increases** as the fitted values increase (takes a form of a horizontal cone) this is the sign of **heteroscedasticity**.  \n",
    "\n",
    "Homoscedasticity describes a situation in which the error term (that is, the “noise” or random disturbance in the relationship between the independent variables and the dependent variable) is the same across all values of the independent variables.  \n",
    "\n",
    "Heteroscedasticity (the violation of homoscedasticity) is present when **the size of the error term differs across values** of an independent variable.  \n",
    "Examining the scatterplot of the residuals against the predicted values of the dependent variable would show the classic cone-shaped pattern of heteroscedasticity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![heteroscedasticity](../images/lme_flu_residuals_fitted5.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Other patterns could be:  \n",
    "- curvilinear  (indicate is non-linear / missing higher-order term)\n",
    "- a single point is far away from zero (probably an outlier)\n",
    "- a single point is far away from the others in the x-direction (probably an influential point)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Residuals should be normally distributed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **sum of the residuals is expected to be zero** (when there is an intercept).  \n",
    "This [follows directly from the normal equation](http://stats.stackexchange.com/questions/189584/why-do-residuals-in-linear-regression-always-sum-to-zero-when-an-intercept-is-in), i.e. the equation that the OLS estimator solves. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-7.3896444519050419e-13"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(residuals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **mean of the residuals is expected to be zero**.  \n",
    "This [comes directly from the fact that OLS minimises the sum of square residuals](http://math.stackexchange.com/questions/494181/why-the-sum-of-residuals-equals-0-when-we-do-a-sample-regression-by-ols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.5395092608135503e-14"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(residuals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is one of the assumptions for regression analysis: residuals should have a normal (or Gaussian) distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  1.,   1.,   3.,   9.,   8.,  12.,   5.,   6.,   2.,   1.]),\n",
       " array([-85.15856609, -68.70574005, -52.25291401, -35.80008797,\n",
       "        -19.34726193,  -2.89443589,  13.55839015,  30.01121619,\n",
       "         46.46404223,  62.91686826,  79.3696943 ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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EnA00gCOAOyNiZb5rIqW0pez5JEndpA9wBbBTVXGg4ZuBXYHvAje33I6sYC5JklSSKr6n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\nAYYCSZKUGQokSRJgKJAkSdn/A0zW1gxXlS2JAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fb57a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(residuals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It looks normal but we can verify better with a Q-Q plot."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Q-Q plot to verify the residuals distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q-Q plots (stands for a \"quantile-quantile plot\") can be used to check whether the data is distributed normally or not.  \n",
    "\n",
    "It is a plot where the axes are purposely transformed in order to make a normal (or Gaussian) distribution appear in a straight line. In other words, a **perfectly normal distribution would exactly follow a line with slope = 1 and intercept = 0.**\n",
    "\n",
    "Therefore, if the plot does not appear to be - roughly - a straight line, then the underlying distribution is not normal. If it bends up, then there are more \"high flyer\" values than expected, for instance.  \n",
    "The theoretical quantiles are placed along the x-axis. That is, the x-axis is not your data, it is simply an expectation of where your data should have been, if it were normal.\n",
    "\n",
    "The actual data is plotted along the y-axis.\n",
    "\n",
    "The values are the standard deviations from the mean. So, 0 is the mean of the data, 1 is 1 standard deviation above, etc. This means, for instance, that 68.27% of all your data should be between -1 & 1, if you have a normal distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*statsmodels* offers a handy *qqplot()* function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gPoLZIRBMNV1OaemNzJ9/aQ6jFZHulFGCYWaHmNk4M7vIzKLhsbeZWb/shici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\ntxJseS/jKQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10aa9c8d0>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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gPoLZIRBMNV1OaemNzJ9/aQ6jFZHulFGCYWaHmNk4M7vIzKLhsbeZWb/shici\neckd6urghBM5ZcfrlFMfVi2OJZgZ8kfgNPr2HcuwYeOZMWMda9dqqqlIT9LhHgwzeyewAngHcCjB\nBmcx4Kvh44uzGaCI5JmmJqishIYGfttnMJ9rsRonBDND5gFzOfroj/HMMw/mJk4RyalMKhjVwJ+A\nI4DXko7fA3w0G0GJSB5KVC1GjIDVq/G6Oi484gMpyUUyY8+evrR3OwIRKS6ZJBinA/Pd/c2U41uA\nIZ2OSETyT1MTlJfDtGkwdixs3IhNnUpJyS7291qkckpKdqmRU6SHyiTBiBDsrJrq7QRDJSJSLFKq\nFtTXw+LFMHgw7s6kSWOIRFamvTUSWcHkyad1c8Aiki8yWQfjAeD/gAvDxx42d14F3J+twEQkx5J6\nLSgrw2+5hZ19+jB75lyWLVtDc3NfevXazsCB9/Lqq3Hi8TNJ3aBs/nxtUCbSU2WSYFwKrDSzTcBh\nwM8I9iV5kWCPEhEpZO5QX49XVeHAHRPL+NafXuGNUy7gxRf/QXPzjQRNnEEyYXYPRxxxBdHoDSkr\nc2rWiEhPlsleJP80s/cC04CTCHZS/RFwt7u/1ubNIpLXdm7ezFPjP8nJTzfyq8PewsV7DueFFZ8j\n2DdkHjADODPpDsP9HF599XDOO28dN900Vz0XIgJkuFS4u+8BfprlWEQkV9x57c47af7ChQzZezjl\n1LHk9Y3AB9ifUKwhSDIOFI9PZOnShVRXK7kQkUC7Egwzm9zeJ3T3pZmHIyLdLuy1OLyhgfs4jUoa\nwmW+x7E/oXCgL8GwSDpGc3Mf7S0iIvu0t4Lxy3Ze56SfYSIiecbjcV6/6y72/m8lb7y5h8rIO6iP\n/5ZEb0XLhMKAxJTUdAmEpqSKSEvtmqbq7pF2fim5EMljsViMmTPncuo7PsR9fYdw+PTpLH99JKXx\n56iPn0z6hCJhDKApqSLSPtquXaSHiMVijP7AObyn8VRW+Cac1yjnCpbw7fCK1ApFIqGYGD6+DJgS\nXjMRTUkVkbZkutnZR83sV2b2dPj1KzMbl+3gRCR7rp51FVdteo1av4aHGMsJjGIJC5KuSK1QXAYs\nBJYTJBVRYAlQR0nJSbz1rZO0kZmItCqTzc4qCfYjWRL+CUGr+f1mNsvda7IYn4h0ljteV8dlt9/M\nXgZQTj1LKAPOpmU/RWqFIpFQzKSk5HKOPnoYhx76erjGxXfp16+fei5EpFWZDJFcCcxy91uSjt1s\nZmvCc0qWNhCpAAAe5UlEQVQwRPJALBbj6llXcXrtXZy5exurOIpKNoYzRODAIZEowVbrNwAL6dVr\nN0OH9lFCISIZySTBGEiwXXuqB4BrOxeOiGRDbMcOrjrhNL72z2dxeodVi1uBQUlXpfZYQGKr9Uhk\nOZWVj3LzzVd1Z9giUkQy6cFYCnwqzfGzgF91LhwR6bSmJp4e+QGu/+fjPMR4TmAjSyjn4D0WEDRt\nLqe09EYWLLismwMXkWKSSQVjEzDbzD4MrA2PfYDgt9cNZjYzcaG739zpCEWkfcI9RKiqYugrO4LV\nOJmadEGixyJOsDpnWz0WatoUkc7JJMH4PPAKMCL8Sng1PJfggBIMkS7m7ti2bTRfeCEl997LfX0G\n8Xk/ma0tkgtI7rHo1etrvOUtw5M2JlOPhYhkVyabnQ3vikBEpP1isRizZ1/PsqWPMHHHdhZs/wtx\nd6rsCup3LwA+RvpVN6PAXIYOfYTNm3+phEJEuowW2hIpMLFYjNGjp/Dipgu4xTdSxioWU0oVc9jG\nZ8Kr0jVwBhKrbiq5EJGulMk6GAaUAWOBwaQ0irr7OdkJTUTSmX3ld3jPplF812fgWNIMkYqkq7Tq\npojkViazSG4CfgIMB3YC21O+RKQD3L3dx33rVs788feTVuPcGC6albrTaaLfYh0wnkjkNK26KSLd\nKpMhkv8BznH3+7MdjEhPsa+HYtkampv7UlKyi0mTxvC1r13ENdfc2uL4hAmngsMh99zHN1/6G6fG\nPaxalCc9Y7qdToM1LcAZOnQczzzzYDf+hCLS02WSYGwHNmc7EJGeItFD0dh4CfH4PBLDF7fccg8/\n+MGHaW6uTjq+g1/c+hEWcShlPMFiyqhiK9soS3nWtnsuzjrr9K79oUREUmQyRDIPmGtmh2c5FpGi\n5+7Mnn19mFwkeiMADPe/8MYbNxGPn0ki6ZjKBWziH5zB3ymnnqksZhtjOXDb9MSiWfeRbtGs+fMv\n7Y4fT0Rkn0wqGPUE3WRNZrYFaE4+6e4jsxCXSNFIHQ75z3+eDisUqdYQ5O8wiCYWUUkZDWHVoiZp\nD5HUBbMM6IfZRRxxxJVEowvZs6df0hoX6rkQke6XSYJxJzAK+Cmwlf3/XBKRFAcOh8CBu5hC8Neo\nLwBTqaOGqnCGyKksYXHKtW0tmPUI0Wg0WHxL01BFJIcySTA+AUxw90eyHYxIMUj+cG85HJKQriHT\nGMQrLKI8pWrxmTTXwsEWzFJyISK5lkmC8TywI9uBiBSy1maF3Hvv79IMh6Q2ZDpTqaeGP+GUpMwQ\n0YJZIlKYMkkwLgWuM7OL3X1LluMRKRiJSkXrs0JW0KvXrzmw+rB/EaxBjGQRVWHV4gNc0nsrL+zp\nC/FE1eJSYAKwh6B4qAWzRKQwZJJg/BToAzxtZrs5sMnzyGwEJpJr6foY0lUq+vfvRWPjrJRhEMP9\nTPbsmU/69SmWMJULqGESFulF1VHv4ZCKCaz76oVce+0PWLr0Rpqb+1BSspuJEz8M/J4VK6r3HVPz\npojku0wSjP/LehQieaK1oY4FCy4DSFupgNNIN4QR+ChwP0H1IZA8Q+Sx40Zw8iOrqDnmmH3nq6vn\nUV2dPsFR86aIFIpMdlO9sysCEcm11oY6ampWsmrVFE47bVSahk2AozlwGCThKxxyyKnE40Y8PpGp\nLA5niDRzydtP4qoNj2CtVCHSJRJKLkSkUHRqN1UzOwzonXzM3dUAKgUp/YyPIDFobHSee+7rxOPf\nTrnLSD8rJKEfb3vbWzlv/EOc9rPpnLm7ifv7DGbNZ6Zz1cJvaIhDRIpWh1fyNLO+ZnaLmTUR/GZ9\nJeWry5jZlWa2xsx2mdnLXfla0vMsW7aGeHxC2nPx+AR27z6U9ElEYqbHgSK2nKtGHM2Ce27nzD5x\nvK6Oj+/ayoLbvqPkQkSKWiZLhV8HfAT4X+AN4AvAXOAF4PzshZZWCcFKot/r4teRHsbdaW5O3ZE0\nWQSIkX5ducQy3b8ieZnuY6yW5dHpfG5FA4wdCxs3YlOnZj12EZF8lMkQySTgfHd/2MxuB37n7k+Z\n2bPAucDdWY0wibtfBWBmn+2q15CeycwoKWlrqMPp0wd27VqZpgcjitlFnHTSLWzfXk3zm4dzdvMz\nXBN7mj4lfaC+HsrL0zyniEjxyqSCcST7d1PdET4GeAT4UDaCEsmFSZPGEIm0MtQRWcG5536C0tKF\nRCLLSd1QbMSIW/nd7xbzzLq7eX50b27Z9gT9PvkJIps2KbkQkR4pkwrGZmA48BzwJDAV+ANBZePV\n7IUm0r0WLLiMVaum0NjoSTud7l/U6rrrgkWt5sy5gaVLF7Zck+JbS4jefz9UVQUzPVS1EJEeztw7\ntleZmc0C9rr7zWY2DlhG8Ju4BLjE3as7+HxXA19t4xIHSt3970n3fBa4sT2LepnZSGD9hz70IQYM\nGNDiXEVFBRUVFR0JV4pcLBYLE4g1KYtaXXpAU+a+NSmamqCyEhoaoKwMampg8OBWXkFEJD/V1tZS\nW1vb4tj27dv57W9/CzDK3Td05Pk6nGAc8ARmw4CRwFPu/tcM7j8KOOogl2129z1J93Q4wVi/fj0j\nR2oneWm/gy5q5R5UKqqqwAwWLVLVQkSKyoYNGxg1ahRkkGB0ah0MgHA/ki2duP8l4KXOxiGSbW0m\nF6paiIi0qd1NnmY22sw+mXLsfDN7xsyazOwHZnZo9kNs8XpDzey9wDuBXmb23vCrb1e+rsg+7lBX\nByNGwOrVQQVj8WIlFyIiKToyi+QbwAmJB2b2HuBHwK+BawiaPK/IanQH+iawgWDdjX7h9xuAUV38\nuiJB1aK8HKZN27euhYZERETS68gQycnA15MeTwPWufsXAczseeAqYF7Wokvh7tOB6V31/CJppfZa\naIaIiMhBdaSCcQSwNenxGcDypMd/BIZmIyiRvKGqhYhIRjqSYGwlWP8CM+tNMHPk0aTzUaA5e6GJ\n5JB6LUREOqUjCcb9wDVmdjpwNbAb+F3S+ZOAp7MYm0iXanWKtqoWIiKd1pEE4+vAHmA18EXgi+7+\nZtL5C4AHshibSNbFYjFmzpzL8OHjGDr0bIYPH8fMmXOJxWKqWoiIZFG7mzzd/UXgQ2Y2ANjp7ntT\nLikHdmYzOJFsisVijB49hcbGS4jH55FYCrymZiV/fmASv3n3QEruvVfrWoiIZEGHF9py9+2tHH+5\n8+GIdJ3Zs68Pk4uWu6GWxbdT87fHeOPZPZRohoiISFZkspuqSEFatmwN8fiEfY8H0cRiyqljGg8x\nng8POlXJhYhIlijBkB7B3Wlu7ktiWGQqdWxiBGewmnLqmcpi/hMf2Hrjp4iIdIgSDOkRzIySkl0M\nYmtS1WIsJ7CRJZQDTknJrrb3HxERkXbr9GZnIvkm7S6o7nzj/w1k0pbjcXpTTn2YWAQikRVMnnxa\nN0cqIlK8VMGQotDm9NNwXYvpKxtY3/8w3mM3s4Sy8E4nEllOaemNzJ9/aU5/BhGRYqIKhhS8Vqef\n3rKC3veM4brd/yISiUB9PR+cOJFPz7mBpUvH09zch5KS3UyePIb58xuIRqM5/klERIqHEgwpCGmH\nPULppp8OYhuL/EeU/fNxHjtuBKeseQgGDyYKVFfPo7q67ecUEZHO0RCJ5K02hz2StJx+mjpDpI5z\n9rwt7aJZSi5ERLqOEgzJS4lhj5qa0WzZ8iD/+te9bNnyIDU1oxk9esq+JCN5+mnLdS0SM0Sm0tzc\nR9NPRUS6mRIMyUsthz0SlQYjHp9IY+Ms5sy5IThiRskhO5nKzw9Y12Ibg9H0UxGR3FCCIXkpddXN\nZPH4RJYuXRM8aGrinkP+TR0VKetaBDT9VEQkN5RgSN5puepmOkbzm4fjP/85jBjBSa82ccnbT2Ja\n5AK2MSjxLJp+KiKSQ0owJO8kVt2E9H0Tg9jKba8+ilVUwNixRDZt4qpNjzBjxjqGDRvPkCFnMWzY\neGbMWMfatZp+KiKSC5qmKnlp0qQx1NSsTNn51JlKPTVcyGHxPZC086mmn4qI5BdVMCQvLVhwGaWl\nC4lElgPeYobI+v6HwcbHW935VMmFiEjuKcGQvBSNRlm7toEZVY9SNegknowMZWxkKbefWcYH//kU\n/Y49NtchiohIGzREInkr+tprVL+wEbY9gZeVYTU1TE+zYJaIiOQfJRiSf9yD/oqqKjCD+nqsleEQ\nERHJTxoikfwS7nzKtGkwdixs3Nhqr4WIiOQvVTAkP6SpWiixEBEpXKpgSO6paiEiUnRUwZDcUdVC\nRKRoqYIhuaGqhYhIUVMFQ7qNuwe7i6hqISJS9FTBkC4Vi8WYOXMuw4eP4+S3ncnyfsfAtGk0n366\nqhYiIkVMFQzpMrFYjNGjp9C4aRZlXkoNM3CMT9uVbPzHH1l7+OFoGzIRkeKkCoZ0mdmzr+fFTRdQ\n5z+ijgoeYiwnsJF6X0Bj4yzmzLkh1yGKiEgXUYIhXcOd+M9/wRM+gzNYTTn1TGUx2wiW+o7HJ7J0\n6ZocBykiIl1FCYZkX1MTXl7OLdue2Fe1WEJqr4XR3NwHd89JiCIi0rWUYEj2uENdHYwYga1eTdWg\nE5lK/b6qRcrFlJTs0tbqIiJFSgmGZEeadS16TTuHSGRl2ssjkRVMnnxaNwcpIiLdRQmGdE5S1YLV\nq4N1LRYvhsGDWbDgMkpLFxKJLAcSQyFOJLKc0tIbmT//0lxGLiIiXUgJhmTuIKtxRqNR1q5tYMaM\ndQwbNp4hQ85i2LDxzJixjrVrG4hGNUlVRKRYaR0M6bgO7CESjUaprp5HdXW4kqd6LkREeoSCqWCY\n2TvN7IdmttnMdpvZP8xsnpmV5Dq2HqUTe4gouRAR6TkKqYLxbsCALwJPAycCPwT6AJfnMK6eQTuf\niohIBxRMBcPdV7r75939N+6+xd1/BVwPnJPr2Iqedj4VEZEOKqQKRjoDgZdzHUTRUtVCREQyVDAV\njFRmdhwwA/h+rmMpSqpaiIhIJ+Q8wTCzq80s3sbXXjM7PuWeIcByoM7df5ybyItUG+taiIiItFc+\nDJFcD9x+kGs2J74xs7cBq4BH3P2i9r7IrFmzGDBgQItjFRUVVFRUdCDUItfUBJWV0NAAZWVQU6PE\nQkSkh6itraW2trbFse3bt2f8fFZIm02FlYtVwB+B//F2BG9mI4H169evZ+TIkV0dYmFK7bVYtEjD\nISIiwoYNGxg1ahTAKHff0JF7cz5E0l5h5eJh4FmCaamDzewYMzsmp4EVOvVaiIhIF8iHIZL2+hhw\nbPj1fHjMCDa56JWroAqWZoiIiEgXKpgKhrvf6e69Ur4i7q7koqNUtRARkS5WSBUM6SxVLUREpJsU\nTAVDOklVCxER6UaqYBQ7VS1ERCQHVMEoZqpaiIhIjqiCUYxUtRARkRxTBaPYqGohIiJ5QBWMYqGq\nhYiI5BFVMIqBqhYiIpJnVMEoZKpaiIhInlIFo1CpaiEiInlMFYxCo6qFiIgUAFUwComqFiIiUiBU\nwSgEqlqIiEiBUQUj36lqISIiBUgVjHylqoWIiBQwVTDykaoWIiJS4FTByCeqWoiISJFQBSNfqGoh\nIiJFRBWMXFPVQkREipAqGLmkqoWIiBQpVTByQVULEREpcqpgdDdVLUREpAdQBaO7qGohIiI9iCoY\n3UFVCxER6WFUwehKqlqIiEgPpQpGV1HVQkREejBVMLJNVQsRERFVMLJKVQsRERFAFYzsKJKqhbtj\nZrkOQ0REioAqGJ1V4FWLWCzGzJlzGT58HEOHns3w4eOYOXMusVgs16GJiEgBUwUjU0VQtYjFYowe\nPYXGxkuIx+cBBjg1NStZtWoKa9c2EI1GcxyliIgUIlUwMlHgVYuE2bOvD5OLiQTJBYARj0+ksXEW\nc+bckMvwRESkgCnB6Ah3qKuDESNg9eqgarF4MQwenOvIMrJs2Rri8Qlpz8XjE1m6dE03RyQiIsVC\nCUZ7FUnVIsHdaW7uy/7KRSqjubkP7t6dYYmISJFQD8bBFEGvRTpmRknJLsBJn2Q4JSW7NKtEREQy\nogpGW4qsapFq0qQxRCIr056LRFYwefJp3RyRiIgUCyUY6RRZr0VrFiy4jNLShUQiywkqGQBOJLKc\n0tIbmT//0lyGJyIiBUwJRqoir1oki0ajrF3bwIwZ6xg2bDxDhpzFsGHjmTFjnaaoiohIp6gHI6FI\ney0OJhqNUl09j+pqreQpIiLZowoG9KiqRVuUXIiISLb07ApGD61aiIiIdLWCqmCY2b1m9qyZvWZm\nL5jZXWb21oyeTFWLg6qtrc11CAVH71lm9L51nN6zzOh96z4FlWAAq4By4HjgHOC/gMUdeoYeMkMk\nG/QXseP0nmVG71vH6T3LjN637lNQQyTuXp308Hkzuwa4x8x6ufvegz5BUxNUVkJDA5SVQU2NEgsR\nEZEuUFAJRjIzOxI4F1jTruTigQfg+uvVayEiItINCm2IBDO7xsx2Ai8CQ4Gz23XjFVeo10JERKSb\n5LyCYWZXA19t4xIHSt397+Hj64AfAu8E5gI/AT7Zxv2HATR+6Uvwuc/BP/8ZfMlBbd++nQ0bNuQ6\njIKi9ywzet86Tu9ZZvS+dUxjY2Pi28M6eq/lerdMMzsKOOogl2129z1p7h0CPA+Mdvd1rTz/Z4C7\nOx2oiIhIz3Wuu/+sIzfkvILh7i8BL2V4e6/wz0PbuGYlQa/GFuD1DF9HRESkJzoMGEbwWdohOa9g\ntJeZvR94H/AI8ApwHPBNYBBwors35zA8ERERSVJITZ67Cda++DXwJHAb8Gfgw0ouRERE8kvBVDBE\nRESkcBRSBUNEREQKhBIMERERyboelWBkdbO0HsLM3mlmPzSzzWa228z+YWbzzKwk17HlMzO70szW\nmNkuM3s51/HkKzOrMrNnwr+Tj5rZ+3IdUz4zs9PNbKmZ/cvM4mY2Odcx5Tszu8LM/mBmO8xsq5nd\nY2bH5zqufGdmF5vZX8xse/j1ezOb2JHn6FEJBtnYLK3neTdgwBeBEcAs4GJgQS6DKgAlQD3wvVwH\nkq/M7NPADQQL5p0C/AVYaWZH5zSw/NaXoLm9kmARQjm404HvAv8NjCP4u/mAmR2e06jy3/MEi2CO\nBEYRfH7ea2al7X2CHt3kaWaTgHuAQ9u1n4kAYGaXARe7+3G5jiXfmdlngRvd/chcx5JvzOxRYJ27\nfzl8bAS/1G529+tyGlwBMLM4cLa7L811LIUkTGCbgA+5+yO5jqeQmNlLwGXufnt7ru9pFYx9OrxZ\nmiQbCKjsLxkLh9hGAb9JHPPgXzu/BkbnKi7pEQYSVH/0O6ydzCxiZtOAPsDa9t7X4xKMjDdLEwDM\n7DhgBvD9XMciBe1ogpV4t6Yc3wq8pfvDkZ4grJLdBDzi7ptyHU++M7MTzSwGvAEsAj7l7k+29/6C\nTzDM7Oqw2am1r70pDT3XAScDHwP2EmyW1uNk8L4l9n5ZDtS5+49zE3nuZPKeiUheWUTQSzYt14EU\niCeB9wLvJ+gnu8vM3t3emwu+B6OrN0srVh1938zsbcBDwO/dfXpXx5ePMvl/TT0Y6YVDJLuBKck9\nBGZ2BzDA3T+Vq9gKhXowOsbMbgEmAae7+3O5jqcQmdmDwFPu/r/tuT7nm511VjdsllaUOvK+hYnY\nKuCPwAVdGVc+6+T/a5LE3ZvNbD3wUWAp7CtffxS4OZexSfEJk4uzgDOUXHRKhA58XhZ8gtFebWyW\n9g860LTS04SVi4eBZ4DLgcHB5wC4e+r4uYTMbChwJPBOoJeZvTc89ZS778pdZHllIXBHmGj8gWAK\ndB/gjlwGlc/MrC/B7y4LDx0b/r/1srs/n7vI8peZLQIqgMnALjM7Jjy13d21w3YrzOzbBEPizwFR\ngkkRZwDj2/0chT5E0l5mdiJQDZxEMJf83wRv3gJ3/3cuY8tnYYk/td/CCJr+e6W5RQAzux04P82p\nse7+2+6OJ1+ZWSVB4noMwfoOX3L3P+U2qvxlZmcQDFWm/uK+0917bHWxLeFQUroPuunufld3x1Mo\nzOyHwEeAtwLbgb8C17j7qnY/R09JMERERKT7FPwsEhEREck/SjBEREQk65RgiIiISNYpwRAREZGs\nU4IhIiIiWacEQ0RERLJOCYaIiIhknRIMERERyTolGCIiIpJ1SjBEcsjMzgi3eu+f61g6wsweMrOF\nWXy+283sF9l6vlwys2fMbGbS47iZTc5lTCK5oARDpIuEHyx7wz9Tv/aa2TfCS/N2vf42EqBPAV/v\n5lgiZjbLzP5qZq+Z2ctmdr+ZfbA740iK57Nm9kqaU6cCP+jueETyjRIMka7zFoKNgt4C/B/BhkHH\nJB2/PleBmVlJey8lSIAs+aC7v5qDXWHrgDnAjcC7CXZ2fB54OEcVgsR704K7v6RdOkWUYIh0GXdv\nSnwRJBfu7tuSju9OuvxUM/ujme0yszVm9q7k5zKzs8xsffgv96fM7BtmFkk6P9TM7jWzmJltN7M6\nMxucdH6umT1mZp83s83Aa+FxM7MrzGyzme0Or5kSnnsnkNg58ZWw6vLj8FyLIRIz621m15rZc2b2\nupn93cymh+ciZvbDpNd4MnkIoT3M7NPAFOB/3P12d3/W3R9394uApcAPzezw8NoDhlvM7EYzeyjp\n8QQz+52ZvWJmL5rZMjM7Nun8O8PKzafMbFX43+XPZvaB8PwZBLsMD0itSKUOkaT5Wd4e/vd5xcxe\nMrNfhu914vyHzWydme0Mr/mdmQ3tyPslkg+UYIjkngHzgVnAKGAPwYdXcNLsdOBO9v/L/SLgs8Ds\n8LwRfMgOBE4HxgHHAj9PeZ3jgHMIhjdODo9dCZwHXAiMCF/jJ+FrPkfwoQ7wLoKqy5db+Rl+Anwa\nmBHG+AVgZ3guQlBpmAKUAlcBC8ys7OBvzT4VwN/c/f40524AjgY+dpDnSK429A3vG0mwJfVe4J40\n98wHrgPeC/wd+FmY2P2eoCq1g6Aq1a6KlJkdAqwkSDjHAB8EYsAKMzvEzHqFcTwEnAh8gGC4JW+H\n0URac0iuAxARHLjS3R8BMLNrgF+ZWW93fxP4BnC1u/80vP7Z8F/L1wHfIkgoTgCGufsL4XOcD2w0\ns1Huvj68r4SgAvByeE1v4Argo+6+LrxmS5hcXOTuvzOzl8Pj29x9R7rgzex4oDx8nkSVYMu+H859\nD0FSkfBs2DcxFVjSzvfoeKCxlXONSde0i7unVji+ADSZ2Qh335R06jvuviK8Zi7wBHCcu//dzPZV\npdr7usA0wNz9wqTX/jzwCvBhYD3QH7jP3beEl/ytA88vkjeUYIjkh8eTvv93+Odg4J8E/3r+oJnN\nSbqmF9DbzA4jqBg8n0guANy90cxeJagYJBKMZxPJReg4oA/wYFgFSSgBNnQg9vcSVF1+29oFZlYF\nTAfeARwO9AYe68BrQEofSBpvtvuJzI4Dvgn8N0H1I0KQ6L0DSE4wUv+7GMF/l7+397VSnAS8y8xi\nKccPBf7L3X9tZncCD5jZg8CvgXp3/0+GryeSM0owRPJDc9L3iXJ4YgizH0EVI900zjc68BqpTZn9\nwj8/DryQcq4jz/taWyfNbBrwHYIhoEcJhgQuB97fgdf4B0GylM6I8M/Ev/TjHJiMpDa1/gp4hmAo\n5wWC93ojQeKTrK3/LpnoB/wJ+EyaGLcBuPsFZlYNTCQYdvqWmX3M3f/QidcV6XZKMETy3wbg/7n7\n5nQnzawRGGpmQ9z9X+GxEQQ9GRvbeN5NBInEOxPDM2kkqgK92niexwk+dM9gf1Nosg8Ca9z91qSY\n/6uN50unFrjbzD7h7velnLsU+BfBv/Yh+KA+IeWakwl/FjM7kmA45fPuviY8dlqa1zxY38ObtP2+\npLOBYGhom7vvbO0id/8L8BfgWjP7PUFCogRDCoqaPEVyL13pP/nYN4Hzw5kjI8zs3Wb2aTP7FoC7\n/5qgN+BuMzvFzN5P0BT6kLu3OgwRfsBdD9xoZueb2bHh/TPM7H/Cy54l+KCdZGZHm1nfNM/zLHAX\n8GMLZrsMs2D9jPLwkn8QzJIZb2bvMrNvAu/rwPuDu/8cuBe408wuCGd5nGRmtxJUYM51973h5avC\n1/sfMzvOzOYRNEwmvAK8BFxoZv9lZh8haPhMTSgONiSzBehnZh8xs6MSs1gO4m7gReBeMzstfK8+\nbGbVZva28PG3zewDZvYOMxtP0GC7qe2nFck/SjBEci/dv5T3HXP3B4BPEsyS+AOwlmAGw5ak6ycT\nfHCuBh4AniJoKGz7hd2/TtAo+jWCD7HlBB/Yz4TnXwDmAtcA/wG+28pTXUzQsFlD0HT5A4L+DoBb\nCYZ3fk4wRHJkeF1HlQHfJvjZnwT+TDAr5hR339f/Eb5f3wKuJXi/+hEkXInzTjD0MIqg+nIDcFma\n1zvYf5e1wPcJ1udoAr7Syn3J97wGfIhghk4DwXt+G0EPxg5gN0FPzRKCIZ/vA991dy3cJQXHgr9r\nIiKFxcxOJhgW+ZG7fzXX8YhIS6pgiEhBcvc/Ax8FdpnZ8FzHIyItqYIhIiIiWacKhoiIiGSdEgwR\nERHJOiUYIiIiknVKMERERCTrlGCIiIhI1inBEBERkaxTgiEiIiJZpwRDREREsk4JhoiIiGTd/wck\ntxJseS/jKQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10aa9c8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sm.qqplot(residuals, fit=True, line = '45')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimating residual variation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The residual variation measures how well the regression line fits the data points.  \n",
    "\n",
    "It is the variation in the dependent variable (Price) that is not explained by the regression model and is represented by the residuals. We want the residual variation to be as small as possible.  \n",
    "\n",
    "Each residual is distributed normally with mean 0 and variance = sigma_squared.    \n",
    "\n",
    "We have [previously seen](https://github.com/Mashimo/datascience/blob/master/01-Regression/LinearRegression.ipynb) that the ML Estimate of variance, sigma_squared, is sum(residuals squared) divided by n   and we called it the **Mean Squared Error (MSE)**.  \n",
    "Most people use (n-2) instead of n so that the estimator is unbiased (the -2 is accounting for the degrees of freedom for intercept and slope).      \n",
    "\n",
    "The square root of the estimate, sigma, is called the **Root Mean Squared Error (RMSE)**.\n",
    "We want both MSE and RMSE to be as small as possible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our diamonds example the estimated residual variation (unbiased RMSE) is :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "31.840522265031755"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = len(y)\n",
    "MSE = sum(residuals**2) / (n-2)\n",
    "RMSE = np.sqrt(MSE)\n",
    "RMSE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "RMSE can be used to calculate the standardized residuals too.  \n",
    "They equal the value of a residual divided by an estimate of its standard deviation (so, RMSE).  \n",
    "Large standardized residuals are an indication of an outlier."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4927258932276684"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max(simpleModel.resid / RMSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Summarizing the variation: R-squared"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The total variation is the residual variation (variation after removing predictors) plus the systematic variation (variation explained by regression model).  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**R-squared** is the percentage of variability explained by the regression model:  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "R-squared = explained / total variation = 1 - residual / total variation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "R-squared is always between 0 and 1 (0% and 100%):\n",
    "- 0% indicates that the model explains none of the variability of the response data around its mean.\n",
    "- 100% indicates that the model explains all the variability of the response data around its mean.  \n",
    "\n",
    "In general, the higher the R-squared, the better the model fits your data. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.97826077798603295"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.rsquared"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are quite close to a perfect model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can use a fitted line plot to graphically illustrate different R-squared values.  \n",
    "The more variation that is explained by the model, the closer the data points fall to the line. Theoretically, if a model could explain 100% of the variation, the fitted values would always equal the observed values and all of the data points would fall on the  line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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It4OqyT1W7hWwRYsWBR1C0bGc+89y7r9czvmjyx+NKURqBtcURCGSyznPBCtG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\niZWSEREREYmVkhERERGJ1f8Ai6Sgi+zxPmEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1127bc278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(1,1,1)\n",
    "ax.plot(simpleModel.fittedvalues, diamondData.price, 'o') # as round marks\n",
    "   # identity line\n",
    "plt.plot(simpleModel.fittedvalues, simpleModel.fittedvalues, '--') \n",
    "   # pretty-up the plot\n",
    "fig.suptitle(\"Relation between estimated and actual diamonds' prices\")\n",
    "ax.set_ylabel('Estimated Price [SIN $]')\n",
    "ax.set_xlabel('Actual Price [SIN $]')\n",
    "ax.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " R-squared can be a misleading summary and needs to be carefully taken (deleting data can inflate R-squared for example). \n",
    "\n",
    "In conclusion (residuals distribution, variation) the model is pretty good and the relation is very strong."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because of this, sometimes is preferred to use the **adjusted Rsquared**, which is Rsquared adjusted for the number of observations.  \n",
    "There are several formula that can be used, normally it is the Wherry's formula:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.97778818620312058"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - (1-simpleModel.rsquared)*((n-1)/simpleModel.df_resid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, it is also available from the model results:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.97778818620312058"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.rsquared_adj"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Confidence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How accurately can we predict the diamond prices?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For any given weight in carats, what is our prediction for the price, and what is the accuracy of this prediction?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In statistics, a sequence of random variables is independent and identically distributed (IID) if each random variable has the same probability distribution as the others and all are mutually independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Inference for regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the case of regression with IID sampling assumptions and normal distributed residuals, the statistics for our estimated beta coefficients:  \n",
    "- will follow a finite sample T-distributions and be normally distributed \n",
    "- can be used to test null hypotesis\n",
    "- can be used to create a confidence interval"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In probability and statistics, the [t-distribution](https://en.wikipedia.org/wiki/Student's_t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.  \n",
    "\n",
    "Whereas a normal distribution describes a full population, t-distributions describe samples drawn from a full population.  \n",
    "\n",
    "The t-distribution becomes closer to the normal (Gaussian) distribution as its degrees of freedom (df) increases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![t-distribution](../images/tdistr.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The t-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors.  \n",
    "\n",
    "If the population standard deviation of these errors is unknown and has to be estimated from the data, the t-distribution is often used to account for the extra uncertainty that results from this estimation.  \n",
    "\n",
    "Confidence intervals and hypothesis tests are two statistical procedures in which the quantiles of the sampling distribution of a particular statistic (e.g. the standard score) are required. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Confidence levels are expressed as a percentage (for example, a 95% confidence level). It means that should you take a sample over and over again, 95 percent of the time your results will match the results you get from the population.  \n",
    "\n",
    "When the population standard deviation sigma is not known, an interval estimate for the population with confidence level (1-alfa) is given by:  \n",
    "Xmean +- t * (estimated standard error of the mean)  \n",
    "\n",
    "where t is a critical value determined from the t-distribution in such a way that there is an are (1-alfa) between t and -t."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![interval](../images/tdecrule.gif)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we need to calculate the variance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Estimating the coefficients and the variance "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that our linear regression model is:  \n",
    "Y = Beta0 + Beta1 * X + errors  \n",
    "\n",
    "We can define the beta parameters as:  \n",
    "beta0 = mean(Y) - beta1 \\* mean(X)  \n",
    "beta1 = Cor(Y,X) \\* Sd(Y)/Sd(X)  \n",
    "\n",
    "For our diamonds example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "  # prepare the data\n",
    "y = diamondData.price\n",
    "x = diamondData.carats\n",
    "n = len(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3721.0248515504722"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "  # calculate beta1\n",
    "beta1 = (np.corrcoef (y,x) * np.std(y) / np.std(x))[0][1]\n",
    "beta1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-259.62590719155486"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "  # calculate beta0\n",
    "beta0 = np.mean(y) - beta1 * np.mean(x)\n",
    "beta0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sigma is unknown but its estimate is the squared root of the sum of the errors squared, divided by n-2 (the degrees of freedom)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "31.840522265031737"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = y - beta0 - beta1 * x # the residuals\n",
    "  # unbiased estimate for variance\n",
    "sigma = np.sqrt(sum(e**2) / (n-2))\n",
    "sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Confidence intervals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A 95% confidence interval is defined as a range of values such that with 95% probability, the range will contain the true unknown value of the parameter. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Recall of quantiles and percentiles of a distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you were the 95th percentile on an exam, it means that 95% of people scored worse than you and 5% scored better.  \n",
    "These are sample quantiles."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now for a population: the *i-th* quantile of a distribution with distribution function *F* is simply the point *x_i* so that :  \n",
    "F(x_i) = i  \n",
    "\n",
    "A percentile is simply a quantile with *i* expressed as a percent.  \n",
    "The 95th percentile of a distribution is the point where the probability that a random variable drawn from the population is less than 95%."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Approximately 68%, 95% and 99% of the **normal** density lies respectively within 1,2 and 3 standard deviations from the mean."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Estimating the Standard Errors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we need to calculate the standard errors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17.318856186448148"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ssx = sum((x - np.mean(x))**2)\n",
    "\n",
    "  # calculate standard error for beta0\n",
    "seBeta0 = (1 / n + np.mean(x) ** 2 / ssx) ** 0.5 * sigma\n",
    "seBeta0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "81.785880366042491"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "  # calculate standard error for beta1\n",
    "seBeta1 = sigma / np.sqrt(ssx)\n",
    "seBeta1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The standard error of the parameter measures the precision of the estimate of the parameter.  \n",
    "The smaller the standard error, the more precise the estimate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hypothesis testing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hypothesis testing is concerned with making decisions using data.  \n",
    "A null hypothesis is specified that represents the status quo, usually labeled H0.  \n",
    "The null hypothesis is assumed true and statistical evidence is required to reject it in favour of an alternative hypothesis.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![hypotesis](../images/one-sample-t-test.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider testing H0: mu = mu0  \n",
    "If we take the set of all possible values for which you fail to reject H0, this set is an alfa% confidence interval for mu, alfa depending on the set."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Getting the T-values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Testing for null hypotesis H0: estimated beta0 and beta1 are equal to real coefficients"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dividing the parameter by its standard error calculates a t-value: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-14.990938454394572"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tBeta0 = beta0 / seBeta0\n",
    "tBeta0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "45.497154703190581"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tBeta1 = beta1 / seBeta1\n",
    "tBeta1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P-values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P-values are the most common measure of \"statistical significance\".  \n",
    "\n",
    "The P-value is the probability under the null hypothesis of obtaining evidence as extreme or more extreme than would be observed by chance alone.  \n",
    "If the p-value is small, then either H0 is true and we have observed an extreme rare event or H0 is false.  \n",
    "\n",
    "Let's say a P-value is 0.1: then the probability of seeing evidence as extreme or more extreme than what actually has been obtained under H0 is 0.1 (10%).  \n",
    "\n",
    "By reporting a P-value, any observer can perform the own hypothesis test at whatever *alfa* level they choose. If the P-value is less than *alfa* then you reject the null hypothesis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![p-value](../images/pvalue.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Estimating the P-values for hypotesis contrary beta0 is not equal to zero"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the T-distribution module from [*SciPy.stats*](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.t.html) to calculate the p-values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The survival function of a random variable is the probability that the random variable is bigger than a value x.  \n",
    "The *SciPy.stats* function *sf()* returns this probability: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "degreesOfFreedom = simpleModel.df_resid # The residual degree of freedom\n",
    "pBeta0 = t.sf(abs(tBeta0), df=degreesOfFreedom)*2 # two-sided"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "pBeta1 = t.sf(abs(tBeta1), df=degreesOfFreedom)*2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's summarise the values calculated until now:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "##            Estimate     Std. Error    t-value        p-value\n",
      "Intercept:  -259.625907192 17.3188561864 -14.9909384544 2.52327062853e-19\n",
      "Carats:      3721.02485155 81.785880366 45.4971547032 6.75125983495e-40\n"
     ]
    }
   ],
   "source": [
    "print (\"##            Estimate     Std. Error    t-value        p-value\")\n",
    "print (\"Intercept: \", beta0, seBeta0, tBeta0, pBeta0)\n",
    "print (\"Carats:     \", beta1, seBeta1, tBeta1, pBeta1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the p-value is less than the significance level (0.05 in our case) then the model explains the variation in the response."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### T confidence intervals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For small samples we can use the t-distribution to calculate the confidence intervals.  \n",
    "The t-distribution has been invented by William Gosset in 1908 and is indexed by a degrees of freedom (*df*): it gets more like a standard normal as *df* gets larger."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*ppf()* is the Percent Point Function from *SciPy.stats*, that has as input the quantile (for a 2-sided 95% probability) and the degrees of freedom."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0128955952945886"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha=0.05 # confidence interval for two-sided hypothesis\n",
    "qt = 1 - (alpha/2)  # =0.975 for a 2-sided 95% probability\n",
    "t_value = t.ppf(qt, df=degreesOfFreedom) \n",
    "t_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can calculate the intervals for beta0 and beta1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-294.48695652479677, -224.76485785831295]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limits=[-1,1]\n",
    "[beta0 + i*t_value*seBeta0 for i in limits]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3556.3984132043752, 3885.6512898965693]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[beta1 + i*t_value*seBeta1 for i in limits]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interpretation: **With 95% confidence, we estimate that 1 carat increase in diamond size results in a 3556 to 3886 increase in price in (Singapore) dollars.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot the confidence interval"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We calculate the interval for each x value; will use the isf() function to get the inverse survival function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "predicted = simpleModel.fittedvalues\n",
    "x_1 = simpleModel.model.exog # just the x values plus column of 1\n",
    "  # get standard deviation of predicted values\n",
    "predvar = simpleModel.mse_resid + (x_1 * np.dot(simpleModel.cov_params(),\n",
    "                                                 x_1.T).T).sum(1)\n",
    "predstd = np.sqrt(predvar)\n",
    "\n",
    "tppf = t.isf(alpha/2.0, simpleModel.df_resid)\n",
    "\n",
    "interval_u = predicted + tppf * predstd\n",
    "interval_l = predicted - tppf * predstd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x112e2f128>"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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RiReRZSJS3ye9uIhMF5GjIhInIp+LSGWfPBVEZI6InBKREyLynoiUCnT78jPf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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112add278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(x,y, 'o', label=\"data\")\n",
    "ax.plot(x, simpleModel.fittedvalues, 'g-', label=\"OLS\")\n",
    "\n",
    "ax.plot(x, interval_u, 'c--', label = \"Intervals\")\n",
    "ax.plot(x, interval_l, 'c--')\n",
    "   # pretty-up the plot\n",
    "fig.suptitle(\"OLS Linear Regression with confidence intervals\")\n",
    "ax.set_ylabel('Predicted Price [SIN $]')\n",
    "ax.set_xlabel('Weight [Carat]')\n",
    "ax.grid(True)\n",
    "ax.legend(loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary of statistic values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The *statsmodel* package offers an overview of the model values, similar to what we calculated above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>OLS Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>          <td>price</td>      <th>  R-squared:         </th> <td>   0.978</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                   <td>OLS</td>       <th>  Adj. R-squared:    </th> <td>   0.978</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>             <td>Least Squares</td>  <th>  F-statistic:       </th> <td>   2070.</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>             <td>Sun, 10 Sep 2017</td> <th>  Prob (F-statistic):</th> <td>6.75e-40</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>                 <td>18:18:58</td>     <th>  Log-Likelihood:    </th> <td> -233.20</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Observations:</th>      <td>    48</td>      <th>  AIC:               </th> <td>   470.4</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Residuals:</th>          <td>    46</td>      <th>  BIC:               </th> <td>   474.1</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Df Model:</th>              <td>     1</td>      <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Covariance Type:</th>      <td>nonrobust</td>    <th>                     </th>     <td> </td>   \n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "     <td></td>       <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th> <th>[95.0% Conf. Int.]</th> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>  <td> -259.6259</td> <td>   17.319</td> <td>  -14.991</td> <td> 0.000</td> <td> -294.487  -224.765</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>carats</th> <td> 3721.0249</td> <td>   81.786</td> <td>   45.497</td> <td> 0.000</td> <td> 3556.398  3885.651</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "  <th>Omnibus:</th>       <td> 0.739</td> <th>  Durbin-Watson:     </th> <td>   1.994</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Prob(Omnibus):</th> <td> 0.691</td> <th>  Jarque-Bera (JB):  </th> <td>   0.181</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Skew:</th>          <td> 0.056</td> <th>  Prob(JB):          </th> <td>   0.913</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Kurtosis:</th>      <td> 3.280</td> <th>  Cond. No.          </th> <td>    18.5</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                            OLS Regression Results                            \n",
       "==============================================================================\n",
       "Dep. Variable:                  price   R-squared:                       0.978\n",
       "Model:                            OLS   Adj. R-squared:                  0.978\n",
       "Method:                 Least Squares   F-statistic:                     2070.\n",
       "Date:                Sun, 10 Sep 2017   Prob (F-statistic):           6.75e-40\n",
       "Time:                        18:18:58   Log-Likelihood:                -233.20\n",
       "No. Observations:                  48   AIC:                             470.4\n",
       "Df Residuals:                      46   BIC:                             474.1\n",
       "Df Model:                           1                                         \n",
       "Covariance Type:            nonrobust                                         \n",
       "==============================================================================\n",
       "                 coef    std err          t      P>|t|      [95.0% Conf. Int.]\n",
       "------------------------------------------------------------------------------\n",
       "const       -259.6259     17.319    -14.991      0.000      -294.487  -224.765\n",
       "carats      3721.0249     81.786     45.497      0.000      3556.398  3885.651\n",
       "==============================================================================\n",
       "Omnibus:                        0.739   Durbin-Watson:                   1.994\n",
       "Prob(Omnibus):                  0.691   Jarque-Bera (JB):                0.181\n",
       "Skew:                           0.056   Prob(JB):                        0.913\n",
       "Kurtosis:                       3.280   Cond. No.                         18.5\n",
       "==============================================================================\n",
       "\n",
       "Warnings:\n",
       "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
       "\"\"\""
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Many values can also be accessed directly, for example the standard errors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "const     17.318856\n",
       "carats    81.785880\n",
       "dtype: float64"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simpleModel.bse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can see all the values available using *dir()*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['HC0_se',\n",
       " 'HC1_se',\n",
       " 'HC2_se',\n",
       " 'HC3_se',\n",
       " '_HCCM',\n",
       " '__class__',\n",
       " '__delattr__',\n",
       " '__dict__',\n",
       " '__dir__',\n",
       " '__doc__',\n",
       " '__eq__',\n",
       " '__format__',\n",
       " '__ge__',\n",
       " '__getattribute__',\n",
       " '__gt__',\n",
       " '__hash__',\n",
       " '__init__',\n",
       " '__le__',\n",
       " '__lt__',\n",
       " '__module__',\n",
       " '__ne__',\n",
       " '__new__',\n",
       " '__reduce__',\n",
       " '__reduce_ex__',\n",
       " '__repr__',\n",
       " '__setattr__',\n",
       " '__sizeof__',\n",
       " '__str__',\n",
       " '__subclasshook__',\n",
       " '__weakref__',\n",
       " '_cache',\n",
       " '_data_attr',\n",
       " '_get_robustcov_results',\n",
       " '_is_nested',\n",
       " '_wexog_singular_values',\n",
       " 'aic',\n",
       " 'bic',\n",
       " 'bse',\n",
       " 'centered_tss',\n",
       " 'compare_f_test',\n",
       " 'compare_lm_test',\n",
       " 'compare_lr_test',\n",
       " 'condition_number',\n",
       " 'conf_int',\n",
       " 'conf_int_el',\n",
       " 'cov_HC0',\n",
       " 'cov_HC1',\n",
       " 'cov_HC2',\n",
       " 'cov_HC3',\n",
       " 'cov_kwds',\n",
       " 'cov_params',\n",
       " 'cov_type',\n",
       " 'df_model',\n",
       " 'df_resid',\n",
       " 'diagn',\n",
       " 'eigenvals',\n",
       " 'el_test',\n",
       " 'ess',\n",
       " 'f_pvalue',\n",
       " 'f_test',\n",
       " 'fittedvalues',\n",
       " 'fvalue',\n",
       " 'get_influence',\n",
       " 'get_robustcov_results',\n",
       " 'initialize',\n",
       " 'k_constant',\n",
       " 'llf',\n",
       " 'load',\n",
       " 'model',\n",
       " 'mse_model',\n",
       " 'mse_resid',\n",
       " 'mse_total',\n",
       " 'nobs',\n",
       " 'normalized_cov_params',\n",
       " 'outlier_test',\n",
       " 'params',\n",
       " 'predict',\n",
       " 'pvalues',\n",
       " 'remove_data',\n",
       " 'resid',\n",
       " 'resid_pearson',\n",
       " 'rsquared',\n",
       " 'rsquared_adj',\n",
       " 'save',\n",
       " 'scale',\n",
       " 'ssr',\n",
       " 'summary',\n",
       " 'summary2',\n",
       " 't_test',\n",
       " 'tvalues',\n",
       " 'uncentered_tss',\n",
       " 'use_t',\n",
       " 'wald_test',\n",
       " 'wresid']"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dir(simpleModel)"
   ]
  }
 ],
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