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<h1 class="title toc-ignore">Input-Output Hidden Markov Model applied to financial time series</h1>
<h4 class="author"><em>Luis Damiano, Brian Peterson, Michael Weylandt</em></h4>
<h4 class="date"><em>2017-07-02</em></h4>
<div id="TOC">
<ul>
<li><a href="#the-input-output-hidden-markov-model"><span class="toc-section-number">1</span> The Input-Output Hidden Markov Model</a><ul>
<li><a href="#definitions"><span class="toc-section-number">1.1</span> Definitions</a></li>
<li><a href="#inference"><span class="toc-section-number">1.2</span> Inference</a><ul>
<li><a href="#filtering"><span class="toc-section-number">1.2.1</span> Filtering</a></li>
<li><a href="#smoothing"><span class="toc-section-number">1.2.2</span> Smoothing</a></li>
<li><a href="#most-likely-hidden-path"><span class="toc-section-number">1.2.3</span> Most likely hidden path</a></li>
</ul></li>
<li><a href="#parameter-estimation"><span class="toc-section-number">1.3</span> Parameter estimation</a></li>
</ul></li>
<li><a href="#learning-by-simulation"><span class="toc-section-number">2</span> Learning by simulation</a><ul>
<li><a href="#auxiliary-files"><span class="toc-section-number">2.1</span> Auxiliary files</a><ul>
<li><a href="#math-functions"><span class="toc-section-number">2.1.1</span> Math functions</a></li>
<li><a href="#plot-functions"><span class="toc-section-number">2.1.2</span> Plot functions</a></li>
<li><a href="#simulation-functions"><span class="toc-section-number">2.1.3</span> Simulation functions</a></li>
</ul></li>
<li><a href="#generative-model"><span class="toc-section-number">2.2</span> Generative model</a><ul>
<li><a href="#data-simulation"><span class="toc-section-number">2.2.1</span> Data simulation</a></li>
<li><a href="#estimation"><span class="toc-section-number">2.2.2</span> Estimation</a></li>
<li><a href="#convergence"><span class="toc-section-number">2.2.3</span> Convergence</a></li>
<li><a href="#state-probability"><span class="toc-section-number">2.2.4</span> State probability</a></li>
<li><a href="#fitted-output"><span class="toc-section-number">2.2.5</span> Fitted output</a></li>
</ul></li>
</ul></li>
<li><a href="#stock-market-forecasting-using-hidden-markov-model"><span class="toc-section-number">3</span> Stock Market Forecasting Using Hidden Markov Model</a><ul>
<li><a href="#preamble"><span class="toc-section-number">3.1</span> Preamble</a></li>
<li><a href="#model"><span class="toc-section-number">3.2</span> Model</a></li>
<li><a href="#the-sampler"><span class="toc-section-number">3.3</span> The sampler</a></li>
<li><a href="#dataset-data-transformation"><span class="toc-section-number">3.4</span> Dataset & data transformation</a></li>
<li><a href="#methodology"><span class="toc-section-number">3.5</span> Methodology</a></li>
<li><a href="#southwest-airlines-luv"><span class="toc-section-number">3.6</span> Southwest Airlines (LUV)</a><ul>
<li><a href="#data-exploration"><span class="toc-section-number">3.6.1</span> Data exploration</a></li>
<li><a href="#estimation-1"><span class="toc-section-number">3.6.2</span> Estimation</a></li>
<li><a href="#convergence-1"><span class="toc-section-number">3.6.3</span> Convergence</a></li>
<li><a href="#state-probability-1"><span class="toc-section-number">3.6.4</span> State probability</a></li>
<li><a href="#fitted-output-1"><span class="toc-section-number">3.6.5</span> Fitted output</a></li>
<li><a href="#in-sample-summary"><span class="toc-section-number">3.6.6</span> In-sample summary</a></li>
<li><a href="#forecast"><span class="toc-section-number">3.6.7</span> Forecast</a></li>
</ul></li>
<li><a href="#ryanair-holdings-plc-rya.l"><span class="toc-section-number">3.7</span> Ryanair Holdings Plc (RYA.L)</a><ul>
<li><a href="#in-sample-analysis"><span class="toc-section-number">3.7.1</span> In-sample analysis</a></li>
<li><a href="#out-of-sample-analysis"><span class="toc-section-number">3.7.2</span> Out-of-sample analysis</a></li>
</ul></li>
<li><a href="#further-research"><span class="toc-section-number">3.8</span> Discussion</a><ul>
<li><a href="#the-statistical-model"><span class="toc-section-number">3.8.1</span> The statistical model</a></li>
<li><a href="#the-financial-application"><span class="toc-section-number">3.8.2</span> The financial application</a></li>
</ul></li>
</ul></li>
<li><a href="#original-computing-environment"><span class="toc-section-number">4</span> Original Computing Environment</a></li>
<li><a href="#references">References</a></li>
</ul>
</div>
<style>img{border: 0px !important;}</style>
<p>This work aims at replicating the Input-Output Hidden Markov Model (IOHMM) originally proposed by <span class="citation">Hassan and Nath (2005)</span> to forecast stock prices. The main goal is to produce public programming code in <a href="http://mc-stan.org/">Stan</a> <span class="citation">(Carpenter et al. 2016)</span> for a fully Bayesian estimation of the model parameters and inference on hidden quantities, namely filtered state belief, smoothed state belief, jointly most probable state path and fitted output. The model is introduced only briefly, a more detailed mathematical treatment can be found in our <a href="https://github.com/luisdamiano/gsoc17-hhmm/blob/master/litreview/main.pdf">literature review</a>.</p>
<p>The authors acknowledge Google for financial support via the Google Summer of Code 2017 program.</p>
<hr />
<div id="the-input-output-hidden-markov-model" class="section level1">
<h1><span class="header-section-number">1</span> The Input-Output Hidden Markov Model</h1>
<p>The IOHMM is an architecture proposed by <span class="citation">Bengio and Frasconi (1995)</span> to map input sequences, sometimes called the control signal, to output sequences. It is a probabilistic framework that can deal with general sequence processing tasks such as production, classification, or prediction. The main difference with Hidden Markov Models (HMM), which are part of the unsupervised learning paradigm, is the capability to learn the output sequence itself instead of the distribution of the output sequence.</p>
<div id="definitions" class="section level2">
<h2><span class="header-section-number">1.1</span> Definitions</h2>
<p>As with HMM, IOHMM involves two interconnected models,</p>
<span class="math display">\[\begin{align*}
z_{t} &= f(z_{t-1}, \mat{u}_{t}) \\
\mat{x}_{t} &= g(z_{t }, \mat{u}_{t}).
\end{align*}\]</span>
<p>The first line corresponds to the state model, which consists of discrete-time, discrete-state hidden states <span class="math inline">\(z_t \in \{1, \dots, K\}\)</span> whose transition depends on the previous hidden state <span class="math inline">\(z_{t-1}\)</span> and the input vector <span class="math inline">\(\mat{u}_{t} \in \RR^M\)</span>. Additionally, the observation model is governed by <span class="math inline">\(g(z_{t}, \mat{u}_{t})\)</span>, where <span class="math inline">\(\mat{x}_t \in \RR^R\)</span> is the vector of observations, emissions or output. The corresponding joint distribution is</p>
<p><span class="math display">\[
p(\mat{z}_{1:T}, \mat{x}_{1:T} | \mat{u}_{t}).
\]</span></p>
<p>In the proposed parametrization with continuous inputs and outputs, the state model involves a multinomial regression whose parameters depend on the previous state taking the value <span class="math inline">\(i\)</span>,</p>
<p><span class="math display">\[
p(z_t | \mat{x}_{t}, \mat{u}_{t}, z_{t-1} = i) = \text{softmax}^{-1}(\mat{w}_i \mat{u}_{t}),
\]</span></p>
<p>and the observation model is built upon the Gaussian density with parameters depending on the current state taking the value <span class="math inline">\(j\)</span>,</p>
<p><span class="math display">\[
p(\mat{x}_t | \mat{u}_{t}, z_{t} = j) = \mathcal{N}(\mat{x}_t | \mat{\mu}_j, \mat{\Sigma}_j).
\]</span></p>
<p>IOHMM adapts the logic of HMM to allow for input and output vectors, retaining its fully probabilistic quality. Hidden states are assumed to follow a multinomial distribution that depends on the input sequence. The transition probabilities <span class="math inline">\(\Psi_t(i, j) = p(z_t = j | z_{t-1} = i, \mat{u}_{t})\)</span>, which govern the state dynamics, are driven by the control signal as well.</p>
<p>As for the output sequence, the local evidence at time <span class="math inline">\(t\)</span> now becomes <span class="math inline">\(\psi_t(j) = p(\mat{x}_t | z_t = j, \mat{\eta}_t)\)</span>, where <span class="math inline">\(\mat{\eta}_t = \ev{\mat{x}_t | z_t, \mat{u}_t}\)</span> can be interpreted as the expected location parameter for the probability distribution of the emission <span class="math inline">\(\mat{x}_{t}\)</span> conditional on the input vector <span class="math inline">\(\mat{u}_t\)</span> and the hidden state <span class="math inline">\(z_t\)</span>.</p>
</div>
<div id="inference" class="section level2">
<h2><span class="header-section-number">1.2</span> Inference</h2>
<p>There are several quantities of interest to be estimated via different algorithms. In this section, the discussion assumes that model parameters are known.</p>
<div id="filtering" class="section level3">
<h3><span class="header-section-number">1.2.1</span> Filtering</h3>
<p>A filter infers the belief state at a given step based on all the information available up to that point,</p>
<span class="math display">\[\begin{align*}
\alpha_t(j)
& \triangleq p(z_t = j | \mat{x}_{1:t}, \mat{u}_{1:t}).
\end{align*}\]</span>
<p>It achieves better noise reduction than simply estimating the hidden state based on the current estimate <span class="math inline">\(p(z_t | \mat{x}_{t})\)</span>. The filtering process can be run online, or recursively, as new data streams in. Filtered marginals can be computed recursively by means of the forward algorithm <span class="citation">(Baum and Eagon 1967)</span>.</p>
</div>
<div id="smoothing" class="section level3">
<h3><span class="header-section-number">1.2.2</span> Smoothing</h3>
<p>A smoother infers the belief state at a given step based on all the observations or evidence,</p>
<p><span class="math display">\[
\begin{align*}
\gamma_t(j)
& \triangleq p(z_t = j | \mat{x}_{1:T}, \mat{u}_{1:T}) \\
& \propto \alpha_t(j) \beta_t(j),
\end{align*}
\]</span></p>
<p>where</p>
<span class="math display">\[\begin{align*}
\beta_{t-1}(i)
& \triangleq p(\mat{x}_{t:T} | z_{t-1} = i, \mat{u}_{t:T}).
\end{align*}\]</span>
<p>Although noise and uncertainty are reduced as a result of conditioning on past and future data, the smoothing process can only be run offline. Inference can be run by means of the forwards-backwards algorithm, also know as the Baum-Welch algorithm <span class="citation">(Baum and Eagon 1967, <span class="citation">Baum et al. (1970)</span>)</span>.</p>
</div>
<div id="most-likely-hidden-path" class="section level3">
<h3><span class="header-section-number">1.2.3</span> Most likely hidden path</h3>
<p>It is also of interest to compute the most probable state sequence or path,</p>
<p><span class="math display">\[
\mat{z}^* = \argmax_{\mat{z}_{1:T}} p(\mat{z}_{1:T} | \mat{x}_{1:T}).
\]</span></p>
<p>The jointly most probable sequence of states can be inferred by means of maximum a posterior (MAP) estimation or Viterbi decoding.</p>
</div>
</div>
<div id="parameter-estimation" class="section level2">
<h2><span class="header-section-number">1.3</span> Parameter estimation</h2>
<p>The parameters of the models are <span class="math inline">\(\mat{\theta} = (\mat{\pi}_1, \mat{\theta}_h, \mat{\theta}_o)\)</span>, where <span class="math inline">\(\mat{\pi}_1\)</span> is the initial state distribution, <span class="math inline">\(\mat{\theta}_h\)</span> are the parameters of the hidden model and <span class="math inline">\(\mat{\theta}_o\)</span> are the parameters of the state-conditional density function <span class="math inline">\(p(\mat{x}_t | z_t = j, \mat{u}_t)\)</span>. The form of <span class="math inline">\(\mat{\theta}_h\)</span> and <span class="math inline">\(\mat{\theta}_o\)</span> depends on the specification of each model. For example, state transition may be characterized by a logistic or multinomial regression with parameters <span class="math inline">\(\mat{w}_k\)</span> for <span class="math inline">\(k \in \{1, \dots, K\}\)</span>, while emissions may be modeled by a linear regression with Gaussian error with parameters <span class="math inline">\(\mat{b}_k\)</span> and <span class="math inline">\(\mat{\Sigma}_k\)</span> for <span class="math inline">\(k \in \{1, \dots, K\}\)</span>.</p>
<hr />
</div>
</div>
<div id="learning-by-simulation" class="section level1">
<h1><span class="header-section-number">2</span> Learning by simulation</h1>
<p>Model complexity and limited software availability requires that we implement our sampler from scratch. Following a very simplified version of the methodology proposed by <span class="citation">Cook, Gelman, and Rubin (2006)</span>, we first create a simulation routine that generates data complying with the model assumptions and we then write a MCMC sampler in Stan to recover the parameters and other hidden quantities. Only after the correctness of our software is tested, we feed our model with real data and analyse the results.</p>
<p>We believe that learning by simulation has many benefits, including:</p>
<ul>
<li>Confirmation whether the software developed to implement the algorithms works properly, i.e. it retrieves the parameters used to generate the data.</li>
<li>Enhanced interpretation of the results and deeper understanding of the model behaviour, especially because the generated data meets all the underlying assumptions and the estimates are free of the effects of data contamination originated by unmodeled phenomena.</li>
<li>The possibility of calibrating different combinations of values for the parameters, the inputs and the outputs help the development of intuition and insight. This may prove valuable for the data analysis stage.</li>
</ul>
<div id="auxiliary-files" class="section level2">
<h2><span class="header-section-number">2.1</span> Auxiliary files</h2>
<div id="math-functions" class="section level3">
<h3><span class="header-section-number">2.1.1</span> Math functions</h3>
<p>We write an auxiliary R function to compute the softmax transformation of a given vector. The calculations are run in log scale for greater numerical stability, i.e. to avoid any overflow.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">source</span>(<span class="st">'../common/R/math.R'</span>)</code></pre></div>
</div>
<div id="plot-functions" class="section level3">
<h3><span class="header-section-number">2.1.2</span> Plot functions</h3>
<p>As plots are extensively used, we arrange the code in an auxiliary file.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">source</span>(<span class="st">'../common/R/plots.R'</span>)</code></pre></div>
</div>
<div id="simulation-functions" class="section level3">
<h3><span class="header-section-number">2.1.3</span> Simulation functions</h3>
<p>We arrange the code to generate simulated data, as explained below, in an auxiliary file.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">source</span>(<span class="st">'../iohmm-reg/R/iohmm-sim.R'</span>)</code></pre></div>
</div>
</div>
<div id="generative-model" class="section level2">
<h2><span class="header-section-number">2.2</span> Generative model</h2>
<div id="data-simulation" class="section level3">
<h3><span class="header-section-number">2.2.1</span> Data simulation</h3>
<p>We first write an R function for our generative model. The arguments are the sequence length <span class="math inline">\(T\)</span>, the number of discrete hidden states <span class="math inline">\(K\)</span>, the input matrix <span class="math inline">\(\mat{u}\)</span>, the initial state distribution vector <span class="math inline">\(\mat{\pi}_1\)</span>, a matrix with the parameters of the multinomial regression that rules the hidden states dynamics <span class="math inline">\(\mat{w}\)</span>, the name of a function drawing samples from the observation distribution and its arguments.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">iohmm_sim <-<span class="st"> </span>function(T, K, u, w, p.init, obs.model, obs.pars) {
m <-<span class="st"> </span><span class="kw">ncol</span>(u)
if (<span class="kw">dim</span>(u)[<span class="dv">1</span>] !=<span class="st"> </span>T)
<span class="kw">stop</span>(<span class="st">"The input matrix must have T rows."</span>)
if (<span class="kw">any</span>(<span class="kw">dim</span>(w) !=<span class="st"> </span><span class="kw">c</span>(K, m)))
<span class="kw">stop</span>(<span class="st">"The transition weight matrix must be of size Kxm, where m is the size of the input vector."</span>)
if (<span class="kw">length</span>(p.init) !=<span class="st"> </span>K)
<span class="kw">stop</span>(<span class="st">"The vector p.init must have length K."</span>)
p.mat <-<span class="st"> </span><span class="kw">matrix</span>(<span class="dv">0</span>, <span class="dt">nrow =</span> T, <span class="dt">ncol =</span> K)
p.mat[<span class="dv">1</span>, ] <-<span class="st"> </span>p.init
z <-<span class="st"> </span><span class="kw">vector</span>(<span class="st">"numeric"</span>, T)
z[<span class="dv">1</span>] <-<span class="st"> </span><span class="kw">sample</span>(<span class="dt">x =</span> <span class="dv">1</span>:K, <span class="dt">size =</span> <span class="dv">1</span>, <span class="dt">replace =</span> <span class="ot">FALSE</span>, <span class="dt">prob =</span> p.init)
for (t in <span class="dv">2</span>:T) {
p.mat[t, ] <-<span class="st"> </span><span class="kw">softmax</span>(<span class="kw">sapply</span>(<span class="dv">1</span>:K, function(j) {u[t, ] %*%<span class="st"> </span>w[j, ]}))
z[t] <-<span class="st"> </span><span class="kw">sample</span>(<span class="dt">x =</span> <span class="dv">1</span>:K, <span class="dt">size =</span> <span class="dv">1</span>, <span class="dt">replace =</span> <span class="ot">FALSE</span>, <span class="dt">prob =</span> p.mat[t, ])
}
x <-<span class="st"> </span><span class="kw">do.call</span>(obs.model, <span class="kw">c</span>(<span class="kw">list</span>(<span class="dt">u =</span> u, <span class="dt">z =</span> z), obs.pars))
<span class="kw">list</span>(
<span class="dt">u =</span> u,
<span class="dt">z =</span> z,
<span class="dt">x =</span> x,
<span class="dt">p.mat =</span> p.mat
)
}</code></pre></div>
<p>The initial hidden state is drawn from a multinomial distribution with one trial and event probabilities given by the initial state probability vector <span class="math inline">\(\mat{\pi}_1\)</span>. The transition probabilities for each of the following steps <span class="math inline">\(t \in \{2, \dots, T\}\)</span> are generated from a multinomial regression with vector parameters <span class="math inline">\(\mat{w}_k\)</span>, one set per possible hidden state <span class="math inline">\(k \in \{1, \dots, K\}\)</span>, and covariates <span class="math inline">\(\mat{u}_t\)</span>. The hidden states are subsequently sampled based on these transition probabilities.</p>
<p>The observation at each step may generate from a linear regressions with parameters <span class="math inline">\(\mat{b}_k\)</span> and <span class="math inline">\(\mat{\Sigma}_k\)</span>, one set per possible hidden state.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">obsmodel_reg <-<span class="st"> </span>function(...) {
args <-<span class="st"> </span><span class="kw">list</span>(...)
u <-<span class="st"> </span>args$u
z <-<span class="st"> </span>args$z
b <-<span class="st"> </span>args$b
s <-<span class="st"> </span>args$s
K <-<span class="st"> </span><span class="kw">length</span>(<span class="kw">unique</span>(z))
m <-<span class="st"> </span><span class="kw">ncol</span>(u)
if (<span class="kw">any</span>(<span class="kw">dim</span>(b) !=<span class="st"> </span><span class="kw">c</span>(K, m)))
<span class="kw">stop</span>(<span class="st">"The regressors matrix must be of size Kxm, where m is the size of the input vector."</span>)
T.length <-<span class="st"> </span><span class="kw">nrow</span>(u)
x <-<span class="st"> </span><span class="kw">vector</span>(<span class="st">"numeric"</span>, T.length)
for (t in <span class="dv">1</span>:T.length) {
x[t] <-<span class="st"> </span><span class="kw">rnorm</span>(<span class="dv">1</span>, u[t, ] %*%<span class="st"> </span>b[z[t], ], s[z[t]])
}
<span class="kw">return</span>(x)
}</code></pre></div>
<p>Alternatively, the observation could originate in a Gaussian mixture density where <span class="math inline">\(\lambda_{kl}\)</span> is the component weight for the <span class="math inline">\(l\)</span>-th component within the <span class="math inline">\(k\)</span>-th state, <span class="math inline">\(0 \le \lambda_{kl} \le 1 \ \forall \ l \in \{1, \dots, L\}, k \in \{1, \dots, K\}\)</span> and <span class="math inline">\(\sum_{l=1}^{L}{\lambda_{kl}} = 1 \ \forall \ k\)</span>.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">obsmodel_mix <-<span class="st"> </span>function(...) {
args <-<span class="st"> </span><span class="kw">list</span>(...)
z <-<span class="st"> </span>args$z
lambda <-<span class="st"> </span>args$lambda
mu <-<span class="st"> </span>args$mu
s <-<span class="st"> </span>args$s
if (!<span class="kw">all.equal</span>(<span class="kw">length</span>(<span class="kw">unique</span>(z)), <span class="kw">length</span>(lambda), <span class="kw">length</span>(mu), <span class="kw">length</span>(s)))
<span class="kw">stop</span>(<span class="st">"The size of the vector parameters lambda, mu and s must equal to the</span>
<span class="st"> number of different states."</span>)
T.length <-<span class="st"> </span><span class="kw">length</span>(z)
L <-<span class="st"> </span><span class="kw">ncol</span>(lambda)
x <-<span class="st"> </span><span class="kw">vector</span>(<span class="st">"numeric"</span>, T.length)
for (t in <span class="dv">1</span>:T.length) {
l <-<span class="st"> </span><span class="kw">sample</span>(<span class="dv">1</span>:L, <span class="dv">1</span>, <span class="ot">FALSE</span>, <span class="dt">prob =</span> lambda[z[t], ])
x[t] <-<span class="st"> </span><span class="kw">rnorm</span>(<span class="dv">1</span>, mu[z[t], l], s[z[t], l])
}
<span class="kw">return</span>(x)
}</code></pre></div>
<p>We set up our simulated experiments for a regression observation model with arbitrary values for all the involved parameters. Additionally, we define the settings for the Markov Chain Monte Carlo sampler.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Data</span>
T.length =<span class="st"> </span><span class="dv">300</span>
K =<span class="st"> </span><span class="dv">3</span>
M =<span class="st"> </span><span class="dv">4</span>
R =<span class="st"> </span><span class="dv">1</span>
u.intercept =<span class="st"> </span><span class="ot">FALSE</span>
w =<span class="st"> </span><span class="kw">matrix</span>(
<span class="kw">c</span>(<span class="fl">1.2</span>, <span class="fl">0.5</span>, <span class="fl">0.3</span>, <span class="fl">0.1</span>, <span class="fl">0.5</span>, <span class="fl">1.2</span>, <span class="fl">0.3</span>, <span class="fl">0.1</span>, <span class="fl">0.5</span>, <span class="fl">0.1</span>, <span class="fl">1.2</span>, <span class="fl">0.1</span>),
<span class="dt">nrow =</span> K, <span class="dt">ncol =</span> M, <span class="dt">byrow =</span> <span class="ot">TRUE</span>)
b =<span class="st"> </span><span class="kw">matrix</span>(
<span class="kw">c</span>(<span class="fl">5.0</span>, <span class="fl">6.0</span>, <span class="fl">7.0</span>, <span class="fl">0.5</span>, <span class="fl">1.0</span>, <span class="fl">5.0</span>, <span class="fl">0.1</span>, -<span class="fl">0.5</span>, <span class="fl">0.1</span>, -<span class="fl">1.0</span>, -<span class="fl">5.0</span>, <span class="fl">0.2</span>),
<span class="dt">nrow =</span> K, <span class="dt">ncol =</span> M, <span class="dt">byrow =</span> <span class="ot">TRUE</span>)
s =<span class="st"> </span><span class="kw">c</span>(<span class="fl">0.2</span>, <span class="fl">1.0</span>, <span class="fl">2.5</span>)
p1 =<span class="st"> </span><span class="kw">c</span>(<span class="fl">0.4</span>, <span class="fl">0.2</span>, <span class="fl">0.4</span>)
<span class="co"># Markov Chain Monte Carlo</span>
n.iter =<span class="st"> </span><span class="dv">400</span>
n.warmup =<span class="st"> </span><span class="dv">200</span>
n.chains =<span class="st"> </span><span class="dv">1</span>
n.cores =<span class="st"> </span><span class="dv">1</span>
n.thin =<span class="st"> </span><span class="dv">1</span>
n.seed =<span class="st"> </span><span class="dv">9000</span></code></pre></div>
<p>We anticipate identification issues in our sampler given the nature of the model. As an arguable simplification with obvious drawbacks, we decide to rely only on one chain to avoid between-chain label switching of regression parameters. We refer the reader to <span class="citation">Betancourt (2017)</span> for an in-depth treatment of the diagnostics, causes and possible solutions for label switching in Bayesian Mixture Models.</p>
<p>We draw random inputs from a standard Gaussian distribution and generate the dataset accordingly.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">set.seed</span>(n.seed)
u <-<span class="st"> </span><span class="kw">matrix</span>(<span class="kw">rnorm</span>(T.length*M, <span class="dv">0</span>, <span class="dv">1</span>), <span class="dt">nrow =</span> T.length, <span class="dt">ncol =</span> M)
if (u.intercept)
u[, <span class="dv">1</span>] =<span class="st"> </span><span class="dv">1</span>
dataset <-<span class="st"> </span><span class="kw">iohmm_sim</span>(T.length, K, u, w, p1, obsmodel_reg, <span class="kw">list</span>(<span class="dt">b =</span> b, <span class="dt">s =</span> s))</code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot_inputoutput</span>(<span class="dt">x =</span> dataset$x, <span class="dt">u =</span> dataset$u, <span class="dt">z =</span> dataset$z)</code></pre></div>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAArMAAAHgCAMAAACByxXtAAAAw1BMVEUAAAAAADoAAGYAOjoAOmYAOpAAZmYAZrYAzQAra6M6AAA6ADo6AGY6OgA6OmY6OpA6ZmY6kLY6kNtmAABmADpmAGZmOgBmOjpmOpBmZjpmZrZmkJBmtrZmtv+QOgCQOjqQOmaQkGaQkNuQtpCQ2wCQ29uQ2/+jayu2ZgC2kDq225C2/7a2//++h0y+voe+vr7T09PbkDrbkJDbtmbb25Db/7bb/9vb////AAD/AP//tmb/25D/29v//wD//7b//9v///+9U105AAAACXBIWXMAAA7DAAAOwwHHb6hkAAAgAElEQVR4nO19i6LcOHJdzcTK0JqNx5F212sn0ky8sazEk7Blr63mVXiF//+qNKoK7wcBkGz01eDs7FU3gXoAOCwUwEeDGBh4WYDeDgwMVAJ6OzAwUAno7cDAQCWgtwMDA5WA3g4MDFQCejswMFAJ6O3AwEAloLcDAwOVgN4ODAxUAno7MDBQCejtwMBAJaC3AwMDlYDeDgwMVAJ6OzAwUAno7cDAQCWgtwMDA5WA3g4MDFQCejvwGHj67kOm8J3++OU1ALxLFG4cfH77zqnhfE9Vzlb6jQJ6O/AYyHHWos0nydcvr3+IFm4cdI5uUnFwNQ3o7cBjoIyzX378QAfeRAqjEqmjg7M7AL0deAzcOPv89r+/lfP+lx//fEsA3jBtnt/+t9u3V5+x1kcOsJ9efbYL/5KSeP79/4Dvf71FZ1P6EXMLmWK8+nek5ZMqJOsq+yAtlEBojwYQ0NuBxwBy9hZrP3334UaadzcmvVMMfKdDno6vWNsUJiWe30rq3nTeqrzBUsl6+V1+xlok9oMw1umg+u7oH0BAbwceA8jCNzLKvZP0skOpzVn+cKvlcjYhgSqJ6U/f/yoLfv8BpTVnv75/41q/1TOibEfrv3OnPCqgtwOPAStyYqTzQykizdmEBEVSTJWVCCYDhrMk6ugj0kb137dPHhbQ24HHgM1ZXGjFOZvIDVISPPsT8Ogttf3+f7/2OatEKPICeGmG1t+hYx4R0NuBx8AWZ2kZFa7BSjiruKZi+JeAs1acxZofE+fEvXvlQQG9HXgMOLmBjKYfJS3f2HO6EMwwjLdhvhmRcIgoP2C2+gTRfNaqasVdR/9du+RxAb0deAw4nL0FNMmrr+9ffb7N1O+s7VhzTcEuTEoQB+U+AYdOCrHwhmnv7BuQdYymwb4E6x9AQG8HHgMOZ2lnVH4B+O9ygwp0iDPXbq3Cv6QkOG7KxOIWYDnH+O7Djb+yhrc/y3FWfufNMMNZpX9AAno78HDgBOBUicfS/9IAvR14OAzOPjqgtwMPh8HZRwf0dmBgoBLQ24GBgUpAbwcGBioBvR0YGKgE9HZgYKAS0NuBgYFKQG8HBgYqAb0dGBioBPR2YGCgEtDbgYGBSkBvBwYGKgG9HRgYqAT0dmBgoBLQ24GBgUpAtpSeGR0Pzw08EiBX+PU93rj55afx9NzA4wByhc9/+NX6Z2DgIQC5QhVnfzc4O/A4gGwpPmYa5LMwMHA6Wjkb1N5SNzBwDKCpSMh9g+/+/Bq+91KDvMzAwAGApqJbPvszvg/V3zfIygy8ZEBvBzSgqQg3DL7+8muwb5CVGXgsQFXlx8n7oKlI7xs8/2nE2ZeKShZWVT4V0FQk+EcvgpdI5mUGHgpQVftyjhMNgKaiQ2UGOqGOhYOzA/0xONssM9AJg7PNMgOdMDjbLDPQCYOzzTIDnTA42ywz0AeXwdlmmYFGwD7xwdlxX9e9sbe3L3U0/AY52y4z0AjYJ17H2cqofCagqehQmfsCejtwHEIWXbdEwBEfnG2VuSu+pfQlYNH1ukFap/WDs+0y9wXcReQuGHG2puhQmfuivusfLDSD/hQ2ZZNYF+fz4GyrzH3R0PVwuBM7YJ1BhZwFu4Lz+WzOwmaNJmTUtlhskTkYG/NjQ9c/ToSRAP2pjLNeDmt/PpmzZ01QGa0tBltkjsXWOuRQzkLEvihIK/fgEvmkj4w4e6i6e+GOcTYSSOQpkzptgspNyHI21vpL8vPZnD1pgoKmIpF6x1xepgXVGjc6qr4fM2ML4aF0nD1oqtTORFh0iZ0uDk8v9vHfGGcT75jLyrSgfqBP5aznTIw2aRsQP1ztTs54Ps5ekl+2zb4Yzj7/w+2PfCjcR+Idcxl1jajWWM7ZwrTTGlvvDIqNOnI27sMhQ3jJczZiw6r3m+CsfLL2CSK/t554x1xGXSOq250XsLp+86qRVujG2WtYcrVrn8xZrUZagcBAdmFmN+VS51DlHTXanxMAW0Vf3/8QLU28Y+4Qp2ycx9mGOCulLaqrEvvQXTkbxP1TOHv1RcPSuKudOPvl9bvgJQbx2iX3IjZsAtW2e6Oj3EVIoQfudGzHWf5iHWrkbHHP3C3OTto1PCNTnM3MVp04i++JieWz9epE+Wxs4/E4a77pFllMqOGspkVFz7icvXhlESMuT4s5O02GtL6oA+U4ZFw9FqGlkqJWmfjAZIermrN5iSM4q76aba1tzkLEnkULqw8295erOHst56xn2PjGZyrVhtAhOh7OsSgSHI3YrIxkGZVZa89vKRf4vmoNFmVJPsbUTjDWsMTU7uLsxVlhod8XT5MzvgY4opk4azm9eR2vhrM3baWc9Q1b04lq0zVCTd2h3nEWyaeLVzRaO/1mNOaM3Uj7+1iWm5dJEPCa+JwRyZjQnR3tissl9tEBJDVO7q4ApXlODcXZSXgAEXI2fv5EnLa1eZx1dF5cTaQtxdmrX9czHHIWOxRSjQi7Mxtn8fiFSZuokxNNF+H+a/yHPp5eRX6yJqNOwmrXFD0a8mxPnL2E7vj0ssCW/chggpecy22e6JKAs/as72uxjsXOH/rk9IKjbZOzfrRMcFbPEqq+3x3xOJtuRIKzFxGewIJ7+XpxTp3S/AhyRV/fUwIQY2e1OgnTLmscbk0zYgFlVYO8hk/R6leXP6kci09yp5MnfbqACGZt44PPWT2e6piJs45vkTQXomSirQn31J2cgJzm7DUQvSje+MKaLLq+H08sIdWv3onnrM0i6wi2HZ7Asgys04YkjefqU4K6ED+simp/SymjTiISZ6+3/xluBSer6gqv4fzVG1v51c4+U2tZsufYuim8Gjn1RQWBi/XJcCDKWT20jm+UezqNk05EyBSJs7dj1pZwmrNk8XpxJFOc1c6z6qspos61aqp+dTmrKLfF2dspF9yEQVn2VTUI48jFj7MN9xrJIlpq+b+akJdJwI2Bej7AjAaSE4wbZw1vJ/+aVGgjNh1n4qxDRR5+lcBNVkgwk12eszQEYNrljyrEyDQZv61NXzN6Lmc9lbyJ4Z4rG5zlQ7hYs/rCnU5MnPWvyehui0QIZfsS3OxmB5eLjiMWOS5WewL4dmJFT2/SlYrVuTHQXsjg2TZ5h3U91TL5j5NQ0FLc5Z01BlnOXixbEx/Hwb6aj6xFmqEgrKdC5YbLWbVRa9K/i0mOY5yNkYlTZlnfud6WiLOuShMGzCH7HHOlHXd1h5nOsE9hRVc/O9b+6gFxG3jhhl/dMwm/2Jy9CV7A1qy7NgZIHLeLKhKEpLrr5ep3OH/Av5oGWc7anU8NFfZhVHLRcbKMs2yZhTCcmnCiYrIcGOtyUCzOqmL7pIFsnI0GQBVnL/YS0g2PpucuUc4KP86avMdIg/UdQm8sm7rVMc6Gcda59UK7eHED9MW4IIs4TTqSsxU/tphSR1PPxTTJ46yOdtpPzQubs84YX1R40+F3Mp2qwoa7ltGc1X+mixUzOc6akTG0v1hx1o5AZnyvXNmcNFbGmuMs2A2/aM6awGUPpxUOk5zVgVPHWRV6tSZWjSc+Z/eOYiv+Xq/K/xhntdhFn/dO+xRnrXSZ65jW3by4OBm0iG9RKEDiuLDz2QNygys14eauy04v3Orvk45/F6t+yFls/2R1mzM1Y7EOzjSR6fGQ40CHJu235os+U1wuOX5YnMWaV+uswE4HfWpmOMuXGyKcNZFIjaKzQ5XnLNe0vHFOQVJ9RePgFHmc1dzPc1bLX+1ilR246TLX0a27+Mu+i329JgQkjueL6mVUfLoGXBTmX5MdadJ66akXYzj6XWgIXM6qaGMlwZMzHhRnhdxN1anYHs7K81GdFcpNd6Fx8TfRVJw1Qc7hrGVHLcOsHSqVLKq22VmrrnkxXl5916UWME1IcPbCDSnkrFOsOWtvS/inLg29XXBx02sfkCrgIhloi7ZnN+7rMr0Qj7PU6Mn22+K2Cpd2RnFRe6GCN2L1HCYszto00cHOxGJ1joPRa3HWGqIg3luJnDoBrhcrzlLZVQ8k6bJWcxY9XTJZ/luc5SSQPk+kWC9w6BRXDTVS9uqcQ93VjIEffqs5ezGcta4J6mKbs3aJR8aQzBd33HxAqoCK8Nbu6BWvSnWmF8wg2m6ik1bHXeyKcohogcVNkUWTvX0KajI0BJgcPfg5ztngAkIJZ+0Fs9ZxuVo1JdSUqDmLqaV1MxgXUJusfS519XcyFWkyoVN7mi6as5xwWnFW08y62+DCoY5yKXez2+GsWtGXcVaJ+StQu9FbnHWyiUlLRi9FICBxXGxeu61T53FWh36rIVZOc3U4SwUcZ/HLRBcBnE1TO85e7P2Ai/LM5awVJBw3m+IsWdM5pMbVIQKHxYsd3q6qgRe9zyVMm3WXXHgyIQcnzSJ5Pl/Aa4emGZ/Fqim6l1w22JzVW5LuRJjjrO58j7OTcOurD+B2ubNqsxp8aY2z4uNt/fUUf1KhSp3LWbUV5Z589qaTG2dVAcdZxVt9bpswceGQ4sdZWsNr4ti09dws46xJJYloZM2Ps05zNRWsXHWiIuSnzoxMm504qy9CXFVX8Cl8a1tIQh0FPc5K1bFGq//7cdZE24vTIcaC1/mqXOX2PmeDBDIaZ8NdTwNIllj7BuVXwpLqCuKsdcCNsxdbg1o5CUfQ7jbniWk9N4EOThuctaKJXW6NPX5T/k7+HBbhrBNn7RSA0lsVqZ3z1JLWFDVngDBxVrYtOtk7nNV1EzfwsEdX67yOcNbtsTxnVZz1loY5ylkdkM5m77ZvoHpReMNidYDdRxenGyzOIllcprmcpUhnn7o6usjzXo/8lOGs7Z0hv8NZMIPjk9bjrPGOHcXq6py1rpJOllk3NnmcdXafGLHJ3grnwkxPcutGeNAuYtS3pEHLmxPtYi7a5jlLbvtLwxwZLZ/T2ezd8lmPs3YrLX9Nsb1IMVed0pxV3a4XYnap0XNR4zDxNbMkZ61B5BKbs5hpJAnj6gSbQxxBTG6k22yJQoSzdg9qf1Lzp+oQq6tUOhW988SMzFUYzurrUyyuM03a1Js0Z/UyxOGsCj5enM2R0T7PWuNs7l7EuvfIpDhr1TfhVK+FhY5LwomPUc7SsavRpEuNxOWi7sWY1M2wgZ8OZ4M4ywKQI4xToMbdXcC424+TI+reLaJd91Vvc9bZ8zQcCO88uZju4ZpAhsGKjmp+kGfypHZxuCs1Z63B8WKS7s2yOJsD5IuSIbbyPTI0T5iP/MHOx62mpuMsZX0ZzgpjSJc6nNVRLBpnhZoSI6eG+28pZ8M4qxujoENPGGdNRfBVT1uc9W4pzMDhLK/q8JRw3bSmsclujxVn85z1ppeII4dwNnUvYuV7ZBKctR8kuAq3OKANLVUiZMpxlrthskpMcYKzZlPJa4H9bzFndUKd5KwXZ922cEX3Cq8QOj2PuxDG2RzU9W0rznJH+Y0x01hsqcobIMaJIM4GneM5fslfASNAQVHsXsTK98hYDXA4a2oEt3HEORsjk89ZxwnseOe2MXOiRHrQjrMXt8D9t4Kz+oJeirO+qKV6R5z1/E3jqi+ou42DaA/pi2cRzmLaa8d3j7OXyNNOnuPunQlRZHToomiCUPceGfukM6stJ7UxcdZeaIac9fsx4Ky7AejEWaxkPRWR4CzXdAq91W8FZ82NE5ucvbj/UkUIyq3ClA81nNVx1rqSdjFb2n5lFVxCztLpOZn1XhBcNt7aAsK9MyFda6vogHsRXc5yD8UfFLLTu0hA3eZsJM66I6jLKzhrO6UVRxEqNTdOuPdLhKIxzkZy/sRXp0CVQapOCPuWehVnIyauukrIWfrX7Ks51i+XDXcizzxFq+WLjroXEd1Qo66vS0aXkBc7pJVw9hI56BQnR7AlzmoN5Zw1h+1UOlZrI842cHbzdVQ2vBsvRJY/ZoFmcdb8M4XWN8nId58fEWeLkL+vC1uhN638mcitao1rCWfVuZ6kkFXmObjBWfALHQ334qxdz/MoObiTKYNUnTy2U4uLTVWXs+YA+BIlps/ibOV7ZJw4m/fpTM5Wxdn4GQhW1YTBOGcvQmxyNtJD9ufYqxdisO9DboQylLkAEOGsksqekcW2G8plkVxpxV+LWPceGYd7HTnrl1XHWd9ukVI8fHFEorXCpNnrLc+jTJzdS1k+OzJXrSKctW6dj9mHOtvpCtmiJxlGn99G3plc+R6ZMzkbrlA9y62c3Rj3jMXU4Q3OhklzvrfyE/cuKA8OjLPFqfVWtUw5iK8/Yyz1L3Y1qKvhrP2xgLPWg7MJj07ibGo4d3CWYY8uJGtl75Day9lt6BvFYo2J9g0cZDmjB1IXuzTCG2tT6i6JzxF4DxuH36ribOQtCCn97qFNzianzf2ctboxF5xyjzKdz1lh3wrimcvdCLMfkCvairPFnL0U1GE4b5Qo4mzEhGftFM62xFnImY4Dmsruw9nKOHsUIFuUy2dFI2e30honztpV85zNKT2Hs0lrycNW088mFWzW2I3SSeNwQL4ovW9Qpa4izjpiDr/N3YC1CX56tZTjbOs4lMXZc0e56lJCqw396bE4e5C6xiaBc0tFlrP5OJs8fmfOWt9efJy1J43BWa0svoVe30OQOP4Nc/YOAP3pzo2BpqJamdY4G9VSzabkRHlXzno+vHzOGnyTnG1RFeJFx9nS664vEt8gZw9aEVy8fw9QOOLsIfgGOXtsnH10zkKh7W8IL5SzG++YOwJ35WxzklbYCUWVXgpeKGfbZaqVH9dDIS+9R6JaAEWV7rCBej/Ao1jLFB0qU6zb3Ol2FCK83M/ZMsCp2u+LO5+AGWOZokNlqpV/E5z9pgCPYi1TdKhMRyODsy8R0FR0qEy1jeNmoggtL5mygYcANBUdKtPRyODsSwQ0FYnad8w9KAZnXyKgqaj6HXMPisHZlwhoKqp+x9yDYnD2JQKaiqrfMfegyHEW7ujHQA2gqUjUvmPuQZHh7Ld1oeqbAjQVHSrTEbGHdIIPAw8GaCoSct/guz+/Dt6mnJd5OIScHeH18QFNRUI+Rv7l9Ttr3+AO93Udj1ycHXhUQFMRbhh8/eXXl75vAL0dGGgANBXpfYPnP73k/dkXNy8MSEBT0Q1ffvxwS2pf+L4B9HZgoAHQVHSozMBAFaCpiBF599HAwOk4mLObMgMDOwFNRYzB2YEOgKaiQ2UGDsPa24G7AJqKDpUZOAyDs8eqGzgf62+CtNBUdKjMwGH4bXLWuvXFL2pR1wG/iWFLYP1NtB7cr/atLxCrX6euB34Lo5bCb4Sz7i6tfesLpKXS6o7yqx2/hVFL4TfCWferfesLhLVr1fXAA4xaNxd+k5y1b33xi1rUdcCOUYP+LhCg2fBvkbNlRbtkWvRW4HjOznd0gQCNcg/G2eUctdBUtEumRW8FvnXO5ojwwjgLbWozYi0aS2Ra9FagZNSWeE2I167i7FroQhYJRxB5zj4UaQdnEdtDMjj7MIi6ajkIbWozYi0aS2R2PdDiDUlE1cagYfHRnF3dj4/F2ZNZDOmiwVlElrOL2E7pspwF/wjixXBW2579o5U2DLbbntEzOIv4djlrW4G0yeUxORvV9k1wFv09mrOz862Ms0tEV8qzuYCzRpXhbEosaoSqz8qPh+PsutGcqDb21S0r5GymLPOETU5jgyXZ/sfmLMTd9zmrxKzDhrOr4WxUGezmrBO9wLF0Nmejp4tywdjUHbbB2SwRMmVtRW0yj8DZtZWzxsoa2PU4uxrOBgoh0UcrmjmFs1F7LsvScDkbIS0oF2ZLhFm7xdnjiZkpapM5grNeP4No52xws2mKs3YIFCWc5XaKWs7OJZyFds46kuWcRbXrBmfB5uzt/7LqNmfTY5VhSbrkcTlryUtdR3F2beHsyoeULsNZbGagEDY5Cy2cVW7Q/+c4Z5d2ztI0sESygwxnVXVwBCKcjY3Yo3B2xhlGUSPXZR41YwUgWJUZITmcJZylztzJ2dlwdjVrFJuzcw/Oyj8eZ5XBG2dn+6jpNyPtHGFNBZwFUcrZ1eGsdtnFuoezz2/lCs1/Y2deJolbs1eLs9GBYfcNNd1K67z6nJ1FR86uKrYxZ5dVrIqzDj9B6JGIjcbqcjbSN3pdU8NZObPByvZkq8lJlLE5C1ra0UuaqEHEWUNa/Kq5xR6D9nQ2nHXbm+Ss3axdnEU8vUlXKle3zVl5uICzOuv3OQseZ2PktTnrORvl7K3ijI6a2MUdWsJZ0/HkWoazs6D/VXB2VoRp4qxlgzKsDc6us83ZWXMW1ac5u7jtXTzOSgddzoI4grP+CzszyKgTalQ0Z9U4qHkVBKdjDmfdfRvJWSoBUcVZHi/mrPxv9VYAsvMD/xVnlwhnV4uzeDbmOGuiUjgaKy1x6D/IcxYUZ2FVXenGq4CzqlGLKgg5e/t8a806q3raxRlVFHFWn0LlnJWiyNlZhJxNcylZYBX5P/SRQUbdEnAWMpxVHRBwVsuD4K5akpy1ObksNZxVRUvAWVAdehvllWjrcHaJcVY6LfnsEVlbdjgLOc4uhrNzmrOqS4izHGgTnJVezbfmbHN20cIUKJB/irOylxRnF5ezNiuYs0CdDMjZFVUcxlnKZw/JDRRnQeD/Q84Ce3vrWaa2bMaMX4U6fUWas3OUs+zRIlkHcc6qfyDFWSCGqJJV2cXQs9prdcNZmDU9gVSIHGdFyFnNIKGbb3EWYpyli9eqSyTRopxdsB0WZ0WUs4D9zbNJlLOz7A4QGc7eQoXNvmXBszPKWXUu0sjlOOs9w/iWvn9/+DOMS5Kz60pdiV6uxFKbs5quIsdZUJydV+UKcYfNywEDh7Mwc5nHWRpR5TX7pThL/JCkujVotjmLDsY4C2o0U5xdVz3VpDi7GM4yAw1nF81Z8Dgrp4JFcRY7EztHcVYxm9rncJZ7V3FW6rxxdhVJzs6yR6iVxFlirOYsn2iasyi0IBeIszKoCD2uqzkRQ/j99/z7D6miEqRlJGeRnMDskbGHp+sWzlKrV+Alks/ZBWmnObsyZykKh5wFSqyIs4vmrOz1lXjGw+1y9saORXF2NZy9/Y1wdk1wFlnO57HNWeB+I0f4AHEWfM7OLmexs3Bdpzk7a84uxErmrAxps6BEIcVZeUotW5xFx2fsf9m/M2UGxNl1WShfMpxdHM6C5ixFCGpkIWfF06vPqaICpGVkr6udKuQsxkjZZyv3MU8VNEWmOQsizVlJMcPZ2eKsoJNkRfYGnF2AJinFWQC9EF9WaUQyBFCH5qygU+sWXEBzVo6X5uxK9KSmSgN4NMbZ2eIsCM1ZWpmEnF2Is7LyArPhLKcNqj9XYpzF2ZnvGvA5K4S17Jc1I5yVZ2nI2dnh7IKcpS6WnAXDWczmHc7eOpZDF3EW5mbOukW4ZXDIvgFzdrE4iy3V8yr2Is1sOpwuRFXs3JnYi2MLgrfsc5yVW2sLRaLFcHZZKVGQcuwtOWUtBkiKIuOtz9ebJCyKszINxtFRnKWgjG3g5E9xljJhMcc4S7YpNnuclTZmTp14qb7MLmcBW45J6RLhLBjOyvwbPY9zVhVzoEQfFhAOZ4HGY4uzc4azZgWKnMU4gJyV7dScXQXNausOzn59T9ntq8/pWoXqkCbMWR0sibP68titF9dVzWLFnCX+8ARsOIvn7UJ7roscu9tQyB5WnJ1jnBWcWRFncc4mzgrmrDyrDGfRTRw3Hl3mrIwroNYRxFlJdnkGIWfN2lA2U3EWhM3Z2zjOekrgrAoszs4uZyn4u5wFw9nV5iz1xILrrtu/xNlbI2c+b1zOYgtkHjKvHmfXhTi7CJV/M2dlisScFcTZVWdGkrM4d2G4Ic5SWoEp9crzDKZWO+JsBRLqlkWRSeYwFmflYYezM3OWhnoRtCmY5CwuJWRPUUORs8g02RUeZ4XD2UXnBpJLwPnsCrwEk0W4Q7AiZ1fi7BLj7BrnLMcMl7MzBntsArYS5wacLfOcBcVZ4gjQokeYaEuhl+6jmBcKBpLZK5HS5aycNxRnlTtS0wzY2foYB1hKImQ8xLEEQVoszoJwOSunlJnmPc1Zgd0my28BGxbDWWyNPAlREoMQ5mIzNHL2sGu3tJm5BJyduSsBe3nmrWscJcXZJcdZGjGbs7PirODjOFpySDG6rJSnLBg+0TOYsTMNZ1fFWRkQsGPl6QTSfcw3AAeOOSsMZ/kyA0YOnAFnmi+Zszzl82wK1F/yPHU5y9OmtLvYnOVMj057IojmLHYLn3BiVufgjPFswUYrzq6SbUCLs5nnaCFbpjmL8Y5cpBYiI13O3s6OeVGcnRfFWeLkbbZcmawLkVIOvnQDy2Wclz3LcsxZ9EpYnBVqGqwgmSk64trtSsmedFtuevHsi22SoQ/9w9kZ94+wGYazIs1ZATZncb9mplNfc3bFGB/hLHsbcHbFocacYeVJGKM55c6GsytlDFiXFo4+Z3E7VdpeFGcXLFyV/5KKK0/GICzOAg4jp57ohs1ZWl3OKiPkkBXjLKjZYsbdkMXlrNCcxURCaqA2rqAXmsxZoHBNJ5sM1HIngSYdm7Oy0xdM6Hklh/u2xFkcFslZdJRoaS062Q/m7LqPs0eswXiBArIX55U5u9JacSH2rHIOVpzFjoIIZ4HN4Ciucc5ClLPYY7gnT5xV6x5JDz10nLFanAXBeoizC6nAyguWrjpPoCzP4uyC4UtxmzmLfi9RzuJH8gUJhGkSEGdpWeJwliYB4GWO4Sw2B9OWOcJZ2ZCFl1awbHIW1J4CcXYlzmI0RhO0q6c4O9OIEWeVOuxClYPMq8NZWi3gvSjU98dw9ohrtzJhsThLC4aV93OYs7RgmOkU5fWJ5qy6uwrYjMVZ6nfDWZxwKQThJ8lZ2TV05m9yluIC8NrM4azk17qo3SbFWc7wFGf1cMsNBIxHq8vZWXEWR1hxVurEVHTFmRxnkUVnsTMtpVHK5ixOHhgLV5qYhMVZXENEOUs9OdOENONOlLng4zRi5hMUsBM5h0Ei+5yVY08CFH0AACAASURBVKU4K/Mq2eh5i7Oc7tPJzttisI+zR127xY4HzVlaiVMygFug6DNxVtic1TvUgoLOzJvfSCHsQ5CBD+M2xQxOlmOcXTlzWhfejJRTnABexKpUlmw5nEXPmLNIvYW6l4PjCoqzqxnuFQcuyllhcxajKSjOLrh6X+eFOCsUZwVzlk93m7OCOTsLw1mB7Z1xoaQ4i6OAiYrhLNA+dshZQeFPLqk8zoI62cHm7JLgrOzImc9puhwHyNmFRnKJc3alJCNNJkQN/3JIyADtlgSc5exS5UN0X5rLWd7t4wGcVY7FnF1oPkE+Ks4C7arw8mQhRhHbAs5K2lqclRu5FDbw/OErA5KzuLBdmLMyUANxSTBnUZvNWdwsoZ1HvoaMLFK7OJxrrDjOtAe28C7wTLsnaoEOahca+4Q5S7tlzFkiNiYChrPIvQXvOMDJQ3FW+JxFU5qzCy3mBU0r80ohUdicJfW8vtKcxSnO5ey6Gs6qzkpxFu8UwRUqcnZGc3Uk00Uy0JZvzybVrXhtHnc4cElD87fiLJ5ni89ZjE3E2ZVmpJCzuNTFDFLWwMWLxVm5t4ocotlrpQwFyIxSiFs/RE6MNzikec7OOPRqGBRnKQrKdjFnsUMoCaSwpDmL0RU0Z2mXAJM5NE47fgFn6VKIzVkM07AwZ2noF5+zcteIdFqcXWel0lzpmlUikOcstU7tCeDGmgzIC6l0OCtoCYCNBbqNhzk7q9My4KxQnE1P2nn+4TuUn/ZfU4hzlvyiEwtvLIpxFscc9/qAsjHkrFk0K84K5uytBLfZibO8GUGcXYiz7mJ/wfhnOLvQ6TEzZ7E3ZynJ+3QrNcZwVlFNLqDpypLgURAhZ3EJw5ylLV3p6xJyVi3CQUQ4K5izlGjTnr7HWYpjyNnV5iwlOSnOEh+Zs7y4NZylHCXNWexJ3FXGTYKVLi0SZ2nyx6kVh0DwhW8h+ITFYowKmBvRkqyKZFx01LVbHGbKDSkPAOTaSisiuTaYibMLclZuji/C4iwyELOx1eMsp40rratxYxMX9DNnnStFccVZGiqLs5ih8JVcNEZXOlFEcVYYzuKqBuOsme44aAKvmSjVtDi7aM7ygk3HQeQsqqbLHbRnzJwVwJd0KRjhvM1bzbguZ84CT1TMWdoYpZWP2pJlzmICJRN8YM5yaqDugtGcxWZQBi08zmIQwJ0+dIc4C9iNIsVZQYHK5qzsedrPJc6uirN06jBnK0mmij6+if1EaF4mglk/6xTnrMBuo4nc4izdooeLF+KsyiKYs4viLO/TAC3F19VwVoScxRlJdlOMs7TFT1vIGFc1Z5GJ+Mj0zLc+cWt5g5f98TiLd+Usa4Sza8hZTkyEw9nV46xwOUtXUAQzYFl0XGPOUsrFl3HlOTgTZ2dhcRYvmdMuBtD+Nc1UGEiZs4vD2VVxlhaAmrPLrO505L3qxeUsbXMozlI7ad6ge4hpI3/Rm/YVJOMi+37aEqTUzTzhCN6TXWeLs+SyxVm8CLmsvIcieN290MWgeaELry5nV81ZvBmHFgbU6llxlg47nJU3QOCpjzklcpZudvE4OwufsyLBWYrsePcdVpKcXTzO6vUX7vDT7hXwXbdgcZaW0sRZJI3aOlmRnLR/QCsC5KzsJObsAnxb3Ep5PHN2wXNkXVfF2dnlLCUOzFnaRSfOgvA5O5MVtWkhH+rVnIVF5URyKGfDWVllpcWo5ixeo1ktzlJ+uCYXYSmS5YuqZZizghy3OSu4eRnO0vy00PYQJ6EYM4DzfbrPGpciK10OZs6iP3SvAXcaDgdwwikHacX4htccZorFmrMUvhRnaUeAOMtt5daGnJ0dzmLbgDb2VnwGkTY1ZhpqrKY4S+mk4Swm+LDwnpjY5iwozuLJYXMW7S8z3ewjcpxlehJHadMNOFEGda8cXc5YeeFEDs+YqSJnyTG+80BxFncDQZGROYv7sYdy9tMx9xtEOCuYsyizCnVa4k1BdAXV4qwwnF3pXqOVT2+suWjO6og+84xGnAV+SnZW9Ak4y3y2OLsCmNzA5iw/CaE5K0DFbs6gcY+Ni3KcXULOYh39TARde4pylnMDul8Sq6Nx3KgWguglDGeFz1mBO1QWZ4kxNmcFX1ChnNriLC3i0pwFHhSPs9jfs77Rw+EsPQthcza92ZUiGRdZjy0UIaVu1p+Ys4LWJivfK6I5izPygtdbhcdZPvP5jgKLsxhqVpezKjrQFUnFWXUrPN/uJScyeokG8tZwlnbeQKiO3eKs+l+Os0LR0nBWpgYmpaOL+ciegLMzrmU0Z5VVm7NLlLOUXCU5O8c5K+M0XXejqA8BZ1feUWOnNWdnkjKcFWuKs+g0LiBllxBneclNy7hCzvrPgx10L+LqcFZkOIuZJgAFP4ezK68KiLPC4yyem3Tjic9Z02f6QRVcOuNTlerZ5YVvnVF7PgttDM9CBwNh9hpinOVrWT5nRZyzdPPUrGOk4az8Zp7xFcSOWUZ9bqzgm9y5tav2UKhNBsCP7JzNWUGc1QX4dBBtMgo8MQPOCnP/vWDaxzgrDGfJ8ELjoTnLV97XOs7GyRSQzH8e7OO7hGCZOkaEs9Izs6HBnKVrg6t6RAoHWgScnUWMs7hcLeWsGgMedMVZXMgnOSsMZ9WePkHd6QVEbY5+usH4N89ZwTm9z1lMA3DzZ501Z9WsBaonhXnLEDBbXc6uvA6Nc5a9pVxfcxYwvaZHT1kZheo14Kz2aFZpjUhxFpciqt94QHzO0uX6DGf9a7fu82AH3T9rm6fPK7Uq5CzNhtyE1ezcu5xdI3FWpDk7O5zVjs5rwFmxJjiLXq4c5tdNzlrt5VbLUIpZCC4oNWfdjktyFptN7uc5S8tFMatYhn2T4KyIcBa4kK9bcW7AylBplrNWFmg4Sx7xfzHOsqMOZ+2E0gUkjueLamUynFVHFfGYsxJ4Gz1YnKWLEkLdrGStYRzOcveA5qwiso5qEc4Kn7PqNRmCe5BtrHMszqpXUxVwVuQ4S8Npc1ZozoooZ4XdLkrcQahMQPmMAdTjLBuyOcs35lAJ6DqzGVb5YV3BPA3vcnYVS4SzqxoA+m+1QqS6P4IdnZs5y9cQYkVbSMgUcHZVRxc9YaqX+5BSxV7DWTFbnOU6zNlZbHCWYF5Sk+csi61G+VrJWeFyViQ4y+eSTljRHD6VbnFWOQ3BoIKaUMRsDwUTC9C2/WKeeWGa0JcoZx2oPZCVjeU4K3zO8jAkOKuaturOa+Js7RWFNM9TnDVH1V+Ls5z+RzmLnElxVliPHc/CGSXPrShnUe+sXu2S5Kzpe8XZRVks4CyPp/bc/tdag9EJxH6JTc4qNaWcFUnOxodSrScdzprIr94wZar7nBUuZ61G66atuzhbj5TMJme5TDjLF9qJDTgrVjWbiRRntYQ6uRNu+Zzle6rkHMrXcEWKs6vwObuIGGeVKcxrkpzVLXDdMpxdlJemOSB82Jy1YOVNzjto/c5ZZh3zMpxVC0ZqkfKRNXic5c2gNs6mxi3q2WZRrcwcfImN1zZnVy2qOCs8zgbf8pzVtgLOirmcs0JzFuYyzsKa4CyZcFPOIznr9tciKjlrG9ZbE2nOCpuzqlXhzS+u14Oz5Zw1CxyPs24wn1UDLGuL3mm0xs8zJSmjOLvQKib4RQc9TdvmrJ2LxS/1cABntTyIJKiogrPeMV+15/UDchZxN87GzYcFi5Us5jmrO9bhLG/bJJPnKGfDagFntfqo1x4gfriUs8Lm7CbinA0sBydmIiEyVl8WZ7EkzdnVq7DF2byvIslZwZ/mGGeN0BmcDfxS6osAmbI5wVnXbgFXjUL9ySwLjuCsSHwNqlcV1co0cpZePaE4q0UcSZezJc7E/Qo5KzY56zixbEzcZoF+HmdzKOFsZoslptB81AojnN3WBInjL4eznqCKZgnOZswnnIn7tfjJosVZUcTZQKVnS3NWLLzD2Z2zYaXjObsDKc76126togYrCZlmzoo9nN2GpekozibhclZkOXsGiub9Gs46uhm5X+M9DNBUdIBMZM3MBQmdkKZ5g/nA1hKJk/FhjnC2BNYWWBfOFqWqg7M5PARnLZRzNm7t05tPb778FD6g/Pynz7LI4+zt45N8Bv/5D79+fQ/vylzcNOHjZuI/3gP8QCZK6PjPf11v4vNHgHfcigLOtrTisxTTHQVp5ZmiA2SS/OvF2cgxn7NLBKrs05un6FDcjsmiCGd/kLfNycvi9v2eawTFJnzcTHz6AZ/3/2hxdo5ASfzzX9ebuP33f//wK7VC38IeYk8rbqWv3+iOgrCOQqboAJnyeZF19udsjLKatE/vnt59+ekvf/sWfhBf/uaP8u9Pn7/+/E/v4dW/3YrsMIcF//M2FF9/+Zc/uJyNUVaTdtOEcE184Ivwt+BlcTZGWU3a/9NmwuNsjLKatE2t+Prz378ozhqJ++Z/YXaXi7MSt6H43a+36f7Ljx9uwY16XIeU+fntdx+e/1FYnOXnQErj7KYJ4ZpgQj0zoUri7Nxk4pOctImzVCkXZ5taQVnD4Ow2NlYk4B+4DYXufkzaHM7+66+UCmQ5u4WsCeGaIELhkycf3xWuwZpMoH6bs0e3AvOGJGef4Ls/v6abD/2iEpTLNPAvuWw7C7Wr6GAobnwxnLWqyYKDOGubsKspE7dQJnZydsuE0OlHM2c3TODT328SnJV6Xr8TKA++igK0yBTj3pRt5yxNeb+Tc1+Ms1iwk7MxE1Y1ZYLfHdzE2UITsiGtcbbQhMCzIsFZuffyy6/UlV5REVpkinF3ztZcfZfQQ4FLC/ER/urvPjy/Dd/QhwU7OVto4qOJUNWcLW3FR3xDcRNnizsqw1l8DyLulQ3OimrOKsS2cgKYt6BVPtvcaqK2MS2tqLyk0NZR/qIO05+n72QOBKIeLTLF+MY4y+/1Lb+msNPEOZx1W3EKZ/2OgnTdTNGhMsV4MZz97eIel26jJDvhebBDMDg7IAGRY4/K2ftjcPYRAZFjg7MDjwxoKjpUZmCgCtBUlJYZGDgdx3L2XEVD/YOpP1d7g/p6ibMVDfUPpv5c7YOzQ/1L0z44O9S/NO2Ds0P9S9M+ODvUvzTtPTk7MHAnQG8HBgYqAb0dGBioBPR2YGCgEtDbgYGBSkBvBwYGKgG9HRgYqAT0dmBgoBLQ24GBgUrAMWqeoOW50jJ8xN/bO8kCvnuAdJ9hgV5tcFIDvspXeJ7mPWs/rfc/Wm7XqYdD7Mu32vyu+Hcc6/D15w/nWcB3RpHuMyyg+tMaQK/wPMt70n6a808/yBO6yXk4xAF6T9ghqkLdvzNvIjtaN71nU7/+7GgLrP7EBtgvbztB/U37ic7Ldya2OA+HWOe3OZ2Cp//0Gn44zYKcvEn3KRZQ/YkNuOk/1/sTnZcvB21yHg4xfyZnxf/7LD6dMybiLpw9sQHyFZ7neY8vCD2z95XbXTh7Zm4goWbA4zU/n5kbMGfFWQ3Ad1id5j29IFSc2PuNiQ0cYvzMNRi+AfrdWRYkqc5bg+kwfkoDSN9Z3pO605yn2NpxDXb2Xteb0yzcaa/rlAbwKzxP8l69IPSs3rcV99jrGhi4H6C3AwMDlYDeDgwMVAJ6OzAwUAno7cDAQCWgtwMDA5WA3g4MDFQCejswMFAJ6O3AwEAloLcDAwOVgN4ODAxUAno7MDBQCejtwMBAJaC3AwMDlYDeDgwMVAJ6OzAwUAno7cDAQCWgtwPfKL68lrf5f/fn8Hfgn/9U8NPwAxlAbwe+XXx6Ez38JaTxQBWgtwPfLiRnv/z0l7/5I/zX9/TYlvxZ7K/v4dUg7S5Abwe+XTBnf/zw5fbfT3+5hVd8hn/E2b2A3g58u2DO/vT5i/zv3+gx1sHZ/YDeDny78Dj7Ax8enN0L6O3AtwuXs7cc4ev7kRscAejtwLcLl7OfeQ0mnt+ONdg+QG8HBgYqAb0dGBioBPR2YGCgEtDbgYGBSkBvBwYGKgG9HRgYqAT0dmBgoBLQ24GBgUpAbwcGBioBvR0YGKgE9HZgYKASkC19AnxC5MRfURoYKIJFRcjVoxuRxp1IA91hUxFyFfkX2dQPsw0M9IJNRchVVOQ+6+fqBgYKYVMRsjX5iWcvn4WBgdORpmKesz7i6u6GboZbAL0dqAH0diBAhmrRgxpP8N2fX8P3XmqQlzkR3Qy3AHo7UAPo7UAA8L5bVPSLHMiffH79Ltg3yMqciW6GWwC9HagB9HYgALhfbSpCrL7CbZX29Zdfg32DrMxe5JSfavhoiztEzwI0lJxqtlzIpmJWHy/W/DdMNflQiGy2fKbhuMEduXu75FnItCZZcK7ZnJT71aZiXt2XH2/rtKdg36DBhWJARv+phqMGd1jcIXoK4OA4Wy7SoDwUsqjYoq/Jhwr1SaacbDi09w1xNt+WXFmLvv3IaG8x3CJziP6zDR9qcI/sGYDmwsNEDlLfYrlF5hB0M9wC6O1ADaC3AwGgqehQmUPQzXALoLcDNYDeDgSApqJDZQ5BN8MtgN4O1AB6OxAAmooOlTkE3Qy3AHo7UAPo7UAAaCo6VOYQdDPcAujtQA2gtwMBoKnoUJlD0M1wC6C3AzWA3g4EgKaiQ2UOQTfDLYDeDtQAejsQAJqKDpU5BN0MtwB6O1AD6O1AAGgqOlTmEHQz3ALo7UANoLcDAaCp6FCZQzQeb/hEk+2SR6HCg42qFZqOQsZkpihW+5znFEpVHm5422J7Y5sFj0KN7/maPR5NaX5OIaFujy97NB5v+EST7ZJHocKDjaoVmo5CxmSm6FCZQ9DNcAugtwM1gN4OBICmokNlthTcDl3PN9yAuM34UcZ1u8qWcAFK9RfVS1dqHJUiq9Va23W3yDjyYZpyO3S9bnbPXsMNiKdU2QSPG5KpkUFBLxS4UF0vWWfbn6iB/RlwRr5FdalMsl6kAB4ozkLyy8ZRxkuKs7BV6eA4mzZUXrFYR73M8avNo/UlrBzj9yFKzoZqK9zZXkHNpqLdMi26s5qOU7hpbb+p/RruqPIEzVl72zYzpXnB42R2KZJnJxxpOGIitFhTv6lGLYxTcLT+Q5WVqd7o4kxhVu5AGQk/NaqZKLBuq+ECA74rG6birjsN3NDQAqUSz+F05xWnxKHmVqRNZp5IzQ5/piwndqSM8Jag4Cm6io2+hnbDJQhUb9jCYuUw/+uusTcUWCjhmFsHcvrRjXLrWmEDdAf4uwtgitOqs0YzhVm542Su+g+r8M4x2eiCbZV6w60omQOUw/TvtTXO5tt9LagTitSuIytqW44Yr3zKovkqpys8yhQdKBN6D94X2c999mcTSmGzhhtnE6GmBFnKqvPCL9hQD3U+FFdEh7TypOdUYwdl2zkb/z2FvEwU+Wkfz8qC0NBgeBPbZgtjVvasbEei54qcOmRnyce1fLz2wFe+8/cUsjItKN0VONywVprV3Ga2TWqnesh+rdWWr70pU6c0L7z39xR2+RJF4Tl7vOE6+3U6D9dYYLO1HY1ieZ27OtWT3ft7CtWuHHVVttpwMZo055vVpHIvjNG6XBI2azQgqrTUL0949+8ppAWiHmXWj3WX5kvr7UPOytX6tLEszqmJamyqlbJSuWZPqdm0Xyvm+ZW265ec9nsKiZ66Jj5z/eJJpLDaPiS9uTqbO3JzLq+nyFqaW9eSWumuOzTONu5axcRcyhr3Ixec7vScwkbTrldqxmS+1zhRWs/BtF2lyMr1qrynr41qArWp4/Z4xwZ/cq3s2VfacHYK99c18mY3ua7t1mwVyqLnf7j9kW/+LkRGXQLMG03ZSZPWqnRA1hvhp7G1D1eftFuA3QbtL4Fd2a4NVlcAMmWqB6NR0zt29Yag3KmKrUIsevruwxO8Cwuf31KA/n7nGoxbDUK1MEYjzhNyerYMR/kZpWxUU049eudf6soip60AXlvCa7eTF4p3OZQrE7EAo806ktdrKkRkDRQI2FSkoq/vf4gKPv8+9vPM5fZVK7EdVys70Rd3rvo7cGqb0b5pOBVSfcHoLTEx0+o2iJoAm7C5DZeBYVt0ORBT1IXDsml1u2c3ks+c1qtyTCcSrrxlPGEq6pEFi4pY9OX1u+BHEwhPryI/z+yrS8K58GiPOl+h13Pt7R9ulUXasF3Fhn25YLggVsE5qPy68q0Q1aT1TW7CTpZjF/LdBSA6rZ10nE6QM+cQuPZZ2Xabr3T28IoUu/Hq++kaV3U2FIfeGirKIvwxkKPy2ciMphlqirmzzQk66Y5W/0balTUc80U5ExfUrkauw3ln1eYuQYi4zRwiM73VadPkxi9w4qwJC02ZLXj2RXZnZML/WY7CVd9N5pzcUGaqMj5lilpkgnxGLzjN6ss7CzlkTIHWonbkFlhTfv2lSyPXz6/c9/xvVSKbdnYLAWnRB2qGaoyq424FWX0ZpAqtzibVTAQhVOdQ7GSZqxHNzZmuqar4lClqkLnZnvwDihxUcKPspPOBq/p7o2xJR4eG06zEWJDfMrDirKVdnWA6YLSlBi1xNkwObt05cZ/KxoA+lVC/ZYKYEiRhoY2dzmIuMPEZpPImb240neUej0HvKPnuZTwqdjYvk1xaXlWpIq6MxCpDNMSFsJ0RexnDwXEnyN4ErTg+Waomrfcqrip0GOdMKxrjbJGYntvtQ9RzkyTtxFPx7dSWS3N11lt2gPvSUgPRRVWCPGD7Ev2s5bkqjeIk7BMI8IDa0LR1U+wK7Xr7aLEEOO4s3njg3wiTQagumhH4dWQQnujUnCRLMbZe9fRr5/GuuSDVLcFkdZFMPSZNWums0omO66XDdbpaaOCp47b8UxSenTDuZU4Tt4UTg2nyzkZynpZjTv4JsTdGJN2BWI1YbUlZnbfoTBDzlCt1s5lcr8ZBXBBEpr2rml/r4uzX95QaxXYIUjIBJudfr7H6XILrRAEVZBI06WCG/wGPXKgdgg/F4CEHK85OsTgLgk8lFfybVzMaZKEqzgovtisfJwpoekUm/6dCGZ1vtBwDWxSicTLrrF/DHUU9HfFGuoqzyhiGYFyvEGlpoCdaKqayq2T/QKrAirMVSKubuBGOJ6DzdeCTH8MCbeTp1XA0LnjzSdpwQg5DkhuYrM9KHbulPLj550yyjahw1sY1+DJRNoWJFSf+OF9ZEwYVq8njeg0ZvNvZaTLTkYhOh1egCUFx+UYGjLy0yMG1WlSxf0CnNjln6RrD9wfsdcVYp7YRyWfKxkBNK3LKBvITYp3rJh1pwyk5TVrTNTo0WDnHVWZiNAdgzIr6UolyZ+Mwq0DpmeCcRpF1khyYbB5Y2Q9R1mXwTmfp/BBmgWVtD2gdMPEMoFfdgJfr1HXQ2MZxsIr2zrics09vttwOZUJEAuVkxU9s8ERxTPYwCIy92+vKTcNJOTchVIpAUdbk0xPHEPZnN2krnDVwkkmO+uphFm6KSg8oqaJ0llaZCQf2x1nOQ+xrXjrnt2pxIAZiqs6OrnozKWqlLc4idq3BQpPGJe3zVVMWIy/wlIy52BX292wOKs7SsCNfwd4lwK5W094hlG1yNlwB8Xnkq7zqYr38EalNlN09qyMruAftTrqqfBaTrIkzQcF5r0g61+KRLvIfRmhTFywicBZTq8mJpw2pAP9K0lLztFiuaVnDPiZLm5Uq6HxE7zASXfkEUo2osXSAswrXyF89FZtwi305UawFbkLqEsq+3MBcx/AoqvzTq270AymLjpBPQs0D1ffXJT2y89lDcgNruFUypv/QsouIC/pWL7meAJ3zOlco6wx7UPTkb/qoEGrZrYM/BVagbq8wkYfrbIXeWJTHqWqSUwNvJ3FTroK3DzNX/Zrj7NVlmk1Z7SFvaQFtY2Lqwo5MatYSwSRQwt6oR5tFsdruFcMIrP62N2twZ5xPV5qc+QqDcBYPIh8Wqpy146w+oFZ+mK9Y+0ccIewMpXIuyztbcTLEExM6BXmr+eoUqAvm5mJjPSDuSFKhM7Z6DaAOCd7ssLXblFUnwjXjboZqeFAG2vLt2XgDgw/mgJzGaMOJpxS+uYD3cNzr6LmwEDOcQtAd/hVkNYNd1VbRFa8z+elEM1xna+Js9CN3mVzfTNYdf9a9CCYsenfT1DhrNzopaxdceSNT30eiYv6kNj6tYOEaUfsJUTsQO2iK8JnG6F2HaRkPViZok++qMkZauF9V9goY5OhU1Lu3TYZTsPrIIiGfOPRRTHoJxsbpQo6XTvifS1HhLCLSfp4DlG9WX9FMoQKaufhPlJ38BK3Y2eSZetUOOveaT7SRdTWhXu3FGUpOao5QrdQm7CslKY8SRfuv3cbCpNmsma6ct2LT5EbopNoBemY7dN8A7M0u1WN8deHKCxj1WcVZlHP6VzgqalHsLCE6clerAIS+K+6qO1MIc8nkqi6ETlb6sCfOut6p9Qn6ADrKoy+UbwveNlbxQV1gsjJtZrRrtiXOio+39ddT/EmFSnU2rioUqJvVKA7w0oh28OR1HTurgUP2Ddw8yFqL4RlzFXyvlNo3sOtbSa+notB2tbMKV+e8B7vA+KGnhAmuah2rTjO19kldUtzlrB1nhb6Q6U37vA7DAKrSFemzFUCudlfmlkYZj2RR4rGvFnUuripnRJGrcV7ticuVsL3HBZANacWGg4qqpyg4mfUBTLz/6bA0SMBaQm2xswp2fpUYTXt9BibrsTIEtZeYXN7Ej9c6C+b+GJuyk7Ut5uQJCfOQjgUZj2qdrZQxZ5vaBjHH6SR0lzxgjUOz4ags83TiIG/5ZyYzl6UFSnOA4MhW6EvE2awKiy8q7ArVyhgSxxPmMjDBc7L+qITbxNlpyp7uycKMR7LoiHw26RMtcalLwZz+E82F8nQldermLy6ONqXMcKIb1JX6K50+vEc7sdYp031ue8JPcQTOls7X+hpCshD0zoC1YrQpOznX/RyUxdlr8itaAf+EdlauPJnpCTQXSts4e8i9iGlMfA+STMFAb4te1fKHqCysJTGL7TDszkVKiDZfaQmGeS2vvAiHowAAD/JJREFUec00XEJZZyLMInS2kLKJ2yBBF2IGxfdec06j5wmVIBgni6x6znrmrVFBdvoZnKEs4C1c5IlZOyZsZq43l8XZCmTURTCZeQRUcn6d1I1JeEUBuJ5Q43o1B9oNC+o9RUm6pQFo7cKRnxYUeedTB+o5WwiOsx7X7BtU5MrVOtUUVy3KmrDHF8k2bGpnJ8sHzyU+Bdw4OzkrAdwsxIikz+nMOitZsFEoi467FzEloJe0QqW4tCKSjJosfarHrtdoIKs1jGHd3JNHOsBbtIjUUmAjXdgKtLXOeggiLSidV7rJx+hXuaQiq5tBqnS9yNmNipPvlp+t8rOomrKT9lk7Ksq6JlNHF0XvRWx8z7dXQW3VWxkXgiY6m7KTfZofEWeNfp8DVngA3q5w1UeWZS6OjrOeQGROV1uG0xTRHuthrbQ2zqbhueVmq6A02dtwQnV+QeiNeMSIvec7liDk3vOdaVzgEwjNEWajlQIEESOjOF8c94WvGUzuqe7rnXxH3EmvAbBZw6tfMJKASUN8H4tnMpMUOEqP7Fm/Y/Rnb5/7aorUJQj6BJHzccOj6Hu+Y/ciZt7zXb6PerXr86zvBD0o3vqEzRoh+Nqsd+JHlZpPGxs0RYDNGi0C9hPK6hD+dRs4qeu3rPTIRIYDOtjfY2rCdSSvx4t2UDyP/Pd8J+9FzL3nO98H5lTEDXL6JptJcZaWQ6ouWFJgKdhqRxHoHgdlxepeThrU6nWyDEwivUFTDGiuD/Y37wPeQeXIqUWWtdlkfxfmcGmU2QLqUvc5+XpB/edtAYE90bbH2RPe882wc0F9e7t+xg4TSEMWM39N+vOe/VkP6lonLr7MsksvY0DouMpH9sdYpbemup5X5Se7q9wPoVpOFT2ndcBV37PNqnTW3qHwFCnoy0VUBdTdpyJ77qQ9an7Pd1xdBNHT2gRUXPMAfeDDII6Ks2Et0HOZTVq74qQuw6WN1yJ0o7Q+mG+WkxF91qFwYlA5gmHOUXGWd9eMFc950LszVhXQFC8LCRmPZJGkcPy1iNXqSqqS8zQyTnA5xnCoCfgk19YDd4STGhyDI5RRWyChrqTP7DibM1TlVrAxEVyXibkGYk+cdYue5M7s89vIO5PD2mZKLbIIbjRDqJBmhsPzce8MFtbK+cy7XDx/HUjb3ZpA/V86T3/2mHAur8Zs1QJlrASLjICr0Bgr4o2lPU01EF9/xhDrb2hJNL/n22RhKpHRmJzpJKIqM3VsG07VjUvqbW++GGnnkRWmShyoF7HmIP5TFi6SmNQUF+nfBr0q+TcaJuExExxjdTa82u57vhMbWlSz9T3fVhX34rR/2SQyCoesFGzFKTHji9lug1C8EfUKPKMQ6NhJ2tS6SeyIs9Z1AnXemwoA7ReG/GrOe74zcXbve74ZU/JLYabRYtjUjSS4gS+ZXdtWNGgATyzteSOcddMxer046/ofqV864EEt5z3fFfls2pUmWS+B3MzNGw37Yqlua1RfaLWsputbuY4yHJJ1ZbE9cW6agq1qsii/bxA+dLNpNcCUWE+6X8wOSgL1hvVOS4Ee9C1mv23rK24kZVcLFYulkWhDboe2zuqU+QqBicrtbm9VZ4OpGCuKVrS1lhk3FzsmZ1s0oYEok21f1nD82ore0Q6sRdXH7DdeYogY2KyZGa8IEleTEm3I7tAWWhTq0QNPNzjfAuN1NiBZ+3TO2heVTW+lN244p8+pzBlOvfErFmczq5kOcdYSYcpupHxcmrxqn4yzmyq3EXtRx+Q4HLvnDE3UrR9jlYs5W6YugniPAv9pWQPnBPLqgrL9+wJbaDHAXuVFdUvrHq3N6yyuuf1g987lAmzWLlS0WyaUbyBttn6uMGeq1olSlOs1Ncu6pFxzgwsH1CtVFhfePm9b9NfJgPfvjn37aoEtSaiesnabjFR01mBHaKxHoerKyazegKmZESjX1Sijzxuzr5WQ31ZbZVjEThOvnPLqWrWVxgtryr9ZTpx2dolyZyH3LTeZHTmn1qqqlwH978ZZut2sOsO2xoQkFBpuQbFSss9/D9FYjybV3l5ycwyuFajW1SiTE4RStbWGoUymVm258Zqa568KN12olQHv+5HIaGsx1CKTU7cZDncYjpz9DVpaEDOTM33m1L+JtOlkycn+ZpS32G2RyenS+go3JmsNuGL34kbEStw0eP/2QNJ2prciBcm69cioarHSIpNS5XbJhuY2w363t2mpN1t6rGtSwICGkkjdA5uS0dRipEXmEF2NhhvFdqLYanHFEwEPpWZDVYuVFpl9gvvk95q9r9VWuV04zOg9FNXZAEKrG3unjjbxTpNvo9E+3h5l8zjvM1RrMdEikxEsVxetuS1ebuBItFqFXdK7jJ6gqFlxRrBFZ4tMqEJpqTg1IWK/6syuqLobGVuZIlUj2apt2Rbs1grev+pr+6TcVHSojKfBjvvl6kAJN3pz13k3barEjVSN3hftUvLsVuBes+KMYIvOFplQRYMW2Gu/XfJQW5miPWp7agXv393IKGqx0SKzV7JQfKeBIwFdRF+IxW17mRrbwsfIkODOiW1T+iH25xnQLnn/VtzZ4L77oTZlD5LZK1kovtPAkYAuoi/E4ra9TI1t4WNkDkE3wy2A3g7UAHo7EACaig6VOQTdDLcAejtQA+jtQABoKjpU5hB0M9wC6O1ADaC3AwGgqUg0/wbIWehmuAXQ24EaQG8HAoD3PfYbIDHkfgOkB7oZbgH0dqAG0NuBAOB+jf4GSAyZ3wA5xJFzxHcaOQpweMUTAQ+iI6nN/w2QJHK/AdLkx9n7s0cYOQpQWu8R/IUDVBzbDk/Z+b8BkvKgQbLeMEQ/3h0FtqG04umA43QcoSqi5vzfAPEljjkHa5U86GOBusYjRFgCHKfpoEZltOQNPMF3f34d/BJuXqbBzGlajjHbhgLbBVXuBHg4Vb4ai4pZC19//vDl9Ttr32Dfcwq70c1wC6C3AzWA3g4E8KhmUxHSUrhK+/rLr0fsGzSINGg5xspBgBNqngZ4KDURVTYVs1Z4sfb8p737s/fJZx8oPxQ1e10P4PYxHpz4rLhNxbyRLz/e1mlPx+0b7MSWlmOsHAQ4oeZpgIdSE1NlUbHFSovMIehmuAXQ24EaQG8HAkBTEeOI3wA5CN0MtwB6O1AD6O1AAIgcO/83QI5HN8MtgN4O1AB6OxAAIsf2cHZg4HTs4WwGEeGyQzskt8qSMrUCRx3fLs9INhW16TvEQLKkRdlxEnnhskODs4WSTUVt+g4xkCxpUXacRF647NDgbKFkU1GbvkMMJEtalB0nkRcuOzQ4WyjZVNSm7xADyZIWZcdJ5IXLDg3OFko2FbXpO8RAsqRF2XESeeGyQ4OzhZJNRW36DjGQLGlRdpzEwEBfQG8HBgYqAb0dGBioBPR2YGCgEtDbgYGBSkBvBwYGKgG9HRgYqAT0dmBgoBKwR/gjwDv3yNf3AMG9i/4zkPJ9Ye82K0VV5WxvSQUWUq6ka6fVJ9wpqZFtaNyNjN9pka0eraidMJF2KuVTnUsMqBWw8PRD4MunH9TTZgbPb70XJDz//oP/OqWwUlRV1vaGVGgh4Uq6dlp9yp2SGrmGJtxI+50W2erRitoJE2mnkj7VucSAyvq+M5Fh+PTG+fr1l3/xKn356fPXnz9sVIqp2radlopbiLiS8yfn1AZn8zUSOpNupPzOe542VFU7ZSLpVN6nOpfEXs5+iswF4cj4R+Tt5h99wdiAZmkQs52Vih2PupK3nYqWW7lBrka6oalZOOl3vte2T6yi2vGCjFMZu3UuSUBlfR8h934fnGmtnI2oytvOSx3D2bRTKQ5t18g09FjObvVoae0DOVvnEgJqBRgfOXe2AjsewsfQvVq+w9E5JGyVqypvO+VAzo2kKwl/8k7l5riUwxmXc06LnN85fmScL3Ur51XGqaRPxS5ZgGoJg8h5Fc3BfYejuXrQqtQaI2l7SyqafaSXDfFhSajPxb2tGtmGJgY7swbL8CPbozW14yZyC8OET3UuMaBexOB20oWhLjjWuNcVVZWzvSV1xF5XUn3CnZIa2YYeude10aMVtQ/b66pziQGV9QcGegN6OzAwUAno7cDAQCWgtwMDA5WA3g4MDFQCejswMFAJ6O3AwEAloLcDAwOVgN4ODAxUAno7MDBQCejtwMBAJaC3AwMDlYDeDgwMVAJ6OzAwUAno7cDAQCWgtwMDA5WA3g4MDFQCejswMFAJOFzjpzef3nz56XNw/PlPn2VRKPAErz7jb52/33zc+o4u/Yd8w8l5LiXR1H3yHTV38rVteG9ix/kHx6ix8OnNU7RRt2OyKBSQT/fRi0a2n7a+n0v8hpOzXEqixVd+R81dfG0aXvHl9ZvjhheOUWPh6d3Tuy8//eVv38rHjf/mj/KvfIb4n97Dq3+7FdlV6eHiW6P4RSNndXqLSxK3sHF3zjb6ejfONvn39ee/f2jOStwa9btfb9345ccPt2BFvpuT8/ntdx+e/1E4nf58JmfbXLofD/b7iu+ouZuv1f5R1vDwnNUNwfTHadS//srvHLszZ2tdkm846cXZSl/R0Xtytso/zBteHGfl6xrCJIgKOnF2yyV6w8ljcHaz++6bx9T694neYvBCOEuTx+/kLBJrFBbcmbOFLvEbTvpyttBXfkfN/TlbPLziBcVZTNLFR/irv/vw/PZV0CosuDNnC136eGxsONVXfkfN/TlbPLwvgLMKsU2RAE/6Rc/36PQHdCmJR/e1k39wjJoEyhoFdIbeZ1P8AV1K4tF97eQfHKNmYOBugN4ODAxUAno7MDBQCejtwMBAJaC3AwMDlYDeDgwMVAJ6OzAwUAno7cDAQCWgtwMDA5WA3g4MDFQCejtwJuS1xdv/8RIj/RRw8leH//TZ+0Dfin/Xcqepmt/X3tuq7Z9Af3BAbwf2AiJQZTy6+Dk/uvrKuXMJ3fs57GsEB5nyf197iuAgU9s/gf7ogN4O7ESMspq0JiJ9gr/64ztBf5/whzHxgSZBEU4+E/rqVgcftH31+Yljnvdz2DHKatLuNCVhPbQao6wm7X5Tg7N9kY2zr29fXn3Gx5e+vMabk1/Lx5fER/mZRo7CDnFAf1C3zbmDm42ze025trJxdq+p7Z9Af2xAbwfOhI5I/wvvOKb7jp/wZm79e8I3BvzANV/Dd/J2+yfze5YN+WyzqeLf195tqvNdwbsBvR04E/HR/YGL1G9g30ZVhqmn73+lx5rMTcq7OVtuquL3tXe3KvcT6C8B0NuBM6FHVz7a/BZnzttf6wFngasf8fENDeoTRiSqgAoaONtmqub3tXea2v4J9EcH9HbgTNirle/+i1ytyL+0WtER6aNMDp/fvvr3t/Cf376TjzWZ1UrLXleTqZrf197bqu2fQH9wQG8HBgYqAb0dGBioBPR2YGCgEtDbgYGBSkBvBwYGKgG9HRgYqAT0dmBgoBL/H3SH405xW7NrAAAAAElFTkSuQmCC" width="\textwidth" /></p>
<p>We observe how the chosen values for the parameters affect the generated data. For example, the relationship between the third input <span class="math inline">\(\mat{u}_3\)</span> and the output <span class="math inline">\(\mat{x}\)</span> is positive, indifferent and negative for the hidden states 1 through 3, the true slopes being 7, 0.1 and -5 respectively.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot_inputprob</span>(<span class="dt">u =</span> dataset$u, <span class="dt">p.mat =</span> dataset$p.mat, <span class="dt">z =</span> dataset$z)</code></pre></div>
<p><img 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" width="\textwidth" /></p>
<p>We then analyse the relationship between the input and the state probabilities, which are usually hidden in applications with real data. The pairs <span class="math inline">\(\{ \mat{u}_1, p(z_t = 1) \}\)</span>, <span class="math inline">\(\{ \mat{u}_2, p(z_t = 2) \}\)</span> and <span class="math inline">\(\{ \mat{u}_3, p(z_t = 3) \}\)</span> are most related due to the choice of values for the true regression parameters: those inputs take the largest weight in each state, namely <span class="math inline">\(w_{11} = 1.2\)</span>, <span class="math inline">\(w_{22} = 1.2\)</span> and <span class="math inline">\(w_{33} = 1.2\)</span>.</p>
</div>
<div id="estimation" class="section level3">
<h3><span class="header-section-number">2.2.2</span> Estimation</h3>
<p>Adopting a fully Bayesian approach, we run our software to draw samples from the posterior density of model parameters and other hidden quantities.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(rstan)</code></pre></div>
<pre><code>## Loading required package: ggplot2</code></pre>
<pre><code>## Loading required package: StanHeaders</code></pre>
<pre><code>## rstan (Version 2.14.2, packaged: 2017-03-19 00:42:29 UTC, GitRev: 5fa1e80eb817)</code></pre>
<pre><code>## For execution on a local, multicore CPU with excess RAM we recommend calling
## rstan_options(auto_write = TRUE)
## options(mc.cores = parallel::detectCores())</code></pre>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(shinystan)</code></pre></div>
<pre><code>## Loading required package: shiny</code></pre>
<pre><code>##
## This is shinystan version 2.3.0</code></pre>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">rstan_options</span>(<span class="dt">auto_write =</span> <span class="ot">TRUE</span>)
<span class="kw">options</span>(<span class="dt">mc.cores =</span> parallel::<span class="kw">detectCores</span>())
stan.model =<span class="st"> '../iohmm-reg/stan/iohmm-reg.stan'</span>
stan.data =<span class="st"> </span><span class="kw">list</span>(
<span class="dt">T =</span> T.length,
<span class="dt">K =</span> K,
<span class="dt">M =</span> M,
<span class="dt">u_tm =</span> <span class="kw">as.array</span>(u),
<span class="dt">x_t =</span> dataset$x
)
stan.fit <-<span class="st"> </span><span class="kw">stan</span>(<span class="dt">file =</span> stan.model,
<span class="dt">data =</span> stan.data, <span class="dt">verbose =</span> T,
<span class="dt">iter =</span> n.iter, <span class="dt">warmup =</span> n.warmup,
<span class="dt">thin =</span> n.thin, <span class="dt">chains =</span> n.chains,
<span class="dt">cores =</span> n.cores, <span class="dt">seed =</span> n.seed)</code></pre></div>
<pre><code>##
## TRANSLATING MODEL 'iohmm-reg' FROM Stan CODE TO C++ CODE NOW.
## successful in parsing the Stan model 'iohmm-reg'.
##
## CHECKING DATA AND PREPROCESSING FOR MODEL 'iohmm-reg' NOW.
##
## COMPILING MODEL 'iohmm-reg' NOW.
##
## STARTING SAMPLER FOR MODEL 'iohmm-reg' NOW.
##
## SAMPLING FOR MODEL 'iohmm-reg' NOW (CHAIN 1).
##
## Chain 1, Iteration: 1 / 400 [ 0%] (Warmup)
## Chain 1, Iteration: 40 / 400 [ 10%] (Warmup)
## Chain 1, Iteration: 80 / 400 [ 20%] (Warmup)
## Chain 1, Iteration: 120 / 400 [ 30%] (Warmup)
## Chain 1, Iteration: 160 / 400 [ 40%] (Warmup)
## Chain 1, Iteration: 200 / 400 [ 50%] (Warmup)
## Chain 1, Iteration: 201 / 400 [ 50%] (Sampling)
## Chain 1, Iteration: 240 / 400 [ 60%] (Sampling)
## Chain 1, Iteration: 280 / 400 [ 70%] (Sampling)
## Chain 1, Iteration: 320 / 400 [ 80%] (Sampling)
## Chain 1, Iteration: 360 / 400 [ 90%] (Sampling)
## Chain 1, Iteration: 400 / 400 [100%] (Sampling)
## Elapsed Time: 99.975 seconds (Warm-up)
## 47.487 seconds (Sampling)
## 147.462 seconds (Total)</code></pre>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">n.samples =<span class="st"> </span>(n.iter -<span class="st"> </span>n.warmup) *<span class="st"> </span>n.chains</code></pre></div>
</div>
<div id="convergence" class="section level3">
<h3><span class="header-section-number">2.2.3</span> Convergence</h3>
<p>We rely on several diagnostic statistics and plots provided by rstan <span class="citation">(Stan Development Team 2017a)</span> and shinystan <span class="citation">(Stan Development Team 2017b)</span> to assess mixing, convergence and the inexistence of divergences. We then extract the samples for some quantities of interest, namely the filtered probabilities vector <span class="math inline">\(\mat{\alpha}_t\)</span>, the smoothed probability vector <span class="math inline">\(\mat{\gamma}_t\)</span>, the most probable hidden path <span class="math inline">\(\mat{z}^*\)</span> and the fitted output <span class="math inline">\(\hat{x}\)</span>.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># MCMC Diagnostics --------------------------------------------------------</span>
<span class="kw">summary</span>(stan.fit,
<span class="dt">pars =</span> <span class="kw">c</span>(<span class="st">'p_1k'</span>, <span class="st">'w_km'</span>, <span class="st">'b_km'</span>, <span class="st">'s_k'</span>),
<span class="dt">probs =</span> <span class="kw">c</span>(<span class="fl">0.50</span>))$summary
<span class="co"># launch_shinystan(stan.fit)</span>
<span class="co"># Extraction --------------------------------------------------------------</span>
alpha_tk <-<span class="st"> </span><span class="kw">extract</span>(stan.fit, <span class="dt">pars =</span> <span class="st">'alpha_tk'</span>)[[<span class="dv">1</span>]]
gamma_tk <-<span class="st"> </span><span class="kw">extract</span>(stan.fit, <span class="dt">pars =</span> <span class="st">'gamma_tk'</span>)[[<span class="dv">1</span>]]
zstar_t <-<span class="st"> </span><span class="kw">extract</span>(stan.fit, <span class="dt">pars =</span> <span class="st">'zstar_t'</span>)[[<span class="dv">1</span>]]
hatx_t <-<span class="st"> </span><span class="kw">extract</span>(stan.fit, <span class="dt">pars =</span> <span class="st">'hatx_t'</span>)[[<span class="dv">1</span>]]</code></pre></div>
<table>
<caption>Estimated parameters and hidden quantities. <em>MCSE = Monte Carlo Standard Error, SE = Standard Error, Med = Median, ESS = Effective Sample Size</em>.</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">Mean</th>
<th align="right">MCSE</th>
<th align="right">SE</th>
<th align="right">Med</th>
<th align="right">ESS</th>
<th align="right"><span class="math inline">\(\hat{R}\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>p_1k[1]</td>
<td align="right">0.24</td>
<td align="right">0.01</td>
<td align="right">0.20</td>
<td align="right">0.20</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>p_1k[2]</td>
<td align="right">0.27</td>
<td align="right">0.02</td>
<td align="right">0.21</td>
<td align="right">0.23</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>p_1k[3]</td>
<td align="right">0.49</td>
<td align="right">0.02</td>
<td align="right">0.24</td>
<td align="right">0.47</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>w_km[1,1]</td>
<td align="right">-0.23</td>
<td align="right">0.30</td>
<td align="right">2.84</td>
<td align="right">-0.33</td>
<td align="right">89.03</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>w_km[1,2]</td>
<td align="right">0.10</td>
<td align="right">0.27</td>
<td align="right">2.95</td>
<td align="right">0.47</td>
<td align="right">122.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>w_km[1,3]</td>
<td align="right">0.39</td>
<td align="right">0.29</td>
<td align="right">2.66</td>
<td align="right">0.46</td>
<td align="right">84.81</td>
<td align="right">1.02</td>
</tr>
<tr class="odd">
<td>w_km[1,4]</td>
<td align="right">-0.18</td>
<td align="right">0.53</td>
<td align="right">3.30</td>
<td align="right">-0.17</td>
<td align="right">38.12</td>
<td align="right">1.01</td>
</tr>
<tr class="even">
<td>w_km[2,1]</td>
<td align="right">-0.09</td>
<td align="right">0.30</td>
<td align="right">2.85</td>
<td align="right">-0.19</td>
<td align="right">88.67</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>w_km[2,2]</td>
<td align="right">0.27</td>
<td align="right">0.27</td>
<td align="right">2.95</td>
<td align="right">0.64</td>
<td align="right">122.88</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>w_km[2,3]</td>
<td align="right">0.32</td>
<td align="right">0.29</td>
<td align="right">2.65</td>
<td align="right">0.36</td>
<td align="right">84.84</td>
<td align="right">1.02</td>
</tr>
<tr class="odd">
<td>w_km[2,4]</td>
<td align="right">-0.53</td>
<td align="right">0.53</td>
<td align="right">3.30</td>
<td align="right">-0.54</td>
<td align="right">38.28</td>
<td align="right">1.01</td>
</tr>
<tr class="even">
<td>w_km[3,1]</td>
<td align="right">-0.28</td>
<td align="right">0.30</td>
<td align="right">2.86</td>
<td align="right">-0.38</td>
<td align="right">90.44</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>w_km[3,2]</td>
<td align="right">0.14</td>
<td align="right">0.27</td>
<td align="right">2.96</td>
<td align="right">0.56</td>
<td align="right">122.08</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>w_km[3,3]</td>
<td align="right">0.17</td>
<td align="right">0.29</td>
<td align="right">2.65</td>
<td align="right">0.15</td>
<td align="right">86.48</td>
<td align="right">1.02</td>
</tr>
<tr class="odd">
<td>w_km[3,4]</td>
<td align="right">-0.37</td>
<td align="right">0.53</td>
<td align="right">3.30</td>
<td align="right">-0.43</td>
<td align="right">38.08</td>
<td align="right">1.01</td>
</tr>
<tr class="even">
<td>b_km[1,1]</td>
<td align="right">0.01</td>
<td align="right">0.03</td>
<td align="right">0.39</td>
<td align="right">0.00</td>
<td align="right">181.98</td>
<td align="right">1.02</td>
</tr>
<tr class="odd">
<td>b_km[1,2]</td>
<td align="right">-1.05</td>
<td align="right">0.02</td>
<td align="right">0.29</td>
<td align="right">-1.06</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>b_km[1,3]</td>
<td align="right">-4.91</td>
<td align="right">0.03</td>
<td align="right">0.36</td>
<td align="right">-4.90</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>b_km[1,4]</td>
<td align="right">-0.09</td>
<td align="right">0.02</td>
<td align="right">0.33</td>
<td align="right">-0.09</td>
<td align="right">200.00</td>
<td align="right">1.01</td>
</tr>
<tr class="even">
<td>b_km[2,1]</td>
<td align="right">0.86</td>
<td align="right">0.01</td>
<td align="right">0.13</td>
<td align="right">0.86</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>b_km[2,2]</td>
<td align="right">5.00</td>
<td align="right">0.01</td>
<td align="right">0.08</td>
<td align="right">5.00</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>b_km[2,3]</td>
<td align="right">0.15</td>
<td align="right">0.01</td>
<td align="right">0.10</td>
<td align="right">0.15</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>b_km[2,4]</td>
<td align="right">-0.28</td>
<td align="right">0.01</td>
<td align="right">0.13</td>
<td align="right">-0.28</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>b_km[3,1]</td>
<td align="right">5.04</td>
<td align="right">0.00</td>
<td align="right">0.03</td>
<td align="right">5.04</td>
<td align="right">200.00</td>
<td align="right">1.01</td>
</tr>
<tr class="odd">
<td>b_km[3,2]</td>
<td align="right">6.01</td>
<td align="right">0.00</td>
<td align="right">0.02</td>
<td align="right">6.01</td>
<td align="right">200.00</td>
<td align="right">1.01</td>
</tr>
<tr class="even">
<td>b_km[3,3]</td>
<td align="right">6.99</td>
<td align="right">0.00</td>
<td align="right">0.02</td>
<td align="right">6.99</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>b_km[3,4]</td>
<td align="right">0.46</td>
<td align="right">0.00</td>
<td align="right">0.03</td>
<td align="right">0.47</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>s_k[1]</td>
<td align="right">2.74</td>
<td align="right">0.01</td>
<td align="right">0.21</td>
<td align="right">2.74</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="odd">
<td>s_k[2]</td>
<td align="right">1.00</td>
<td align="right">0.01</td>
<td align="right">0.09</td>
<td align="right">0.99</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
<tr class="even">
<td>s_k[3]</td>
<td align="right">0.21</td>
<td align="right">0.00</td>
<td align="right">0.02</td>
<td align="right">0.21</td>
<td align="right">200.00</td>
<td align="right">1.00</td>
</tr>
</tbody>
</table>
<p>While mixing and convergence is extremely efficient, as expected when dealing with generated data, we note that the regression parameters for the latent states are the worst performers. The small effective size translates into a high Monte Carlo standard error and broader credibility intervals. One possible reason is that softmax is invariant to change in location, thus the parameters of a multinomial regression do not have a natural center and become harder to estimate.</p>
</div>
<div id="state-probability" class="section level3">
<h3><span class="header-section-number">2.2.4</span> State probability</h3>
<p>We assess that our software recover the hidden states correctly. Due to label switching, the states generated under the labels 1 through 3 were recovered in inverse order. In consequence, we decide to relabel the observations based on the best fit. This would not prove to be a problem with real data considering that the hidden states are never observed.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Relabelling (ugly hack edition) -----------------------------------------</span>
dataset$zrelab <-<span class="st"> </span><span class="kw">rep</span>(<span class="dv">0</span>, T)
hard <-<span class="st"> </span><span class="kw">sapply</span>(<span class="dv">1</span>:T.length, function(t, med) {
<span class="kw">which.max</span>(med[t, ])
}, <span class="dt">med =</span> <span class="kw">apply</span>(alpha_tk, <span class="kw">c</span>(<span class="dv">2</span>, <span class="dv">3</span>),
function(x) {
<span class="kw">quantile</span>(x, <span class="kw">c</span>(<span class="fl">0.50</span>)) }))
tab <-<span class="st"> </span><span class="kw">table</span>(<span class="dt">hard =</span> hard, <span class="dt">original =</span> dataset$z)
for (k in <span class="dv">1</span>:K) {
dataset$zrelab[<span class="kw">which</span>(dataset$z ==<span class="st"> </span>k)] <-<span class="st"> </span><span class="kw">which.max</span>(tab[, k])
}
<span class="kw">print</span>(<span class="st">"Label re-imputation (relabeling due to switching labels)"</span>)</code></pre></div>
<pre><code>## [1] "Label re-imputation (relabeling due to switching labels)"</code></pre>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">table</span>(<span class="dt">new =</span> dataset$zrelab, <span class="dt">original =</span> dataset$z)</code></pre></div>
<pre><code>## original
## new 1 2 3
## 1 0 0 102
## 2 0 108 0
## 3 90 0 0</code></pre>
<p>Point estimates and credibility intervals are provided by rstan’s <code>{r}summary</code> function.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">print</span>(<span class="st">"Estimated initial state probabilities"</span>)
<span class="kw">summary</span>(stan.fit,
<span class="dt">pars =</span> <span class="kw">c</span>(<span class="st">'p_1k'</span>),
<span class="dt">probs =</span> <span class="kw">c</span>(<span class="fl">0.10</span>, <span class="fl">0.50</span>, <span class="fl">0.90</span>))$summary[, <span class="kw">c</span>(<span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>)]
<span class="kw">print</span>(<span class="st">"Estimated probabilities in the transition matrix"</span>)
<span class="kw">head</span>(<span class="kw">summary</span>(stan.fit,
<span class="dt">pars =</span> <span class="kw">c</span>(<span class="st">'A_ij'</span>),
<span class="dt">probs =</span> <span class="kw">c</span>(<span class="fl">0.10</span>, <span class="fl">0.50</span>, <span class="fl">0.90</span>))$summary[, <span class="kw">c</span>(<span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>)])
<span class="kw">print</span>(<span class="st">"Estimated regressors of hidden states"</span>)
<span class="kw">summary</span>(stan.fit,
<span class="dt">pars =</span> <span class="kw">c</span>(<span class="st">'w_km'</span>),
<span class="dt">probs =</span> <span class="kw">c</span>(<span class="fl">0.10</span>, <span class="fl">0.50</span>, <span class="fl">0.90</span>))$summary[, <span class="kw">c</span>(<span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>)]
<span class="kw">print</span>(<span class="st">"Estimated regressors and standard deviation of observations in each state"</span>)
<span class="kw">summary</span>(stan.fit,
<span class="dt">pars =</span> <span class="kw">c</span>(<span class="st">'b_km'</span>, <span class="st">'s_k'</span>),
<span class="dt">probs =</span> <span class="kw">c</span>(<span class="fl">0.10</span>, <span class="fl">0.50</span>, <span class="fl">0.90</span>))$summary[, <span class="kw">c</span>(<span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">4</span>, <span class="dv">5</span>, <span class="dv">6</span>)]</code></pre></div>
<table>
<caption>Estimated initial state probabilities.</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">Mean</th>
<th align="right">SE</th>
<th align="right">10%</th>
<th align="right">Med</th>
<th align="right">90%</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>p_1k[1]</td>
<td align="right">0.24</td>
<td align="right">0.20</td>
<td align="right">0.03</td>
<td align="right">0.20</td>
<td align="right">0.52</td>
</tr>
<tr class="even">
<td>p_1k[2]</td>
<td align="right">0.27</td>
<td align="right">0.21</td>
<td align="right">0.03</td>
<td align="right">0.23</td>
<td align="right">0.55</td>
</tr>
<tr class="odd">
<td>p_1k[3]</td>
<td align="right">0.49</td>
<td align="right">0.24</td>
<td align="right">0.16</td>
<td align="right">0.47</td>
<td align="right">0.85</td>
</tr>
</tbody>
</table>
<table>
<caption>Estimated probabilities in the transition matrix.</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">Mean</th>
<th align="right">SE</th>
<th align="right">10%</th>
<th align="right">Med</th>
<th align="right">90%</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>A_ij[1,1]</td>
<td align="right">0.24</td>
<td align="right">0.20</td>
<td align="right">0.03</td>
<td align="right">0.20</td>
<td align="right">0.52</td>
</tr>
<tr class="even">
<td>A_ij[1,2]</td>
<td align="right">0.27</td>
<td align="right">0.21</td>
<td align="right">0.03</td>
<td align="right">0.23</td>
<td align="right">0.55</td>
</tr>
<tr class="odd">
<td>A_ij[1,3]</td>
<td align="right">0.49</td>
<td align="right">0.24</td>
<td align="right">0.16</td>
<td align="right">0.47</td>
<td align="right">0.85</td>
</tr>
<tr class="even">
<td>A_ij[2,1]</td>
<td align="right">0.33</td>
<td align="right">0.02</td>
<td align="right">0.31</td>
<td align="right">0.33</td>
<td align="right">0.36</td>
</tr>
<tr class="odd">
<td>A_ij[2,2]</td>
<td align="right">0.36</td>
<td align="right">0.02</td>
<td align="right">0.33</td>
<td align="right">0.36</td>
<td align="right">0.39</td>
</tr>
<tr class="even">
<td>A_ij[2,3]</td>
<td align="right">0.31</td>
<td align="right">0.02</td>
<td align="right">0.29</td>
<td align="right">0.31</td>
<td align="right">0.33</td>
</tr>
</tbody>
</table>
<table>
<caption>Estimated regressors of hidden states.</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">Mean</th>
<th align="right">SE</th>
<th align="right">10%</th>
<th align="right">Med</th>
<th align="right">90%</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>w_km[1,1]</td>
<td align="right">-0.23</td>
<td align="right">2.84</td>
<td align="right">-3.50</td>
<td align="right">-0.33</td>
<td align="right">3.30</td>
</tr>
<tr class="even">
<td>w_km[1,2]</td>
<td align="right">0.10</td>
<td align="right">2.95</td>
<td align="right">-3.56</td>
<td align="right">0.47</td>
<td align="right">3.64</td>
</tr>
<tr class="odd">
<td>w_km[1,3]</td>
<td align="right">0.39</td>
<td align="right">2.66</td>
<td align="right">-3.27</td>
<td align="right">0.46</td>
<td align="right">3.51</td>
</tr>
<tr class="even">
<td>w_km[1,4]</td>
<td align="right">-0.18</td>
<td align="right">3.30</td>
<td align="right">-4.51</td>
<td align="right">-0.17</td>
<td align="right">4.22</td>
</tr>
<tr class="odd">
<td>w_km[2,1]</td>
<td align="right">-0.09</td>
<td align="right">2.85</td>
<td align="right">-3.57</td>
<td align="right">-0.19</td>
<td align="right">3.43</td>
</tr>
<tr class="even">
<td>w_km[2,2]</td>
<td align="right">0.27</td>
<td align="right">2.95</td>
<td align="right">-3.49</td>
<td align="right">0.64</td>
<td align="right">3.84</td>
</tr>
<tr class="odd">
<td>w_km[2,3]</td>
<td align="right">0.32</td>
<td align="right">2.65</td>
<td align="right">-3.23</td>
<td align="right">0.36</td>
<td align="right">3.52</td>
</tr>
<tr class="even">
<td>w_km[2,4]</td>
<td align="right">-0.53</td>
<td align="right">3.30</td>
<td align="right">-4.89</td>
<td align="right">-0.54</td>
<td align="right">3.83</td>
</tr>
<tr class="odd">
<td>w_km[3,1]</td>
<td align="right">-0.28</td>
<td align="right">2.86</td>
<td align="right">-3.69</td>
<td align="right">-0.38</td>
<td align="right">3.36</td>
</tr>
<tr class="even">
<td>w_km[3,2]</td>
<td align="right">0.14</td>
<td align="right">2.96</td>
<td align="right">-3.65</td>
<td align="right">0.56</td>
<td align="right">3.68</td>
</tr>
<tr class="odd">
<td>w_km[3,3]</td>
<td align="right">0.17</td>
<td align="right">2.65</td>
<td align="right">-3.46</td>
<td align="right">0.15</td>
<td align="right">3.37</td>
</tr>
<tr class="even">
<td>w_km[3,4]</td>
<td align="right">-0.37</td>
<td align="right">3.30</td>
<td align="right">-4.72</td>
<td align="right">-0.43</td>
<td align="right">4.01</td>
</tr>
</tbody>
</table>
<table>
<caption>Estimated regression parameters and standard deviation of observations in each state.</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">Mean</th>
<th align="right">SE</th>
<th align="right">10%</th>
<th align="right">Med</th>
<th align="right">90%</th>
</tr>
</thead>
<tbody>
<tr class="odd">