Weibull Forecasting using MLE Method

gistfile1.txt

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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h1 align='center'>Weibull Analysis</h1><br>\n",
      "<center>Created by Pybokeh</center>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Part 3 - Estimating Weibull parameters using Maximum Likelihood Estimation (MLE)"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "To solve for our scale parameter (k) using MLE, we need to solve for this equation below:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<center>$\\huge{\\sum\\limits_{i=1}^r \\frac{ln(x_i)}{r}=\\frac{\\sum\\limits_{i=1}^n (x_i)^kln(x_i)}{\\sum\\limits_{i=1}^n (x_i)^k}-\\frac{1}{k}\\approx0}$</center> ,<br><br>\n",
      "<center>Where $x_i$ is the ith failure time,</center><br>\n",
      "<center>r is the number of failures,</center><br>\n",
      "<center>and n is the total number of failure times, both failed and suspended</center>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "The equation above basically means we need to find a value of k such that the left of the equation equals the right equation or in other words, the value of k such that the difference between the left of the equation and right of the equation is as close to zero as possible.  Once we find the value of k, the scale parameter can be calculated as:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<center>$\\large{\\lambda=\\left(\\sum\\limits_{i=1}^n \\frac{(x_i)^k}{r}\\right)^\\frac{1}{k}}$</center>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Let's get data from an external file and calculate the left side of the equation $\\left(\\large{\\sum\\limits_{i=1}^r \\frac{ln(x_i)}{r}}\\right)$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pandas\n",
      "import openpyxl\n",
      "from numpy import log as ln\n",
      "\n",
      "# Open csv file\n",
      "xls = pandas.ExcelFile(r'D:\\weibull\\CRV_18190.xlsx')\n",
      "df = xls.parse(\"Weibull\")\n",
      "\n",
      "df_failed = df[df.STATUS=='FAILED']\n",
      "df_suspended = df[df.STATUS==\"SUSPENDED\"]\n",
      "\n",
      "mtf_failed = df_failed[\"MTF\"].values\n",
      "\n",
      "def ln_x_div_r(series):\n",
      "    return ln(series[\"MTF\"])/len(df_failed)\n",
      "\n",
      "left_eq_sum = np.sum( ln(mtf_failed)/len(df_failed) )\n",
      "\n",
      "df_failed[\"ln_x_div_r\"] = df_failed.apply(ln_x_div_r, axis=1)\n",
      "\n",
      "print(\"number of failures:\", len(df_failed))\n",
      "print(\"number of suspensions:\", len(df_suspended))\n",
      "print(\"failures + suspensions:\", len(df_failed)+len(df_suspended))\n",
      "print(df_failed.to_string(),\"\\n\")\n",
      "print(\"Sum of \\\"ln_x_div_r\\\" column =\", left_eq_sum)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "number of failures: 241\n",
        "number of suspensions: 153664\n",
        "failures + suspensions: 153905\n",
        "                VIN_NO   DTF     MTF  STATUS  ln_x_div_r\n",
        "0    5J6RE4H79AL081186   991     197  FAILED    0.021922\n",
        "1    5J6RE3H33AL047462   970     996  FAILED    0.028646\n",
        "2    5J6RE3H53AL026824  1094    3611  FAILED    0.033991\n",
        "3    5J6RE3H32AL021340  1092    4908  FAILED    0.035264\n",
        "4    5J6RE3H50AL030815  1067    5776  FAILED    0.035940\n",
        "5    5J6RE3H38AL016109  1098    5807  FAILED    0.035962\n",
        "6    5J6RE4H76AL029207  1189    5952  FAILED    0.036064\n",
        "7    5J6RE3H32AL005932  1255    6310  FAILED    0.036307\n",
        "8    5J6RE4H50AL064274  1054    6479  FAILED    0.036416\n",
        "9    5J6RE3H30AL005900  1212    6607  FAILED    0.036497\n",
        "10   5J6RE3H33AL005924  1231    6940  FAILED    0.036701\n",
        "11   5J6RE4H31AL080207  1018    7047  FAILED    0.036765\n",
        "12   5J6RE3H36AL002371  1276    7740  FAILED    0.037154\n",
        "13   5J6RE3H34AL005916  1266    8015  FAILED    0.037299\n",
        "14   5J6RE3H37AL005893  1267    8173  FAILED    0.037380\n",
        "15   5J6RE3H32AL005946  1208    8262  FAILED    0.037425\n",
        "16   5J6RE3H39AL005913  1145    9243  FAILED    0.037891\n",
        "17   5J6RE3H74AL047196   953   10237  FAILED    0.038314\n",
        "18   5J6RE3H37AL005909  1269   10440  FAILED    0.038396\n",
        "19   5J6RE3H36AL005903  1245   10962  FAILED    0.038598\n",
        "20   5J6RE3H71AL006315  1246   11101  FAILED    0.038651\n",
        "21   5J6RE3H30AL025757  1090   12106  FAILED    0.039010\n",
        "22   5J6RE3H39AL005894  1230   12525  FAILED    0.039151\n",
        "23   5J6RE3H50AL012783  1184   13335  FAILED    0.039411\n",
        "24   5J6RE4H54AL093647   944   15130  FAILED    0.039935\n",
        "25   5J6RE4H57AL000409  1284   15403  FAILED    0.040010\n",
        "26   5J6RE3H3XAL005919  1252   15514  FAILED    0.040039\n",
        "27   5J6RE4H54AL063662  1064   15660  FAILED    0.040078\n",
        "28   5J6RE3H34AL046272   919   16656  FAILED    0.040334\n",
        "29   5J6RE3H56AL037011  1039   16992  FAILED    0.040417\n",
        "30   5J6RE3H33AL005941  1239   17420  FAILED    0.040520\n",
        "31   5J6RE4H30AL080148  1010   19599  FAILED    0.041009\n",
        "32   5J6RE4H79AL097291   958   19717  FAILED    0.041034\n",
        "33   5J6RE3H74AL006356  1269   19948  FAILED    0.041083\n",
        "34   5J6RE3H50AL034668  1033   20033  FAILED    0.041100\n",
        "35   5J6RE4H32AL079731   976   22965  FAILED    0.041667\n",
        "36   5J6RE3H32AL005901  1254   23852  FAILED    0.041824\n",
        "37   5J6RE3H73AL006364  1261   24711  FAILED    0.041971\n",
        "38   5J6RE3H39AL019391  1067   24907  FAILED    0.042004\n",
        "39   5J6RE3H39AL014319  1182   25133  FAILED    0.042041\n",
        "40   5J6RE3H53AL040089  1016   25308  FAILED    0.042070\n",
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        "51   5J6RE3H39AL034196  1013   32450  FAILED    0.043101\n",
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        "54   5J6RE4H39AL076518  1019   33884  FAILED    0.043281\n",
        "55   5J6RE4H3XAL028560  1185   34591  FAILED    0.043367\n",
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        "74   5J6RE3H72AL050727   885   41717  FAILED    0.044144\n",
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        "76   5J6RE4H58AL021981  1165   42126  FAILED    0.044184\n",
        "77   5J6RE4H57AL040022  1120   42241  FAILED    0.044196\n",
        "78   5J6RE4H36AL028278  1185   42560  FAILED    0.044227\n",
        "79   5J6RE4H52AL005694  1269   42689  FAILED    0.044239\n",
        "80   5J6RE4H52AL028876  1189   42885  FAILED    0.044258\n",
        "81   5J6RE4H5XAL045103  1074   43365  FAILED    0.044305\n",
        "82   5J6RE3H54AL032793  1064   43808  FAILED    0.044347\n",
        "83   5J6RE4H54AL049423  1113   43917  FAILED    0.044357\n",
        "84   5J6RE4H77AL017261  1255   43970  FAILED    0.044362\n",
        "85   5J6RE3H76AL021182  1095   44460  FAILED    0.044408\n",
        "86   5J6RE4H54AL045131  1138   44640  FAILED    0.044425\n",
        "87   5J6RE4H35AL098077   945   44788  FAILED    0.044439\n",
        "88   5J6RE4H34AL030837  1180   45368  FAILED    0.044492\n",
        "89   5J6RE3H37AL015713  1144   46038  FAILED    0.044553\n",
        "90   5J6RE4H35AL034556  1035   46773  FAILED    0.044619\n",
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        "107  5J6RE4H74AL001230  1297   51663  FAILED    0.045031\n",
        "108  5J6RE4H3XAL043804  1134   52484  FAILED    0.045097\n",
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        "199  5J6RE4H73AL051326  1115   76275  FAILED    0.046648\n",
        "200  5J6RE3H79AL033584   944   76302  FAILED    0.046649\n",
        "201  5J6RE4H39AL063199  1084   76434  FAILED    0.046656\n",
        "202  5J6RE4H7XAL046379  1133   76549  FAILED    0.046663\n",
        "203  5J6RE3H37AL009331  1245   76722  FAILED    0.046672\n",
        "204  5J6RE4H72AL004708  1289   76941  FAILED    0.046684\n",
        "205  5J6RE3H75AL003711  1270   77550  FAILED    0.046717\n",
        "206  5J6RE4H76AL038196  1164   77711  FAILED    0.046725\n",
        "207  5J6RE3H30AL020302  1140   78011  FAILED    0.046741\n",
        "208  5J6RE3H38AL029863  1053   78017  FAILED    0.046741\n",
        "209  5J6RE3H36AL018540  1147   78055  FAILED    0.046743\n",
        "210  5J6RE4H3XAL040160  1149   78079  FAILED    0.046745\n",
        "211  5J6RE3H78AL021927  1135   78724  FAILED    0.046779\n",
        "212  5J6RE4H76AL017185  1222   78783  FAILED    0.046782\n",
        "213  5J6RE4H32AL055249  1092   78954  FAILED    0.046791\n",
        "214  5J6RE4H30AL035579  1163   79141  FAILED    0.046801\n",
        "215  5J6RE4H79AL016516  1226   79248  FAILED    0.046806\n",
        "216  5J6RE4H58AL035170  1133   79289  FAILED    0.046809\n",
        "217  5J6RE4H75AL033605  1096   79362  FAILED    0.046812\n",
        "218  5J6RE3H78AL010913  1214   79892  FAILED    0.046840\n",
        "219  5J6RE4H77AL059896   944   79900  FAILED    0.046840\n",
        "220  5J6RE4H73AL009111  1271   79909  FAILED    0.046841\n",
        "221  5J6RE4H34AL036914  1075   79943  FAILED    0.046843\n",
        "222  5J6RE4H7XAL014550  1243   82029  FAILED    0.046949\n",
        "223  5J6RE4H57AL004086  1215   82193  FAILED    0.046958\n",
        "224  5J6RE4H51AL058032  1092   82938  FAILED    0.046995\n",
        "225  5J6RE4H75AL040411  1116   83451  FAILED    0.047021\n",
        "226  5J6RE4H78AL035705  1145   83649  FAILED    0.047031\n",
        "227  5J6RE4H53AL029048  1196   87312  FAILED    0.047208\n",
        "228  5J6RE4H5XAL062712  1064   88840  FAILED    0.047280\n",
        "229  5J6RE4H30AL048090  1113   92842  FAILED    0.047463\n",
        "230  5J6RE4H78AL031847  1166   93106  FAILED    0.047475\n",
        "231  5J6RE3H38AL045190   958   93188  FAILED    0.047479\n",
        "232  5J6RE4H73AL037670  1085   93450  FAILED    0.047490\n",
        "233  5J6RE4H35AL043516  1131   96552  FAILED    0.047626\n",
        "234  5J6RE4H75AL024385  1212   97685  FAILED    0.047674\n",
        "235  5J6RE4H76AL035413  1168   99812  FAILED    0.047764\n",
        "236  5J6RE4H72AL055982  1102  103924  FAILED    0.047931\n",
        "237  5J6RE4H7XAL041862  1126  105316  FAILED    0.047986\n",
        "238  5J6RE3H75AL003370  1277  113744  FAILED    0.048306\n",
        "239  5J6RE4H37AL096069   952  132520  FAILED    0.048940\n",
        "240  5J6RE4H79AL025507  1215  260685  FAILED    0.051747 \n",
        "\n",
        "Sum of \"ln_x_div_r\" column = 10.6783493147\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<center>left side = $\\large{\\sum\\limits_{i=1}^r \\frac{ln(x_i)}{r}}$ = sum of ln_x_div_r column</center>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "The left side of the equation was calculated to be 6.9254.  Next we are going to calculate the right side of the equation using 100,000 values of k ranging from 0.1 to 10 and then substract the result from 4.9138.  Basically, what we are trying to achieve is throw a lot of k values into the right side of the equation, and subtract the result from it from 4.9138 until we find the difference as small or close to zero as possible. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<center>sum of ln_x_div_r - $\\large{\\left(\\frac{\\sum\\limits_{i=1}^n (x_i)^kln(x_i)}{\\sum\\limits_{i=1}^n (x_i)^k}-\\frac{1}{k}\\right) \\approx 0}$</center>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "NOTE: If your data set is large (> 50,000), reduce the # of k values to about 50000 or smaller.  &nbsp;Otherwise, it will take several minutes, perhaps longer than 30 minutes.<br>This script below will calculate the 100,000 values that represent the difference of the left side and right side of the equation:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mtf_all = df[\"MTF\"].values\n",
      "\n",
      "# Generate 10,0000 k values that is between 0.1 and 10\n",
      "# Can't use 0 because it is not a valid value in the right-side equation formula, so we're using 0.1 instead\n",
      "k = np.linspace(0.1,8,20000,endpoint=True)\n",
      "\n",
      "right = []\n",
      "for value in k:\n",
      "    right_eq = np.sum( mtf_all**value * ln(mtf_all) ) / np.sum( mtf_all**value ) - (1/value)\n",
      "    right.append(right_eq)\n",
      "    \n",
      "right_values = np.array(right)\n",
      "\n",
      "diff = right_values - left_eq_sum"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "When it is done, at this point we should have 100,000 difference values and their corresponding k values.  Now let's plot them and see what they look like:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(k,diff)\n",
      "ylabel(\"Difference\")\n",
      "xlabel(\"k\")\n",
      "grid()\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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NTkFEZH5utYkyMjKQl5eH6dOn4/Tp0ygtLUWIh+eAvpU20YQJwLhxYsyAiKgpkjZmMHfu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\ngcWAiIjAYkBERGAxICIisBgQERGA/wfj+e1OvV6aKgAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "As I've already mentioned, we need to choose a k value where the difference between the left equation and right equation is close to 0 as possible.  From the plot above, it looks like it would be approximately less than 3, maybe 2.8.  But instead of trying to \"eye-ball it\", let's calculate a more precise value for k where the difference is as close to zero as possible:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ks = np.array(k)\n",
      "diffs = np.array(diff)\n",
      "\n",
      "d = {'k' : ks,\n",
      "     'diff' : diffs}\n",
      "    \n",
      "df_k_diff = pandas.DataFrame(d, index=np.arange(1,ks.size+1)) # Create one-based index instead of zero-based\n",
      "df_k_diff = df_k_diff.sort_index(axis=1, ascending=False)  # Re-sorting the columns so that k column in on the left\n",
      "\n",
      "print(\"What the k values and their diffs look like:\")\n",
      "print(df_k_diff.head())\n",
      "print(df_k_diff.tail(), \"\\n\")\n",
      "\n",
      "print(\"The row with the first smallest difference:\")\n",
      "k_with_smallest_diff = df_k_diff[ (df_k_diff[\"diff\"] >= 0.000001)][0:1]\n",
      "print(k_with_smallest_diff, \"\\n\")\n",
      "\n",
      "final_k = k_with_smallest_diff[\"k\"].values\n",
      "k_final = final_k[0]\n",
      "print(\"Therefore, the k value we should use where the difference is as close to zero is:\", k_final)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "What the k values and their diffs look like:\n",
        "          k       diff\n",
        "1  0.100000 -10.126434\n",
        "2  0.100395 -10.087083\n",
        "3  0.100790 -10.048040\n",
        "4  0.101185 -10.009302\n",
        "5  0.101580  -9.970865\n",
        "              k      diff\n",
        "19996  7.998420  1.577376\n",
        "19997  7.998815  1.577441\n",
        "19998  7.999210  1.577506\n",
        "19999  7.999605  1.577571\n",
        "20000  8.000000  1.577636 \n",
        "\n",
        "The row with the first smallest difference:\n",
        "              k      diff\n",
        "13216  5.320186  0.000109 \n",
        "\n",
        "Therefore, the k value we should use where the difference is as close to zero is: 5.3201860093\n"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "OK so now that we solved for k, we can solve for the scale ($\\lambda$) parameter next:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<center>$\\large{\\lambda=\\left(\\sum\\limits_{i=1}^n \\frac{(x_i)^k}{r}\\right)^\\frac{1}{k}}$</center>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "l = (np.sum((mtf_all**k_final)/len(df_failed)))**(1/k_final)\n",
      "print(\"scale parameter:\",l)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "scale parameter: 136443.859106\n"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Let's revisit the difference vs k value plot again:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = []\n",
      "for value in range(0,20000):\n",
      "    x.append(k_final)\n",
      "\n",
      "plot(k,diff)\n",
      "plot(x,diff)\n",
      "title(\"The difference is zero when k=2.762\")\n",
      "ylabel(\"Difference\")\n",
      "xlabel(\"k\")\n",
      "grid()\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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dxeCRRx7BU089hatXr2LVqlV48MEHMXv27AYJ8/bbb8Pa2hpXrlyp92MlJQE9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\nAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "We're done!  So using MLE method, we calculated the shape parameter to be 2.762 and the scale parameter to be 4452.77.  With the usual Weibull CDF plot below, we are equipped to make failure predictions."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "shape = k_final\n",
      "scale = l\n",
      "\n",
      "x2 = np.arange(0,130000)\n",
      "y2 = 1-exp(-(x2/scale)**shape) # This is the equation for Weibull CDF\n",
      "\n",
      "plot(x2,y2)\n",
      "title(\"Weibull CDF - Prediction\",weight='bold')\n",
      "grid()\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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PdhYUf39/oqOjiY2NBc5WbatOl71nlTzVKX9sbKyl8kj+s99v2zaWIUNgxIhM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\npBAIIUQ1J4VACCGqOSkEQghRzf1/+30oqT/vXwcAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"Coordinates ( x, y ) of the CDF:\")\n",
      "for value in zip(x2,y2):\n",
      "    print(str(value[0]) + \" , \" + str(value[1]*100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Coordinates ( x, y ) of the CDF:\n",
        "0 , 0.0"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "1 , 0.0\n",
        "2 , 0.0\n",
        "3 , 0.0\n",
        "4 , 0.0\n",
        "5 , 0.0\n",
        "6 , 0.0\n",
        "7 , 0.0\n",
        "8 , 0.0\n",
        "9 , 0.0\n",
        "10 , 0.0\n",
        "11 , 0.0\n",
        "12 , 0.0\n",
        "13 , 0.0\n",
        "14 , 0.0\n",
        "15 , 0.0\n",
        "16 , 0.0\n",
        "17 , 0.0\n",
        "18 , 0.0\n",
        "19 , 0.0\n",
        "20 , 0.0\n",
        "21 , 0.0\n",
        "22 , 0.0\n",
        "23 , 0.0\n",
        "24 , 0.0\n",
        "25 , 0.0\n",
        "26 , 0.0\n",
        "27 , 0.0\n",
        "28 , 0.0\n",
        "29 , 0.0\n",
        "30 , 0.0\n",
        "31 , 0.0\n",
        "32 , 0.0\n",
        "33 , 0.0\n",
        "34 , 0.0\n",
        "35 , 0.0\n",
        "36 , 0.0\n",
        "37 , 0.0\n",
        "38 , 0.0\n",
        "39 , 0.0\n",
        "40 , 0.0\n",
        "41 , 0.0\n",
        "42 , 0.0\n",
        "43 , 0.0\n",
        "44 , 0.0\n",
        "45 , 0.0\n",
        "46 , 0.0\n",
        "47 , 0.0\n",
        "48 , 0.0\n",
        "49 , 0.0\n",
        "50 , 0.0\n",
        "51 , 0.0\n",
        "52 , 0.0\n",
        "53 , 0.0\n",
        "54 , 0.0\n",
        "55 , 0.0\n",
        "56 , 0.0\n",
        "57 , 0.0\n",
        "58 , 0.0\n",
        "59 , 0.0\n",
        "60 , 0.0\n",
        "61 , 0.0\n",
        "62 , 0.0\n",
        "63 , 0.0\n",
        "64 , 0.0\n",
        "65 , 0.0\n",
        "66 , 0.0\n",
        "67 , 0.0\n",
        "68 , 0.0\n",
        "69 , 0.0\n",
        "70 , 0.0\n",
        "71 , 0.0\n",
        "72 , 0.0\n",
        "73 , 0.0\n",
        "74 , 0.0\n",
        "75 , 0.0\n",
        "76 , 0.0\n",
        "77 , 0.0\n",
        "78 , 0.0\n",
        "79 , 0.0\n",
        "80 , 0.0\n",
        "81 , 0.0\n",
        "82 , 0.0\n",
        "83 , 0.0\n",
        "84 , 0.0\n",
        "85 , 0.0\n",
        "86 , 0.0\n",
        "87 , 0.0\n",
        "88 , 0.0\n",
        "89 , 0.0\n",
        "90 , 0.0\n",
        "91 , 0.0\n",
        "92 , 0.0\n",
        "93 , 0.0\n",
        "94 , 0.0\n",
        "95 , 0.0\n",
        "96 , 0.0\n",
        "97 , 0.0\n",
        "98 , 0.0\n",
        "99 , 0.0\n",
        "100 , 0.0\n",
        "101 , 0.0\n",
        "102 , 0.0\n",
        "103 , 0.0\n",
        "104 , 0.0\n",
        "105 , 0.0\n",
        "106 , 0.0\n",
        "107 , 0.0\n",
        "108 , 0.0\n",
        "109 , 0.0\n",
        "110 , 0.0\n",
        "111 , 0.0\n",
        "112 , 0.0\n",
        "113 , 0.0\n",
        "114 , 0.0\n",
        "115 , 0.0\n",
        "116 , 0.0\n",
        "117 , 0.0\n",
        "118 , 0.0\n",
        "119 , 0.0\n",
        "120 , 0.0\n",
        "121 , 1.11022302463e-14\n",
        "122 , 1.11022302463e-14\n",
        "123 , 1.11022302463e-14\n",
        "124 , 1.11022302463e-14\n",
        "125 , 1.11022302463e-14\n",
        "126 , 1.11022302463e-14\n",
        "127 , 1.11022302463e-14\n",
        "128 , 1.11022302463e-14\n",
        "129 , 1.11022302463e-14\n",
        "130 , 1.11022302463e-14\n",
        "131 , 1.11022302463e-14\n",
        "132 , 1.11022302463e-14\n",
        "133 , 1.11022302463e-14\n",
        "134 , 1.11022302463e-14\n",
        "135 , 1.11022302463e-14\n",
        "136 , 1.11022302463e-14\n",
        "137 , 1.11022302463e-14\n",
        "138 , 1.11022302463e-14\n",
        "139 , 1.11022302463e-14\n",
        "140 , 1.11022302463e-14\n",
        "141 , 1.11022302463e-14\n",
        "142 , 1.11022302463e-14\n",
        "143 , 1.11022302463e-14\n",
        "144 , 1.11022302463e-14\n",
        "145 , 1.11022302463e-14\n",
        "146 , 1.11022302463e-14\n",
        "147 , 1.11022302463e-14\n",
        "148 , 2.22044604925e-14\n",
        "149 , 2.22044604925e-14\n",
        "150 , 2.22044604925e-14\n",
        "151 , 2.22044604925e-14\n",
        "152 , 2.22044604925e-14\n",
        "153 , 2.22044604925e-14\n",
        "154 , 2.22044604925e-14\n",
        "155 , 2.22044604925e-14\n",
        "156 , 2.22044604925e-14\n",
        "157 , 2.22044604925e-14\n",
        "158 , 2.22044604925e-14\n",
        "159 , 2.22044604925e-14\n",
        "160 , 2.22044604925e-14\n",
        "161 , 2.22044604925e-14\n",
        "162 , 2.22044604925e-14\n",
        "163 , 3.33066907388e-14\n",
        "164 , 3.33066907388e-14\n",
        "165 , 3.33066907388e-14\n",
        "166 , 3.33066907388e-14\n",
        "167 , 3.33066907388e-14\n",
        "168 , 3.33066907388e-14\n",
        "169 , 3.33066907388e-14\n",
        "170 , 3.33066907388e-14\n",
        "171 , 3.33066907388e-14\n",
        "172 , 3.33066907388e-14\n",
        "173 , 3.33066907388e-14\n",
        "174 , 4.4408920985e-14\n",
        "175 , 4.4408920985e-14\n",
        "176 , 4.4408920985e-14\n",
        "177 , 4.4408920985e-14\n",
        "178 , 4.4408920985e-14\n",
        "179 , 4.4408920985e-14\n",
        "180 , 4.4408920985e-14\n",
        "181 , 4.4408920985e-14\n",
        "182 , 5.55111512313e-14\n",
        "183 , 5.55111512313e-14\n",
        "184 , 5.55111512313e-14\n",
        "185 , 5.55111512313e-14\n",
        "186 , 5.55111512313e-14\n",
        "187 , 5.55111512313e-14\n",
        "188 , 5.55111512313e-14\n",
        "189 , 6.66133814775e-14\n",
        "190 , 6.66133814775e-14\n",
        "191 , 6.66133814775e-14\n",
        "192 , 6.66133814775e-14\n",
        "193 , 6.66133814775e-14\n",
        "194 , 6.66133814775e-14\n",
        "195 , 7.77156117238e-14\n",
        "196 , 7.77156117238e-14\n",
        "197 , 7.77156117238e-14\n",
        "198 , 7.77156117238e-14\n",
        "199 , 7.77156117238e-14\n",
        "200 , 8.881784197e-14\n",
        "201 , 8.881784197e-14\n",
        "202 , 8.881784197e-14\n",
        "203 , 8.881784197e-14\n",
        "204 , 8.881784197e-14\n",
        "205 , 9.99200722163e-14\n",
        "206 , 9.99200722163e-14\n",
        "207 , 9.99200722163e-14\n",
        "208 , 9.99200722163e-14\n",
        "209 , 1.11022302463e-13\n",
        "210 , 1.11022302463e-13\n",
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        "1899 , 1.32877708836e-08\n",
        "1900 , 1.33250410705e-08\n",
        "1901 , 1.33624000753e-08\n",
        "1902 , 1.33998367957e-08\n",
        "1903 , 1.34373623339e-08\n",
        "1904 , 1.34749655878e-08\n",
        "1905 , 1.35126687617e-08\n",
        "1906 , 1.35504496512e-08\n",
        "1907 , 1.35883082564e-08\n",
        "1908 , 1.36262667816e-08\n",
        "1909 , 1.36643030224e-08\n",
        "1910 , 1.37024280811e-08\n",
        "1911 , 1.37406308554e-08\n",
        "1912 , 1.37789335497e-08\n",
        "1913 , 1.38173139597e-08\n",
        "1914 , 1.38557831875e-08\n",
        "1915 , 1.38943412331e-08\n",
        "1916 , 1.39329880966e-08\n",
        "1917 , 1.39717237779e-08\n",
        "1918 , 1.40105371749e-08\n",
        "1919 , 1.40494393897e-08\n",
        "1920 , 1.40884415245e-08\n",
        "1921 , 1.4127521375e-08\n",
        "1922 , 1.41666900433e-08\n",
        "1923 , 1.42059475294e-08\n",
        "1924 , 1.42452938334e-08\n",
        "1925 , 1.42847289553e-08\n",
        "1926 , 1.43242528949e-08\n",
        "1927 , 1.43638656525e-08\n",
        "1928 , 1.44035672278e-08\n",
        "1929 , 1.4443357621e-08\n",
        "1930 , 1.44832368321e-08\n",
        "1931 , 1.4523204861e-08\n",
        "1932 , 1.45632617077e-08\n",
        "1933 , 1.46034073722e-08\n",
        "1934 , 1.46436529569e-08\n",
        "1935 , 1.46839762571e-08\n",
        "1936 , 1.47243994775e-08\n",
        "1937 , 1.47649004134e-08\n",
        "1938 , 1.48055012694e-08\n",
        "1939 , 1.48461909433e-08\n",
        "1940 , 1.4886969435e-08\n",
        "1941 , 1.49278478467e-08\n",
        "1942 , 1.49688039741e-08\n",
        "1943 , 1.50098600216e-08\n",
        "1944 , 1.50510048869e-08\n",
        "1945 , 1.509223857e-08\n",
        "1946 , 1.51335721732e-08\n",
        "1947 , 1.5174983492e-08\n",
        "1948 , 1.52164947309e-08\n",
        "1949 , 1.52581058899e-08\n",
        "1950 , 1.52997947644e-08\n",
        "1951 , 1.53415835591e-08\n",
        "1952 , 1.53834722738e-08\n",
        "1953 , 1.54254387041e-08\n",
        "1954 , 1.54675050545e-08\n",
        "1955 , 1.5509671325e-08\n",
        "1956 , 1.55519264133e-08\n",
        "1957 , 1.55942703195e-08\n",
        "1958 , 1.56367141457e-08\n",
        "1959 , 1.56792467898e-08\n",
        "1960 , 1.57218682517e-08\n",
        "1961 , 1.57646007359e-08\n",
        "1962 , 1.58074109358e-08\n",
        "1963 , 1.58503210557e-08\n",
        "1964 , 1.58933310956e-08\n",
        "1965 , 1.59364299535e-08\n",
        "1966 , 1.59796176291e-08\n",
        "1967 , 1.60229163271e-08\n",
        "1968 , 1.60662927406e-08\n",
        "1969 , 1.61097801765e-08\n",
        "1970 , 1.61533564302e-08\n",
        "1971 , 1.61970215018e-08\n",
        "1972 , 1.62407975957e-08\n",
        "1973 , 1.62846625074e-08\n",
        "1974 , 1.63286162369e-08\n",
        "1975 , 1.63726698865e-08\n",
        "1976 , 1.64168234562e-08\n",
        "1977 , 1.6461076946e-08\n",
        "1978 , 1.65054192536e-08\n",
        "1979 , 1.65498614813e-08\n",
        "1980 , 1.6594403629e-08\n",
        "1981 , 1.66390345946e-08\n",
        "1982 , 1.66837765825e-08\n",
        "1983 , 1.67286073882e-08\n",
        "1984 , 1.6773538114e-08\n",
        "1985 , 1.68185687599e-08\n",
        "1986 , 1.68636882236e-08\n",
        "1987 , 1.69089187096e-08\n",
        "1988 , 1.69542380135e-08\n",
        "1989 , 1.69996572374e-08\n",
        "1990 , 1.70451763815e-08\n",
        "1991 , 1.70907954455e-08\n",
        "1992 , 1.71365144297e-08\n",
        "1993 , 1.71823333339e-08\n",
        "1994 , 1.72282521582e-08\n",
        "1995 , 1.72742709026e-08\n",
        "1996 , 1.7320389567e-08\n",
        "1997 , 1.73665970493e-08\n",
        "1998 , 1.74129155539e-08\n",
        "1999 , 1.74593339786e-08\n",
        "2000 , 1.75058523233e-08\n",
        "2001 , 1.75524705881e-08\n",
        "2002 , 1.7599188773e-08\n",
        "2003 , 1.76460068779e-08\n",
        "2004 , 1.76929249029e-08\n",
        "2005 , 1.7739942848e-08\n",
        "2006 , 1.77870718154e-08\n",
        "2007 , 1.78342896007e-08\n",
        "2008 , 1.78816184082e-08\n",
        "2009 , 1.79290471358e-08\n",
        "2010 , 1.79765757835e-08\n",
        "2011 , 1.80242154535e-08\n",
        "2012 , 1.80719439413e-08\n",
        "2013 , 1.81197834515e-08\n",
        "2014 , 1.81677228817e-08\n",
        "2015 , 1.82157733342e-08\n",
        "2016 , 1.82639126045e-08\n",
        "2017 , 1.83121628972e-08\n",
        "2018 , 1.83605131099e-08\n",
        "2019 , 1.84089743449e-08\n",
        "2020 , 1.84575355e-08\n",
        "2021 , 1.85061965752e-08\n",
        "2022 , 1.85549686726e-08\n",
        "2023 , 1.86038406902e-08\n",
        "2024 , 1.865282373e-08\n",
        "2025 , 1.870190669e-08\n",
        "2026 , 1.87510895699e-08\n",
        "2027 , 1.88003834722e-08\n",
        "2028 , 1.88497772946e-08\n",
        "2029 , 1.88992821393e-08\n",
        "2030 , 1.8948886904e-08\n",
        "2031 , 1.89986026911e-08\n",
        "2032 , 1.90484183982e-08\n",
        "2033 , 1.90983451276e-08\n",
        "2034 , 1.91483828793e-08\n",
        "2035 , 1.91985205511e-08\n",
        "2036 , 1.9248758143e-08\n",
        "2037 , 1.92991178594e-08\n",
        "2038 , 1.93495774958e-08\n",
        "2039 , 1.94001370524e-08\n",
        "2040 , 1.94508076312e-08\n",
        "2041 , 1.95015892324e-08\n",
        "2042 , 1.95524818558e-08\n",
        "2043 , 1.96034743993e-08\n",
        "2044 , 1.96545779652e-08\n",
        "2045 , 1.97057925533e-08\n",
        "2046 , 1.97571070615e-08\n",
        "2047 , 1.9808532592e-08\n",
        "2048 , 1.9860080247e-08\n",
        "2049 , 1.99117167199e-08\n",
        "2050 , 1.99634753173e-08\n",
        "2051 , 2.0015344937e-08\n",
        "2052 , 2.00673144768e-08\n",
        "2053 , 2.01193950389e-08\n",
        "2054 , 2.01715866233e-08\n",
        "2055 , 2.022388923e-08\n",
        "2056 , 2.02763028589e-08\n",
        "2057 , 2.03288275102e-08\n",
        "2058 , 2.03814631838e-08\n",
        "2059 , 2.04342098797e-08\n",
        "2060 , 2.04870564957e-08\n",
        "2061 , 2.05400252362e-08\n",
        "2062 , 2.0593104999e-08\n",
        "2063 , 2.06462846819e-08\n",
        "2064 , 2.06995864893e-08\n",
        "2065 , 2.0752999319e-08\n",
        "2066 , 2.0806523171e-08\n",
        "2067 , 2.08601580454e-08\n",
        "2068 , 2.0913903942e-08\n",
        "2069 , 2.09677719631e-08\n",
        "2070 , 2.10217399044e-08\n",
        "2071 , 2.10758188679e-08\n",
        "2072 , 2.1130019956e-08\n",
        "2073 , 2.11843320663e-08\n",
        "2074 , 2.1238755199e-08\n",
        "2075 , 2.12933004562e-08\n",
        "2076 , 2.13479456335e-08\n",
        "2077 , 2.14027129353e-08\n",
        "2078 , 2.14575912594e-08\n",
        "2079 , 2.1512591708e-08\n",
        "2080 , 2.15676920767e-08\n",
        "2081 , 2.162291457e-08\n",
        "2082 , 2.16782591878e-08\n",
        "2083 , 2.17337037256e-08\n",
        "2084 , 2.1789270388e-08\n",
        "2085 , 2.18449591749e-08\n",
        "2086 , 2.19007589841e-08\n",
        "2087 , 2.19566698156e-08\n",
        "2088 , 2.20127027717e-08\n",
        "2089 , 2.206884675e-08\n",
        "2090 , 2.21251128529e-08\n",
        "2091 , 2.21814899781e-08\n",
        "2092 , 2.22379781256e-08\n",
        "2093 , 2.22945994999e-08\n",
        "2094 , 2.23513207942e-08\n",
        "2095 , 2.24081753153e-08\n",
        "2096 , 2.24651297565e-08\n",
        "2097 , 2.25222174244e-08\n",
        "2098 , 2.25794161146e-08\n",
        "2099 , 2.26367258271e-08\n",
        "2100 , 2.26941687664e-08\n",
        "2101 , 2.2751722728e-08\n",
        "2102 , 2.28093877119e-08\n",
        "2103 , 2.28671748204e-08\n",
        "2104 , 2.29250840533e-08\n",
        "2105 , 2.29831154108e-08\n",
        "2106 , 2.30412688929e-08\n",
        "2107 , 2.30995333972e-08\n",
        "2108 , 2.31579200261e-08\n",
        "2109 , 2.32164287794e-08\n",
        "2110 , 2.32750485551e-08\n",
        "2111 , 2.33337904554e-08\n",
        "2112 , 2.33926655824e-08\n",
        "2113 , 2.34516517317e-08\n",
        "2114 , 2.35107600055e-08\n",
        "2115 , 2.35699904039e-08\n",
        "2116 , 2.36293318245e-08\n",
        "2117 , 2.3688806472e-08\n",
        "2118 , 2.37484032439e-08\n",
        "2119 , 2.38081110382e-08\n",
        "2120 , 2.38679520592e-08\n",
        "2121 , 2.39279041025e-08\n",
        "2122 , 2.39879893726e-08\n",
        "2123 , 2.4048185665e-08\n",
        "2124 , 2.41085151842e-08\n",
        "2125 , 2.41689668279e-08\n",
        "2126 , 2.42295405961e-08\n",
        "2127 , 2.42902253866e-08\n",
        "2128 , 2.43510434039e-08\n",
        "2129 , 2.4411994648e-08\n",
        "2130 , 2.44730569143e-08\n",
        "2131 , 2.45342413052e-08\n",
        "2132 , 2.45955589229e-08\n",
        "2133 , 2.46569986651e-08\n",
        "2134 , 2.47185605318e-08\n",
        "2135 , 2.4780244523e-08\n",
        "2136 , 2.4842061741e-08\n",
        "2137 , 2.49040010836e-08\n",
        "2138 , 2.49660625506e-08\n",
        "2139 , 2.50282461423e-08\n",
        "2140 , 2.50905629606e-08\n",
        "2141 , 2.51530019035e-08\n",
        "2142 , 2.5215562971e-08\n",
        "2143 , 2.52782572652e-08\n",
        "2144 , 2.53410736839e-08\n",
        "2145 , 2.54040233294e-08\n",
        "2146 , 2.54670950994e-08\n",
        "2147 , 2.55303000962e-08\n",
        "2148 , 2.55936161153e-08\n",
        "2149 , 2.56570764634e-08\n",
        "2150 , 2.5720658936e-08\n",
        "2151 , 2.57843635332e-08\n",
        "2152 , 2.58482013571e-08\n",
        "2153 , 2.59121724078e-08\n",
        "2154 , 2.5976265583e-08\n",
        "2155 , 2.6040491985e-08\n",
        "2156 , 2.61048405115e-08\n",
        "2157 , 2.61693222647e-08\n",
        "2158 , 2.62339372448e-08\n",
        "2159 , 2.62986743493e-08\n",
        "2160 , 2.63635446807e-08\n",
        "2161 , 2.64285482388e-08\n",
        "2162 , 2.64936739214e-08\n",
        "2163 , 2.65589328308e-08\n",
        "2164 , 2.66243249669e-08\n",
        "2165 , 2.66898503298e-08\n",
        "2166 , 2.67554978173e-08\n",
        "2167 , 2.68212785315e-08\n",
        "2168 , 2.68871924725e-08\n",
        "2169 , 2.69532396402e-08\n",
        "2170 , 2.70194200347e-08\n",
        "2171 , 2.70857225537e-08\n",
        "2172 , 2.71521694017e-08\n",
        "2173 , 2.72187383743e-08\n",
        "2174 , 2.72854516759e-08\n",
        "2175 , 2.73522871019e-08\n",
        "2176 , 2.74192557548e-08\n",
        "2177 , 2.74863687366e-08\n",
        "2178 , 2.7553603843e-08\n",
        "2179 , 2.76209721761e-08\n",
        "2180 , 2.76884848383e-08\n",
        "2181 , 2.77561196249e-08\n",
        "2182 , 2.78238987406e-08\n",
        "2183 , 2.78917999808e-08\n",
        "2184 , 2.79598455499e-08\n",
        "2185 , 2.80280243459e-08\n",
        "2186 , 2.80963363686e-08\n",
        "2187 , 2.8164781618e-08\n",
        "2188 , 2.82333600943e-08\n",
        "2189 , 2.83020828995e-08\n",
        "2190 , 2.83709389315e-08\n",
        "2191 , 2.84399281902e-08\n",
        "2192 , 2.85090506758e-08\n",
        "2193 , 2.85783174903e-08\n",
        "2194 , 2.86477064293e-08\n",
        "2195 , 2.87172507996e-08\n",
        "2196 , 2.87869172944e-08\n",
        "2197 , 2.88567281181e-08\n",
        "2198 , 2.89266721687e-08\n",
        "2199 , 2.89967605482e-08\n",
        "2200 , 2.90669821545e-08\n",
        "2201 , 2.91373480898e-08\n",
        "2202 , 2.92078472519e-08\n",
        "2203 , 2.92784796407e-08\n",
        "2204 , 2.93492563586e-08\n",
        "2205 , 2.94201774054e-08\n",
        "2206 , 2.94912316789e-08\n",
        "2207 , 2.95624191793e-08\n",
        "2208 , 2.96337510086e-08\n",
        "2209 , 2.97052271669e-08\n",
        "2210 , 2.9776836552e-08\n",
        "2211 , 2.98485902661e-08\n",
        "2212 , 2.99204883092e-08\n",
        "2213 , 2.9992519579e-08\n",
        "2214 , 3.00646951779e-08\n",
        "2215 , 3.01370040034e-08\n",
        "2216 , 3.02094682603e-08\n",
        "2217 , 3.02820657438e-08\n",
        "2218 , 3.03547964542e-08\n",
        "2219 , 3.04276825958e-08\n",
        "2220 , 3.05007019641e-08\n",
        "2221 , 3.05738656614e-08\n",
        "2222 , 3.06471736877e-08\n",
        "2223 , 3.0720626043e-08\n",
        "2224 , 3.07942227273e-08\n",
        "2225 , 3.08679637406e-08\n",
        "2226 , 3.09418379807e-08\n",
        "2227 , 3.1015867652e-08\n",
        "2228 , 3.109003055e-08\n",
        "2229 , 3.11643377771e-08\n",
        "2230 , 3.12388004353e-08\n",
        "2231 , 3.13133963203e-08\n",
        "2232 , 3.13881365344e-08\n",
        "2233 , 3.14630321796e-08\n",
        "2234 , 3.15380610516e-08\n",
        "2235 , 3.16132453548e-08\n",
        "2236 , 3.16885628848e-08\n",
        "2237 , 3.1764035846e-08\n",
        "2238 , 3.18396531362e-08\n",
        "2239 , 3.19154147554e-08\n",
        "2240 , 3.19913207036e-08\n",
        "2241 , 3.20673820831e-08\n",
        "2242 , 3.21435877915e-08\n",
        "2243 , 3.22199267266e-08\n",
        "2244 , 3.22964321953e-08\n",
        "2245 , 3.23730708907e-08\n",
        "2246 , 3.24498650173e-08\n",
        "2247 , 3.25268034729e-08\n",
        "2248 , 3.26038862575e-08\n",
        "2249 , 3.26811244733e-08\n",
        "2250 , 3.27585070181e-08\n",
        "2251 , 3.28360449942e-08\n",
        "2252 , 3.29137272992e-08\n",
        "2253 , 3.29915539332e-08\n",
        "2254 , 3.30695359985e-08\n",
        "2255 , 3.31476623927e-08\n",
        "2256 , 3.32259442182e-08\n",
        "2257 , 3.33043703726e-08\n",
        "2258 , 3.33829519583e-08\n",
        "2259 , 3.34616889752e-08\n",
        "2260 , 3.35405703211e-08\n",
        "2261 , 3.3619595996e-08\n",
        "2262 , 3.36987771021e-08\n",
        "2263 , 3.37781136395e-08\n",
        "2264 , 3.3857605608e-08\n",
        "2265 , 3.39372419056e-08\n",
        "2266 , 3.40170336344e-08\n",
        "2267 , 3.40969696921e-08\n",
        "2268 , 3.41770722834e-08\n",
        "2269 , 3.42573192036e-08\n",
        "2270 , 3.4337721555e-08\n",
        "2271 , 3.44182682355e-08\n",
        "2272 , 3.44989814494e-08\n",
        "2273 , 3.45798389922e-08\n",
        "2274 , 3.46608519664e-08\n",
        "2275 , 3.47420203717e-08\n",
        "2276 , 3.48233442082e-08\n",
        "2277 , 3.4904823476e-08\n",
        "2278 , 3.4986458175e-08\n",
        "2279 , 3.5068237203e-08\n",
        "2280 , 3.51501827645e-08\n",
        "2281 , 3.52322837571e-08\n",
        "2282 , 3.53145290788e-08\n",
        "2283 , 3.53969409339e-08\n",
        "2284 , 3.54795082202e-08\n",
        "2285 , 3.55622309378e-08\n",
        "2286 , 3.56451090866e-08\n",
        "2287 , 3.57281426666e-08\n",
        "2288 , 3.58113316778e-08\n",
        "2289 , 3.58946872225e-08\n",
        "2290 , 3.59781870962e-08\n",
        "2291 , 3.60618535034e-08\n",
        "2292 , 3.61456753417e-08\n",
        "2293 , 3.62296637135e-08\n",
        "2294 , 3.63137964143e-08\n",
        "2295 , 3.63980956486e-08\n",
        "2296 , 3.64825503141e-08\n",
        "2297 , 3.6567171513e-08\n",
        "2298 , 3.66519370409e-08\n",
        "2299 , 3.67368691023e-08\n",
        "2300 , 3.68219676972e-08\n",
        "2301 , 3.69072217232e-08\n",
        "2302 , 3.69926311805e-08\n",
        "2303 , 3.70782071712e-08\n",
        "2304 , 3.71639496954e-08\n",
        "2305 , 3.72498365486e-08\n",
        "2306 , 3.73359010375e-08\n",
        "2307 , 3.74221209576e-08\n",
        "2308 , 3.75084963089e-08\n",
        "2309 , 3.75950381937e-08\n",
        "2310 , 3.76817466119e-08\n",
        "2311 , 3.77686104613e-08\n",
        "2312 , 3.78556408442e-08\n",
        "2313 , 3.79428266584e-08\n",
        "2314 , 3.80301901082e-08\n",
        "2315 , 3.81177089892e-08\n",
        "2316 , 3.82053833015e-08\n",
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       "text": [
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       "text": [
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      {
       "output_type": "stream",
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       "text": [
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        "11560 , 0.000198052622069\n",
        "11561 , 0.000198143787522\n",
        "11562 , 0.000198234987059\n",
        "11563 , 0.000198326220668\n",
        "11564 , 0.000198417488373\n",
        "11565 , 0.000198508790172\n",
        "11566 , 0.0001986001261\n",
        "11567 , 0.000198691496145\n",
        "11568 , 0.000198782900318\n",
        "11569 , 0.000198874338631\n",
        "11570 , 0.000198965811105\n",
        "11571 , 0.000199057317729\n",
        "11572 , 0.000199148858537\n",
        "11573 , 0.000199240433518\n",
        "11574 , 0.000199332042694\n",
        "11575 , 0.000199423686065\n",
        "11576 , 0.000199515363652\n",
        "11577 , 0.000199607075457\n",
        "11578 , 0.000199698821501\n",
        "11579 , 0.000199790601774\n",
        "11580 , 0.000199882416296\n",
        "11581 , 0.000199974265092\n",
        "11582 , 0.000200066148148\n",
        "11583 , 0.000200158065478\n",
        "11584 , 0.000200250017113\n",
        "11585 , 0.000200342003032\n",
        "11586 , 0.000200434023268\n",
        "11587 , 0.00020052607782\n",
        "11588 , 0.000200618166701\n",
        "11589 , 0.000200710289922\n",
        "11590 , 0.000200802447492\n",
        "11591 , 0.000200894639413\n",
        "11592 , 0.000200986865706\n",
        "11593 , 0.000201079126383\n",
        "11594 , 0.000201171421443\n",
        "11595 , 0.000201263750899\n",
        "11596 , 0.00020135611476\n",
        "11597 , 0.000201448513037\n",
        "11598 , 0.000201540945755\n",
        "11599 , 0.0002016334129\n",
        "11600 , 0.000201725914495\n",
        "11601 , 0.00020181845054\n",
        "11602 , 0.000201911021058\n",
        "11603 , 0.000202003626049\n",
        "11604 , 0.000202096265522\n",
        "11605 , 0.000202188939502\n",
        "11606 , 0.000202281647987\n",
        "11607 , 0.000202374390978\n",
        "11608 , 0.000202467168497\n",
        "11609 , 0.000202559980556\n",
        "11610 , 0.000202652827164\n",
        "11611 , 0.000202745708322\n",
        "11612 , 0.000202838624042\n",
        "11613 , 0.000202931574345\n",
        "11614 , 0.000203024559231\n",
        "11615 , 0.000203117578712\n",
        "11616 , 0.000203210632799\n",
        "11617 , 0.000203303721491\n",
        "11618 , 0.000203396844822\n",
        "11619 , 0.000203490002781\n",
        "11620 , 0.000203583195379\n",
        "11621 , 0.000203676422639\n",
        "11622 , 0.000203769684559\n",
        "11623 , 0.000203862981152\n",
        "11624 , 0.000203956312428\n",
        "11625 , 0.000204049678409\n",
        "11626 , 0.000204143079086\n",
        "11627 , 0.000204236514478\n",
        "11628 , 0.000204329984588\n",
        "11629 , 0.000204423489436\n",
        "11630 , 0.000204517029023\n",
        "11631 , 0.000204610603372\n",
        "11632 , 0.000204704212481\n",
        "11633 , 0.000204797856362\n",
        "11634 , 0.000204891535027\n",
        "11635 , 0.000204985248486\n",
        "11636 , 0.000205078996751\n",
        "11637 , 0.000205172779821\n",
        "11638 , 0.000205266597719\n",
        "11639 , 0.000205360450445\n",
        "11640 , 0.00020545433802\n",
        "11641 , 0.000205548260446\n",
        "11642 , 0.000205642217732\n",
        "11643 , 0.000205736209902\n",
        "11644 , 0.000205830236943\n",
        "11645 , 0.000205924298879\n",
        "11646 , 0.00020601839571\n",
        "11647 , 0.000206112527468\n",
        "11648 , 0.000206206694142\n",
        "11649 , 0.000206300895744\n",
        "11650 , 0.000206395132296\n",
        "11651 , 0.000206489403798\n",
        "11652 , 0.00020658371026\n",
        "11653 , 0.000206678051695\n",
        "11654 , 0.000206772428102\n",
        "11655 , 0.000206866839514\n",
        "11656 , 0.000206961285931\n",
        "11657 , 0.000207055767354\n",
        "11658 , 0.000207150283793\n",
        "11659 , 0.000207244835271\n",
        "11660 , 0.000207339421787\n",
        "11661 , 0.000207434043364\n",
        "11662 , 0.000207528699991\n",
        "11663 , 0.000207623391701\n",
        "11664 , 0.000207718118483\n",
        "11665 , 0.000207812880371\n",
        "11666 , 0.000207907677352\n",
        "11667 , 0.000208002509439\n",
        "11668 , 0.000208097376653\n",
        "11669 , 0.000208192279005\n",
        "11670 , 0.000208287216485\n",
        "11671 , 0.000208382189126\n",
        "11672 , 0.000208477196928\n",
        "11673 , 0.000208572239901\n",
        "11674 , 0.000208667318058\n",
        "11675 , 0.000208762431408\n",
        "11676 , 0.000208857579964\n",
        "11677 , 0.000208952763725\n",
        "11678 , 0.000209047982702\n",
        "11679 , 0.000209143236918\n",
        "11680 , 0.000209238526383\n",
        "11681 , 0.000209333851087\n",
        "11682 , 0.000209429211062\n",
        "11683 , 0.000209524606298\n",
        "11684 , 0.000209620036828\n",
        "11685 , 0.000209715502641\n",
        "11686 , 0.000209811003771\n",
        "11687 , 0.000209906540194\n",
        "11688 , 0.000210002111956\n",
        "11689 , 0.000210097719033\n",
        "11690 , 0.000210193361472\n",
        "11691 , 0.000210289039249\n",
        "11692 , 0.000210384752397\n",
        "11693 , 0.000210480500906\n",
        "11694 , 0.000210576284809\n",
        "11695 , 0.000210672104095\n",
        "11696 , 0.000210767958786\n",
        "11697 , 0.000210863848893\n",
        "11698 , 0.000210959774427\n",
        "11699 , 0.000211055735388\n",
        "11700 , 0.000211151731788\n",
        "11701 , 0.000211247763648\n",
        "11702 , 0.000211343830969\n",
        "11703 , 0.000211439933762\n",
        "11704 , 0.000211536072037\n",
        "11705 , 0.000211632245806\n",
        "11706 , 0.000211728455068\n",
        "11707 , 0.000211824699858\n",
        "11708 , 0.000211920980164\n",
        "11709 , 0.000212017296009\n",
        "11710 , 0.000212113647391\n",
        "11711 , 0.000212210034334\n",
        "11712 , 0.000212306456826\n",
        "11713 , 0.000212402914912\n",
        "11714 , 0.000212499408569\n",
        "11715 , 0.000212595937821\n",
        "11716 , 0.000212692502677\n",
        "11717 , 0.000212789103149\n",
        "11718 , 0.000212885739248\n",
        "11719 , 0.000212982410974\n",
        "11720 , 0.00021307911835\n",
        "11721 , 0.000213175861374\n",
        "11722 , 0.000213272640071\n",
        "11723 , 0.000213369454438\n",
        "11724 , 0.000213466304488\n",
        "11725 , 0.000213563190232\n",
        "11726 , 0.000213660111681\n",
        "11727 , 0.000213757068857\n",
        "11728 , 0.000213854061748\n",
        "11729 , 0.000213951090366\n",
        "11730 , 0.000214048154745\n",
        "11731 , 0.000214145254873\n",
        "11732 , 0.000214242390773\n",
        "11733 , 0.000214339562432\n",
        "11734 , 0.000214436769896\n",
        "11735 , 0.000214534013143\n",
        "11736 , 0.000214631292206\n",
        "11737 , 0.000214728607073\n",
        "11738 , 0.000214825957778\n",
        "11739 , 0.000214923344311\n",
        "11740 , 0.000215020766703\n",
        "11741 , 0.000215118224933\n",
        "11742 , 0.000215215719046\n",
        "11743 , 0.00021531324903\n",
        "11744 , 0.000215410814897\n",
        "11745 , 0.000215508416668\n",
        "11746 , 0.000215606054343\n",
        "11747 , 0.000215703727946\n",
        "11748 , 0.000215801437464\n",
        "11749 , 0.00021589918292\n",
        "11750 , 0.000215996964337\n",
        "11751 , 0.000216094781702\n",
        "11752 , 0.000216192635039\n",
        "11753 , 0.000216290524346\n",
        "11754 , 0.000216388449659\n",
        "11755 , 0.000216486410953\n",
        "11756 , 0.000216584408264\n",
        "11757 , 0.000216682441601\n",
        "11758 , 0.000216780510953\n",
        "11759 , 0.000216878616355\n",
        "11760 , 0.000216976757805\n",
        "11761 , 0.000217074935316\n",
        "11762 , 0.000217173148898\n",
        "11763 , 0.00021727139855\n",
        "11764 , 0.000217369684308\n",
        "11765 , 0.000217468006158\n",
        "11766 , 0.000217566364125\n",
        "11767 , 0.000217664758206\n",
        "11768 , 0.000217763188415\n",
        "11769 , 0.000217861654772\n",
        "11770 , 0.000217960157278\n",
        "11771 , 0.000218058695955\n",
        "11772 , 0.000218157270793\n",
        "11773 , 0.000218255881823\n",
        "11774 , 0.000218354529036\n",
        "11775 , 0.000218453212464\n",
        "11776 , 0.000218551932096\n",
        "11777 , 0.000218650687944\n",
        "11778 , 0.000218749480041\n",
        "11779 , 0.000218848308375\n",
        "11780 , 0.000218947172959\n",
        "11781 , 0.000219046073813\n",
        "11782 , 0.000219145010938\n",
        "11783 , 0.000219243984345\n",
        "11784 , 0.000219342994046\n",
        "11785 , 0.000219442040061\n",
        "11786 , 0.000219541122382\n",
        "11787 , 0.000219640241028\n",
        "11788 , 0.000219739396023\n",
        "11789 , 0.000219838587356\n",
        "11790 , 0.000219937815038\n",
        "11791 , 0.000220037079091\n",
        "11792 , 0.000220136379525\n",
        "11793 , 0.000220235716342\n",
        "11794 , 0.000220335089551\n",
        "11795 , 0.000220434499176\n",
        "11796 , 0.000220533945217\n",
        "11797 , 0.000220633427683\n",
        "11798 , 0.000220732946588\n",
        "11799 , 0.00022083250194\n",
        "11800 , 0.000220932093764\n",
        "11801 , 0.000221031722036\n",
        "11802 , 0.000221131386802\n",
        "11803 , 0.000221231088049\n",
        "11804 , 0.0002213308258\n",
        "11805 , 0.000221430600056\n",
        "11806 , 0.000221530410838\n",
        "11807 , 0.000221630258146\n",
        "11808 , 0.000221730142003\n",
        "11809 , 0.000221830062408\n",
        "11810 , 0.000221930019373\n",
        "11811 , 0.000222030012897\n",
        "11812 , 0.000222130043015\n",
        "11813 , 0.000222230109725\n",
        "11814 , 0.00022233021304\n",
        "11815 , 0.00022243035297\n",
        "11816 , 0.000222530529514\n",
        "11817 , 0.000222630742697\n",
        "11818 , 0.000222730992516\n",
        "11819 , 0.000222831278995\n",
        "11820 , 0.000222931602134\n",
        "11821 , 0.000223031961955\n",
        "11822 , 0.000223132358457\n",
        "11823 , 0.000223232791652\n",
        "11824 , 0.000223333261551\n",
        "11825 , 0.000223433768176\n",
        "11826 , 0.000223534311516\n",
        "11827 , 0.000223634891594\n",
        "11828 , 0.00022373550842\n",
        "11829 , 0.000223836162005\n",
        "11830 , 0.000223936852362\n",
        "11831 , 0.000224037579488\n",
        "11832 , 0.000224138343408\n",
        "11833 , 0.00022423914412\n",
        "11834 , 0.000224339981647\n",
        "11835 , 0.000224440855989\n",
        "11836 , 0.000224541767158\n",
        "11837 , 0.000224642715174\n",
        "11838 , 0.000224743700028\n",
        "11839 , 0.000224844721752\n",
        "11840 , 0.000224945780347\n",
        "11841 , 0.000225046875824\n",
        "11842 , 0.000225148008182\n",
        "11843 , 0.000225249177455\n",
        "11844 , 0.000225350383631\n",
        "11845 , 0.000225451626723\n",
        "11846 , 0.000225552906763\n",
        "11847 , 0.00022565422374\n",
        "11848 , 0.000225755577665\n",
        "11849 , 0.000225856968561\n",
        "11850 , 0.000225958396427\n",
        "11851 , 0.000226059861275\n",
        "11852 , 0.000226161363126\n",
        "11853 , 0.000226262901981\n",
        "11854 , 0.00022636447784\n",
        "11855 , 0.000226466090736\n",
        "11856 , 0.000226567740669\n",
        "11857 , 0.00022666942765\n",
        "11858 , 0.000226771151679\n",
        "11859 , 0.000226872912779\n",
        "11860 , 0.00022697471096\n",
        "11861 , 0.000227076546222\n",
        "11862 , 0.000227178418588\n",
        "11863 , 0.000227280328069\n",
        "11864 , 0.000227382274653\n",
        "11865 , 0.000227484258386\n",
        "11866 , 0.000227586279244\n",
        "11867 , 0.000227688337262\n",
        "11868 , 0.000227790432439\n",
        "11869 , 0.000227892564786\n",
        "11870 , 0.000227994734303\n",
        "11871 , 0.000228096941024\n",
        "11872 , 0.000228199184948\n",
        "11873 , 0.000228301466088\n",
        "11874 , 0.000228403784441\n",
        "11875 , 0.000228506140043\n",
        "11876 , 0.000228608532871\n",
        "11877 , 0.000228710962968\n",
        "11878 , 0.000228813430325\n",
        "11879 , 0.000228915934952\n",
        "11880 , 0.000229018476872\n",
        "11881 , 0.000229121056083\n",
        "11882 , 0.00022922367261\n",
        "11883 , 0.000229326326451\n",
        "11884 , 0.000229429017606\n",
        "11885 , 0.00022953174611\n",
        "11886 , 0.000229634511961\n",
        "11887 , 0.000229737315172\n",
        "11888 , 0.000229840155752\n",
        "11889 , 0.000229943033714\n",
        "11890 , 0.000230045949068\n",
        "11891 , 0.000230148901814\n",
        "11892 , 0.000230251891975\n",
        "11893 , 0.000230354919561\n",
        "11894 , 0.000230457984574\n",
        "11895 , 0.000230561087033\n",
        "11896 , 0.000230664226941\n",
        "11897 , 0.000230767404308\n",
        "11898 , 0.000230870619156\n",
        "11899 , 0.000230973871496\n",
        "11900 , 0.000231077161317\n",
        "11901 , 0.000231180488652\n",
        "11902 , 0.000231283853491\n",
        "11903 , 0.000231387255867\n",
        "11904 , 0.000231490695779\n",
        "11905 , 0.000231594173228\n",
        "11906 , 0.000231697688247\n",
        "11907 , 0.000231801240824\n",
        "11908 , 0.000231904830983\n",
        "11909 , 0.000232008458734\n",
        "11910 , 0.000232112124088\n",
        "11911 , 0.000232215827045\n",
        "11912 , 0.000232319567617\n",
        "11913 , 0.000232423345825\n",
        "11914 , 0.000232527161681\n",
        "11915 , 0.000232631015185\n",
        "11916 , 0.000232734906347\n",
        "11917 , 0.000232838835179\n",
        "11918 , 0.000232942801703\n",
        "11919 , 0.00023304680592\n",
        "11920 , 0.000233150847839\n",
        "11921 , 0.000233254927473\n",
        "11922 , 0.000233359044832\n",
        "11923 , 0.000233463199928\n",
        "11924 , 0.000233567392771\n",
        "11925 , 0.000233671623362\n",
        "11926 , 0.000233775891734\n",
        "11927 , 0.000233880197875\n",
        "11928 , 0.000233984541809\n",
        "11929 , 0.000234088923534\n",
        "11930 , 0.000234193343074\n",
        "11931 , 0.000234297800439\n",
        "11932 , 0.000234402295629\n",
        "11933 , 0.000234506828656\n",
        "11934 , 0.000234611399541\n",
        "11935 , 0.000234716008285\n",
        "11936 , 0.000234820654899\n",
        "11937 , 0.000234925339404\n",
        "11938 , 0.000235030061801\n",
        "11939 , 0.000235134822102\n",
        "11940 , 0.000235239620316\n",
        "11941 , 0.000235344456456\n",
        "11942 , 0.000235449330521\n",
        "11943 , 0.000235554242545\n",
        "11944 , 0.000235659192527\n",
        "11945 , 0.000235764180467\n",
        "11946 , 0.0002358692064\n",
        "11947 , 0.000235974270313\n",
        "11948 , 0.000236079372218\n",
        "11949 , 0.000236184512137\n",
        "11950 , 0.000236289690081\n",
        "11951 , 0.00023639490605\n",
        "11952 , 0.000236500160067\n",
        "11953 , 0.000236605452131\n",
        "11954 , 0.000236710782264\n",
        "11955 , 0.000236816150467\n",
        "11956 , 0.000236921556751\n",
        "11957 , 0.000237027001138\n",
        "11958 , 0.000237132483616\n",
        "11959 , 0.000237238004219\n",
        "11960 , 0.000237343562948\n",
        "11961 , 0.000237449159812\n",
        "11962 , 0.000237554794813\n",
        "11963 , 0.000237660467983\n",
        "11964 , 0.000237766179323\n",
        "11965 , 0.000237871928843\n",
        "11966 , 0.000237977716544\n",
        "11967 , 0.000238083542448\n",
        "11968 , 0.000238189406565\n",
        "11969 , 0.000238295308908\n",
        "11970 , 0.000238401249475\n",
        "11971 , 0.00023850722829\n",
        "11972 , 0.000238613245351\n",
        "11973 , 0.000238719300683\n",
        "11974 , 0.000238825394283\n",
        "11975 , 0.000238931526175\n",
        "11976 , 0.000239037696359\n",
        "11977 , 0.000239143904845\n",
        "11978 , 0.000239250151657\n",
        "11979 , 0.000239356436793\n",
        "11980 , 0.000239462760265\n",
        "11981 , 0.000239569122085\n",
        "11982 , 0.000239675522273\n",
        "11983 , 0.00023978196082\n",
        "11984 , 0.000239888437759\n",
        "11985 , 0.000239994953077\n",
        "11986 , 0.000240101506799\n",
        "11987 , 0.000240208098945\n",
        "11988 , 0.000240314729505\n",
        "11989 , 0.000240421398501\n",
        "11990 , 0.000240528105944\n",
        "11991 , 0.000240634851834\n",
        "11992 , 0.000240741636193\n",
        "11993 , 0.000240848459032\n",
        "11994 , 0.000240955320363\n",
        "11995 , 0.000241062220185\n",
        "11996 , 0.00024116915851\n",
        "11997 , 0.00024127613536\n",
        "11998 , 0.000241383150745\n",
        "11999 , 0.000241490204667\n",
        "12000 , 0.000241597297135\n",
        "129999 , 53.8392137768\n"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Let's print out defect rates at 12K-mile intervals:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"MTF , defect%\")\n",
      "print(\"12K ,\",y2[12000]*100)\n",
      "print(\"24K ,\",y2[24000]*100)\n",
      "print(\"36K ,\",y2[36000]*100)\n",
      "print(\"48K ,\",y2[48000]*100)\n",
      "print(\"60K ,\",y2[60000]*100)\n",
      "print(\"72K ,\",y2[72000]*100)\n",
      "print(\"84K ,\",y2[84000]*100)\n",
      "print(\"96K ,\",y2[96000]*100)\n",
      "print(\"108K ,\",y2[108000]*100)\n",
      "print(\"120K ,\",y2[120000]*100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "MTF , defect%\n",
        "12K , 0.000241597297135\n",
        "24K , 0.00965177541458\n",
        "36K , 0.0834228853136\n",
        "48K , 0.384881145426\n",
        "60K , 1.25603346141\n",
        "72K , 3.2793050139\n",
        "84K , 7.29179061233\n",
        "96K , 14.2782277181\n",
        "108K , 25.0461114074\n",
        "120K , 39.6485461142\n"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "AMMENDMENT:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "After doing MLE calculations on large data sets, I discovered that the iterative method that I used to calculate the shape parameter is very inefficient.  It can take several minutes to complete.  So after some research, I discovered that I can use Newton-Raphson method which will find the shape parameter MUCH faster."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<center><h3>Newton-Raphson Method:</h3></center><br>\n",
      "<center>$\\huge{X_{n+1}=X_n+\\frac{f(x)}{f'(x)}}$</center><br>\n",
      "<center>where $X_n$ is an arbitrary initial value,</center><br>\n",
      "<center>f(x) in our case is equal to our MLE equation above:</center><br>\n",
      "<center>$\\large{f(x)=\\sum\\limits_{i=1}^r \\frac{ln(x_i)}{r}+\\frac{1}{k}-\\frac{\\sum\\limits_{i=1}^n (x_i)^kln(x_i)}{\\sum\\limits_{i=1}^n (x_i)^k}}$</center><br>\n",
      "Due to the tediousness of the equation for f(x), some arbitrary variables(A,B,C, and H) were used to simplify the Newton-Raphson equation:<br>\n",
      "<center>$\\huge{k_{n+1}=k_n+\\frac{A+\\frac{1}{k_n}-\\frac{C_n}{B_n}}{\\frac{1}{k_n^2}+\\frac{(B_nH_n-C_n^2)}{B_n^2}}}$</center><br><br>\n",
      "<center>where</center><br><center>$A=\\sum\\limits_{i=1}^r \\frac{ln(x_i)}{r}$ &nbsp;,&nbsp;&nbsp;&nbsp;&nbsp; $B=\\sum\\limits_{i=1}^n x_i^{k_n}$&nbsp;,&nbsp;&nbsp; $C=\\sum\\limits_{i=1}^n (x_i)^{k_n}\\space ln(x_i)$</center><br>\n",
      "<center>and</center><br>\n",
      "<center>$H=\\sum\\limits_{i=1}^n x_i^{k_n}(lnx_i)^2$</center><br><br>\n",
      "<h3>But what do we use for the initial k value($k_n$)?</h3><br>\n",
      "It has been suggested to use the following:<br>\n",
      "<center>$\\large{k_n=\\left[\\frac{\\frac{6}{\\pi^2}\\left[\\sum\\limits_{i=1}^n (lnx_i)^2-\\left(\\sum\\limits_{i=1}^n lnx_i\\right)^2/n\\right]}{n-1}\\space \\right]^{-\\frac{1}{2}}}$</center><br>\n",
      "Supposedly using this equation, we will get the shape parameter on average after 3.5 iterations to achieve 4-place accuracy.  If it doesn't work, we can just use k = 1 as the initial value."
     ]
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "Below is the translation of those equations above into Python equivalent code and interating Newton-Raphson method 10 times:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mtf_all = df[\"MTF\"].values\n",
      "\n",
      "# give initial value for the shape paramter, use shape = 1 if the iteration does not find the root for the equation above\n",
      "shape = 1\n",
      "#shape = (( (6.0/np.pi**2)*(np.sum(ln(mtf_all)**2) - ((np.sum(ln(mtf_all)))**2)/mtf_all.size) ) / (mtf_all.size-1))**-0.5\n",
      "for i in xrange(1,11):\n",
      "    A = np.sum(ln(mtf_failed))*1.0/mtf_failed.size\n",
      "    B = np.sum(mtf_all**shape)\n",
      "    C = np.sum( (mtf_all**shape) * ln(mtf_all) )\n",
      "    H = np.sum( (mtf_all**shape) * (ln(mtf_all))**2 )\n",
      "    shape = shape + (A+(1.0/shape) - (C/B)) / ( (1.0/shape**2) + ( (B*H)-C**2 ) / B**2 )\n",
      "    print(\"shape\"+str(i)+\" \"+str(shape))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "shape1 2.10341949076\n",
        "shape2 4.52411673696\n",
        "shape3 5.81360169599"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "shape4 5.4142172346\n",
        "shape5 5.3247106663"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "shape6 5.3199527415\n",
        "shape7 5.31993976685"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "shape8 5.31993976674\n",
        "shape9 5.31993976675"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "shape10 5.31993976675\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"So it looks like the Newton-Raphson method calculated the shape value to be:\")\n",
      "print(shape)\n",
      "print(\"after only 6 iterations. Very nice!\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "So it looks like the Newton-Raphson method calculated the shape value to be:\n",
        "5.31993976675\n",
        "after only 6 iterations. Very nice!\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 4,
     "metadata": {},
     "source": [
      "This is very close to the shape parameter value obtained earlier using the inefficient iterative method."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "scale = (np.sum((mtf_all**shape)/len(df_failed)))**(1/shape)\n",
      "print(\"scale parameter:\", scale)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "scale parameter: 136449.905085\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This IPython notebook Weibull analysis was made possible based on reference material from the <a href=\"http://www.weibull.nl/weibullstatistics.htm\">weibull.nl</a> website and also from <a href=\"http://www.qualitydigest.com/jan99/html/body_weibull.html\">this</a> example of doing Weibull linear regression analysis using Excel.<br><br>\n",
      "Here is a <a href=\"http://books.google.com/books?id=5GzgBpdJkHwC&pg=PA37&lpg=PA37&dq=weibull+using+Excel+solver&source=bl&ots=T2FaOvB7jj&sig=kSDeIWwQ_xj8OQ5MDFkxO5VquT8&hl=en&sa=X&ei=bAEqUqS4H8mG2gWV4oDoDA&ved=0CHwQ6AEwCQ#v=onepage&q=weibull%20using%20Excel%20solver&f=false\">book</a> that shows how to use Excel's Solver add-on to calculate Weibull parameters using MLE estimation.\n",
      "I also found this <a href=\"http://www.weibull.com/hotwire/issue16/relbasics16.htm\">reference</a> which suggests that when the number of failures is small and the number of suspensions is high, the MLE method should be used."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}