SpatialStatisticsBasics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's generate 200 uniformly random datapoints in a rectangle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "library(spatstat)\n",
    "library(dplyr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n <- 200 \n",
    "# number of points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# let's set the range\n",
    "xmin <- 0\n",
    "xmax <- 1\n",
    "\n",
    "ymin <- 0\n",
    "ymax <- 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# create a uniform sample distribution\n",
    "x <- runif(n, 0, 1)\n",
    "y <- runif(n, 0, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.468452134402469</li>\n",
       "\t<li>0.162395716644824</li>\n",
       "\t<li>0.192506315885112</li>\n",
       "\t<li>0.0139029056299478</li>\n",
       "\t<li>0.846057121874765</li>\n",
       "\t<li>0.205207568127662</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.468452134402469\n",
       "\\item 0.162395716644824\n",
       "\\item 0.192506315885112\n",
       "\\item 0.0139029056299478\n",
       "\\item 0.846057121874765\n",
       "\\item 0.205207568127662\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.468452134402469\n",
       "2. 0.162395716644824\n",
       "3. 0.192506315885112\n",
       "4. 0.0139029056299478\n",
       "5. 0.846057121874765\n",
       "6. 0.205207568127662\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.46845213 0.16239572 0.19250632 0.01390291 0.84605712 0.20520757"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1.13213354721665</li>\n",
       "\t<li>0.659006745088845</li>\n",
       "\t<li>1.51062953472137</li>\n",
       "\t<li>0.944892221130431</li>\n",
       "\t<li>1.92811814090237</li>\n",
       "\t<li>1.59218589495867</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1.13213354721665\n",
       "\\item 0.659006745088845\n",
       "\\item 1.51062953472137\n",
       "\\item 0.944892221130431\n",
       "\\item 1.92811814090237\n",
       "\\item 1.59218589495867\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1.13213354721665\n",
       "2. 0.659006745088845\n",
       "3. 1.51062953472137\n",
       "4. 0.944892221130431\n",
       "5. 1.92811814090237\n",
       "6. 1.59218589495867\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1.1321335 0.6590067 1.5106295 0.9448922 1.9281181 1.5921859"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When plotting spatial data, make sure the one X unit is the same size as one Y unit. We don't want any misrepresentation. The Y axis scale to X axis scale is called the aspect ratio. Spatial data should be represented with an aspect ratio of 1:1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "angle :  num [1:300] 2.47 5.76 3.96 5.2 4.81 ...\n",
       "disc10 : List of 5\n",
       " $ type  : chr \"polygonal\"\n",
       " $ xrange: num [1:2] -10 10\n",
       " $ yrange: num [1:2] -10 10\n",
       " $ bdry  :List of 1\n",
       " $ units :List of 3\n",
       "lamba :  num 1.59\n",
       "lambda :  num 1.59\n",
       "mapxy : function (a = NA)  \n",
       "n :  num 200\n",
       "n_points :  num 300\n",
       "p_cluster : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 332\n",
       " $ x         : num [1:332] 8.934 1.243 -0.349 0.351 1.133 ...\n",
       " $ y         : num [1:332] 0.402 -2.997 -0.612 -0.483 -2.52 ...\n",
       " $ markformat: chr \"none\"\n",
       "ppois : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 497\n",
       " $ x         : num [1:497] -3.153 8.513 -0.794 2.806 2.125 ...\n",
       " $ y         : num [1:497] 8.128 -0.465 6.899 0.279 -8.242 ...\n",
       " $ markformat: chr \"none\"\n",
       "ppxy : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 300\n",
       " $ x         : num [1:300] -4.497 6.119 -2.546 0.766 0.87 ...\n",
       " $ y         : num [1:300] 3.58 -3.56 -2.72 -1.45 -8.67 ...\n",
       " $ markformat: chr \"none\"\n",
       "qt : List of 11\n",
       " $ statistic  : Named num 12.1\n",
       " $ parameter  : Named num 24\n",
       " $ p.value    : num 0.0434\n",
       " $ method     : chr [1:2] \"Chi-squared test of CSR using quadrat counts\" \"Pearson X2 statistic\"\n",
       " $ data.name  : chr \"ppxy\"\n",
       " $ alternative: chr \"two.sided\"\n",
       " $ observed   : int [1:25] 1 11 23 11 1 14 16 10 17 12 ...\n",
       " $ expected   : num [1:25] 2.09 11.75 15.02 11.75 2.09 ...\n",
       " $ residuals  : num [1:25] -0.752 -0.219 2.059 -0.219 -0.752 ...\n",
       " $ CR         : num 1\n",
       " $ method.key : chr \"Chisq\"\n",
       "radius :  num 10\n",
       "raidus :  num 10\n",
       "rSquared :  num [1:300] 33.02 50.1 13.87 2.69 75.85 ...\n",
       "strauss : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 300\n",
       " $ x         : num [1:300] -9.32 -8.98 -9.9 -9.74 -9.03 ...\n",
       " $ y         : num [1:300] -3.513 -2.87 -0.864 -1.642 -0.203 ...\n",
       " $ markformat: chr \"none\"\n",
       "thomas : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 379\n",
       " $ x         : num [1:379] -3.262 -0.262 1.058 2.326 2.246 ...\n",
       " $ y         : num [1:379] 1.76 9.18 9.59 -9.71 9.16 ...\n",
       " $ markformat: chr \"none\"\n",
       "x :  num [1:200] 0.4685 0.1624 0.1925 0.0139 0.8461 ...\n",
       "xmax :  num 1\n",
       "xmin :  num 0\n",
       "y :  num [1:200] 1.132 0.659 1.511 0.945 1.928 ...\n",
       "ymax :  num 2\n",
       "ymin :  num 0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ls.str() # show me previously assigned variables. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "mapxy <- function(a = NA){\n",
    "    plot(x, y, asp = a)\n",
    "    rect(xmin, ymin, xmax, ymax)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ROSpX8QSUyOSLa9q3yxVGuTooFIWnJEUrIS3LZanKN/PhBJTYpISvYmNe5fStE/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\nSUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mapxy(1) # aspect ratio = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's create a uniform density pattern inside a circle. The spatstat libary contains the disc() function that creates a circular window for you"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_points <- 300\n",
    "radius <- 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rSquared <- runif(n_points, 0, radius^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>82.638123491779</li>\n",
       "\t<li>51.5189690049738</li>\n",
       "\t<li>53.2897343160585</li>\n",
       "\t<li>2.59653693065047</li>\n",
       "\t<li>63.3745149243623</li>\n",
       "\t<li>19.4733514683321</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 82.638123491779\n",
       "\\item 51.5189690049738\n",
       "\\item 53.2897343160585\n",
       "\\item 2.59653693065047\n",
       "\\item 63.3745149243623\n",
       "\\item 19.4733514683321\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 82.638123491779\n",
       "2. 51.5189690049738\n",
       "3. 53.2897343160585\n",
       "4. 2.59653693065047\n",
       "5. 63.3745149243623\n",
       "6. 19.4733514683321\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 82.638123 51.518969 53.289734  2.596537 63.374515 19.473351"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(rSquared)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "angle <- runif(n_points, 0, 2*pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>3.60708036247272</li>\n",
       "\t<li>3.2708122353269</li>\n",
       "\t<li>4.34278647126521</li>\n",
       "\t<li>0.712797120346877</li>\n",
       "\t<li>2.00412720766829</li>\n",
       "\t<li>5.41497669389165</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 3.60708036247272\n",
       "\\item 3.2708122353269\n",
       "\\item 4.34278647126521\n",
       "\\item 0.712797120346877\n",
       "\\item 2.00412720766829\n",
       "\\item 5.41497669389165\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 3.60708036247272\n",
       "2. 3.2708122353269\n",
       "3. 4.34278647126521\n",
       "4. 0.712797120346877\n",
       "5. 2.00412720766829\n",
       "6. 5.41497669389165\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 3.6070804 3.2708122 4.3427865 0.7127971 2.0041272 5.4149767"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(angle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x <- sqrt(rSquared) * cos(angle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-8.12334175454742</li>\n",
       "\t<li>-7.11782959260997</li>\n",
       "\t<li>-2.63708055229358</li>\n",
       "\t<li>1.21906440979348</li>\n",
       "\t<li>-3.34271390048446</li>\n",
       "\t<li>2.85156815899366</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -8.12334175454742\n",
       "\\item -7.11782959260997\n",
       "\\item -2.63708055229358\n",
       "\\item 1.21906440979348\n",
       "\\item -3.34271390048446\n",
       "\\item 2.85156815899366\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -8.12334175454742\n",
       "2. -7.11782959260997\n",
       "3. -2.63708055229358\n",
       "4. 1.21906440979348\n",
       "5. -3.34271390048446\n",
       "6. 2.85156815899366\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -8.123342 -7.117830 -2.637081  1.219064 -3.342714  2.851568"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y <- sqrt(rSquared) * sin(angle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-4.08037280534579</li>\n",
       "\t<li>-0.924916696540626</li>\n",
       "\t<li>-6.80702141004224</li>\n",
       "\t<li>1.05376415550413</li>\n",
       "\t<li>7.22501063693821</li>\n",
       "\t<li>-3.36777530469977</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -4.08037280534579\n",
       "\\item -0.924916696540626\n",
       "\\item -6.80702141004224\n",
       "\\item 1.05376415550413\n",
       "\\item 7.22501063693821\n",
       "\\item -3.36777530469977\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -4.08037280534579\n",
       "2. -0.924916696540626\n",
       "3. -6.80702141004224\n",
       "4. 1.05376415550413\n",
       "5. 7.22501063693821\n",
       "6. -3.36777530469977\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -4.0803728 -0.9249167 -6.8070214  1.0537642  7.2250106 -3.3677753"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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NZkZSTklBYtr5TVt3aPS2DLtnP4PliPkR4ebcqmgxmTMzkZl5HmPDJWP/5+b6Jg\nOyfvq3HoZlOJlYblcxczJmdyciAjzZk2jzUb+kx9cjt1XY7xNi1GlrvvZMbkzHnDJSPNWVR+\nhdbndnacdNzCne/ft7gX6gafMeGGrTLSnFZGOtz+6f6rBOaUzuw1QxsJOWyVkea0MdLJDjyN\nZC4dWiMh+hJkzy4jzWkzRzorzrP91xjJ/EbOORKkL4E2EjLSnDardqdGOtt/cXEf7nnnu88o\nV+0g9paR3Hil1vc8Ut6y3NGfb2XFfVA6+9skPI+EsYCM5EYjefWtPP5LzU/MnbXDZpYDWUBz\nJDda6Yv9IpJdH+0lIO9zBO0GgSgjYRTfv8m0diPWfTbenxV2fbXsls6JkQ7+ZnsRCNS+IGP4\nW6v2Y77Xxvuzwq6vnpKvr83qAJoulG+r92JfLMVeqOz6/DiqBz4jrY0Tt7Y4zQS03y0t7Pr8\nOCpFRiOldi4j0cCuz5PTO+QQzZGSuw7ZuYyUhF1fEGyrdptdr57b71tzpBX0AoNAn0fCrg2E\nGkmrdilcBBJeGxALvPhw3ULBsZoCji99QTkI5LxaLRT4cAiV4/3zbCcXbVXv2Qh9OXkYyWvD\n3eIxEsP0CjvHKvMy4obQlxNeYOiyMCe0KdkTdtoWykhrZKQG0KZkR9i5XhlpjYzUAtbRroyE\nw8tJ/JEXME1lK1a06y/pY3VqpPaREOZuhYORMqqmywXyP58euH+EoMhKlAHvHKuztlBG2uKh\n8KxqaBvoY6bZf8OQPFZnB0hG2tJM4eaeV/y5WfJ00GBOSnPcFspIWxopnLdxnS5HXMpIx8hI\nW1oZafPYQ3IWyEgvZKQETSQuvJNhJMq5+ZBzpBIC4u8g5e2NdL4oxLkYUbNqNxQyUooII2Ut\nChHmrvQ80mDISCnaSFx5I+NrXrpIXjOo/CsjpWhkJMNoTUZaQzbWlZFStJKY36bKSGvIxroy\nUhI+jWR1Ew5ZyxIhgyT0Qz6XpHkGDnQjmXBkJJbQD5kIKxf4FY1MYRUiI7GEfsjU+1jq6AZ1\nbC1EGa7Hx9zUyEhpJrYWz8ahV/puIV44tgf2TYdks4tDOPVtpNlj+m+dBrbg84qKVByVY1d7\nUyMj7dG1kQ619xzYmnTXUdtXFWRIRtpjchoBNZnnX8dIs8ez31q3KiNBmFzG4I3m+ceVEDVH\nwrch6TirWwoZCcnkd+T9E3C4n6AbVXu0S6vno9/at8vuo36M5LZN/y7puGojziN9NkydGMlq\nehlpn66NxHfO1WmoPHs8+61twzahMtI+nRuJDZ/FG59VuwIdDfcVvVczfk7qJAFQnnWNrg8m\nu6oAABKUSURBVG+X80hmES13Fr5bKy5GCpnnU/BsQ0YMXkY6wkfmaVvJNrlB8WxDRoxORjoi\n5tTAuF3W55rdmKFdardmInQOPIkat42IiqmXXAboHHtZb9hR68X2a6VeZ/GnWnpJUQh0fpSR\nTsgTuntcC8YyMtIpfCPEMC1MSTgkR+jBcS2Z73QwRwruEPgyJCOdkWWk3VcW9S587e2KaIGE\nfbaMdEaG0IPjWnjI6WYAS6I7BBmJYMdmzpXijcRNeFDhAjbENyr8VBkpvPH2IL6O6bIanYoe\nyB7bpRcb0NMJglEfgZHYZpEyUgYZXdLRcYVWPkcFFXcIObnIyxdBezIjUAtTGk5gOq4cY5pC\nO+e8zbBpIi/JSFnwaH0NqqJrqEhATiuQ3VJw9M2fRKrgyEAePFofZXa/gppHVSY5U6v86RdH\n3/yJjJQHj9Zn9Uw3JlWZ7Jvk3b9lGyl+wWOGjJQJj9hPBzG1xvns1f58kNalkUI1MCQgGx6x\nn4O6Po205//Fr3NDk5EYdm6FSex2GNQNO7f7eU76Dl6T3BjNZ21lpGy41HbaId3Si33Twzu2\nFclp3jfHEquBIQP5cKllWvmtZ5rmT9nvevZJ8chIBsjkkpTQkyo5RUYiGt5Gn9GL3b0Vcrmh\nxqrsINdDu7w3rZ4DkZEsUMu1VjLYdpVTtmkqWDeQkUh2b4ZZr62S0TOsypp+XqZhfDvNgku0\nhOj9WyHWa6xkdAnWGulWtP5Gs+ASLSF6/2Z4BdsqGT4oqtvg/V3T/jcqH7yTwUbxZREuwAqv\n4EIjwQqxqot7y7lx9DBGwhWHC7ASNpfP2OHsMfPFyMKt2tSyFeiuKuIVhwswk1uo7ZtW2y6n\nzSNAQPmGlkJ6K4t4vfEKrATN5fN2alr8vr1tF38Ylq1AvB4b8XrjFVgJmss7MJseMcicZovf\nSD0thtjx+YtXYCZLMlOFHkAnE96RNxliE6SPQIKVkYxEN7kH1f26r5WRGDE4qU14FYMXvuVm\nwEhsFlSTBo0hfQwarOQZqVmFQi4WHYpZIyYjEZPUvK3Ha93jjoi5eWQkZraiA0dIvczG2rHI\nSINmhiLzFCKsJIw0f2x8Gnb1LJZG8m/iKDJPIcLMZhT3fm7eN8lIG1ZXSTgfDY7Ec6iwcmSk\n5Av8xWz2GLmIEP2NmE0bM44S5lBhZd9IAf1Dsmwil7UJltRbOpmjhDlUmEk7abrFDLQSZRO5\nlHetZUSSOElkWNkudT8eMozU8tqviPRG7TtoPElSwSQyrKQWwBcrd9s/vP75fPDkekYKG0+S\nVDCJDDO7ulcHdHN824x7LmikgH3G7DENiw4ruUO39fFtVWVXmyOB85o/TGQpYBYdZvKEb45v\nMyNdbNUOmldDADT1SyPECrmRrnYeCWuk/E3R1C+NEDOmPK/HdmRhnxY+/yXiwLwaTMmTFR4l\nVtLK1xWXWMR7PtBwKolQ8wbk7ZBWzxkvjYdHiZmE9O3RTBxfutY91ZgfrphwAr5BX07ARCkh\nkmIlZaTEH+h8syZRN+lb+pDHgSO74SDKCJEUMxvtfVZcykjJP/QVVgW5w0SmhDBpsdKpkdLT\nuIOTXX2EBSXzezfddeTDpMXMWnwPFZeatM0eFz9Oey8Qf6BKCJUYKztdEnVMCYk7lzHNng+G\nOvRTQD+oAqcSY2bTJdGvE6c7zYxF+z0b3cgD9oMrai41Vrbq2RvovNFn2h9kH3uKhitqLjVm\nupOfO41L3V3s8TD7Uw+TQi/IYiaTY6U/+cV9yOONyy+xmK7bJZHFTCbHTHf6S2c1iRW85/cn\nd5iFethCZtNjpr8AyqZxrz7o/c9p9t/VoAuZTpCV7gPIZGukxJzpOtCFTCfITP8R5LG5bGg+\naboafDHzKbLSfwR3Mj6SNHvVfGI0SAJM8MXMp8jMACHkLUHMFhc+Fxtu82+rHIP+LrJ7QCjJ\nCjSE0Juz5e16ei9+387d1xWZETGGzKjJCi6GqNIsG6WN1yNltieMITNqMgML4vRAOnVYRUYa\nb46UGRFlwJSirKCCODuQbh1WrZEGWbqTkaIBRXFqJOTOqrf8UjvMXCnPSJyBcqqy0sZIjmOp\nIi883efn73JqLt/osUMiVWUG6qTjDskpZwWVtzizlCer0Riw+ILCjPeRViypLDOYOI4PJN/s\nfjr6JpuDT2K460oKynrn6cm0go22gFWXFdh6w+GNGqG7QpE2Uso0rfR7tjhs2X/CqstMi0BI\nZ/VJfyR+2axHxe1o067RJf8JrTArbQI56rDK5x+VM5f9zmeZlu6MtHvHJUJohZkJjqS8twL0\nc4np0Op573c+gMaQ283wliuvMivRRioW4TJzSZpmZ0/4pTzMGHgbA3G1EkuzEhpKeWvv1E+k\nTJOsb/DE7+FKhDllpCAiY+EzUtIf3nf0grpykxnmYmXWZkVGWm7Y9NEezCIKdpi63hpzsTJr\nMxPvJJo5knHv+/s39THgRmG1b+papRZnJdRIkat2FZwa6fCvxo2ZSXzfGinU4swgoimeJoed\nR6rj2Ck2a7gusHOXKrc6M/WnLkK7hwig1xc6DlPJjwm5PCsO5wDH5/ByjdXz6aZuXu0Q+TEh\nl2emMh7XsUmXZLcswBNI+zp4YddnZqo6ljLSmtwb+zgPiekPCL1AM1WH9OpGSrVBWe2S85CY\n/3jwK7RSd0ivOEd6UbGEv3iGDe/et/DDbM8RfoVGKm9hMOqqnXO3MjcSLIXvDXVwPDqQaGN6\nxFQ+TSo8q8qcybzarhjXLoxUupH0Rqc+irQHjSam2/xwmt9d5gf2fmxR27sx1kwQ33uATTNf\nG+LN64wuRJqYyhuxYj+Qz6wyB15VRtoMw2SkzpnKF8BL/cC+1pc78KqcXS7fXrj8t1bDm9Ul\nfai0MVX5qCAnjId8noOZvkOpyA+2brZi3zh5P7+kE5lm6tdwW7zRjeQHEDIGXpAPtqYdY7cF\n+8xzQScyzdjimipH96FtZ/Ik6uzx5jKDKVNk3HHp2CKAXnSaMQT2KrNiPwS2nen7MKyeZ7Ud\n5flCB3dTn90INZMf2au2KvwQ1nQmjXFQtVGeLzNSP+XZj1IzuaEtF4c7S0i6Pk+WFGK6ztlj\nrpSODkZHUs1kxsa3VGBgRzzhgte2KzztHKn0n9CTVjN5wVmNRNVr7RmJccFrnbgzt5PJP6Yr\nsVZMXVLui11L1G7SPfFUdk9y2oCxB7CgK7FmMrskizc8B00nQpLeqDZ28DLJfi77Ks2+1JrJ\n7mZKVibgHJp01zHn3811dE+GKXj1YT/eZkIgdCbXDDo+TyMdb7u0Kzy7S9DSSi1NddxuNJOB\noTe9ZsABhhmpeMdn5Xr/y/OC06ZrFEd7664uuxNsxadLchzZgY101s1N8+fWq+b7/V93ddmd\nYDPoLsmx0T6q4xZGcp0AmohXYKU/xWbgfZLj4vdt36SFvcXZlP7ZE1EZKVyAnQ4lm+koxgOT\nlnaFxwacPtca5iO7+IRF77+EHjWbGSTI0tuy3A4N+LTS/efZYxzR+y+iS9FmrhHlHqeXhs4W\nvwk+AtTnwepTtZXmUYZXYyF3G9Vprw6908xFC2hD2zCP7tQzxTf5B1SP7eoXNXmTc0inss2U\nxYm+G9Fz6MRqpfrVhnonlr81lF51mzkIdM8udetkqV1Os/8YqTYSbAPd0a1wM/sLwLcdu5S2\nrrvV9HQQrZPCjcSZlhz6VW7GeqKzuCj6NVL1yCwZuv3a+g7pWLqZ9Ox/94/lreteOXZgpNq1\ngm3ohk2SJiWLnrWb2R+/gY20Vzvsc6Rb9ep14s4Ms8eT99bsOJquxZuxDeAqxjk75ci+aodg\nFXp+c9R3TvpWb8ZkF48LvcnPI+HJNlLnSelcvpnECO7ALteqeRdyjdR7onvXbyZ5mpQrC2x6\n6sgbIHcfcfcBmGGPmPKWdBWk49mZSfVL/xGYIQ+59lQOH9sedm2uAaIdIAQz1DFXX1zQA6vG\nYoRYR4jBDHPQ8Ubyn6KtYmQ+HNkMEYQZ4qi9jJRrjxZTtGWMxAfDwBhRmCEO26W68u3RorgX\nRiI+FBYGCcMMb9wuXUK2PdqMLGdyeA+EjVHiMEMcOH6SYr9Ox9lIr8aC+DDYGCYQM1eKnM1I\nr8ZinIMwTiRmYKHzX4lgsEfTBQD2vBkYKBQzmNi7uBIh3x4twyFPmomRYjGTDN7av3Sxhmux\nR7MOljxnNoYKxsy2ZMwNcvwJ1Dzoxp9seioZKxo7m6vAkr/N2MDVE2lltHyNFo+ZKfmv/LTI\nSEUMl67hAjKTvJzf3CUpjybGS9d4EZmZEj9bjNTDqh0ZA2ZrwJDMzF1Q0r/QzePJGTJdI8Zk\n550F3v6F3K6XuAvkAWNGZWbeJ1EWLK/B71zkLpAHDBqWGUAeXB1IvqRhuHLCV0gYo8Zlpvrr\nsVy7DPJF9mx5vH1qLcMGZqcuFb5dxiBGIpWPYODQzBjOwm7vi2PehokxjESqHsLIsZnJTEZq\nFOdd6SPMkWjVIxg6ODOZl0cnXutupP5X7WjFQxg7OjM5pZr2jHuXwbks/+JMHrf6egYPz855\nQnaMxN1lRDN8YoYP0Mz5Eu7OC8m7jFDGz8z4EZo59QP5xD+fVta/QgtzgRDtnK0+jTGKaxZG\n95nK4RJBmjntk/q3UbuOdYBUZXCNKM2MYJQTGp3jvUAm71wkTDvDJ6aNkYZP45PLBGqmJjM9\nDP2aGIk/DSiuE6mZYjN0shjhP0fizwGOC4VqpzA5nSyPu/udPgNILhWsmaIyI79Se4bvRxH5\n40dyrWjtFJRDP0Zy5GI2uvjRzsKcIRnpeja69NHOxVwVncyR/Lhi6FeM2Yz1+yn6WLXz4pqR\nXzJoO2YrNc8ry6krEhnNuWjYdrgTRdMJEkiI4bKBm2Eo011IpmXUOfLlupHb4S0TjoVC3vw0\n4Mqx22HNFoORLm0j3tIghbRaCIzEmZh2XD1+M5xWip4jcWalJZdPgB3GooldtWPMSGuUggJa\n3OfAWpyB55FUQzcloQzvoqU5LZRBJzK9URbK8C0f7JTHs7OSjR4oD6V4VufquW5jjr2bbPRC\nmSinj68Vg/RuyU5NNpqhXNTg1dCvns3vn+lCmDLZqclGC5SNOnzKqaoXWZY9xEjbLchGK5SP\nWjxKqmpesyx7gJG2m5CNNigj9fhYqXhcl3yu75BmzyqaLcoJApZP1f0hWfZV+hZbZAqVCWUF\nBE2BpQZitWt2z0eaIPlQYnCQVBn+AtZHp0YSHyfKDRKKWvM4AzvJRicoO2AY6g1f9AxRcaME\nwRmu7R4uIA+UIg8GKr2BQnFFWXLC9YrrZkdNLspFiXLDqd7bfVZJnZEBpcoTl2seZo+eyEUm\nlC1f4K069CMW+3uRjYwoX+5gi7KBkeSiApSyFgAr091IclERylobcK286xxJnVEpylszQFfZ\nuK3a6SqgGpS6pkBq1aPgZaJKlL7W8DX8fIo6RBmMgKd0eZR0jrIYRXgJ66MRSJTJSMJqWR5C\no3SGMzW1U9u9XQellAT/ApeFPFFiqZg8qt1lo2KJ0ssIqPTloHYoy8RML4repGPbECW7C6Z8\noqVeFOVdCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQ\nkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAy\nkhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBG\nEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhI\nQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJ\nCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJ\nAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQh\nAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQE\nABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQA\nICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAA\nZCQhAMhIQgCQkYQAICMJAUBGEgKAjCQEABlJCAAykhAAZCQhAMhIQgCQkYQAICMJAUBGEgKA\njCQEABlJCAD/BzTZVK1MRk3rAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"disc(radius)\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(disc(radius))\n",
    "points(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Point = location in a planar space (x, y)\n",
    "- Event = observed data point\n",
    "\n",
    "For example: trees can be in any point but the actual locations are considered events. \n",
    "\n",
    "Spatial studies occur in a confined area called a \"window\". Events occcuring outside the \"window\" are considered unobserved. Spatial point pattern is a set of observed events and the window. Spatial point process is a random number generator for events in a window. Most spatial pattern analysis is spent making inferences about the spatial point process. \n",
    "\n",
    "- Marks = other characteristics (height, width, species, etc) about the events. Also called a \"Mark Point Pattern\".\n",
    "\n",
    "spatstats stores information in its ppp object (planar point pattern). PPPs contain the (x, y) coords, the window, the marks (optional). \n",
    "\n",
    "- Complete Spatial Randomness (CSR) = This just means that no part of the window is any more special than another. And the chance of finding an event somewhere is the same as another location. \n",
    "\n",
    "Testing a hypothesis would be interesting for a scientist since rejecting the hypothesis might show that some parts of the forest are producing more trees than others or trees are clumping together or spacing themselves out. The spatstat library has a bunch of tests:\n",
    "\n",
    "- The Quadrat Test = It can be used to check if the points are completely spatially random. The window is divided into parts (~20 squares) and the number of points in each square are counted. If the process is completely spatially random, then the number of points in the subregions follows a poisson distribution (a simple distribution for counting). The difference between the observed counts and the expected counts lead to a test statistic with a \"chi squared\" distribution. Larger/smaller values of the test statistic indicate the deviation from complete spatial randomness.\n",
    "\n",
    "### Create PPP objects and perform a Quadrat Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's run a quadrat count test on some points to verify whether they were uniformly generated on a circle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "angle :  num [1:300] 3.607 3.271 4.343 0.713 2.004 ...\n",
       "disc10 : List of 5\n",
       " $ type  : chr \"polygonal\"\n",
       " $ xrange: num [1:2] -10 10\n",
       " $ yrange: num [1:2] -10 10\n",
       " $ bdry  :List of 1\n",
       " $ units :List of 3\n",
       "lamba :  num 1.59\n",
       "lambda :  num 1.59\n",
       "mapxy : function (a = NA)  \n",
       "n :  num 200\n",
       "n_points :  num 300\n",
       "p_cluster : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 332\n",
       " $ x         : num [1:332] 8.934 1.243 -0.349 0.351 1.133 ...\n",
       " $ y         : num [1:332] 0.402 -2.997 -0.612 -0.483 -2.52 ...\n",
       " $ markformat: chr \"none\"\n",
       "ppois : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 497\n",
       " $ x         : num [1:497] -3.153 8.513 -0.794 2.806 2.125 ...\n",
       " $ y         : num [1:497] 8.128 -0.465 6.899 0.279 -8.242 ...\n",
       " $ markformat: chr \"none\"\n",
       "ppxy : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 300\n",
       " $ x         : num [1:300] -4.497 6.119 -2.546 0.766 0.87 ...\n",
       " $ y         : num [1:300] 3.58 -3.56 -2.72 -1.45 -8.67 ...\n",
       " $ markformat: chr \"none\"\n",
       "qt : List of 11\n",
       " $ statistic  : Named num 12.1\n",
       " $ parameter  : Named num 24\n",
       " $ p.value    : num 0.0434\n",
       " $ method     : chr [1:2] \"Chi-squared test of CSR using quadrat counts\" \"Pearson X2 statistic\"\n",
       " $ data.name  : chr \"ppxy\"\n",
       " $ alternative: chr \"two.sided\"\n",
       " $ observed   : int [1:25] 1 11 23 11 1 14 16 10 17 12 ...\n",
       " $ expected   : num [1:25] 2.09 11.75 15.02 11.75 2.09 ...\n",
       " $ residuals  : num [1:25] -0.752 -0.219 2.059 -0.219 -0.752 ...\n",
       " $ CR         : num 1\n",
       " $ method.key : chr \"Chisq\"\n",
       "radius :  num 10\n",
       "raidus :  num 10\n",
       "rSquared :  num [1:300] 82.6 51.5 53.3 2.6 63.4 ...\n",
       "strauss : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 300\n",
       " $ x         : num [1:300] -9.32 -8.98 -9.9 -9.74 -9.03 ...\n",
       " $ y         : num [1:300] -3.513 -2.87 -0.864 -1.642 -0.203 ...\n",
       " $ markformat: chr \"none\"\n",
       "thomas : List of 5\n",
       " $ window    :List of 5\n",
       " $ n         : int 379\n",
       " $ x         : num [1:379] -3.262 -0.262 1.058 2.326 2.246 ...\n",
       " $ y         : num [1:379] 1.76 9.18 9.59 -9.71 9.16 ...\n",
       " $ markformat: chr \"none\"\n",
       "x :  num [1:300] -8.12 -7.12 -2.64 1.22 -3.34 ...\n",
       "xmax :  num 1\n",
       "xmin :  num 0\n",
       "y :  num [1:300] -4.08 -0.925 -6.807 1.054 7.225 ...\n",
       "ymax :  num 2\n",
       "ymin :  num 0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ls.str()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ppxy <- ppp(x = x, y = y, window = disc(radius))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Planar point pattern: 300 points\n",
       "window: polygonal boundary\n",
       "enclosing rectangle: [-10, 10] x [-10, 10] units"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ppxy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"Some expected counts are small; chi^2 approximation may be inaccurate\""
     ]
    }
   ],
   "source": [
    "# testing the point pattern\n",
    "qt <- quadrat.test(ppxy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "\tChi-squared test of CSR using quadrat counts\n",
      "\tPearson X2 statistic\n",
      "\n",
      "data:  ppxy\n",
      "X2 = 29.408, df = 24, p-value = 0.4103\n",
      "alternative hypothesis: two.sided\n",
      "\n",
      "Quadrats: 25 tiles (irregular windows)\n"
     ]
    }
   ],
   "source": [
    "print(qt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7EOyeILCSPFVqxDsvhCwkixFeuQLL6QMFJsxTokiy8kjBRbsQ7J4gsJI8VWrEOy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\njASAADASAALASAAIACMBIMB/5fD+hPayeBcAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"qt\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(qt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* p-value >= 0.05 = we cannot reject the null hypothesis that the points are spatially random.\n",
    "* p-value < 0.05 = we reject the null hypothesis that the points are spatially random."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Poisson point process creates events according to a Poisson distribution. The intensity parameter specifies the expected events per unit area. The total number of events generated is a single number from a poisson distribution. Multiple realizations of the same process can have different numbers of events. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# create a disc of radius = 10\n",
    "disc10 <- disc(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# compute the rate as count/area\n",
    "lambda <- 500/area(disc10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "1.59218876930438"
      ],
      "text/latex": [
       "1.59218876930438"
      ],
      "text/markdown": [
       "1.59218876930438"
      ],
      "text/plain": [
       "[1] 1.592189"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lambda"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# create ppp\n",
    "ppois <- rpoispp(lambda = lambda, win = disc10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Planar point pattern: 494 points\n",
       "window: polygonal boundary\n",
       "enclosing rectangle: [-10, 10] x [-10, 10] units"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ppois"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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wnHhZS7tPFIxACvnkRIhFYu7y77+Xy/ifwMk76OBFdVSyl1K1KO/GVJX9r9i/GT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\n",
      "text/plain": [
       "Plot with title \"ppois\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(ppois)\n",
    "# now we have a poisson process that generates completely spatially random points. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Other point patterns:\n",
    "- Clustered Processes = Points occur together more than under a uniform poisson process\n",
    "- Regular (inhibitory) Processes = Points are more spaced apart than under a uniform intensity poisson process. \n",
    "\n",
    "The quadrat.test() function can test against clustered or regular alternative hypotheses. \n",
    "\n",
    "### Thomas and Strauss Process:\n",
    "\n",
    "- Thomas Process = Clustered pattern where a number of parent points (uniformly distributed) create a number of child points in their neighborhood. The child points themselves form a pattern. This is an atractive point pattern. It makes sense for modelling things like trees, since new trees will grow near the original tree.\n",
    "\n",
    "- Strauss Process = Lowers the probability of finding another point nearby. No 2 points can be closer than a set threshold. This is a repulsive point pattern and it makes sense for modelling things like territorial animals since the other animal will avoid the territory of a given animal. Strauss process points are also called \"Regularly spaced\" points. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "disc10 <- disc(10) \n",
    "# create a disc with radius = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "set.seed(123)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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TzrTEmw9NKxgH2HlRRpIxZdcXJ8srPPvpaNJp0TTJ/zPuykd89zpy6XIqEQFak0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\nq6YZlgAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"thomas\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# generate clustered points from a thomas process\n",
    "thomas <- rThomas(kappa = 0.35, scale = 1, mu = 3, win = disc10)\n",
    "plot(thomas)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"Some expected counts are small; chi^2 approximation may be inaccurate\""
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "\tChi-squared test of CSR using quadrat counts\n",
       "\tPearson X2 statistic\n",
       "\n",
       "data:  thomas\n",
       "X2 = 53.387, df = 24, p-value = 0.0005142\n",
       "alternative hypothesis: clustered\n",
       "\n",
       "Quadrats: 25 tiles (irregular windows)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# run a quadrat test \n",
    "quadrat.test(thomas, alternative = \"clustered\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"Simulation will be performed in the containing rectangle and clipped to the original window.\""
     ]
    },
    {
     "data": {
      "image/png": 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rgZgtMikY7gReq+jhQxUSJ9QLUylEhH2sMbdlJj07jg+a5QrQwl0pEOkdpOattN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\nQgCQSEIAkEhCAJBIQgCQSEIA+H8XmLa0JIe68QAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"strauss\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "strauss <- rStrauss(beta = 2.9, gamma = 0.025, R = 0.5, W = disc10)\n",
    "plot(strauss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"Some expected counts are small; chi^2 approximation may be inaccurate\""
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "\tChi-squared test of CSR using quadrat counts\n",
       "\tPearson X2 statistic\n",
       "\n",
       "data:  strauss\n",
       "X2 = 11.653, df = 24, p-value = 0.01646\n",
       "alternative hypothesis: regular\n",
       "\n",
       "Quadrats: 25 tiles (irregular windows)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "quadrat.test(strauss, alternative = \"regular\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The P-Values in the quadrat.tests determined the properties of the pattern"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One problem with the quadrat.test function is that you would have to choose the number of subregions. Too few and you don't have many numbers for the statistical tests to have much power. Too many and your regions won't have many points in them - you'll lose statistical powers too. \n",
    "\n",
    "Another way to assess clustering and regularity is to consider each point and how it relates to the other points. One simple measure is the distribution of the distances from each point to its nearest neighbor. \n",
    "\n",
    "Nearest Neighbors distribution looks at each events in the pattern and finds the distance to the nearest event. Once we do this for every event, we plot a histogram to get an estimate of the probability density function (PDF) of the nearest neighbors distribution. The cumulative distribution function (CDF) is the probability of finding a nearest neighbor within a distance D. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Other point pattern distribution functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are other point patterns that have been developed. \n",
    "\n",
    "* The K function is defined as the expected number of points within a distance of a process of the process (scaled by intensity). Deviations can indicate clustering or point inhibition. K function calculations also need edge corrections. The default edge correction in spatstat is generally the best but slow. We can use \"border\" correction for speed. \n",
    "*  Uncertainties on the K function estimates can be assessed by randomly sampling points from a uniform Poisson process in the area and computing the K function of the simulated data. Repeat this 99 times and take the minimum and maximum value of K over each of the distance values. This gives an \"envelope\". If the K function from the data goes above the top of the envelope then we have evidence for clustering. If the K function goes below the envelope then there is evidence for inhibitory process causing points to be spaced out. Envelopes are useful where theoretical distribution is hard or impossible to calculate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.4.1"
  }
 },
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}