TD2.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tutorial 2\n",
    "Question proposed by teacher and answers done by Chensheng LUO of CentraleSupélec"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first load the data needed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "[ 98474.4  12373.4 107318.4   9739.7  13000.1  12207.7  63589.4  31893.8\n   5784.9   9662.7  61731.1  15269.2   4730.   11269.2  26947.1  27838.1\n  90682.5   8086.6   7905.6  48162.7]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "array1 = np.loadtxt('Projectdata.txt')\n",
    "print(array1)\n",
    "array2 = [31.7, 39.2, 57.5, 65.8, 70.0, 101.7, 109.2, 130.0]\n",
    "array3 = [6.21, 5.12, 2.99, 2.00, 9.79, 5.78, 5.54, 6.44, 8.92, 6.19]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We study the failure dates of pressure transmitters (PT) recorded in the file Projectdata.txt.   \n",
    "1. Demonstrate that it is reasonable to assume an exponential probability density function for the lifetime of the PT.   \n",
    "2. Estimate the parameter of the pdf with two methods.   \n",
    "3. Propose a test to accept or reject the assumption that the exponential pdf obtained with the estimated failure rate is the good one."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer**\n",
    "1. We calculate the mean and the standard deviation, then we calculate the sample coefficient of variation CV, which is close to 1 and proves that it is reasonable to assume an exponential probability density function for the lifetime of the PT."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statistics\n",
    "m=statistics.mean(array1)\n",
    "s=statistics.stdev(array1)\n",
    "print(m/s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. We first do with the **MLE**. With the calculation, we have $\\lambda_{\\text{MLE}}=\\frac{1}{E(x)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The estimation of lambda by MLE method is 3.0000003000000297e-05\n"
     ]
    }
   ],
   "source": [
    "lambda_MLE = 1/m\n",
    "print('The estimation of lambda by MLE method is', lambda_MLE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then use the **method of moment**: we have that the theorical expectation is $E[X]=\\frac{1}{\\lambda}$ so $\\lambda_{MM}=\\frac{1}{E[x]}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The estimation of lambda by MM is 3.0000003000000297e-05\n"
     ]
    }
   ],
   "source": [
    "lambda_MM = 1/m\n",
    "print('The estimation of lambda by MM is', lambda_MM)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then verify the quality of estimation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import expon\n",
    "\n",
    "fig, ax = plt.subplots(1, 1, figsize=(4, 4))\n",
    "\n",
    "## Statistic-Empirical\n",
    "ax.hist(array1,density=True, alpha=0.5, label='Empirical distribution')\n",
    "\n",
    "## Theoretical distribution\n",
    "xlim = ax.get_xlim()\n",
    "vx = np.arange(xlim[0], xlim[1],500)\n",
    "vy = expon.pdf(vx, scale=1/lambda_MLE)\n",
    "ax.plot(vx, vy, 'r--', label='Theoretical distribution')\n",
    "\n",
    "ax.grid(True)\n",
    "ax.legend(loc='upper right')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. We do the Kolmogorov-Smirnov test:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "KstestResult(statistic=1.0, pvalue=1.4508889103849681e-11)"
      ]
     },
     "metadata": {},
     "execution_count": 16
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.kstest(array1, expon.cdf(array1, scale=1/lambda_MLE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "We study the failure dates of pressure transmitters (PT) recorded in the file Projectdata.txt.\n",
    "1. Determine the empirical cumulative distribution corresponding to this data set and plot it.\n",
    "2. Compare with the cumulative distribution found in Exercise 1.\n",
    "3. Comment and explain how you could improve your results."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. By the following code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x24656708700>"
      ]
     },
     "metadata": {},
     "execution_count": 17
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 288x288 with 1 Axes>",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import expon\n",
    "\n",
    "fig, ax = plt.subplots(1, 1, figsize=(4, 4))\n",
    "\n",
    "## Statistic-Empirical\n",
    "ax.hist(array1,density=True,cumulative=True,alpha=0.5, label='Empirical cumulative distribution')\n",
    "\n",
    "## Theoretical distribution\n",
    "xlim = ax.get_xlim()\n",
    "vx = np.arange(xlim[0], xlim[1],500)\n",
    "vy = expon.cdf(vx, scale=1/lambda_MLE)\n",
    "ax.plot(vx, vy, 'r--', label='Theoretical distribution')\n",
    "\n",
    "ax.grid(True)\n",
    "ax.legend(loc='upper right')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Let's plot the Q-Q plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 288x288 with 1 Axes>",
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "# We can use probplot function\n",
    "from scipy.stats import probplot\n",
    "fig, ax = plt.subplots(1, 1, figsize=(4, 4))\n",
    "probplot(array1, dist=expon, sparams=(1/lambda_MLE), plot=ax)\n",
    "ax.grid(True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "# We can also try to write by ourselves\n",
    "\n",
    "## First we sort the array\n",
    "array1 = np.sort(array1)\n",
    "\n",
    "## Find the corresponding theorical point: emperical cumulative pdf\n",
    "theorical=[]\n",
    "for i in range(len(array1)):\n",
    "    theorical.append(expon.ppf((i+1)/len(array1), scale=1/lambda_MLE))\n",
    "\n",
    "## Plot it!\n",
    "fig, ax = plt.subplots(1, 1, figsize=(4, 4))\n",
    "ax.scatter(array1, theorical)\n",
    "xlim = ax.get_xlim()\n",
    "ylim = ax.get_ylim()\n",
    "lim = [np.min([xlim[0], ylim[0]]), np.max([xlim[1], ylim[1]])]\n",
    "ax.plot(lim, lim, 'r')\n",
    "ax.set_xlim(lim)\n",
    "ax.set_ylim(lim)\n",
    "ax.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise3 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We consider a complete sample of lifetimes that are independent and identical to Weibull distribution.\n",
    "1. Propose a method to estimate the parameters of the distribution\n",
    "2. Apply it to the follow data set: (31.7, 39.2, 57.5, 65.8, 70.0, 101.7, 109.2, 130.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer**\n",
    "We use the methode of **maximisation of likelihood function**, here we use ```optimize``` package:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "Optimization terminated successfully.\n         Current function value: 78851333551228976.000000\n         Iterations: 116\n         Function evaluations: 264\n[85.5741035  2.543016 ]\n"
     ]
    }
   ],
   "source": [
    "import scipy\n",
    "\n",
    "def weib(x,n,a):\n",
    "    return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)\n",
    "\n",
    "def likelihood(x): \n",
    "    likelihood=1\n",
    "    for i in array2:\n",
    "        likelihood *= weib(i,x[0],x[1])\n",
    "    return likelihood\n",
    "max_x = scipy.optimize.fmin(lambda x: 1/likelihood(x), [120,1])\n",
    "print(max_x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We consider the following sample: (6.21, 5.12, 2.99, 2.00, 9.79, 5.78, 5.54, 6.44, 8.92, 6.19)  \n",
    "Propose a method to accept or reject the assumption that the data are from the Normal distribution $N(5, 2^{2})$. We want a probability of error equals to 0.243 (i.e. the probability to decide that the data are not from the Normal distribution $N(5, 2^{2})$ whereas they are)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer** Let's do Kolmogorov-Smirnov test.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "KstestResult(statistic=0.5, pvalue=0.062288669185221024)"
      ]
     },
     "metadata": {},
     "execution_count": 21
    }
   ],
   "source": [
    "from scipy import stats\n",
    "new=[]\n",
    "for i in array3:\n",
    "    new.append(i-5)\n",
    "scipy.stats.kstest(new, stats.norm.cdf(array1, scale=2))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "Maximum of difference is 0.3239221826541068\n",
      "threshold is  0.3245023948069677\n",
      "The answer of the test is: True .\n"
     ]
    }
   ],
   "source": [
    "## First we sort the array\n",
    "new = np.sort(new)\n",
    "\n",
    "## Find the corresponding theorical point: emperical cumulative pdf\n",
    "difference=[]\n",
    "for i in range(len(new)):\n",
    "    difference.append(abs(stats.norm.cdf(new[i], scale=2)-(i+1)/len(new)))\n",
    "    difference.append(abs(stats.norm.cdf(new[i], scale=2)-i/len(new)))\n",
    "print('Maximum of difference is', max(difference))\n",
    "\n",
    "## Threshold\n",
    "import math\n",
    "alpha = 0.243\n",
    "threshold = 1/ math.sqrt(len(array3)) *  stats.kstwobign.ppf(1-alpha)\n",
    "print('threshold is ',threshold)\n",
    "\n",
    "## Answer\n",
    "print('The answer of the test is:',max(difference)<threshold,'.')"
   ]
  }
 ],
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