3_scipy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction à scipy\n",
    "Nous passons maintenant à scipy. \n",
    "Scipy est une collection de fonction diverse telle que des fonction pour l'optimisation, les distribution ou l'integration.\n",
    "Pour plus d'info voir https://docs.scipy.org/doc/scipy-1.0.0/reference/tutorial/general.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scipy se montre très pratique pour minimiser des fonction. \n",
    "Par exemple la fonction dites 'la banane':\n",
    "$$\n",
    "f(x) = \\sum_{i=1}^{N-1} 100(x_i - x_{i-1}^2)^2 + (1-x_{i-1})^2\n",
    "$$. \n",
    "Le minimum se trouve à $x^\\star = [1,..,1]$ et $f(x^\\star) = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def rosen(x):\n",
    "    \"\"\"The Rosenbrock function\"\"\"\n",
    "    return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On visualise la fonction pour avoir une meilleure idée de son comportment en dimension 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.colorbar.Colorbar at 0x7fa3b5179250>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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azCxRDmgzs0SVOqDdhzazKit1QJtZenwmR3Ec0GYV4jM5qqX0AV2XGye5zWFWHpKWS7pL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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa3e00b8450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X1 = np.linspace(-0.5, 0.5, 100)\n",
    "X2 = np.linspace(-0.5, 0.5, 100)\n",
    "\n",
    "# On construit une grille pour visualiser la fonction\n",
    "param_grid = np.array([[X1i, X2i] for X1i in X1 for X2i in X2])\n",
    "function_value = [rosen(params[:,np.newaxis]) for params in param_grid]\n",
    "\n",
    "C, G = np.meshgrid(X1, X2)\n",
    "plt.figure()\n",
    "cp = plt.contourf(C, G, np.array(function_value).reshape(C.shape)) # pour visuliser des fonction en 3d\n",
    "plt.colorbar(cp)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.000000\n",
      "         Iterations: 79\n",
      "         Function evaluations: 150\n"
     ]
    }
   ],
   "source": [
    "from scipy.optimize import minimize\n",
    "\n",
    "#x0 = np.array([1.3, 0.7, 0.8, 1.9, 1.2])\n",
    "x0 = np.array([1.3, 0.7])\n",
    "res = minimize(rosen, x0, method='nelder-mead', options={'xtol': 1e-8, 'disp': True})\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['status', 'success', 'final_simplex', 'nfev', 'fun', 'x', 'message', 'nit']\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([1., 1.])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print res.keys() # les resultats de la procedure d'optimisation\n",
    "res['x']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise: \n",
    "Chercher l'optimum $(a,b)$ pour le problème de moindre carré \n",
    "$$\n",
    "\\min \\| y_i - a+b*x_i \\|\n",
    "$$\n",
    "On utilise le code de la derniere question sur de matplotlib (a priori vous connaisez les \"vraies\" valeurs de $(a,b)$). \n",
    "\n",
    "On procède par:\n",
    "\n",
    "a) Coder la fonction a minimiser associé. Important: assurer que $y$ et $x$ sont fixés !\n",
    "\n",
    "b) Visualiser la fonction à minimiser comme ci-dessus\n",
    "\n",
    "c) Appliquer la procedure de minimisation (attention: vous devez eventuellement regarder la documentation pour donner des arguments en plus à la fonction)\n",
    "\n",
    "d) Le resultat correspond il à ce que vous vous attendez ? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Scipy stats\n",
    "Numpy vient avec pas mal de fonctionnalité pour simuler des variables aléatoire. \n",
    "Pourtant la bibliothèque de scipy et plus riches et permet de faire plus de choses outils pour un data scientist. On introduit maintenant les fonctionalités pour des distributions statistiques"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import gamma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on gamma_gen in module scipy.stats._continuous_distns object:\n",
      "\n",
      "class gamma_gen(scipy.stats._distn_infrastructure.rv_continuous)\n",
      " |  A gamma continuous random variable.\n",
      " |  \n",
      " |  %(before_notes)s\n",
      " |  \n",
      " |  See Also\n",
      " |  --------\n",
      " |  erlang, expon\n",
      " |  \n",
      " |  Notes\n",
      " |  -----\n",
      " |  The probability density function for `gamma` is:\n",
      " |  \n",
      " |  .. math::\n",
      " |  \n",
      " |      f(x, a) = \\frac{x^{a-1} \\exp(-x)}{\\Gamma(a)}\n",
      " |  \n",
      " |  for :math:`x \\ge 0`, :math:`a > 0`. Here :math:`\\Gamma(a)` refers to the\n",
      " |  gamma function.\n",
      " |  \n",
      " |  `gamma` has a shape parameter `a` which needs to be set explicitly.\n",
      " |  \n",
      " |  When :math:`a` is an integer, `gamma` reduces to the Erlang\n",
      " |  distribution, and when :math:`a=1` to the exponential distribution.\n",
      " |  \n",
      " |  %(after_notes)s\n",
      " |  \n",
      " |  %(example)s\n",
      " |  \n",
      " |  Method resolution order:\n",
      " |      gamma_gen\n",
      " |      scipy.stats._distn_infrastructure.rv_continuous\n",
      " |      scipy.stats._distn_infrastructure.rv_generic\n",
      " |      __builtin__.object\n",
      " |  \n",
      " |  Methods defined here:\n",
      " |  \n",
      " |  fit(self, data, *args, **kwds)\n",
      " |      Return MLEs for shape (if applicable), location, and scale\n",
      " |      parameters from data.\n",
      " |      \n",
      " |      MLE stands for Maximum Likelihood Estimate.  Starting estimates for\n",
      " |      the fit are given by input arguments; for any arguments not provided\n",
      " |      with starting estimates, ``self._fitstart(data)`` is called to generate\n",
      " |      such.\n",
      " |      \n",
      " |      One can hold some parameters fixed to specific values by passing in\n",
      " |      keyword arguments ``f0``, ``f1``, ..., ``fn`` (for shape parameters)\n",
      " |      and ``floc`` and ``fscale`` (for location and scale parameters,\n",
      " |      respectively).\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      data : array_like\n",
      " |          Data to use in calculating the MLEs.\n",
      " |      args : floats, optional\n",
      " |          Starting value(s) for any shape-characterizing arguments (those not\n",
      " |          provided will be determined by a call to ``_fitstart(data)``).\n",
      " |          No default value.\n",
      " |      kwds : floats, optional\n",
      " |          Starting values for the location and scale parameters; no default.\n",
      " |          Special keyword arguments are recognized as holding certain\n",
      " |          parameters fixed:\n",
      " |      \n",
      " |          - f0...fn : hold respective shape parameters fixed.\n",
      " |            Alternatively, shape parameters to fix can be specified by name.\n",
      " |            For example, if ``self.shapes == \"a, b\"``, ``fa``and ``fix_a``\n",
      " |            are equivalent to ``f0``, and ``fb`` and ``fix_b`` are\n",
      " |            equivalent to ``f1``.\n",
      " |      \n",
      " |          - floc : hold location parameter fixed to specified value.\n",
      " |      \n",
      " |          - fscale : hold scale parameter fixed to specified value.\n",
      " |      \n",
      " |          - optimizer : The optimizer to use.  The optimizer must take ``func``,\n",
      " |            and starting position as the first two arguments,\n",
      " |            plus ``args`` (for extra arguments to pass to the\n",
      " |            function to be optimized) and ``disp=0`` to suppress\n",
      " |            output as keyword arguments.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      mle_tuple : tuple of floats\n",
      " |          MLEs for any shape parameters (if applicable), followed by those\n",
      " |          for location and scale. For most random variables, shape statistics\n",
      " |          will be returned, but there are exceptions (e.g. ``norm``).\n",
      " |      \n",
      " |      Notes\n",
      " |      -----\n",
      " |      This fit is computed by maximizing a log-likelihood function, with\n",
      " |      penalty applied for samples outside of range of the distribution. The\n",
      " |      returned answer is not guaranteed to be the globally optimal MLE, it\n",
      " |      may only be locally optimal, or the optimization may fail altogether.\n",
      " |      \n",
      " |      When the location is fixed by using the argument `floc`, this\n",
      " |      function uses explicit formulas or solves a simpler numerical\n",
      " |      problem than the full ML optimization problem.  So in that case,\n",
      " |      the `optimizer`, `loc` and `scale` arguments are ignored.\n",
      " |      \n",
      " |      Examples\n",
      " |      --------\n",
      " |      \n",
      " |      Generate some data to fit: draw random variates from the `beta`\n",
      " |      distribution\n",
      " |      \n",
      " |      >>> from scipy.stats import beta\n",
      " |      >>> a, b = 1., 2.\n",
      " |      >>> x = beta.rvs(a, b, size=1000)\n",
      " |      \n",
      " |      Now we can fit all four parameters (``a``, ``b``, ``loc`` and ``scale``):\n",
      " |      \n",
      " |      >>> a1, b1, loc1, scale1 = beta.fit(x)\n",
      " |      \n",
      " |      We can also use some prior knowledge about the dataset: let's keep\n",
      " |      ``loc`` and ``scale`` fixed:\n",
      " |      \n",
      " |      >>> a1, b1, loc1, scale1 = beta.fit(x, floc=0, fscale=1)\n",
      " |      >>> loc1, scale1\n",
      " |      (0, 1)\n",
      " |      \n",
      " |      We can also keep shape parameters fixed by using ``f``-keywords. To\n",
      " |      keep the zero-th shape parameter ``a`` equal 1, use ``f0=1`` or,\n",
      " |      equivalently, ``fa=1``:\n",
      " |      \n",
      " |      >>> a1, b1, loc1, scale1 = beta.fit(x, fa=1, floc=0, fscale=1)\n",
      " |      >>> a1\n",
      " |      1\n",
      " |      \n",
      " |      Not all distributions return estimates for the shape parameters.\n",
      " |      ``norm`` for example just returns estimates for location and scale:\n",
      " |      \n",
      " |      >>> from scipy.stats import norm\n",
      " |      >>> x = norm.rvs(a, b, size=1000, random_state=123)\n",
      " |      >>> loc1, scale1 = norm.fit(x)\n",
      " |      >>> loc1, scale1\n",
      " |      (0.92087172783841631, 2.0015750750324668)\n",
      " |  \n",
      " |  ----------------------------------------------------------------------\n",
      " |  Methods inherited from scipy.stats._distn_infrastructure.rv_continuous:\n",
      " |  \n",
      " |  __init__(self, momtype=1, a=None, b=None, xtol=1e-14, badvalue=None, name=None, longname=None, shapes=None, extradoc=None, seed=None)\n",
      " |  \n",
      " |  cdf(self, x, *args, **kwds)\n",
      " |      Cumulative distribution function of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      x : array_like\n",
      " |          quantiles\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      cdf : ndarray\n",
      " |          Cumulative distribution function evaluated at `x`\n",
      " |  \n",
      " |  expect(self, func=None, args=(), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)\n",
      " |      Calculate expected value of a function with respect to the\n",
      " |      distribution.\n",
      " |      \n",
      " |      The expected value of a function ``f(x)`` with respect to a\n",
      " |      distribution ``dist`` is defined as::\n",
      " |      \n",
      " |                  ubound\n",
      " |          E[x] = Integral(f(x) * dist.pdf(x))\n",
      " |                  lbound\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      func : callable, optional\n",
      " |          Function for which integral is calculated. Takes only one argument.\n",
      " |          The default is the identity mapping f(x) = x.\n",
      " |      args : tuple, optional\n",
      " |          Shape parameters of the distribution.\n",
      " |      loc : float, optional\n",
      " |          Location parameter (default=0).\n",
      " |      scale : float, optional\n",
      " |          Scale parameter (default=1).\n",
      " |      lb, ub : scalar, optional\n",
      " |          Lower and upper bound for integration. Default is set to the\n",
      " |          support of the distribution.\n",
      " |      conditional : bool, optional\n",
      " |          If True, the integral is corrected by the conditional probability\n",
      " |          of the integration interval.  The return value is the expectation\n",
      " |          of the function, conditional on being in the given interval.\n",
      " |          Default is False.\n",
      " |      \n",
      " |      Additional keyword arguments are passed to the integration routine.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      expect : float\n",
      " |          The calculated expected value.\n",
      " |      \n",
      " |      Notes\n",
      " |      -----\n",
      " |      The integration behavior of this function is inherited from\n",
      " |      `integrate.quad`.\n",
      " |  \n",
      " |  fit_loc_scale(self, data, *args)\n",
      " |      Estimate loc and scale parameters from data using 1st and 2nd moments.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      data : array_like\n",
      " |          Data to fit.\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information).\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      Lhat : float\n",
      " |          Estimated location parameter for the data.\n",
      " |      Shat : float\n",
      " |          Estimated scale parameter for the data.\n",
      " |  \n",
      " |  isf(self, q, *args, **kwds)\n",
      " |      Inverse survival function (inverse of `sf`) at q of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      q : array_like\n",
      " |          upper tail probability\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      x : ndarray or scalar\n",
      " |          Quantile corresponding to the upper tail probability q.\n",
      " |  \n",
      " |  logcdf(self, x, *args, **kwds)\n",
      " |      Log of the cumulative distribution function at x of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      x : array_like\n",
      " |          quantiles\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      logcdf : array_like\n",
      " |          Log of the cumulative distribution function evaluated at x\n",
      " |  \n",
      " |  logpdf(self, x, *args, **kwds)\n",
      " |      Log of the probability density function at x of the given RV.\n",
      " |      \n",
      " |      This uses a more numerically accurate calculation if available.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      x : array_like\n",
      " |          quantiles\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      logpdf : array_like\n",
      " |          Log of the probability density function evaluated at x\n",
      " |  \n",
      " |  logsf(self, x, *args, **kwds)\n",
      " |      Log of the survival function of the given RV.\n",
      " |      \n",
      " |      Returns the log of the \"survival function,\" defined as (1 - `cdf`),\n",
      " |      evaluated at `x`.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      x : array_like\n",
      " |          quantiles\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      logsf : ndarray\n",
      " |          Log of the survival function evaluated at `x`.\n",
      " |  \n",
      " |  nnlf(self, theta, x)\n",
      " |      Return negative loglikelihood function.\n",
      " |      \n",
      " |      Notes\n",
      " |      -----\n",
      " |      This is ``-sum(log pdf(x, theta), axis=0)`` where `theta` are the\n",
      " |      parameters (including loc and scale).\n",
      " |  \n",
      " |  pdf(self, x, *args, **kwds)\n",
      " |      Probability density function at x of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      x : array_like\n",
      " |          quantiles\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      pdf : ndarray\n",
      " |          Probability density function evaluated at x\n",
      " |  \n",
      " |  ppf(self, q, *args, **kwds)\n",
      " |      Percent point function (inverse of `cdf`) at q of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      q : array_like\n",
      " |          lower tail probability\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      x : array_like\n",
      " |          quantile corresponding to the lower tail probability q.\n",
      " |  \n",
      " |  sf(self, x, *args, **kwds)\n",
      " |      Survival function (1 - `cdf`) at x of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      x : array_like\n",
      " |          quantiles\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      sf : array_like\n",
      " |          Survival function evaluated at x\n",
      " |  \n",
      " |  ----------------------------------------------------------------------\n",
      " |  Methods inherited from scipy.stats._distn_infrastructure.rv_generic:\n",
      " |  \n",
      " |  __call__(self, *args, **kwds)\n",
      " |      Freeze the distribution for the given arguments.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution.  Should include all\n",
      " |          the non-optional arguments, may include ``loc`` and ``scale``.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      rv_frozen : rv_frozen instance\n",
      " |          The frozen distribution.\n",
      " |  \n",
      " |  __getstate__(self)\n",
      " |  \n",
      " |  __setstate__(self, state)\n",
      " |  \n",
      " |  entropy(self, *args, **kwds)\n",
      " |      Differential entropy of the RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information).\n",
      " |      loc : array_like, optional\n",
      " |          Location parameter (default=0).\n",
      " |      scale : array_like, optional  (continuous distributions only).\n",
      " |          Scale parameter (default=1).\n",
      " |      \n",
      " |      Notes\n",
      " |      -----\n",
      " |      Entropy is defined base `e`:\n",
      " |      \n",
      " |      >>> drv = rv_discrete(values=((0, 1), (0.5, 0.5)))\n",
      " |      >>> np.allclose(drv.entropy(), np.log(2.0))\n",
      " |      True\n",
      " |  \n",
      " |  freeze(self, *args, **kwds)\n",
      " |      Freeze the distribution for the given arguments.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution.  Should include all\n",
      " |          the non-optional arguments, may include ``loc`` and ``scale``.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      rv_frozen : rv_frozen instance\n",
      " |          The frozen distribution.\n",
      " |  \n",
      " |  interval(self, alpha, *args, **kwds)\n",
      " |      Confidence interval with equal areas around the median.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      alpha : array_like of float\n",
      " |          Probability that an rv will be drawn from the returned range.\n",
      " |          Each value should be in the range [0, 1].\n",
      " |      arg1, arg2, ... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information).\n",
      " |      loc : array_like, optional\n",
      " |          location parameter, Default is 0.\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter, Default is 1.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      a, b : ndarray of float\n",
      " |          end-points of range that contain ``100 * alpha %`` of the rv's\n",
      " |          possible values.\n",
      " |  \n",
      " |  mean(self, *args, **kwds)\n",
      " |      Mean of the distribution.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      mean : float\n",
      " |          the mean of the distribution\n",
      " |  \n",
      " |  median(self, *args, **kwds)\n",
      " |      Median of the distribution.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          Location parameter, Default is 0.\n",
      " |      scale : array_like, optional\n",
      " |          Scale parameter, Default is 1.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      median : float\n",
      " |          The median of the distribution.\n",
      " |      \n",
      " |      See Also\n",
      " |      --------\n",
      " |      stats.distributions.rv_discrete.ppf\n",
      " |          Inverse of the CDF\n",
      " |  \n",
      " |  moment(self, n, *args, **kwds)\n",
      " |      n-th order non-central moment of distribution.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      n : int, n >= 1\n",
      " |          Order of moment.\n",
      " |      arg1, arg2, arg3,... : float\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information).\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |  \n",
      " |  rvs(self, *args, **kwds)\n",
      " |      Random variates of given type.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information).\n",
      " |      loc : array_like, optional\n",
      " |          Location parameter (default=0).\n",
      " |      scale : array_like, optional\n",
      " |          Scale parameter (default=1).\n",
      " |      size : int or tuple of ints, optional\n",
      " |          Defining number of random variates (default is 1).\n",
      " |      random_state : None or int or ``np.random.RandomState`` instance, optional\n",
      " |          If int or RandomState, use it for drawing the random variates.\n",
      " |          If None, rely on ``self.random_state``.\n",
      " |          Default is None.\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      rvs : ndarray or scalar\n",
      " |          Random variates of given `size`.\n",
      " |  \n",
      " |  stats(self, *args, **kwds)\n",
      " |      Some statistics of the given RV.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional (continuous RVs only)\n",
      " |          scale parameter (default=1)\n",
      " |      moments : str, optional\n",
      " |          composed of letters ['mvsk'] defining which moments to compute:\n",
      " |          'm' = mean,\n",
      " |          'v' = variance,\n",
      " |          's' = (Fisher's) skew,\n",
      " |          'k' = (Fisher's) kurtosis.\n",
      " |          (default is 'mv')\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      stats : sequence\n",
      " |          of requested moments.\n",
      " |  \n",
      " |  std(self, *args, **kwds)\n",
      " |      Standard deviation of the distribution.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      std : float\n",
      " |          standard deviation of the distribution\n",
      " |  \n",
      " |  var(self, *args, **kwds)\n",
      " |      Variance of the distribution.\n",
      " |      \n",
      " |      Parameters\n",
      " |      ----------\n",
      " |      arg1, arg2, arg3,... : array_like\n",
      " |          The shape parameter(s) for the distribution (see docstring of the\n",
      " |          instance object for more information)\n",
      " |      loc : array_like, optional\n",
      " |          location parameter (default=0)\n",
      " |      scale : array_like, optional\n",
      " |          scale parameter (default=1)\n",
      " |      \n",
      " |      Returns\n",
      " |      -------\n",
      " |      var : float\n",
      " |          the variance of the distribution\n",
      " |  \n",
      " |  ----------------------------------------------------------------------\n",
      " |  Data descriptors inherited from scipy.stats._distn_infrastructure.rv_generic:\n",
      " |  \n",
      " |  __dict__\n",
      " |      dictionary for instance variables (if defined)\n",
      " |  \n",
      " |  __weakref__\n",
      " |      list of weak references to the object (if defined)\n",
      " |  \n",
      " |  random_state\n",
      " |      Get or set the RandomState object for generating random variates.\n",
      " |      \n",
      " |      This can be either None or an existing RandomState object.\n",
      " |      \n",
      " |      If None (or np.random), use the RandomState singleton used by np.random.\n",
      " |      If already a RandomState instance, use it.\n",
      " |      If an int, use a new RandomState instance seeded with seed.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(gamma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rv = gamma(1, scale=2.) # on genere la variable aléatoire, on fixe la variable aléatoire"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa3a32e2710>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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2c3L8f2NRUfOjuNjvqFVU5L+2dmTJLweFu0i2MGsaGz/55OjX9+9v6nE3Hjt3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\nNsVUxQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa3a323c510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#x = np.linspace(rv.ppf(0.01), rv.ppf(0.99), 100)\n",
    "x = np.linspace(0.1, 9, 100)\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(x, rv.pdf(x), 'r-', lw=5, alpha=0.6, label='gamma pdf')\n",
    "ax.legend()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.0, 2.0)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rv.mean(), rv.std() # les valeur théorique"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.032586211783519 2.0571413337008937\n"
     ]
    }
   ],
   "source": [
    "# on compare avec des simulations\n",
    "realizations = rv.rvs(size=1000)\n",
    "print realizations.mean(), realizations.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8zjlXkdioIj2Um0vr8JG0Dh9JY+x4Wxs5hw+Rc+gAOYcOkXP4INmHD5Fz+DDZ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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa3a31c1850>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, rv.pdf(x), 'r-', lw=5, alpha=0.6, label='gamma pdf')\n",
    "plt.hist(realizations, density=True, histtype='stepfilled', alpha=0.2, bins=20)\n",
    "plt.legend(loc='best', frameon=False)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9770108728450158, 0.001881871309078058, 2.0788291604211357)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# on peut aussi 'fiter' des données\n",
    "gamma.fit(realizations)\n",
    "# cela renvoie les EMV pour shape, location, scale "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise\n",
    "1. Reproduisez ce qu'on vient de faire pour une distribution gaussienne avec les paramètres $ \\mu = 1.5, \\sigma = 2 $     \n",
    "2. Utiliser le module scipy integrate pour calculer $ \\mathbb{P}(X\\leq 1) $ où $ X $ suit une loi exponentiel de paramètre $\\lambda = 2$. Faites la même chose avec les distributions stats de scipy, à la main et par Monte Carlo. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solution exercise 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fa3a2e8ed50>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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QMtH06W66abicHFezyJggtWwJl13mbV+wAL780ttu4soSQqbZsgU+/NDb/q1v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\n/H8d+yvrQOXa5wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa3a2fe5650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import norm\n",
    "rv_norm = norm(1.5, scale=2.) # on genere la variable aléatoire, on fixe la variable aléatoir\n",
    "x = np.linspace(rv_norm.ppf(0.01), rv_norm.ppf(0.99), 100)\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "ax.plot(x, rv_norm.pdf(x), 'r-', lw=5, alpha=0.6, label='gamma pdf')\n",
    "ax.legend()\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5 2.0\n",
      "1.487841152506433 1.9605588881536116\n"
     ]
    }
   ],
   "source": [
    "print rv_norm.mean(), rv_norm.std() # les valeur théorique\n",
    "realizations = rv_norm.rvs(size=1000)\n",
    "print realizations.mean(), realizations.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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VBJNYqqthrccjJa+8EtIimnif0Bp796F25Gj3hvfecxdJY9rICoJJLKtXQ11d81h6Osya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\nCMYYYxxWEIwxxgDw/wMduyIDdAd0pgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa3a2fda710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, rv_norm.pdf(x), 'r-', lw=5, alpha=0.6, label='normal pdf')\n",
    "plt.hist(realizations, density=True, histtype='stepfilled', alpha=0.2, bins=20)\n",
    "plt.legend(loc='best', frameon=False)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.487841152506433, 1.9605588881536116)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm.fit(realizations)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}