CW6.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Zad 1.\n",
    "Wygeneruj dane z rozkładu jednostajnego na odcinku [-1,1].\n",
    "Narysuj na jednym wykresie histogram z danych oraz gęstość rozkładu normalnego o parametrach m=0 oraz sigma =1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa2d4ec28d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.mlab as mlab\n",
    "import math\n",
    "\n",
    "N = 100000\n",
    "mu = 0 # mean of distribution\n",
    "sigma = 1 # standard deviation of distribution\n",
    "num_bins = 50\n",
    "\n",
    "#X = np.concatenate((np.random.normal(0, 1, N) ,np.random.normal(5, 1, N)))\n",
    "#X = np.random.normal(mu, sigma, N)\n",
    "X = np.random.uniform(-1, 1, N)\n",
    "n, bins, patches = plt.hist(X, num_bins, normed=1, facecolor='green', alpha=0.5)\n",
    "y = mlab.normpdf(bins, mu, sigma)\n",
    "plt.plot(bins, y, 'r--')\n",
    "plt.show()   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zad 2.\n",
    "Wygeneruj dane z rozkładu jednostajnego na odcinku [-1,1].\n",
    "Narysuj histogram z danych oraz optymalny (najlepszy z punktu widzenia MLE MLE) rozkład normalny opisujący dane.\n",
    "Użyj estymatorów największej wiarygodności dla m i sigma. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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nK5PmOphIumVaAeG3K9kUmleWa4DZVQkVbcVjNh/z9Tbeu6K3CruIVMrmE1px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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa2a1e544a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 100000\n",
    "X = np.random.uniform(-1, 1, N)\n",
    "\n",
    "mu = sum(X) / N\n",
    "sigma = 0\n",
    "for i in range(N):\n",
    "    sigma += (X[i] - mu)**2\n",
    "sigma = math.sqrt(sigma / N)\n",
    "    \n",
    "num_bins = 50\n",
    "\n",
    "gauss = np.random.normal(mu, sigma, N)\n",
    "n, bins, patches = plt.hist(X, num_bins, normed=1, facecolor='green', alpha=0.5)\n",
    "y = mlab.normpdf(bins, mu, sigma)\n",
    "plt.plot(bins, y, 'r--')\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zad 3.\n",
    "Wygeneruj dane z rozkładu normalnego o parametrach m=0 oraz sigma =1.\n",
    "Narysuj histogram z danych oraz optymalny (najlepszy z punktu widzenia MLE) rozkład normalny opisujący dane.\n",
    "Użyj estymatorów największej wiarygodności dla m i sigma. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.000826801646325 1.0002307571\n"
     ]
    },
    {
     "data": {
      "image/png": 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cMpposGULdOoE9eq5TmLOUVl8HOuu6+c6hqkhNuCaOXeqcOON0LYtLFrkOo0J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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fe570441940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.mlab as mlab\n",
    "import math\n",
    "\n",
    "N = 1000000\n",
    "X = np.random.normal(0, 1, N)\n",
    "mu = sum(X) / N\n",
    "sigma = 0\n",
    "for i in range(N):\n",
    "    sigma += (X[i] - mu)**2\n",
    "sigma = sigma / N\n",
    "\n",
    "print(mu, sigma)\n",
    "    \n",
    "num_bins = 50\n",
    "\n",
    "gauss = np.random.normal(mu, sigma, N)\n",
    "n, bins, patches = plt.hist(X, num_bins, normed=1, facecolor='green', alpha=0.5)\n",
    "y = mlab.normpdf(bins, mu, sigma)\n",
    "plt.plot(bins, y, 'r--')\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Zad 4.\n",
    "Wygeneruj dane z rozkładu jednostajnego na odcinku [-1,1].\n",
    "Napisz funkcję wyliczającą wartość logarytmicznej funkcji wiarygodności względem rodziny rozkładów normalnych dla zadanego zbioru danych. Policz jej wartość dla próbki oraz:\n",
    "\n",
    "    m = 0, sigma =1\n",
    "    m = 1, sigma =1.1\n",
    "    m = 0, sigma =0.9\n",
    "    m = 1, sigma =1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "m = 0, sigma =1    583.813676907\n",
      "m = 1, sigma =1.1  -6763.67061515\n",
      "m = 0, sigma =0.9  1955.4978049\n",
      "m = 1, sigma =1    -4410.25369963\n"
     ]
    }
   ],
   "source": [
    "N = 10000\n",
    "X = np.random.uniform(-1, 1, N)\n",
    "\n",
    "def loglike(data, mu, sigma):\n",
    "    N = len(data)\n",
    "    summ = 0\n",
    "    for i in range(N):\n",
    "        summ += (X[i] - mu)**2\n",
    "    return -N*np.log(sigma) - N/2*np.log(2/math.pi) -1/2*sigma**2 * summ\n",
    "    \n",
    "print(\"m = 0, sigma =1   \", loglike(X, 0, 1))\n",
    "print(\"m = 1, sigma =1.1 \", loglike(X, 1, 1.1))\n",
    "print(\"m = 0, sigma =0.9 \", loglike(X, 0, 0.9))\n",
    "print(\"m = 1, sigma =1   \", loglike(X, 1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zad 5.\n",
    "Wygeneruj dane z rozkładu normalnego o parametrach m=0 oraz sigma =1.\n",
    "Zminimalizuj zlogarytmowaną funkcję wiarygodności dla rodziny rozkładów normalnych względem parametrów m i sigma. Narysuj histogram z danych oraz optymalny (najlepszy z punktu widzenia MLE) rozkład normalny opisujący dane.\n",
    "Użyj wyliczonych wartości m i sigma. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/lib/python3.6/site-packages/ipykernel/__main__.py:12: RuntimeWarning: invalid value encountered in log\n",
      "/usr/lib/python3.6/site-packages/ipykernel/__main__.py:35: RuntimeWarning: invalid value encountered in log\n"
     ]
    },
    {
     "data": {
      "image/png": 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dPlyvJZC1ad062LzjjTd8J5EIqLhLymu2Yy87c1tRXNDFd5TElpMDvXtrvXuS0AVVSXm7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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa2d9723c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import fmin\n",
    "import random\n",
    "\n",
    "N = 100000\n",
    "X = np.random.normal(0, 1, N)\n",
    "\n",
    "def loglike(data, mu, sigma):\n",
    "    N = len(data)\n",
    "    summ = 0\n",
    "    for i in range(N):\n",
    "        summ += (X[i] - mu)**2\n",
    "    return -N*np.log(sigma) - N/2*np.log(2/math.pi) -1/2*sigma**2 * summ\n",
    "\n",
    "max_func = 0\n",
    "min_func = 0\n",
    "maximal_x, maximal_y = None, None\n",
    "minimal_x, minimal_y = None, None\n",
    "\n",
    "for i in range(100):\n",
    "    randx = random.random()*2 - 1\n",
    "    randy = random.random()*2 - 1\n",
    "    result = loglike(X, randx, randy)\n",
    "    max_func = max(max_func, result)\n",
    "    if max_func == result:\n",
    "        maximal_x, maximal_y = randx, randy\n",
    "    min_func = min(min_func, result)\n",
    "    if min_func == result:\n",
    "        minimal_x, minimal_y = randx, randy\n",
    "\n",
    "        \n",
    "num_bins = 50\n",
    "\n",
    "gauss = np.random.normal(-minimal_x, minimal_y, N)\n",
    "for i in range(N):\n",
    "    gauss[i] = np.log(gauss[i])\n",
    "    \n",
    "n, bins, patches = plt.hist(X, num_bins, normed=1, facecolor='green', alpha=0.5)\n",
    "y = mlab.normpdf(bins, minimal_x, minimal_y)\n",
    "plt.plot(bins, y, 'r--')\n",
    "plt.show() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zad 6.\n",
    "Wygeneruj dane z rozkładu jednostajnego na odcinku [-1,1]. Zminimalizuj zlogarytmowaną funkcję wiarygodności dla rodziny rozkładów Split Normal Distribution link względem parametrów m, sigma i tau. Narysuj histogram z danych oraz optymalny (najlepszy z punktu widzenia MLE) rozkład Split Normal Distribution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/lib/python3.6/site-packages/ipykernel/__main__.py:12: RuntimeWarning: invalid value encountered in log\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fa2a1c685f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import fmin\n",
    "import random\n",
    "\n",
    "N = 1000000\n",
    "X = np.random.uniform(-1, 1, N)\n",
    "\n",
    "def loglike(data, mu, sigma):\n",
    "    N = len(data)\n",
    "    summ = 0\n",
    "    for i in range(N):\n",
    "        summ += (X[i] - mu)**2\n",
    "    return -N*np.log(sigma) - N/2*np.log(2/math.pi) -1/2*sigma**2 * summ # заменить на сплит\n",
    "\n",
    "max_func = 0\n",
    "min_func = 0\n",
    "maximal_x, maximal_y = None, None\n",
    "minimal_x, minimal_y = None, None\n",
    "\n",
    "for i in range(100):\n",
    "    randx = random.random()*2 - 1\n",
    "    randy = random.random()*2 - 1\n",
    "    result = loglike(X, randx, randy)\n",
    "    max_func = max(max_func, result)\n",
    "    if max_func == result:\n",
    "        maximal_x, maximal_y = randx, randy\n",
    "    min_func = min(min_func, result)\n",
    "    if min_func == result:\n",
    "        minimal_x, minimal_y = randx, randy\n",
    "\n",
    "        \n",
    "num_bins = 50\n",
    "\n",
    "    \n",
    "def SN(xlist, m,s,t):\n",
    "    c=math.sqrt(2./math.pi)*(1./s)*(1./(1+t))\n",
    "    if x<=m:\n",
    "        return (c*math.exp(-1./(2* s**2) * (x-m)**2))\n",
    "    else:\n",
    "        return (c*math.exp(-1./(t**2 * s**2) * (x-m)**2))\n",
    "    \n",
    "res = []\n",
    "X = []\n",
    "for i in range (10000):\n",
    "    #I.append(i)\n",
    "    x = np.random.normal(0, 1)\n",
    "    X.append(x)\n",
    "    y = SN(x, 0, 1, 1)\n",
    "    res.append(y)\n",
    "\n",
    "\n",
    "plt.scatter(X, res, color='blue', alpha=0.7)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Zad 7.\n",
    "Pobierz dowolny plik z dźwiękiem pliki. Zminimalizuj zlogarytmowaną funkcję wiarygodności dla rodziny rozkładów:\n",
    "\n",
    "    Rozkładów normalnych\n",
    "    Split Normal Distribution\n",
    "\n",
    "Narysuj wyniki ora porównaj wartości zlogarytmowanej funkcji wiarygodności. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}