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<!DOCTYPE html>
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<title>Hidden Markov Model | Model Estimation by Example</title>
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<ul class="summary">
<li class='before'><a href="./">Model Estimation by Example</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="introduction.html"><a href="introduction.html"><i class="fa fa-check"></i>Introduction</a></li>
<li class="part"><span><b>Models</b></span></li>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html"><i class="fa fa-check"></i>Linear Regression</a>
<ul>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html#data-setup"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html#functions"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html#estimation"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html#comparison"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html#python"><i class="fa fa-check"></i>Python</a></li>
<li class="chapter" data-level="" data-path="linear-regression.html"><a href="linear-regression.html#source"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html"><i class="fa fa-check"></i>Logistic Regression</a>
<ul>
<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html#data-setup-1"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html#functions-1"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html#estimation-1"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html#comparison-1"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html#python-1"><i class="fa fa-check"></i>Python</a></li>
<li class="chapter" data-level="" data-path="logistic-regression.html"><a href="logistic-regression.html#source-1"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html"><i class="fa fa-check"></i>Mixed Models</a>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#one-factor-mixed-model"><i class="fa fa-check"></i>One-factor Mixed Model</a>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#data-setup-2"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#function"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#estimation-2"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#comparison-2"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#source-2"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#two-factor-mixed-model"><i class="fa fa-check"></i>Two-factor Mixed Model</a>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#data-setup-3"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#function-1"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#estimation-3"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#comparison-3"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#source-3"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#mixed-model-via-ml"><i class="fa fa-check"></i>Mixed Model via ML</a>
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<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#introduction-1"><i class="fa fa-check"></i>Introduction</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#data-setup-4"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#function-2"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#estimation-4"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#comparison-4"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#link-with-penalized-regression"><i class="fa fa-check"></i>Link with penalized regression</a></li>
<li class="chapter" data-level="" data-path="mixed-models.html"><a href="mixed-models.html#source-4"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="probit.html"><a href="probit.html"><i class="fa fa-check"></i>Probit & Bivariate Probit</a>
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<li class="chapter" data-level="" data-path="probit.html"><a href="probit.html#standard-probit"><i class="fa fa-check"></i>Standard Probit</a>
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<li class="chapter" data-level="" data-path="probit.html"><a href="probit.html#examples"><i class="fa fa-check"></i>Examples</a></li>
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<li class="chapter" data-level="" data-path="probit.html"><a href="probit.html#bivariate-probit"><i class="fa fa-check"></i>Bivariate Probit</a>
<ul>
<li class="chapter" data-level="" data-path="probit.html"><a href="probit.html#example"><i class="fa fa-check"></i>Example</a></li>
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<li class="chapter" data-level="" data-path="probit.html"><a href="probit.html#source-5"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html"><i class="fa fa-check"></i>Heckman Selection</a>
<ul>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#data-setup-5"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#two-step-approach"><i class="fa fa-check"></i>Two step approach</a>
<ul>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#step-1-probit-model"><i class="fa fa-check"></i>Step 1: Probit Model</a></li>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#step-2-estimate-via-linear-regression"><i class="fa fa-check"></i>Step 2: Estimate via Linear Regression</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#maximum-likelihood"><i class="fa fa-check"></i>Maximum Likelihood</a>
<ul>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#comparison-5"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="heckman-selection.html"><a href="heckman-selection.html#source-6"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="marginal-structural.html"><a href="marginal-structural.html"><i class="fa fa-check"></i>Marginal Structural Model</a>
<ul>
<li class="chapter" data-level="" data-path="marginal-structural.html"><a href="marginal-structural.html#data-setup-6"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="marginal-structural.html"><a href="marginal-structural.html#function-4"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="marginal-structural.html"><a href="marginal-structural.html#estimation-5"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="marginal-structural.html"><a href="marginal-structural.html#comparison-6"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="marginal-structural.html"><a href="marginal-structural.html#source-7"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html"><i class="fa fa-check"></i>Tobit Regression</a>
<ul>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#censoring-with-an-upper-limit"><i class="fa fa-check"></i>Censoring with an Upper Limit</a>
<ul>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#data-setup-7"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#function-5"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#estimation-6"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#comparison-7"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#censoring-with-a-lower-limit"><i class="fa fa-check"></i>Censoring with a Lower Limit</a>
<ul>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#comparison-8"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="tobit.html"><a href="tobit.html#source-8"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html"><i class="fa fa-check"></i>Cox Survival</a>
<ul>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#standard-proportional-hazards"><i class="fa fa-check"></i>Standard Proportional Hazards</a>
<ul>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#data-setup-8"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#function-6"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#estimation-7"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#comparison-9"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#time-varying-coefficients"><i class="fa fa-check"></i>Time-varying coefficients</a>
<ul>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#data-setup-9"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#function-7"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#estimation-8"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#comparison-10"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#stratified-cox-model"><i class="fa fa-check"></i>Stratified Cox Model</a>
<ul>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#data-setup-10"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#function-8"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#estimation-9"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#comparison-11"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cox.html"><a href="cox.html#source-9"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html"><i class="fa fa-check"></i>Hurdle Model</a>
<ul>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#poisson"><i class="fa fa-check"></i>Poisson</a>
<ul>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#data-setup-11"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#function-9"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#estimation-10"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#comparison-12"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#negative-binomial"><i class="fa fa-check"></i>Negative Binomial</a>
<ul>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#function-10"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#estimation-11"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#comparison-13"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="hurdle.html"><a href="hurdle.html#source-10"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html"><i class="fa fa-check"></i>Zero-Inflated Model</a>
<ul>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#poisson-1"><i class="fa fa-check"></i>Poisson</a>
<ul>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#data-setup-12"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#function-11"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#estimation-12"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#comparison-14"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#negative-binomial-1"><i class="fa fa-check"></i>Negative Binomial</a>
<ul>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#function-12"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#estimation-13"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#comparison-15"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#supplemental-example"><i class="fa fa-check"></i>Supplemental Example</a></li>
<li class="chapter" data-level="" data-path="zi.html"><a href="zi.html#source-11"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="naive-bayes.html"><a href="naive-bayes.html"><i class="fa fa-check"></i>Naive Bayes</a>
<ul>
<li class="chapter" data-level="" data-path="naive-bayes.html"><a href="naive-bayes.html#initialization"><i class="fa fa-check"></i>Initialization</a></li>
<li class="chapter" data-level="" data-path="naive-bayes.html"><a href="naive-bayes.html#comparison-16"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="naive-bayes.html"><a href="naive-bayes.html#source-12"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html"><i class="fa fa-check"></i>Multinomial Regression</a>
<ul>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#standard-categorical-model"><i class="fa fa-check"></i>Standard (Categorical) Model</a>
<ul>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#data-setup-13"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#function-13"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#comparison-17"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#alternative-specific-and-constant-variables"><i class="fa fa-check"></i>Alternative specific and constant variables</a>
<ul>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#comparison-18"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="multinomial.html"><a href="multinomial.html#source-13"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="ordinal.html"><a href="ordinal.html"><i class="fa fa-check"></i>Ordinal Regression</a>
<ul>
<li class="chapter" data-level="" data-path="ordinal.html"><a href="ordinal.html#data"><i class="fa fa-check"></i>Data</a></li>
<li class="chapter" data-level="" data-path="ordinal.html"><a href="ordinal.html#function-14"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="ordinal.html"><a href="ordinal.html#estimation-14"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="ordinal.html"><a href="ordinal.html#comparison-19"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="ordinal.html"><a href="ordinal.html#source-14"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html"><i class="fa fa-check"></i>Markov Model</a>
<ul>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#data-setup-14"><i class="fa fa-check"></i>Data Setup</a>
<ul>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#data-functions"><i class="fa fa-check"></i>Data Functions</a></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#two-states-demo"><i class="fa fa-check"></i>Two states Demo</a></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#three-states-demo"><i class="fa fa-check"></i>Three states demo</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#function-15"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#estimation-15"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#comparison-20"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="markov.html"><a href="markov.html#source-15"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="hmm.html"><a href="hmm.html"><i class="fa fa-check"></i>Hidden Markov Model</a>
<ul>
<li class="chapter" data-level="" data-path="hmm.html"><a href="hmm.html#data-setup-15"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="hmm.html"><a href="hmm.html#function-16"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="hmm.html"><a href="hmm.html#estimation-16"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="hmm.html"><a href="hmm.html#supplemental-demo"><i class="fa fa-check"></i>Supplemental demo</a></li>
<li class="chapter" data-level="" data-path="hmm.html"><a href="hmm.html#source-16"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html"><i class="fa fa-check"></i>Quantile Regression</a>
<ul>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#data-setup-16"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#function-17"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#estimation-17"><i class="fa fa-check"></i>Estimation</a>
<ul>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#other-quantiles"><i class="fa fa-check"></i>Other quantiles</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#comparison-21"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#visualize"><i class="fa fa-check"></i>Visualize</a></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#python-2"><i class="fa fa-check"></i>Python</a></li>
<li class="chapter" data-level="" data-path="quantile-regression.html"><a href="quantile-regression.html#source-17"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cubic-spline.html"><a href="cubic-spline.html"><i class="fa fa-check"></i>Cubic Spline Model</a>
<ul>
<li class="chapter" data-level="" data-path="cubic-spline.html"><a href="cubic-spline.html#data-setup-17"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="cubic-spline.html"><a href="cubic-spline.html#functions-2"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="cubic-spline.html"><a href="cubic-spline.html#example-1"><i class="fa fa-check"></i>Example 1</a></li>
<li class="chapter" data-level="" data-path="cubic-spline.html"><a href="cubic-spline.html#example-2"><i class="fa fa-check"></i>Example 2</a></li>
<li class="chapter" data-level="" data-path="cubic-spline.html"><a href="cubic-spline.html#source-18"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html"><i class="fa fa-check"></i>Gaussian Processes</a>
<ul>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#noise-free-demonstration"><i class="fa fa-check"></i>Noise-Free Demonstration</a>
<ul>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#data-setup-18"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#functions-3"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#visualize-the-prior-distribution"><i class="fa fa-check"></i>Visualize the prior distribution</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#generate-the-posterior-predictive-distribution"><i class="fa fa-check"></i>Generate the posterior predictive distribution</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#visualize-the-posterior-predictive-distribution"><i class="fa fa-check"></i>Visualize the Posterior Predictive Distribution</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#noisy-demonstration"><i class="fa fa-check"></i>Noisy Demonstration</a>
<ul>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#data-setup-19"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#functions-4"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#visualize-the-prior-distribution-1"><i class="fa fa-check"></i>Visualize the prior distribution</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#generate-the-posterior-predictive-distribution-1"><i class="fa fa-check"></i>Generate the posterior predictive distribution</a></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#visualize-the-posterior-predictive-distribution-1"><i class="fa fa-check"></i>Visualize the Posterior Predictive Distribution</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="gaussian-process.html"><a href="gaussian-process.html#source-19"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html"><i class="fa fa-check"></i>Neural Network</a>
<ul>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#example-1-1"><i class="fa fa-check"></i>Example 1</a>
<ul>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#python-3"><i class="fa fa-check"></i>Python</a></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#r"><i class="fa fa-check"></i>R</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#example-2-1"><i class="fa fa-check"></i>Example 2</a>
<ul>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#python-4"><i class="fa fa-check"></i>Python</a></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#r-1"><i class="fa fa-check"></i>R</a></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#comparison-22"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#example-3"><i class="fa fa-check"></i>Example 3</a>
<ul>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#python-5"><i class="fa fa-check"></i>Python</a></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#r-2"><i class="fa fa-check"></i>R</a></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#comparison-23"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="neural-network.html"><a href="neural-network.html#source-20"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html"><i class="fa fa-check"></i>Extreme Learning Machine</a>
<ul>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html#data-setup-20"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html#function-18"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html#estimation-18"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html#comparison-24"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html#supplemental-example-1"><i class="fa fa-check"></i>Supplemental Example</a></li>
<li class="chapter" data-level="" data-path="elm.html"><a href="elm.html#source-21"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html"><i class="fa fa-check"></i>Reproducing Kernel Hilbert Space Regression</a>
<ul>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#data-setup-21"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#functions-5"><i class="fa fa-check"></i>Functions</a>
<ul>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#function-to-find-the-inverse-of-a-matrix"><i class="fa fa-check"></i>Function to find the inverse of a matrix</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#reproducing-kernel"><i class="fa fa-check"></i>Reproducing Kernel</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#gram-matrix"><i class="fa fa-check"></i>Gram matrix</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#ridge-regression"><i class="fa fa-check"></i>Ridge regression</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#estimation-19"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#comparison-25"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#example-cubic-spline"><i class="fa fa-check"></i>Example: Cubic Spline</a>
<ul>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#data-setup-22"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#functions-6"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#estimation-20"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#comparison-26"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#source-22"><i class="fa fa-check"></i>Source</a>
<ul>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#current-supplemental-code"><i class="fa fa-check"></i>Current Supplemental Code</a></li>
<li class="chapter" data-level="" data-path="rkhs.html"><a href="rkhs.html#original-supplemental-code"><i class="fa fa-check"></i>Original Supplemental Code</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html"><i class="fa fa-check"></i>Confirmatory Factor Analysis</a>
<ul>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#data-setup-23"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#functions-7"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#estimation-21"><i class="fa fa-check"></i>Estimation</a>
<ul>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#raw"><i class="fa fa-check"></i>Raw</a></li>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#standardized"><i class="fa fa-check"></i>Standardized</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#comparison-27"><i class="fa fa-check"></i>Comparison</a>
<ul>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#mplus"><i class="fa fa-check"></i>Mplus</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="cfa.html"><a href="cfa.html#source-23"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="part"><span><b>Bayesian</b></span></li>
<li class="chapter" data-level="" data-path="bayesian.html"><a href="bayesian.html"><i class="fa fa-check"></i>Introduction to Bayesian Methods</a></li>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html"><i class="fa fa-check"></i>Basics</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html#prior-likelihood-posterior-distributions"><i class="fa fa-check"></i>Prior, likelihood, & posterior distributions</a></li>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html#prior"><i class="fa fa-check"></i>Prior</a></li>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html#likelihood"><i class="fa fa-check"></i>Likelihood</a></li>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html#posterior"><i class="fa fa-check"></i>Posterior</a></li>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html#posterior-predictive"><i class="fa fa-check"></i>Posterior predictive</a></li>
<li class="chapter" data-level="" data-path="bayesian-basics.html"><a href="bayesian-basics.html#source-24"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html"><i class="fa fa-check"></i>Bayesian t-test</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html#data-setup-24"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html#model-code"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html#estimation-22"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html#comparison-28"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html#visualization"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="bayesian-t-test.html"><a href="bayesian-t-test.html#source-25"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html"><i class="fa fa-check"></i>Bayesian Linear Regression</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html#data-setup-25"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html#model-code-1"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html#estimation-23"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html#comparison-29"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html#visualize-1"><i class="fa fa-check"></i>Visualize</a></li>
<li class="chapter" data-level="" data-path="bayesian-linear-regression.html"><a href="bayesian-linear-regression.html#source-26"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html"><i class="fa fa-check"></i>Bayesian Beta Regression</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html#data-setup-26"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html#model-code-2"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html#estimation-24"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html#comparison-30"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html#visualization-1"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="bayesian-beta-regression.html"><a href="bayesian-beta-regression.html#source-27"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html"><i class="fa fa-check"></i>Bayesian Mixed Model</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html#data-setup-27"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html#model-code-3"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html#estimation-25"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html#comparison-31"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html#visualize-2"><i class="fa fa-check"></i>Visualize</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-model.html"><a href="bayesian-mixed-model.html#source-28"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html"><i class="fa fa-check"></i>Bayesian Multilevel Mediation</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html#data-setup-28"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html#model-code-4"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html#estimation-26"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html#comparison-32"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html#visualization-2"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="bayesian-mixed-mediation.html"><a href="bayesian-mixed-mediation.html#source-29"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html"><i class="fa fa-check"></i>Bayesian IRT</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#one-parameter-irt"><i class="fa fa-check"></i>One Parameter IRT</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#data-setup-29"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#model-code-5"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#estimation-27"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#comparison-33"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#two-parameter-irt"><i class="fa fa-check"></i>Two Parameter IRT</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#model-code-6"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#estimation-28"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#comparison-34"><i class="fa fa-check"></i>Comparison</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#three-parameter-irt"><i class="fa fa-check"></i>Three Parameter IRT</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#four-parameter-irt"><i class="fa fa-check"></i>Four Parameter IRT</a></li>
<li class="chapter" data-level="" data-path="bayesian-irt.html"><a href="bayesian-irt.html#source-30"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-cfa.html"><a href="bayesian-cfa.html"><i class="fa fa-check"></i>Bayesian CFA</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-cfa.html"><a href="bayesian-cfa.html#data-setup-30"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-cfa.html"><a href="bayesian-cfa.html#model-code-7"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-cfa.html"><a href="bayesian-cfa.html#estimation-29"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-cfa.html"><a href="bayesian-cfa.html#comparison-35"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-cfa.html"><a href="bayesian-cfa.html#source-31"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-non-parametric.html"><a href="bayesian-non-parametric.html"><i class="fa fa-check"></i>Bayesian Nonparametric Models</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-non-parametric.html"><a href="bayesian-non-parametric.html#chinese-restaurant-process"><i class="fa fa-check"></i>Chinese Restaurant Process</a></li>
<li class="chapter" data-level="" data-path="bayesian-non-parametric.html"><a href="bayesian-non-parametric.html#indian-buffet-process"><i class="fa fa-check"></i>Indian Buffet Process</a></li>
<li class="chapter" data-level="" data-path="bayesian-non-parametric.html"><a href="bayesian-non-parametric.html#source-32"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html"><i class="fa fa-check"></i>Bayesian Stochastic Volatility Model</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html#data-setup-31"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html#model-code-8"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html#estimation-30"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html#results"><i class="fa fa-check"></i>Results</a></li>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html#visualization-3"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="bayesian-stochastic-volatility.html"><a href="bayesian-stochastic-volatility.html#source-33"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html"><i class="fa fa-check"></i>Bayesian Multinomial Models</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html#data-setup-32"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html#model-code-9"><i class="fa fa-check"></i>Model Code</a></li>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html#estimation-31"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html#comparison-36"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html#adding-complexity"><i class="fa fa-check"></i>Adding Complexity</a></li>
<li class="chapter" data-level="" data-path="bayesian-multinomial-model.html"><a href="bayesian-multinomial-model.html#source-34"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html"><i class="fa fa-check"></i>Variational Bayes Regression</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#data-setup-33"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#function-19"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#estimation-32"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#comparison-37"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#visualization-4"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#supplemental-example-2"><i class="fa fa-check"></i>Supplemental Example</a></li>
<li class="chapter" data-level="" data-path="bayesian-variational.html"><a href="bayesian-variational.html#source-35"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="bayesian-topic-model.html"><a href="bayesian-topic-model.html"><i class="fa fa-check"></i>Topic Model</a>
<ul>
<li class="chapter" data-level="" data-path="bayesian-topic-model.html"><a href="bayesian-topic-model.html#data-setup-34"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="bayesian-topic-model.html"><a href="bayesian-topic-model.html#function-20"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="bayesian-topic-model.html"><a href="bayesian-topic-model.html#estimation-33"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="bayesian-topic-model.html"><a href="bayesian-topic-model.html#comparison-38"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="bayesian-topic-model.html"><a href="bayesian-topic-model.html#source-36"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="part"><span><b>Estimation</b></span></li>
<li class="chapter" data-level="" data-path="maximum-likelihood.html"><a href="maximum-likelihood.html"><i class="fa fa-check"></i>Maximum Likelihood</a>
<ul>
<li class="chapter" data-level="" data-path="maximum-likelihood.html"><a href="maximum-likelihood.html#example-4"><i class="fa fa-check"></i>Example</a></li>
<li class="chapter" data-level="" data-path="maximum-likelihood.html"><a href="maximum-likelihood.html#linear-model"><i class="fa fa-check"></i>Linear Model</a></li>
<li class="chapter" data-level="" data-path="maximum-likelihood.html"><a href="maximum-likelihood.html#source-37"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html"><i class="fa fa-check"></i>Penalized Maximum Likelihood</a>
<ul>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#introduction-2"><i class="fa fa-check"></i>Introduction</a>
<ul>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#data-setup-35"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#functions-8"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#estimation-34"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#comparison-39"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#source-38"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#l1-lasso-regularization"><i class="fa fa-check"></i>L1 (lasso) regularization</a>
<ul>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#data-setup-36"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#function-21"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#estimation-35"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#comparison-40"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#source-39"><i class="fa fa-check"></i>Source</a></li>
</ul></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#l2-ridge-regularization"><i class="fa fa-check"></i>L2 (ridge) regularization</a>
<ul>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#data-setup-37"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#function-22"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#estimation-36"><i class="fa fa-check"></i>Estimation</a></li>
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<li class="chapter" data-level="" data-path="penalized-maximum-likelihood.html"><a href="penalized-maximum-likelihood.html#source-40"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html"><i class="fa fa-check"></i>Newton and IRLS</a>
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<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html#data-setup-38"><i class="fa fa-check"></i>Data Setup</a></li>
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<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html#newtons-method"><i class="fa fa-check"></i>Newton’s Method</a></li>
<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html#irls"><i class="fa fa-check"></i>IRLS</a></li>
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<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html#estimation-37"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html#comparison-42"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="newton-irls.html"><a href="newton-irls.html#source-41"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html"><i class="fa fa-check"></i>Nelder-Mead</a>
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<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#functions-10"><i class="fa fa-check"></i>Functions</a>
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<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#first-version"><i class="fa fa-check"></i>First Version</a></li>
<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#example-5"><i class="fa fa-check"></i>Example</a></li>
<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#a-regression-model"><i class="fa fa-check"></i>A Regression Model</a></li>
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<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#second-version"><i class="fa fa-check"></i>Second Version</a>
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<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#example-6"><i class="fa fa-check"></i>Example</a></li>
<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#a-regression-model-1"><i class="fa fa-check"></i>A Regression Model</a></li>
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<li class="chapter" data-level="" data-path="nelder-mead.html"><a href="nelder-mead.html#source-42"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html"><i class="fa fa-check"></i>Expectation-Maximization</a>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#mixture-model"><i class="fa fa-check"></i>Mixture Model</a>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#data-setup-40"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="em.html"><a href="em.html#function-23"><i class="fa fa-check"></i>Function</a></li>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#visualization-5"><i class="fa fa-check"></i>Visualization</a></li>
<li class="chapter" data-level="" data-path="em.html"><a href="em.html#supplemental-example-3"><i class="fa fa-check"></i>Supplemental Example</a></li>
<li class="chapter" data-level="" data-path="em.html"><a href="em.html#source-43"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#multivariate-mixture-model"><i class="fa fa-check"></i>Multivariate Mixture Model</a>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#function-24"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="em.html"><a href="em.html#example-1-old-faithful"><i class="fa fa-check"></i>Example 1: Old Faithful</a></li>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#data-setup-43"><i class="fa fa-check"></i>Data Setup</a></li>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#probabilistic-pca"><i class="fa fa-check"></i>Probabilistic PCA</a>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#data-setup-45"><i class="fa fa-check"></i>Data Setup</a></li>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#state-space-model"><i class="fa fa-check"></i>State Space Model</a>
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<li class="chapter" data-level="" data-path="em.html"><a href="em.html#data-setup-47"><i class="fa fa-check"></i>Data Setup</a></li>
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<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#visualize-estimates"><i class="fa fa-check"></i>Visualize Estimates</a></li>
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<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#comparison-55"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#visualize-estimates-1"><i class="fa fa-check"></i>Visualize Estimates</a></li>
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<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#sgd-variants"><i class="fa fa-check"></i>SGD Variants</a>
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<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#data-setup-51"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#function-33"><i class="fa fa-check"></i>Function</a></li>
<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#estimation-50"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#comparison-56"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#visualize-estimates-2"><i class="fa fa-check"></i>Visualize Estimates</a></li>
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<li class="chapter" data-level="" data-path="stochastic-gradient-descent.html"><a href="stochastic-gradient-descent.html#source-50"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="metropolis-hastings.html"><a href="metropolis-hastings.html#data-setup-52"><i class="fa fa-check"></i>Data Setup</a></li>
<li class="chapter" data-level="" data-path="metropolis-hastings.html"><a href="metropolis-hastings.html#functions-11"><i class="fa fa-check"></i>Functions</a></li>
<li class="chapter" data-level="" data-path="metropolis-hastings.html"><a href="metropolis-hastings.html#estimation-51"><i class="fa fa-check"></i>Estimation</a></li>
<li class="chapter" data-level="" data-path="metropolis-hastings.html"><a href="metropolis-hastings.html#comparison-57"><i class="fa fa-check"></i>Comparison</a></li>
<li class="chapter" data-level="" data-path="metropolis-hastings.html"><a href="metropolis-hastings.html#source-51"><i class="fa fa-check"></i>Source</a></li>
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<li class="chapter" data-level="" data-path="hamiltonian-monte-carlo.html"><a href="hamiltonian-monte-carlo.html"><i class="fa fa-check"></i>Hamiltonian Monte Carlo</a>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Model Estimation by Example</a>
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<div id="hidden-markov-model" class="section level1">
<h1>Hidden Markov Model</h1>
<p>This function duplicates <a href="supplemental.html#python-hmm">hmm_viterbi.py</a>, which comes from the <a href="https://en.wikipedia.org/wiki/Viterbi_algorithm">Viterbi algorithm wikipedia page</a> (at least as it was when I stumbled across it, see it in the <a href="supplemental.html#python-hmm">supplemental section</a>). This first function is just to provide R code that is similar, in case anyone is interested in a more direct comparison, but the original used lists of tuples and thus was very inefficient R-wise, and provided output that wasn’t succinct. The second function takes a vectorized approach and returns a matrix in a much more straightforward fashion. Both will provide the same result as the Python code. See The <a href="markov.html#markov-model">Markov Model chapter</a> also.</p>
<div id="data-setup-15" class="section level2">
<h2>Data Setup</h2>
<div class="sourceCode" id="cb244"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb244-1"><a href="hmm.html#cb244-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
<span id="cb244-2"><a href="hmm.html#cb244-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb244-3"><a href="hmm.html#cb244-3" aria-hidden="true" tabindex="-1"></a>obs <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'normal'</span>, <span class="st">'cold'</span>, <span class="st">'dizzy'</span>) <span class="co"># observed state</span></span>
<span id="cb244-4"><a href="hmm.html#cb244-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb244-5"><a href="hmm.html#cb244-5" aria-hidden="true" tabindex="-1"></a>states <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Healthy'</span>, <span class="st">'Fever'</span>) <span class="co"># latent states</span></span>
<span id="cb244-6"><a href="hmm.html#cb244-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb244-7"><a href="hmm.html#cb244-7" aria-hidden="true" tabindex="-1"></a>start_p <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Healthy'</span> <span class="ot">=</span> <span class="fl">0.6</span>, <span class="st">'Fever'</span> <span class="ot">=</span> <span class="fl">0.4</span>) <span class="co"># starting probabilities</span></span>
<span id="cb244-8"><a href="hmm.html#cb244-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb244-9"><a href="hmm.html#cb244-9" aria-hidden="true" tabindex="-1"></a><span class="co"># transition matrix</span></span>
<span id="cb244-10"><a href="hmm.html#cb244-10" aria-hidden="true" tabindex="-1"></a>trans_p <span class="ot">=</span> <span class="fu">list</span>(</span>
<span id="cb244-11"><a href="hmm.html#cb244-11" aria-hidden="true" tabindex="-1"></a> <span class="st">'Healthy'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Healthy'</span> <span class="ot">=</span> <span class="fl">0.7</span>, <span class="st">'Fever'</span> <span class="ot">=</span> <span class="fl">0.3</span>), </span>
<span id="cb244-12"><a href="hmm.html#cb244-12" aria-hidden="true" tabindex="-1"></a> <span class="st">'Fever'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Healthy'</span> <span class="ot">=</span> <span class="fl">0.4</span>, <span class="st">'Fever'</span> <span class="ot">=</span> <span class="fl">0.6</span>)</span>
<span id="cb244-13"><a href="hmm.html#cb244-13" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb244-14"><a href="hmm.html#cb244-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb244-15"><a href="hmm.html#cb244-15" aria-hidden="true" tabindex="-1"></a><span class="co"># emission matrix</span></span>
<span id="cb244-16"><a href="hmm.html#cb244-16" aria-hidden="true" tabindex="-1"></a>emit_p <span class="ot">=</span> <span class="fu">list</span>(</span>
<span id="cb244-17"><a href="hmm.html#cb244-17" aria-hidden="true" tabindex="-1"></a> <span class="st">'Healthy'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'normal'</span> <span class="ot">=</span> <span class="fl">0.5</span>, <span class="st">'cold'</span> <span class="ot">=</span> <span class="fl">0.4</span>, <span class="st">'dizzy'</span> <span class="ot">=</span> <span class="fl">0.1</span>),</span>
<span id="cb244-18"><a href="hmm.html#cb244-18" aria-hidden="true" tabindex="-1"></a> <span class="st">'Fever'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'normal'</span> <span class="ot">=</span> <span class="fl">0.1</span>, <span class="st">'cold'</span> <span class="ot">=</span> <span class="fl">0.3</span>, <span class="st">'dizzy'</span> <span class="ot">=</span> <span class="fl">0.6</span>)</span>
<span id="cb244-19"><a href="hmm.html#cb244-19" aria-hidden="true" tabindex="-1"></a>)</span></code></pre></div>
</div>
<div id="function-16" class="section level2">
<h2>Function</h2>
<p>This first function takes a Python-esque approach in the manner of the original, allowing for easier comparison.</p>
<div class="sourceCode" id="cb245"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb245-1"><a href="hmm.html#cb245-1" aria-hidden="true" tabindex="-1"></a>viterbi <span class="ot"><-</span> <span class="cf">function</span>(obs, states, start_p, trans_p, emit_p) {</span>
<span id="cb245-2"><a href="hmm.html#cb245-2" aria-hidden="true" tabindex="-1"></a> V <span class="ot">=</span> <span class="fu">vector</span>(<span class="st">'list'</span>, <span class="fu">length</span>(obs))</span>
<span id="cb245-3"><a href="hmm.html#cb245-3" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-4"><a href="hmm.html#cb245-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (st <span class="cf">in</span> <span class="fu">seq_along</span>(states)) {</span>
<span id="cb245-5"><a href="hmm.html#cb245-5" aria-hidden="true" tabindex="-1"></a> V[[<span class="dv">1</span>]][[states[st]]] <span class="ot">=</span> <span class="fu">list</span>(<span class="st">"prob"</span> <span class="ot">=</span> start_p[st] <span class="sc">*</span> emit_p[[st]][obs[<span class="dv">1</span>]], </span>
<span id="cb245-6"><a href="hmm.html#cb245-6" aria-hidden="true" tabindex="-1"></a> <span class="st">"prev"</span> <span class="ot">=</span> <span class="cn">NULL</span>)</span>
<span id="cb245-7"><a href="hmm.html#cb245-7" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb245-8"><a href="hmm.html#cb245-8" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-9"><a href="hmm.html#cb245-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (t <span class="cf">in</span> <span class="dv">2</span><span class="sc">:</span><span class="fu">length</span>(obs)) {</span>
<span id="cb245-10"><a href="hmm.html#cb245-10" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-11"><a href="hmm.html#cb245-11" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (st <span class="cf">in</span> <span class="fu">seq_along</span>(states)) {</span>
<span id="cb245-12"><a href="hmm.html#cb245-12" aria-hidden="true" tabindex="-1"></a> max_tr_prob <span class="ot">=</span> <span class="fu">numeric</span>()</span>
<span id="cb245-13"><a href="hmm.html#cb245-13" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-14"><a href="hmm.html#cb245-14" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (prev_st <span class="cf">in</span> states) {</span>
<span id="cb245-15"><a href="hmm.html#cb245-15" aria-hidden="true" tabindex="-1"></a> max_tr_prob[prev_st] <span class="ot">=</span> V[[t<span class="dv">-1</span>]][[prev_st]][[<span class="st">"prob"</span>]] <span class="sc">*</span> trans_p[[prev_st]][[st]]</span>
<span id="cb245-16"><a href="hmm.html#cb245-16" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb245-17"><a href="hmm.html#cb245-17" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-18"><a href="hmm.html#cb245-18" aria-hidden="true" tabindex="-1"></a> max_tr_prob <span class="ot">=</span> <span class="fu">max</span>(max_tr_prob)</span>
<span id="cb245-19"><a href="hmm.html#cb245-19" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-20"><a href="hmm.html#cb245-20" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (prev_st <span class="cf">in</span> states) {</span>
<span id="cb245-21"><a href="hmm.html#cb245-21" aria-hidden="true" tabindex="-1"></a> flag <span class="ot">=</span> V[[t<span class="dv">-1</span>]][[prev_st]][[<span class="st">"prob"</span>]] <span class="sc">*</span> trans_p[[prev_st]][[st]] <span class="sc">==</span> max_tr_prob</span>
<span id="cb245-22"><a href="hmm.html#cb245-22" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (flag) {</span>
<span id="cb245-23"><a href="hmm.html#cb245-23" aria-hidden="true" tabindex="-1"></a> max_prob <span class="ot">=</span> max_tr_prob <span class="sc">*</span> emit_p[[st]][obs[t]]</span>
<span id="cb245-24"><a href="hmm.html#cb245-24" aria-hidden="true" tabindex="-1"></a> V[[t]][[states[st]]] <span class="ot">=</span> <span class="fu">list</span>(<span class="st">'prob'</span> <span class="ot">=</span> max_prob, <span class="st">'prev'</span> <span class="ot">=</span> prev_st)</span>
<span id="cb245-25"><a href="hmm.html#cb245-25" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb245-26"><a href="hmm.html#cb245-26" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb245-27"><a href="hmm.html#cb245-27" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-28"><a href="hmm.html#cb245-28" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb245-29"><a href="hmm.html#cb245-29" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-30"><a href="hmm.html#cb245-30" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb245-31"><a href="hmm.html#cb245-31" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-32"><a href="hmm.html#cb245-32" aria-hidden="true" tabindex="-1"></a> <span class="co"># I don't bother duplicating the text output code of the original</span></span>
<span id="cb245-33"><a href="hmm.html#cb245-33" aria-hidden="true" tabindex="-1"></a> df_out <span class="ot">=</span> <span class="fu">rbind</span>(</span>
<span id="cb245-34"><a href="hmm.html#cb245-34" aria-hidden="true" tabindex="-1"></a> <span class="at">Healthy =</span> <span class="fu">sapply</span>(V, <span class="cf">function</span>(x) x<span class="sc">$</span>Healthy<span class="sc">$</span>prob),</span>
<span id="cb245-35"><a href="hmm.html#cb245-35" aria-hidden="true" tabindex="-1"></a> <span class="at">Fever =</span> <span class="fu">sapply</span>(V, <span class="cf">function</span>(x) x<span class="sc">$</span>Fever<span class="sc">$</span>prob)</span>
<span id="cb245-36"><a href="hmm.html#cb245-36" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb245-37"><a href="hmm.html#cb245-37" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-38"><a href="hmm.html#cb245-38" aria-hidden="true" tabindex="-1"></a> <span class="fu">colnames</span>(df_out) <span class="ot">=</span> obs</span>
<span id="cb245-39"><a href="hmm.html#cb245-39" aria-hidden="true" tabindex="-1"></a> <span class="fu">print</span>(df_out)</span>
<span id="cb245-40"><a href="hmm.html#cb245-40" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-41"><a href="hmm.html#cb245-41" aria-hidden="true" tabindex="-1"></a> m <span class="ot">=</span> <span class="fu">paste0</span>(</span>
<span id="cb245-42"><a href="hmm.html#cb245-42" aria-hidden="true" tabindex="-1"></a> <span class="st">'The steps of states are: '</span>, </span>
<span id="cb245-43"><a href="hmm.html#cb245-43" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(<span class="fu">rownames</span>(df_out)[<span class="fu">apply</span>(df_out, <span class="dv">2</span>, which.max)], <span class="at">collapse =</span> <span class="st">' '</span>), </span>
<span id="cb245-44"><a href="hmm.html#cb245-44" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(<span class="st">'</span><span class="sc">\n</span><span class="st">Highest probability: '</span>, <span class="fu">max</span>(df_out[, <span class="fu">ncol</span>(df_out)]))</span>
<span id="cb245-45"><a href="hmm.html#cb245-45" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb245-46"><a href="hmm.html#cb245-46" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-47"><a href="hmm.html#cb245-47" aria-hidden="true" tabindex="-1"></a> <span class="fu">message</span>(m)</span>
<span id="cb245-48"><a href="hmm.html#cb245-48" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb245-49"><a href="hmm.html#cb245-49" aria-hidden="true" tabindex="-1"></a> V</span>
<span id="cb245-50"><a href="hmm.html#cb245-50" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
<p>This approach is much more R-like.</p>
<div class="sourceCode" id="cb246"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb246-1"><a href="hmm.html#cb246-1" aria-hidden="true" tabindex="-1"></a>viterbi_2 <span class="ot"><-</span> <span class="cf">function</span>(obs, states, start_p, trans_mat, emit_mat) {</span>
<span id="cb246-2"><a href="hmm.html#cb246-2" aria-hidden="true" tabindex="-1"></a> prob_mat <span class="ot">=</span> <span class="fu">matrix</span>(<span class="cn">NA</span>, <span class="at">nrow =</span> <span class="fu">length</span>(states), <span class="at">ncol =</span> <span class="fu">length</span>(obs))</span>
<span id="cb246-3"><a href="hmm.html#cb246-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">colnames</span>(prob_mat) <span class="ot">=</span> obs</span>
<span id="cb246-4"><a href="hmm.html#cb246-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">rownames</span>(prob_mat) <span class="ot">=</span> states</span>
<span id="cb246-5"><a href="hmm.html#cb246-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb246-6"><a href="hmm.html#cb246-6" aria-hidden="true" tabindex="-1"></a> prob_mat[,<span class="dv">1</span>] <span class="ot">=</span> start_p <span class="sc">*</span> emit_mat[,<span class="dv">1</span>]</span>
<span id="cb246-7"><a href="hmm.html#cb246-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb246-8"><a href="hmm.html#cb246-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (t <span class="cf">in</span> <span class="dv">2</span><span class="sc">:</span><span class="fu">length</span>(obs)) {</span>
<span id="cb246-9"><a href="hmm.html#cb246-9" aria-hidden="true" tabindex="-1"></a> prob_tran <span class="ot">=</span> prob_mat[,t<span class="dv">-1</span>] <span class="sc">*</span> trans_mat</span>
<span id="cb246-10"><a href="hmm.html#cb246-10" aria-hidden="true" tabindex="-1"></a> max_tr_prob <span class="ot">=</span> <span class="fu">apply</span>(prob_tran, <span class="dv">2</span>, max)</span>
<span id="cb246-11"><a href="hmm.html#cb246-11" aria-hidden="true" tabindex="-1"></a> prob_mat[,t] <span class="ot">=</span> max_tr_prob <span class="sc">*</span> emit_mat[, obs[t]]</span>
<span id="cb246-12"><a href="hmm.html#cb246-12" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb246-13"><a href="hmm.html#cb246-13" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb246-14"><a href="hmm.html#cb246-14" aria-hidden="true" tabindex="-1"></a> <span class="fu">print</span>(prob_mat)</span>
<span id="cb246-15"><a href="hmm.html#cb246-15" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb246-16"><a href="hmm.html#cb246-16" aria-hidden="true" tabindex="-1"></a> m <span class="ot">=</span> <span class="fu">paste0</span>(</span>
<span id="cb246-17"><a href="hmm.html#cb246-17" aria-hidden="true" tabindex="-1"></a> <span class="st">'The steps of states are: '</span>, </span>
<span id="cb246-18"><a href="hmm.html#cb246-18" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(states[<span class="fu">apply</span>(prob_mat, <span class="dv">2</span>, which.max)], <span class="at">collapse =</span> <span class="st">' '</span>), </span>
<span id="cb246-19"><a href="hmm.html#cb246-19" aria-hidden="true" tabindex="-1"></a> <span class="fu">paste</span>(<span class="st">'</span><span class="sc">\n</span><span class="st">Highest probability: '</span>, <span class="fu">max</span>(prob_mat[, <span class="fu">ncol</span>(prob_mat)]))</span>
<span id="cb246-20"><a href="hmm.html#cb246-20" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb246-21"><a href="hmm.html#cb246-21" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb246-22"><a href="hmm.html#cb246-22" aria-hidden="true" tabindex="-1"></a> <span class="fu">message</span>(m)</span>
<span id="cb246-23"><a href="hmm.html#cb246-23" aria-hidden="true" tabindex="-1"></a>}</span></code></pre></div>
</div>
<div id="estimation-16" class="section level2">
<h2>Estimation</h2>
<p>First we demo the initial function.</p>
<div class="sourceCode" id="cb247"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb247-1"><a href="hmm.html#cb247-1" aria-hidden="true" tabindex="-1"></a>test <span class="ot">=</span> <span class="fu">viterbi</span>(</span>
<span id="cb247-2"><a href="hmm.html#cb247-2" aria-hidden="true" tabindex="-1"></a> obs,</span>
<span id="cb247-3"><a href="hmm.html#cb247-3" aria-hidden="true" tabindex="-1"></a> states,</span>
<span id="cb247-4"><a href="hmm.html#cb247-4" aria-hidden="true" tabindex="-1"></a> start_p,</span>
<span id="cb247-5"><a href="hmm.html#cb247-5" aria-hidden="true" tabindex="-1"></a> trans_p,</span>
<span id="cb247-6"><a href="hmm.html#cb247-6" aria-hidden="true" tabindex="-1"></a> emit_p</span>
<span id="cb247-7"><a href="hmm.html#cb247-7" aria-hidden="true" tabindex="-1"></a>)</span></code></pre></div>
<pre><code> normal cold dizzy
Healthy 0.30 0.084 0.00588
Fever 0.04 0.027 0.01512</code></pre>
<div class="sourceCode" id="cb249"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb249-1"><a href="hmm.html#cb249-1" aria-hidden="true" tabindex="-1"></a><span class="co"># test</span></span>
<span id="cb249-2"><a href="hmm.html#cb249-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb249-3"><a href="hmm.html#cb249-3" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">123</span>)</span>
<span id="cb249-4"><a href="hmm.html#cb249-4" aria-hidden="true" tabindex="-1"></a>obs <span class="ot">=</span> <span class="fu">sample</span>(obs, <span class="dv">6</span>, <span class="at">replace =</span> <span class="cn">TRUE</span>)</span>
<span id="cb249-5"><a href="hmm.html#cb249-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb249-6"><a href="hmm.html#cb249-6" aria-hidden="true" tabindex="-1"></a>test <span class="ot">=</span> <span class="fu">viterbi</span>(</span>
<span id="cb249-7"><a href="hmm.html#cb249-7" aria-hidden="true" tabindex="-1"></a> obs,</span>
<span id="cb249-8"><a href="hmm.html#cb249-8" aria-hidden="true" tabindex="-1"></a> states,</span>
<span id="cb249-9"><a href="hmm.html#cb249-9" aria-hidden="true" tabindex="-1"></a> start_p,</span>
<span id="cb249-10"><a href="hmm.html#cb249-10" aria-hidden="true" tabindex="-1"></a> trans_p,</span>
<span id="cb249-11"><a href="hmm.html#cb249-11" aria-hidden="true" tabindex="-1"></a> emit_p</span>
<span id="cb249-12"><a href="hmm.html#cb249-12" aria-hidden="true" tabindex="-1"></a>)</span></code></pre></div>
<pre><code> dizzy dizzy dizzy cold dizzy cold
Healthy 0.06 0.0096 0.003456 0.00497664 0.0003483648 0.0003224863
Fever 0.24 0.0864 0.031104 0.00559872 0.0020155392 0.0003627971</code></pre>
<div class="sourceCode" id="cb251"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb251-1"><a href="hmm.html#cb251-1" aria-hidden="true" tabindex="-1"></a><span class="co"># test</span></span></code></pre></div>
<p>Now the vectorized approach.</p>
<div class="sourceCode" id="cb252"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb252-1"><a href="hmm.html#cb252-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">123</span>)</span>
<span id="cb252-2"><a href="hmm.html#cb252-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb252-3"><a href="hmm.html#cb252-3" aria-hidden="true" tabindex="-1"></a>obs <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'normal'</span>, <span class="st">'cold'</span>, <span class="st">'dizzy'</span>)</span>
<span id="cb252-4"><a href="hmm.html#cb252-4" aria-hidden="true" tabindex="-1"></a>obs <span class="ot">=</span> <span class="fu">sample</span>(obs, <span class="dv">6</span>, <span class="at">replace =</span> T)</span>
<span id="cb252-5"><a href="hmm.html#cb252-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb252-6"><a href="hmm.html#cb252-6" aria-hidden="true" tabindex="-1"></a><span class="co"># need matrices now</span></span>
<span id="cb252-7"><a href="hmm.html#cb252-7" aria-hidden="true" tabindex="-1"></a>emit_mat <span class="ot">=</span> <span class="fu">do.call</span>(rbind, emit_p)</span>
<span id="cb252-8"><a href="hmm.html#cb252-8" aria-hidden="true" tabindex="-1"></a>trans_mat <span class="ot">=</span> <span class="fu">do.call</span>(rbind, trans_p)</span>
<span id="cb252-9"><a href="hmm.html#cb252-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb252-10"><a href="hmm.html#cb252-10" aria-hidden="true" tabindex="-1"></a><span class="fu">viterbi_2</span>(</span>
<span id="cb252-11"><a href="hmm.html#cb252-11" aria-hidden="true" tabindex="-1"></a> obs, </span>
<span id="cb252-12"><a href="hmm.html#cb252-12" aria-hidden="true" tabindex="-1"></a> states, </span>
<span id="cb252-13"><a href="hmm.html#cb252-13" aria-hidden="true" tabindex="-1"></a> start_p, </span>
<span id="cb252-14"><a href="hmm.html#cb252-14" aria-hidden="true" tabindex="-1"></a> trans_mat, </span>
<span id="cb252-15"><a href="hmm.html#cb252-15" aria-hidden="true" tabindex="-1"></a> emit_mat</span>
<span id="cb252-16"><a href="hmm.html#cb252-16" aria-hidden="true" tabindex="-1"></a>)</span></code></pre></div>
<pre><code> dizzy dizzy dizzy cold dizzy cold
Healthy 0.30 0.021 0.00216 0.0031104 0.000217728 0.0002015539
Fever 0.04 0.054 0.01944 0.0034992 0.001259712 0.0002267482</code></pre>
</div>
<div id="supplemental-demo" class="section level2">
<h2>Supplemental demo</h2>
<p>This example comes from the <a href="https://en.wikipedia.org/wiki/Hidden_Markov_model">hidden markov model wikipedia page</a>.</p>
<div class="sourceCode" id="cb254"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb254-1"><a href="hmm.html#cb254-1" aria-hidden="true" tabindex="-1"></a>states <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Rainy'</span>, <span class="st">'Sunny'</span>)</span>
<span id="cb254-2"><a href="hmm.html#cb254-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb254-3"><a href="hmm.html#cb254-3" aria-hidden="true" tabindex="-1"></a>observations <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'walk'</span>, <span class="st">'shop'</span>, <span class="st">'clean'</span>)</span>
<span id="cb254-4"><a href="hmm.html#cb254-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb254-5"><a href="hmm.html#cb254-5" aria-hidden="true" tabindex="-1"></a>start_probability <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Rainy'</span> <span class="ot">=</span> <span class="fl">0.6</span>, <span class="st">'Sunny'</span> <span class="ot">=</span> <span class="fl">0.4</span>)</span>
<span id="cb254-6"><a href="hmm.html#cb254-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb254-7"><a href="hmm.html#cb254-7" aria-hidden="true" tabindex="-1"></a>transition_probability <span class="ot">=</span> <span class="fu">rbind</span>(</span>
<span id="cb254-8"><a href="hmm.html#cb254-8" aria-hidden="true" tabindex="-1"></a> <span class="st">'Rainy'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Rainy'</span> <span class="ot">=</span> <span class="fl">0.7</span>, <span class="st">'Sunny'</span> <span class="ot">=</span> <span class="fl">0.3</span>),</span>
<span id="cb254-9"><a href="hmm.html#cb254-9" aria-hidden="true" tabindex="-1"></a> <span class="st">'Sunny'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Rainy'</span> <span class="ot">=</span> <span class="fl">0.4</span>, <span class="st">'Sunny'</span> <span class="ot">=</span> <span class="fl">0.6</span>)</span>
<span id="cb254-10"><a href="hmm.html#cb254-10" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb254-11"><a href="hmm.html#cb254-11" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb254-12"><a href="hmm.html#cb254-12" aria-hidden="true" tabindex="-1"></a>emission_probability <span class="ot">=</span> <span class="fu">rbind</span>(</span>
<span id="cb254-13"><a href="hmm.html#cb254-13" aria-hidden="true" tabindex="-1"></a> <span class="st">'Rainy'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'walk'</span> <span class="ot">=</span> <span class="fl">0.1</span>, <span class="st">'shop'</span> <span class="ot">=</span> <span class="fl">0.4</span>, <span class="st">'clean'</span> <span class="ot">=</span> <span class="fl">0.5</span>),</span>
<span id="cb254-14"><a href="hmm.html#cb254-14" aria-hidden="true" tabindex="-1"></a> <span class="st">'Sunny'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'walk'</span> <span class="ot">=</span> <span class="fl">0.6</span>, <span class="st">'shop'</span> <span class="ot">=</span> <span class="fl">0.3</span>, <span class="st">'clean'</span> <span class="ot">=</span> <span class="fl">0.1</span>)</span>
<span id="cb254-15"><a href="hmm.html#cb254-15" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb254-16"><a href="hmm.html#cb254-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb254-17"><a href="hmm.html#cb254-17" aria-hidden="true" tabindex="-1"></a><span class="fu">viterbi_2</span>(</span>
<span id="cb254-18"><a href="hmm.html#cb254-18" aria-hidden="true" tabindex="-1"></a> observations,</span>
<span id="cb254-19"><a href="hmm.html#cb254-19" aria-hidden="true" tabindex="-1"></a> states,</span>
<span id="cb254-20"><a href="hmm.html#cb254-20" aria-hidden="true" tabindex="-1"></a> start_probability,</span>
<span id="cb254-21"><a href="hmm.html#cb254-21" aria-hidden="true" tabindex="-1"></a> transition_probability,</span>
<span id="cb254-22"><a href="hmm.html#cb254-22" aria-hidden="true" tabindex="-1"></a> emission_probability</span>
<span id="cb254-23"><a href="hmm.html#cb254-23" aria-hidden="true" tabindex="-1"></a>)</span></code></pre></div>
<pre><code> walk shop clean
Rainy 0.06 0.0384 0.013440
Sunny 0.24 0.0432 0.002592</code></pre>
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