{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Markov chain modeling\n",
    "\n",
    "In this tutorial, we implement a Markov chain model that allows for different orders. We demonstrate the model on human navigation data derived from Wikispeedia."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Markov chain model\n",
    "\n",
    "First, we implement the model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# further implementation can be found at https://github.com/psinger/PathTools\n",
    "\n",
    "from __future__ import division\n",
    "\n",
    "import itertools\n",
    "from scipy.sparse import csr_matrix\n",
    "from scipy.special import gammaln\n",
    "from scipy.stats import chi2\n",
    "from collections import defaultdict\n",
    "from sklearn.preprocessing import normalize\n",
    "import numpy as np\n",
    "\n",
    "class MarkovChain():\n",
    "    \"\"\"\n",
    "    Markov Chain Model\n",
    "    \"\"\"\n",
    "\n",
    "    # dummy constant for our reset state\n",
    "    RESET_STATE = \"-1\"\n",
    "\n",
    "    def __init__(self, order=1, reset=True):\n",
    "        \"\"\"\n",
    "        Constructor for class MarkovChain\n",
    "        \n",
    "        Args:\n",
    "            order: Order of Markov Chain model\n",
    "            reset: Boolean for using additional reset states\n",
    "        \"\"\"\n",
    "        self.order = order\n",
    "        self.reset = reset\n",
    "\n",
    "    def fit(self,sequences):\n",
    "        \"\"\"\n",
    "        Function for fitting the Markov Chain model given data\n",
    "        \n",
    "        Args:\n",
    "            sequences: Data of sequences, list of lists\n",
    "        \"\"\"\n",
    "        \n",
    "        # first, we derive all basic states from given sequences\n",
    "        states = set(itertools.chain.from_iterable(sequences))\n",
    "        self.state_count = len(states)\n",
    "        if self.reset:\n",
    "            self.state_count += 1\n",
    "\n",
    "        # dictionary of dictionaries for counting transitions between states\n",
    "        transitions = defaultdict(lambda : defaultdict(float))\n",
    "\n",
    "        # remember all start and target states that we observe\n",
    "        # this is just necessary for memory saving reasons\n",
    "        start_states = set()\n",
    "        target_states = set()\n",
    "\n",
    "        # iterate through sequences\n",
    "        for seq in sequences:\n",
    "            i = 0\n",
    "            # Prepend and append correct amount of reset states if flag set\n",
    "            if self.reset:\n",
    "                seq = self.order*[self.RESET_STATE] + seq + [self.RESET_STATE]\n",
    "            # iterate through elements of a sequence\n",
    "            for j in xrange(self.order, len(seq)):\n",
    "                # start state based on order\n",
    "                elemA = tuple(seq[i:j])\n",
    "                # target state based on order\n",
    "                elemB = tuple(seq[j-self.order+1:j+1])\n",
    "                i += 1\n",
    "\n",
    "                # increase transition count\n",
    "                transitions[elemA][elemB] += 1\n",
    "\n",
    "                # remember start and target states\n",
    "                start_states.add(elemA)\n",
    "                target_states.add(elemB)\n",
    "\n",
    "        # build vocabularies for mapping states to indices\n",
    "        self.start_vocab = dict((v,k) for k,v in enumerate(start_states))\n",
    "        self.target_vocab = dict((v,k) for k,v in enumerate(target_states))\n",
    "\n",
    "        # transform transition dictionary of dictionaries to sparse csr_matrix\n",
    "        i_indices = []\n",
    "        j_indices = []\n",
    "        values = []\n",
    "\n",
    "        for s,ts in transitions.iteritems():\n",
    "            for t,c in ts.iteritems():\n",
    "                i_indices.append(self.start_vocab[s])\n",
    "                j_indices.append(self.target_vocab[t])\n",
    "                values.append(c)\n",
    "        shape = (len(start_states), len(target_states))\n",
    "        self.transitions = csr_matrix((values, (i_indices, j_indices)),\n",
    "                                 shape=shape)\n",
    "\n",
    "        #print \"fit done\"\n",
    "\n",
    "    def loglikelihood(self):\n",
    "        \"\"\"\n",
    "        Returns the loglikelihood given fitted model\n",
    "        \n",
    "        Returns:\n",
    "            loglikelihood\n",
    "        \"\"\"\n",
    "        \n",
    "        # get mle by row-normalizing transition matrix\n",
    "        self.mle = normalize(self.transitions,norm=\"l1\",axis=1)\n",
    "        # log mle\n",
    "        mle_log = self.mle.copy()\n",
    "        mle_log.data = np.log(mle_log.data)\n",
    "        return self.transitions.multiply(mle_log).sum()\n",
    "\n",
    "    @staticmethod\n",
    "    def lrt(ln, pn, lt, pt):\n",
    "        \"\"\"\n",
    "        Performs a likelihood ratio test given a null and alternative model\n",
    "        \n",
    "        Args:\n",
    "            ln: loglikelihood of null model\n",
    "            pn: nr. of parameters of null model\n",
    "            lt: loglikelihood of alternative model\n",
    "            pt: nr. of parameters of alternative model\n",
    "        Returns:\n",
    "            lr: log-likelihood ratio\n",
    "            p_val: p value\n",
    "        \"\"\"\n",
    "        \n",
    "        # likelihood ratio\n",
    "        lr = float(-2*(ln-lt))\n",
    "        # degrees of freedom\n",
    "        dof = float(pt - pn)\n",
    "        # chi2 test\n",
    "        p_val = chi2.sf(lr,dof)\n",
    "        return lr, p_val\n",
    "\n",
    "    def aic(self):\n",
    "        \"\"\"\n",
    "        Determines the AIC given fitted model\n",
    "        \n",
    "        Returns:\n",
    "            aic\n",
    "        \"\"\"\n",
    "        ll = self.loglikelihood()\n",
    "        # parameter count of model\n",
    "        para = self.state_count**self.order*(self.state_count-1)\n",
    "        return 2*para - 2*ll\n",
    "\n",
    "    def bic(self):\n",
    "        \"\"\"\n",
    "        Determines the BIC given fitted model\n",
    "        \n",
    "        Returns:\n",
    "            bic\n",
    "        \"\"\"\n",
    "        ll = self.loglikelihood()\n",
    "        # parameter count of model\n",
    "        para = self.state_count**self.order*(self.state_count-1)\n",
    "        return para*np.log(self.transitions.sum()) - 2*ll\n",
    "\n",
    "    def evidence(self, prior=1.):\n",
    "        \"\"\"\n",
    "        Determines Bayesian evidence given fitted model\n",
    "        \n",
    "        Args:\n",
    "            prior: Dirichlet prior parameter (symmetric)\n",
    "        Returns:\n",
    "            evidence\n",
    "        \"\"\"\n",
    "\n",
    "        i_dim, j_dim = self.transitions.shape\n",
    "\n",
    "        # elegantly calculate evidence\n",
    "        evidence = 0\n",
    "        evidence += i_dim * gammaln(self.state_count*prior)\n",
    "        evidence -= gammaln(self.transitions.sum(axis=1)+self.state_count*prior).sum()\n",
    "        evidence += gammaln(self.transitions.data+prior).sum()\n",
    "        evidence -= len(self.transitions.data) * gammaln(prior)\n",
    "\n",
    "        return evidence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Slides example\n",
    "\n",
    "For a first demonstration, we present the example from the slides consisting of blue and yellow states."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sequences = [[\"b\",\"b\",\"b\",\"y\",\"y\",\"b\",\"y\",\"b\",\"b\",\"b\",\"b\"]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting\n",
    "\n",
    "We fit both a first and second order model and print the transition matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First-order model\n",
      "[[ 1.  2.  0.]\n",
      " [ 2.  5.  1.]\n",
      " [ 0.  1.  0.]]\n",
      "---------\n",
      "Second-order model\n",
      "[[ 0.  0.  0.  0.  1.  1.]\n",
      " [ 1.  3.  0.  1.  0.  0.]\n",
      " [ 0.  1.  0.  0.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.]\n",
      " [ 1.  1.  0.  0.  0.  0.]\n",
      " [ 0.  0.  0.  0.  1.  0.]]\n"
     ]
    }
   ],
   "source": [
    "print \"First-order model\"\n",
    "mc1 = MarkovChain(reset=True, order=1)\n",
    "mc1.fit(sequences)\n",
    "print mc1.transitions.todense()\n",
    "\n",
    "print \"---------\"\n",
    "print \"Second-order model\"\n",
    "mc2 = MarkovChain(reset=True, order=2)\n",
    "mc2.fit(sequences)\n",
    "print mc2.transitions.todense()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Model selection\n",
    "\n",
    "Next, let us compare the two models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First-order model\n",
      "Log-likelihood: -9.11159091503\n",
      "Evidence: -13.4304361395\n",
      "AIC: 30.2231818301\n",
      "BIC: 33.1326217288\n",
      "---------\n",
      "Second-order model\n",
      "Log-likelihood: -7.52394141841\n",
      "Evidence: -14.3059048769\n",
      "AIC: 51.0478828368\n",
      "BIC: 59.776202533\n"
     ]
    }
   ],
   "source": [
    "\n",
    "print \"First-order model\"\n",
    "print \"Log-likelihood:\", mc1.loglikelihood()\n",
    "print \"Evidence:\", mc1.evidence()\n",
    "print \"AIC:\", mc1.aic()\n",
    "print \"BIC:\", mc1.bic()\n",
    "\n",
    "print \"---------\"\n",
    "print \"Second-order model\"\n",
    "\n",
    "print \"Log-likelihood:\", mc2.loglikelihood()\n",
    "print \"Evidence:\", mc2.evidence()\n",
    "print \"AIC:\", mc2.aic()\n",
    "print \"BIC:\", mc2.bic()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The simpler, first-order model is to be preferred."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Synthetic example\n",
    "\n",
    "Now, let us generate synthetic data with given memory. We always start with the state \"A\" and add additional \"B\" states based on a given memory. Then, we repeat this process 1000 times. For example, for a simple Markovian first-order process, we would produce \"ABABABAB...\". For a third-order process we would produce \"ABBBABBB...\". Then, the model comparison criteria correctly determine corresponding Markov chain model as the most appropriate ones."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e4c6533d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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BB5I/jzgi2xxmZpVwsaix9lmFn1lhZvXMB7hraMkS+NKXYOlS+PjHs05jZs3OB7hz6ppr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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e1ea45cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e1ea45d90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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8EDA+Op4OHBIdNwFW1VKf66WXXnrpVf9XQ3/Wp72zqI2Z9QEuA45w9y9Synch\nDFZXmNmewF7Au+5ebmbrzKwnMBcYAtwefWwqcCYwBzgFeLGm723oHrIiItIwFv2m3rAPm5UBTYFP\noqLZ7j7czAYA1wAbgQrganefFn3mIOABoDkwzd0vicqbAROBblF9g9z9/QYHJyIiWZNRshARkeKQ\n2BncZnavma00swW1XHN7NIlvnpl1bcz4UmKoNU4zO9LMylMmKP6msWOM4uhgZi+a2Vtm9i8zu7iG\n62Jr07rEmIT2NLNmZjbHzN6I4hxVw3Wx/v2sS5xJaM+UWLaJYphaw/kk/HuvMcaEteX7ZjY/+n//\nzxquqV97ZjrAnasX0AvoCiyo4fzxwLPR8SGELrAkxnkkMDUB7dkO6BodfwNYDHwvSW1axxiT0p4t\noj+bEOYY9UxSW9YjzkS0ZxTLL4E/VRdPgtqzthiT1JbvAjvXcr7e7ZnYOwt3/xuwtpZL+hIm9eHu\nc4AdzaxtLdfnRB3ihAQ8BuzuH7v7vOj4c2AhX85xqRRrm9YxRkhGe66PDpsRHkGv2p+blL+f6eKE\nBLSnmXUAfgTcU8MlsbdnHWKEBLRlxKi956je7ZnYZFEHWyfxRZZT/Q+WJPh+dKv3bAPmm2Sdme1B\nuBuaU+VUYtq0lhghAe0ZdUe8AXwMzPQvVyWolIi2rEOckID2BG4lPFlZ0yBqEtozXYyQjLaEEONM\nM5trZudWc77e7ZnPySJfvEaYoNgV+AMwJc5gzOwbhOVULol+e0+cNDEmoj3dvcLduxGWuDkkCb8E\nVKcOccbenmZ2ArAyuqs0kvPb+VZ1jDH2tkxxuLt3J9wJXZSyikaD5XOyWA7slvK+Q1SWKO7+eWVX\ngLv/BdgurmVMzGxbwg/hie7+52ouib1N08WYpPaMYviUMOm0T5VTsbdlqpriTEh7Hg6cZGbvApOA\nH5rZg1Wuibs908aYkLasjOWj6M/VwFNAzyqX1Ls9k54savstYyphUh9mdihQ7u4rGyuwKmqMM7Uf\n0MJkRHP3NY0VWBX3AW+7+201nE9Cm9YaYxLa08x2MbMdo+PtgWOBRVUui70t6xJnEtrT3a90947u\nvidhVYcX3X1Ilctibc+6xJiEtoy+u0V0d46Z7QD0Bt6sclm92zOjGdy5ZGYPAyVAGzP7ABhFmADo\n7v5Hd59mZj8ys3eA/wBDkxgncLKZXQhsAjYAp8YU5+HA6cC/oj5sB64EdichbVqXGElGe34LmGBm\n2xB+4XrAZIjoAAAAYElEQVQkarvzSUhb1jVOktGe1Upge35NQtuyLfCUmTnhZ/xD7j4j0/bUpDwR\nEUkr6d1QIiKSAEoWIiKSlpKFiIikpWQhIiJpKVmIiEhaShYiIpKWkoWIiKSlZCEiImn9H7HOATzK\niZKIAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e21e05b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "for o in xrange(1,5):\n",
    "    sample = [([\"A\"]+[\"B\"]*o)*1000]\n",
    "\n",
    "    loglikelihoods = []\n",
    "    aics = []\n",
    "    bics = []\n",
    "    evidences = []\n",
    "    \n",
    "    max_order = 5\n",
    "    fig = plt.figure()\n",
    "\n",
    "    for i in xrange(1,max_order+1):\n",
    "        mc = MarkovChain(reset=True, order=i)\n",
    "        mc.fit(sample)\n",
    "        loglikelihoods.append(mc.loglikelihood())\n",
    "        aics.append(mc.aic())\n",
    "        bics.append(mc.bic())\n",
    "        evidences.append(mc.evidence())\n",
    "    \n",
    "    # let us just plot the evidences here, the other criteria confirm these results\n",
    "    ax = fig.add_subplot(111)\n",
    "    ax.plot(xrange(1,max_order+1), evidences)\n",
    "    max_index = evidences.index(max(evidences))\n",
    "    ax.plot((xrange(1,max_order+1))[max_index], evidences[max_index], 'rD', clip_on = False)\n",
    "    ax.set_title(sample[0][:o+1])\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, the modeling selection techniques also indicate those orders as the most plausible after which the data has been generated."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wikispeedia human Web navigation example\n",
    "\n",
    "We now focus on an example of human Web navigation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data\n",
    "\n",
    "We load our data into a list of lists where each element corresponds to a sequence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import csv\n",
    "sequences = []\n",
    "\n",
    "# from https://snap.stanford.edu/data/wikispeedia.html\n",
    "for line in csv.reader((row for row in open(\"data/paths_finished.tsv\") if not row.startswith('#')), delimiter='\\t'):\n",
    "    if len(line) == 0:\n",
    "        continue\n",
    "    # for simplicity, let us remove back clicks\n",
    "    seq = line[3].split(\";\")\n",
    "    # for simplicity, let us remove back clicks\n",
    "    seq = [x for x in seq if x != \"<\"]\n",
    "    sequences.append(seq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[['14th_century', '15th_century', '16th_century', 'Pacific_Ocean', 'Atlantic_Ocean', 'Accra', 'Africa', 'Atlantic_slave_trade', 'African_slave_trade'], ['14th_century', 'Europe', 'Africa', 'Atlantic_slave_trade', 'African_slave_trade'], ['14th_century', 'Niger', 'Nigeria', 'British_Empire', 'Slavery', 'Africa', 'Atlantic_slave_trade', 'African_slave_trade'], ['14th_century', 'Renaissance', 'Ancient_Greece', 'Greece'], ['14th_century', 'Italy', 'Roman_Catholic_Church', 'HIV', 'Ronald_Reagan', 'President_of_the_United_States', 'John_F._Kennedy'], ['14th_century', 'Europe', 'North_America', 'United_States', 'President_of_the_United_States', 'John_F._Kennedy'], ['14th_century', 'China', 'Gunpowder', 'Fire'], ['14th_century', 'Time', 'Isaac_Newton', 'Light', 'Color', 'Rainbow'], ['14th_century', 'Time', 'Light', 'Rainbow'], ['14th_century', '15th_century', 'Plato', 'Nature', 'Ultraviolet', 'Color', 'Rainbow']]\n"
     ]
    }
   ],
   "source": [
    "print sequences[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "51318"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(sequences)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting\n",
    "\n",
    "We now fit the Markov chain model to the data. We vary the order from 1 to 5. Additionally, we determine all model comparison techniques and store them for each order at hand."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "loglikelihoods = []\n",
    "aics = []\n",
    "bics = []\n",
    "evidences = []\n",
    "lrts = []\n",
    "\n",
    "max_order = 5\n",
    "\n",
    "for i in xrange(1,max_order+1):\n",
    "    mc = MarkovChain(reset=True, order=i)\n",
    "    mc.fit(sequences)\n",
    "    loglikelihoods.append(mc.loglikelihood())\n",
    "    aics.append(mc.aic())\n",
    "    bics.append(mc.bic())\n",
    "    evidences.append(mc.evidence())\n",
    "\n",
    "for i, ln in enumerate(loglikelihoods):\n",
    "    for j, lt in enumerate(loglikelihoods[i+1:]):\n",
    "        lrt = mc.lrt(ln, mc.state_count**(i+1)*(mc.state_count-1), lt, mc.state_count**(i+1+j+1)*(mc.state_count-1))\n",
    "        lrts.append([str(i+1)+\" vs. \"+str(i+1+j+1),lrt[0],lrt[1]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Model Selection\n",
    "\n",
    "Next, we select the most appropriate model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Likelihood ratio test\n",
    "\n",
    "Perform likelihood ratio test between all fits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "orders |  likelihood ratio |  p-value\n",
      "1 vs. 2 | 862481.65175 | 1.0\n",
      "1 vs. 3 | 1405620.38073 | 1.0\n",
      "1 vs. 4 | 1538326.871 | 1.0\n",
      "1 vs. 5 | 1558210.23636 | 1.0\n",
      "2 vs. 3 | 543138.728976 | 1.0\n",
      "2 vs. 4 | 675845.219245 | 1.0\n",
      "2 vs. 5 | 695728.584614 | 1.0\n",
      "3 vs. 4 | 132706.490269 | 1.0\n",
      "3 vs. 5 | 152589.855638 | 1.0\n",
      "4 vs. 5 | 19883.3653686 | 1.0\n"
     ]
    }
   ],
   "source": [
    "print \"orders | \", \"likelihood ratio | \", \"p-value\"\n",
    "for x in lrts:\n",
    "    print \" | \".join([str(k) for k in x])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### AIC, BIC and Evidence\n",
    "\n",
    "Now, we determine the AIC, BIC and Evidence for all fits. Additionally, we plot the simple log-likelihood of the fits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Xw6675qpW55xzLnvvvw/z5sFxx6XdktzI65weSdOA5CiICLuaX2Vm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\nWV1ykmNmZmZ1yUmOmZmZ1aX/D5HEFwXDv+aaAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0e4d361050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2,2,figsize=(8,6))\n",
    "\n",
    "x = xrange(1,max_order+1)\n",
    "\n",
    "ax1.plot(x, loglikelihoods)\n",
    "max_index = loglikelihoods.index(max(loglikelihoods))\n",
    "ax1.plot(x[max_index], loglikelihoods[max_index], 'rD', clip_on = False)\n",
    "ax1.set_ylabel(\"Log-likelihood\")\n",
    "ax2.plot(x, aics)\n",
    "min_index = aics.index(min(aics))\n",
    "ax2.plot(x[min_index], aics[min_index], 'rD', clip_on = False)\n",
    "ax2.set_ylabel(\"AIC\")\n",
    "ax3.plot(x, bics)\n",
    "min_index = bics.index(min(bics))\n",
    "ax3.plot(x[min_index], bics[min_index], 'rD', clip_on = False)\n",
    "ax3.set_ylabel(\"BIC\")\n",
    "ax4.plot(x, evidences)\n",
    "max_index = evidences.index(max(evidences))\n",
    "ax4.plot(x[max_index], evidences[max_index], 'rD', clip_on = False)\n",
    "ax4.set_ylabel(\"Evidence\")\n",
    "\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Summary\n",
    "\n",
    "All modeling comparison techniques indicate that a first-order Markov chain model sufficiently models given data. While higher order models always produce better fits due to nested models, their increasing number of necessary parameters does not fully justify their increasing complexity. However, with less states and/or more data, this might be different."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}