02_Boulder.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Master's Thesis: Investigating Maximum Rainfall Trends in Colorado"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b>Institution:</b> University of Georgia <br>\n",
    "<b>Department:</b> Statistics <br>\n",
    "<b>Student:</b> Jeremy Mulcahey <br>\n",
    "<b>Advisor:</b> Dr. Lynne Seymour <br>\n",
    "<b>File:</b> 02_Boulder<br>\n",
    "<b>Last Edit:</b> 7/4/2016"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Section 0: Preparing data <br>\n",
    "Section 1: Distribution of Max Precipitation and Trends<br>\n",
    "Section 2: Parameter Estimation, Distribution, and Trends"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  Section 0: Preparing data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: fitdistrplus\n",
      "Warning message:\n",
      ": package 'fitdistrplus' was built under R version 3.3.1Loading required package: MASS\n",
      "Loading required package: survival\n",
      "Loading required package: caTools\n",
      "Loading required package: repr\n",
      "Loading required package: IRdisplay\n"
     ]
    }
   ],
   "source": [
    "require(fitdistrplus); # required for parameter estimation\n",
    "require(caTools);      # required for smoothing\n",
    "require(repr);         # required for resizing plots in IRKernel\n",
    "require(\"IRdisplay\");  # graphic embedding\n",
    "require(MASS);         # testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# establish working directorys\n",
    "setwd(\"C:/Users/flunk_000/Desktop/UGA/Master'sThesis/\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data files for individual station names."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = read.csv(file = \"data/BOULDER_MonMax_GHCN.csv\", head=T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adjust the data for later use."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "165"
      ],
      "text/latex": [
       "165"
      ],
      "text/markdown": [
       "165"
      ],
      "text/plain": [
       "[1] 165"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create an index as x - used as needed\n",
    "month = seq(1,nrow(data),by=1)\n",
    "data = cbind(data,month)\n",
    "\n",
    "# first year of the data (changes by station name)\n",
    "firstYear = data$YEAR[1]\n",
    "\n",
    "# station name for plotting\n",
    "stationName = as.character(data$STATION_NAME[1])\n",
    "\n",
    "# -999 messed with the plotting - change to NA\n",
    "data[data==-999] = NA\n",
    "\n",
    "sum(is.na(data)) #print number of missing values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check number of years in the data set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "116"
      ],
      "text/latex": [
       "116"
      ],
      "text/markdown": [
       "116"
      ],
      "text/plain": [
       "[1] 116"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data$YEAR[nrow(data)]-firstYear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a data frame of all the 10 year periods following the first year for data collected at the station."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#create the first decade (first year and the following 9 years)\n",
    "max_PRCP = data.frame(data[which(data$YEAR < (firstYear + 10)),6])\n",
    "\n",
    "# add all the subsequent decades to the data frame incrementing by 1 year at a time\n",
    "# i.e. 1896-1905, 1897-1906, etc\n",
    "for(year in c((data$YEAR[1] + 1):2005)){\n",
    "    max_PRCP = cbind(max_PRCP,data[which(data$YEAR >= year & data$YEAR < year+10),6])\n",
    "}\n",
    "\n",
    "# name the columns as the first year of the observed 10 year periods\n",
    "colnames(max_PRCP) = as.character(c(firstYear:2005))\n",
    "\n",
    "# set the missing values to NA\n",
    "max_PRCP[max_PRCP==-999] = NA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Section 1: Distribution of Max Precipitation and Trends"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "options(repr.plot.width=5, repr.plot.height=5) #reduces the plot sizes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"BOULDER CO US \n",
       "Observed 24 Hour Max Rainfall \n",
       "from 1898 to 2014\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(data$Max.PRCP, xlab = sprintf(\"Observed 24 hour Maximum Rainfall (tenths of mm)\",\n",
    "                                   firstYear), main=sprintf(\"%s \\nObserved 24 Hour Max Rainfall \\nfrom %i to 2014\",\n",
    "                                   stationName,firstYear),col=\"lightblue\")        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Maximum Rainfall Trends"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# create for maximum rainfall\n",
    "maxTS = ts(data$Max.PRCP)\n",
    "\n",
    "# create regression line\n",
    "reg = lm(Max.PRCP. ~ month, data=data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Slope of the regression line:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>month:</strong> 0.031"
      ],
      "text/latex": [
       "\\textbf{month:} 0.031"
      ],
      "text/markdown": [
       "**month:** 0.031"
      ],
      "text/plain": [
       "month \n",
       "0.031 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(reg$coefficients[2],3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>month:</strong> 43.524"
      ],
      "text/latex": [
       "\\textbf{month:} 43.524"
      ],
      "text/markdown": [
       "**month:** 43.524"
      ],
      "text/plain": [
       " month \n",
       "43.524 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length(data$MONTH)*round(reg$coefficients[2],3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Although month isn't very predictive of maximum observed rainfall, there does appear to be an increase over the time data was collected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"BOULDER CO US \n",
       "Observed 24 Hour Max Rainfall \n",
       "from 1898 to 2014 by Monthly Max\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(maxTS, xlab = \"Time (months)\", col = \"black\", lwd=2, ylab = \"Observed Max Precipitation\", \n",
    "     main=sprintf(\"%s \\nObserved 24 Hour Max Rainfall \\nfrom %i to 2014 by Monthly Max\", stationName,firstYear))\n",
    "abline(reg, col=\"red\", lwd=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the time series plot, we also see that the peaks are gradually increasing over time, which is of specific interest to people working with maximum rainfall estimation. These increasing peaks have been wreaking havoc on Texas and the Eastern US this year. (Template statement - change as needed for individual stations)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Produce smoothed plots to get a better understanding of the trends."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Smoothed 24 hour max rainfall from 1896 to 2014\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rollSmall = runmean(data$Max.PRCP., 300)\n",
    "rollLarge = runmean(data$Max.PRCP., 900)\n",
    "plot(rollSmall, type=\"l\", ylim = c(145,200), xlab = \"Time (months)\", \n",
    "    ylab = \"Rolling Average\", col = 2, main=\"Smoothed 24 hour max rainfall from 1896 to 2014\")\n",
    "    points(rollLarge, type=\"l\", col=4)\n",
    "    legend(0,200, c(\"300 window\",\"\", \"900 window\"), col=c(2,0,4), lty=c(1,0,1), bty=\"n\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While the data shows a slight decrease in average rainfall at the tail end of our data (2014), the long term trend shows that average rainfall has been increasing. Empirically, we have had a wet year in 2016 that we would expect to match/exceed the long term observed increase in average maximum rainfall.(Template statement - change as needed for individual stations)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Section 2: Parameter Estimation, Distribution, and Trends"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "mle = data.frame(alpha = double(),\n",
    "                 theta = double(),\n",
    "                 KS = double(),\n",
    "                 KStest = character(),\n",
    "                 CvM = double(),\n",
    "                 CvMtest = character(),\n",
    "                 AD = double(),\n",
    "                 ADtest = character(),\n",
    "                 stringsAsFactors = FALSE)\n",
    "                \n",
    "for (i in 1:length(max_PRCP)) {\n",
    "\n",
    "    testDist = na.omit(max_PRCP[,i])\n",
    "        \n",
    "        for(j in 1:length(testDist)){\n",
    "            if(testDist[j] < .1){\n",
    "            testDist[j]=.0001\n",
    "            }\n",
    "        }\n",
    "\n",
    "   gamma1 = fitdist(as.vector(testDist),\"gamma\",method = \"mle\")\n",
    "   mle[i,1] = gamma1$estimate[1]\n",
    "   mle[i,2] = gamma1$estimate[2]\n",
    "   \n",
    "   gof = gofstat(gamma1)\n",
    "   mle[i,3] = gof$ks\n",
    "   mle[i,4] = gof$kstest\n",
    "   mle[i,5] = gof$cvm\n",
    "   mle[i,6] = gof$cvmtest\n",
    "   mle[i,7] = gof$ad\n",
    "   mle[i,8] = gof$adtest\n",
    "}\n",
    "\n",
    "mle[\"invTheta\"] = 1/(mle$theta)\n",
    "mle[\"mean\"] = mle$invTheta*mle$alpha\n",
    "mle[\"var\"] = mle$alpha*(mle$invTheta^2)\n",
    "mle[\"sd\"] = sqrt(mle$var)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the estimated distributions over time and create a gif (done in R/windows)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# display one pane of the gif - change per station\n",
    "display_png(file = \"plots/baseRSlideShowBoulder/0098plot.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check distribution of estimated means - should be approximately normal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tShapiro-Wilk normality test\n",
       "\n",
       "data:  mle$mean\n",
       "W = 0.98347, p-value = 0.2015\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "shapiro.test(mle$mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obtain values for histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "meanSum = summary(mle$mean)\n",
    "meanDist = fitdistr(mle$mean,\"normal\")\n",
    "x = seq(meanSum[1],meanSum[6],length = 10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Distribution of Mean using MLE\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# distribution of mean parameter\n",
    "hist(mle$mean, freq = FALSE, col = \"light blue\", main = \"Distribution of Mean using MLE\", \n",
    "     xlab = \"Estimated Mean of Gamma Fit for Maximum Rainfall\")\n",
    "\n",
    "# normal fit using estimated parameters\n",
    "curve(dnorm(x,meanDist$estimate[1],meanDist$estimate[2]), col=\"blue\", lwd=2,add=TRUE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Look at the mean over time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "time = seq(firstYear, 2005)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "meanTS = ts(mle$mean, start=c(firstYear))\n",
    "reg = lm(mle$mean ~ time)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>time:</strong> 0.298"
      ],
      "text/latex": [
       "\\textbf{time:} 0.298"
      ],
      "text/markdown": [
       "**time:** 0.298"
      ],
      "text/plain": [
       " time \n",
       "0.298 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(reg$coefficients[2],3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>time:</strong> 32.184"
      ],
      "text/latex": [
       "\\textbf{time:} 32.184"
      ],
      "text/markdown": [
       "**time:** 32.184"
      ],
      "text/plain": [
       "  time \n",
       "32.184 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length(mle$mean)*round(reg$coefficients[2],3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Fitted Gamma Distribution Means over time\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(meanTS, type=\"l\", col=\"red\", lwd = 1.75,\n",
    "     xlab = \"Time (year)\", ylab = \"Estimated Mean of Gamma Fit\", \n",
    "     main = \"Fitted Gamma Distribution Means over time\")\n",
    "abline(reg, col=\"blue\", lwd=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Parameter estimation looks good since it matches the smoothed averages."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Repeat for the variances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tShapiro-Wilk normality test\n",
       "\n",
       "data:  mle$var\n",
       "W = 0.9542, p-value = 0.0009469\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "shapiro.test(mle$var)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "varSum = summary(mle$var)\n",
    "varDist = fitdistr(mle$var,\"normal\")\n",
    "x = seq(varSum[1],varSum[6],length = 10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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4yF3FIgRihYsCsTQgVsAom7coJKlsB0ZiveKqKuLAow1Hi7Xvzoo7lqUNcmex1FN45ddHxZXHsSAWC3SxEvGuviKu3gG/PlodLdbOu1fvWZg0iG2UrRvXk1S2A9MB0q/IQo+UQqyQUa4qVipyiLWTQ8XaejgQSWU7MD4UfnP5xdErHwr3Pnpm19KUQe4slvqGw1PusIJ+kxRiBY1yQbGqVyR7WFX8DrqJI8Xa/Xi/fYsTBrGPsv7IWJLKduBmA6QQi6SyHYBY/kAsA/cSa/+zuXcuTxdkR5S1UvIU6xlf/NsNEIulWM+rf21mt1cQiwxdrCjkPWYGIFboKCvlZCnW1b/zvt8riEWGLlYqSH4LDbGCR1kuKUuxiigJ+02shqPEcvAKYpExubntlTvvEOsfxAoQ1b6yA7UmbZB9URYLy1IsIiBW+CgQq4JYJFGWSstUrDxVX/YLe6P3Y8Ry8gpikTESK2m6VyIKahbEIohyKbFeIimlWC/xCLmJQ8Ry8+q6Yi0VmKVYkSib0fcLnhVCrDYKSWU7ML2kA7FOCrI7irnELMWK2z3W1/+OfjpHiOXoFcQiw9DHygN/ywFi0UQxlpmlWFXajrsH/fXXEWK5egWxyJiPY4k07I90IBZVFFOpmYpFAsQiigKxSKJu1DBVa1IFcYliKDdHsfKH/C1FkoX+ThbEoopyCbGKpP/OTGJ/RedVu7j1aCdysdy9urhYhpKzE6uM5PMv64niHds8Ua4ZQ21lzNYX9UxxIepa9ZK2Jk2Qu4qVaWMMibwzyAZKrExktYtFtj7uBbHoosyKzk6sWAzHv8JiIEuJFTW/vijXR+qpxfLwCmKRod/nvcfiWuHomuL68hCLLspNxXp0Yq12yojF8vHq8mLNSn8DsdLnKxdykL7M1nvvEIswyg3F6n/OI0S0+kNXiEUZZVJ8hmLtfHbl9/t6panqwmfrP6CmFcvLK4hFhrNY9kAsyijcxSKEVCw/r24g1qQGbiDW59l8fSvdurgIsUij3EysUn+G9Pp4KsSijTKqgsuLlYno/VVTRR6dN9zg6RXEIsNVrEh8++nveQOkEOtuYu0Y9yIUy9erW4g1qoXLi8VjjwWxJLcSq+5j5c33Ic7sY0EshVYNlxerSrSzwvikSzreXkEsMjzGsTI1jhWlz9PGsSBWw73EsoZMLH+vbiKWVhMQyzsqxBq4o1jbF6ypxArg1V3EGuoCYnlvF2Jp/IxYZN/BGbYQwqvbiNWbdXexRkv4bsIcFWLpQKxQBPHqPmJ1ZkEsXyDWmNuJtQ1JWcN4BbHI8BPL6oEDEOuQKE2N3EMsuwcOUJT1749Ha4YLArEGLB84ALGOiaLMuoVYlg8cICjr35/5Kfa7gVhU+Ihl+cCB8GX9g1gmpFm3EMvygQMUYhkeK+cExKIiQB9r64EDEOuoKLVZtxDL8oEDwcsqvYJYBm4jlt0DByDWUVHqvudNxLIidFmVVxDLBMTyAWIt09QNB5zFKh9136q9w/uhww1t3TFqTUapXF+sMmpuNaNeQCw2UdiY5f6D1Vdt1ytSJ4RHitXVHKPWZJSK7L8HrW5n3H9ir/4VUVxArBBBQkW5ulidS2WSHCpWX2+MWpNRKv9ExcQsV7Fi0f2sPk4gFqMoVxdr+KpMIZKdYgl3/v76yTDtwCZIMLGYmOU83JD1NuUbv/Cai+Veb9pPfkM0A8Qiw32A9Jt2U8XjKLH0u1+EaIZbisXDrDNG3iEWYZQKYu1ndOu6EM1wT7FYmAWxmASBWLsJJtb49sAhmuGmYnEwC2IxCQKxdhNKrMmzPUI0w13FYmAWxGISBGLtJpBY0+enhWiG24p1vlkQi0kQiLWbMGLNHn4cohnuK9bpZkEsJkEg1m6CiDW7bRGj1mSUiva7wpPNglhMghCIdapZFxFrfp81Rq3JKBX9l9AQywKIZRtFq+dTzbqGWIYbQzJqTUapQKxd/EEs6yh6RZ9p1kXEMtRgiGaAWGRcQSzjHZIZtSajVCa3MTrRLIjFJAiRWKeZdQGxzLd0Z9SajFKZ3ngNYq0AsfZEmdT1aWbxF2vhGRSMWpNRKhDLGtNQQ8B2YBOESKzTzLqAWAs1GKIZIBYZ3MVafBgTo9ZklIrhrsknmcVcrKUDIavWZJSKUaxTzOIt1rJXnFqTUSqm+7xDrDkrTyVk1JqMUjE+QOAUs1iLtfa0S0atySiVBbFOMIuzWCsHQlatySgV8yNPINaY1cfzMmpNRqksPEvnBLMYi7X+2GdGrckolUWxDjeLr1irB0JWrckolaWnf0GsgY3n1DNqTUapLD5W7nCz2Iq14RWn1mSUyopYB5vFVayNAyGr1mSUyvKDMI82i69YWzUYohl+SKyjD4ZMxdr0ilNrMkpl7dG9x5rFU6zNAyGr1mSUyrpYR5rFVaztGgzRDD8l1rFmEYk1egDObOZW7Vh4xak1GaWy/hT7G4ils1ssiwMhq9ZklMq6WEeaxVMsmxoM0Qy/J9ZhZjEUy8orTq3JKJUNsQ40i59YVgdCVq3JKJUtsY47GHIUy64GQzTD74l1mFnsxLL0ilNrMkrFRqxjzOImluWBkFVrMkplW6yjdlnMxLL2ilNrMkrFQqyDzGInlnUNhmiGHxXrCLN4iWXvFafWZJSKjVjHmMVKLPsDIavWZJSKlViHHAw5ibXHK06tySgVO7GOMIuRWLu84tSajFKxFovcLD5i7fOKU2sySsVSrAPMYiPWTq84tSajVGzFojeLi1h7veLUmoxSsRaL3CwmYu32ilNrMkrFXixqs3iItd8rTq3JKJUdYhGbxUIsB684tSajVPaIRWsWB7FcvOLUmoxS2SUW6XAWA7GcvOLUmoxS2ScWpVnni+XmFafWZJTKbrHIzDpdLEevOLUmo1R2ikVo1tliuXrFqTUZpbJXLDqzThbL2StOrckold1iSbNI1DpXLHevOLUmo1T2i0Vl1qlieXjFqTUZpeIgFtHh8EyxfLzi1JqMUnESi2SndaJYXl5xak1GqbiJRWHWeWL5ecWpNRml4igWweHwNLE8veLUmoxScRYruFlnieXrFafWZJSKu1ihzTpJLG+vOLUmo1Q8xAps1ili/fl7xak1GaXiI1ZYs84QK4RXnFqTUSpeYgU9OTxHrBA1GCAGxJq1TDCzzr4I7V6DIYJArCnBzHIX6/NM1T2R0+yzviDEOjKKc3u2hNppuYpVxtr9tpPVRSHWkVEc23MgkFmuYmUien/VVJFHIltbFGIdGcWxPXVOFSsS3376K6K1RSHWkVGC4CiFjqtYo42vZwKxrhblTLGwxwochFOUM8Wq+1h5oabQx2KUyvXFqhLtkByXa0tCrKtFOVWs6pOpcawofWIc62ZRzhXLGoh1tSgQi0cURqncQSxc0gkahFOUM8XCJZ3AQThFOXm4AZd0QgbhFAUDpDyiMErl+mLhkk7gIJyiYI/FIwqjVK4vFi7pBA7CKQou6fCIwiiVG4iFSzphg3CKgpF3HlEYpQKx/AsfIgjEMgcJ0OqeYtl8ixViXS0KxOIRhVEqEMu/8CGCQCxzED8pFBCLSRBOUU4Wy/bXQhDralEYi7X6K7UgP30DhLhL0XPAoRD8IhALkACxAAkQC5BwwCUd8ItALECCn1h5Ko+FaREoGXAfvMRKmjEPEcEsMMFHrJdISinWSzyC5QNugo9YkSib00KcG4IpPmKpwyDEAiZ8xIrbPdZXxMHyATchQB8rj8QrWD7gJnidFabtxfD1m4LMOPGyPbDCR4oW/3Eskb53rhWkRxamW3e3VPgU6IyRdzaFv18qfAoEsZgE4RTlInf0m8Gm8PdLhU+BDrij3ww2hb9fKnwKdMAd/WawKfz9UuFToAPujzWDTeHvlwqfAh1wR7/5uo7bDB/kdqnwKRD2WEyCcIpykTv6zWBT+PulwqdAB9zRbwabwt8vFT4FOuCOfjPYFP5+qfApEH5MAUiAWIAEiAVIgFiABIgFSIBYgASIBUiAWIAEiAVIgFiABIgFSIBYgASIBUiAWIAEiAVIgFiABFKxXt03xrJIRFlpN0mSSGy/fdpUyocQj/bnAmfXSs1HUKVCKda3+/VO8zXm2GqSgkyFj0oGqUQq/Ndq+8Sp1JRR00IEqRCK9Y1asT4i+spXH4tJkkTEo2zvlHp2KplMIhNpdX4qkrRpIYpU6MR6iaQVKxN5/fctnhaTFKRNGjKbs1ORt21tf4d5dioqdtNCFKnQiSWy7oesqZA/FPvKz+nmJCHqlvQsUlG/wzw/laL76FOkQifWt/+FtPZvc5KOUt67hEUqmbq15vmpJKJoglOkQnpWyEqsl9ydM0ilPv5k2ibOS+Up3hXE8qaIUh6pvNJI9VXOTkUd2CCWL2WUcEmlqh7yWHh2KrEcfrm2WNGQ1uYkFUnMJhXZ3YtOT+WhTvSa4BSpHCFWc05RDKcXK5M0FHFSMElFISvm5FT0e29TpHKEWE/14chln3VzkoS8v5nl2ak041iFHMM+ORVdLIpUjhDr7DHmYrhJ6tmpqJH3MpV9rLNTUTQtdK2R92o4JsfDLXA3Jwl4aA9cODmV9lqh3faJU5G0LUSQyiFilerauN0kSRqDWCenor4mEL/stk+dStW3EEEq+D4WIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiAhvFj6XVM18mbe+rr5Usjm/0d7sFksDAv73rpaPk1wuG9d+UrrUqQvv5gSrUpUinrq421OidJXc7/n4pVGi9HdU0qmtxWdBJtmp931fZ2jxIr1O9UvES/M7lcbbq9aCFM1+4olReqfcJVHbTmiwi/qTKxROUfbNK35UBOP6SdVX8Ynpc/0/dHLaXanimX/ruVC/fvP/tOTkdylWojBobxuUlnpn9To8M64yy/1bZrWjJutR3HYJwm00bKNO9ZOs7upWGXfxtFqc7iiJxD1x1r1hJJgcacv12umPhCp57F+6/8UYm01zHQ2F7HypD6M5+2Ot0/rKSL1dMVmr5PXu1t1R97++Ckf4dy2ZRYNzz2s5JPQmtb+qA9av2a9RBmLtO3AaO8WqdpUE6h9OoUWvereiF/d9rtNvYfHMBTZZxJ2XILlEhnqpJ7UN9Nvs0uiK0o/P1dav8S7WaePnaiD2Kc+UppqdXhIziy7cUqTqO0qbb112U1nV9vQivVqMnuNxXrKF7l6zLB66oFop7o6ToebiquF0iFk3vc4cn3NSi3UGDh6V3WTnl0g9VRoLXpD0r0xEiuddD5GYcclWC6RoU4WxOqT6IrSzy+VZWn7bMEhdtPLjOoymWpVF2s8Z5xScygcFU6rtza72WwLDQg777LYcjf+ludyekGTUhqn/kby9Vs9xK8vbC5nlWrf9G6fiDCURH9c0GjNpOzjj999yc2/5dRDVo0WvaHbxruaCjAp1jSsVoLFEk3qpDK0jHqhJdEWZZiruvp13GmZX3XLP5s1TDks1fe0mb7Twun1ZqhoFmKJvv30gn6qrlM4pKiJlarnzZTNw4HUQUgrSaYEeE8/eu3JzSzep51UgVQPTYvekLbPiUlG63eTQ1nmYbsSrJRoUieVoWXanXSfxPg8rZ6b1W90R7xR7ES8RFqZc9iq7y6l5DvJRF/FUCNMDoX1YT39fod3x0Xu/hb5M9FKoZnZt24f8asESERTH5M1l+KN29vkykI1jsQyhN0s0bxOzGJpSYybrX71bvdM7Ywhdn1wNIg0F2ua3bDZOOo++KPCaQtPN8pErOopD9ZyHGiloMno07whVn1gqI9k7UDpdM2leC5i6X2sxbCbJZrXyX6x5DOmktqgbjc1xM7EpEO9V6xPN5owLtxMLNPsDciHG/IsnvaxhqXk30d9NpQXsxbTY+khm56FOn+arbkcb+bRLN+pWO/2NGEjzY0SzTe6Xyz5LLq+26TH9t9j1R+f1FCTU7GMszc4YhxrvaBqSm+xdOhYN5MfPaTcW8WqnzRf0xxP/k20PtbkUlDXvdEeGa0YxrHKxbCbJZrXiVksLYm5WPVJh7R82HbRVdSr7xkac/hGAIBvAAAByElEQVRsivXtOu+jwk3FMs7egFasuDmdUHus5c/Wp/p2B3C5kDpDqlS/NJ+dFaqRhnZnMlqzMsbr3nzJ8xx15NCiNyycFcpThlSNFNUdxWgh7GaJ5nXSLFBM5o3OCqe1WZ+P9TO02PIM5tk9XXqaQ1zPKJNNsdpd1qRwc7EMszegvVb4biY+6oF3kbkKMjFZqD2iqyt0atDpMSpJ3eDtvkRbUxNr9q76O4xjadEbhiGkcZV9umuFymNz2K0S9XUymmzLqc/Tx7HGtdkf8tSMIXYZqXGstvM1zUGNIKbbYpVqlzUp3FQs4+wNiC9Cq5F39bmPl8SSl1eTjzoKNAupUWjxaFr+OR55l0R9qwxramLN3m3+yhPUfuS9i97w6h8hOKmyd1q7lTwL48aWxdIW7etkNNmVU5/XJzEXqy1xM6OP/WhH3hNzDnXFPcxzxillKs1x4aZiGWdvgO9jARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARIgFiABYgESIBYgAWIBEiAWIAFiARL+B4diotyMgrjyAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Distribution of Variance using MLE\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# distribution of mean parameter\n",
    "hist(mle$var, freq = FALSE, col = \"light blue\", main = \"Distribution of Variance using MLE\", \n",
    "     xlab = \"Estimated Variance of Gamma Fit for Maximum Rainfall\")\n",
    "\n",
    "# normal fit using estimated parameters\n",
    "curve(dnorm(x,varDist$estimate[1],varDist$estimate[2]), col=\"blue\", lwd=2,add=TRUE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "varTS = ts(mle$var, start=c(firstYear))\n",
    "reg = lm(mle$var ~ time)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>time:</strong> -99.304"
      ],
      "text/latex": [
       "\\textbf{time:} -99.304"
      ],
      "text/markdown": [
       "**time:** -99.304"
      ],
      "text/plain": [
       "   time \n",
       "-99.304 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(reg$coefficients[2],3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>time:</strong> -10724.832"
      ],
      "text/latex": [
       "\\textbf{time:} -10724.832"
      ],
      "text/markdown": [
       "**time:** -10724.832"
      ],
      "text/plain": [
       "     time \n",
       "-10724.83 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length(mle$var)*round(reg$coefficients[2],3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Plot with title \"Fitted Gamma Distribution Variances over time\""
      ]
     },
     "metadata": {
      "image/svg+xml": {
       "isolated": true
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(varTS, type=\"l\", col=\"red\", lwd = 1.75,\n",
    "     xlab = \"Time (year)\", ylab = \"Estimated Variance of Gamma Fit\", \n",
    "     main = \"Fitted Gamma Distribution Variances over time\")\n",
    "abline(reg, col=\"blue\", lwd=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In case the distributions aren't making sense, check the number of times the Gamma distribution fit passed the KS test."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "not rejected     rejected \n",
       "          71           37 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(mle$KStest)"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.3.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}