radu
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SSMBook
mirror of https://github.com/probml/ssm-book.git


			
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   State Space Models: A Modern Approach
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     HMM filtering (forwards algorithm)
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     HMM smoothing (forwards-backwards algorithm)
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     Viterbi algorithm
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     Parallel HMM  smoothing
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     Forwards-filtering backwards sampling
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   Particle filtering
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   Sequential Monte Carlo
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     Stochastic Gradient Descent (SGD)
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     Variational Bayes (VB)
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     Markov Chain Monte Carlo (MCMC)
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   Multi-target tracking
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   Data assimilation using Ensemble Kalman filter
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   Bayesian non-parametric SSMs
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   Changepoint detection
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   Timeseries forecasting
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   Markovian Gaussian processes
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   Differential equations and SSMs
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   Optimal control
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   What are State Space Models?
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   Hidden Markov Models
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                <h1>What are State Space Models?</h1>
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   Hidden Markov Models
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># meta-data does not work yet in VScode</span>
<span class="c1"># https://github.com/microsoft/vscode-jupyter/issues/1121</span>

<span class="p">{</span>
    <span class="s2">&quot;tags&quot;</span><span class="p">:</span> <span class="p">[</span>
        <span class="s2">&quot;hide-cell&quot;</span>
    <span class="p">]</span>
<span class="p">}</span>


<span class="c1">### Install necessary libraries</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">jax</span>
<span class="k">except</span><span class="p">:</span>
    <span class="c1"># For cuda version, see https://github.com/google/jax#installation</span>
    <span class="o">%</span><span class="k">pip</span> install --upgrade &quot;jax[cpu]&quot; 
    <span class="kn">import</span> <span class="nn">jax</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">jsl</span>
<span class="k">except</span><span class="p">:</span>
    <span class="o">%</span><span class="k">pip</span> install git+https://github.com/probml/jsl
    <span class="kn">import</span> <span class="nn">jsl</span>

<span class="k">try</span><span class="p">:</span>
    <span class="kn">import</span> <span class="nn">rich</span>
<span class="k">except</span><span class="p">:</span>
    <span class="o">%</span><span class="k">pip</span> install rich
    <span class="kn">import</span> <span class="nn">rich</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">{</span>
    <span class="s2">&quot;tags&quot;</span><span class="p">:</span> <span class="p">[</span>
        <span class="s2">&quot;hide-cell&quot;</span>
    <span class="p">]</span>
<span class="p">}</span>


<span class="c1">### Import standard libraries</span>

<span class="kn">import</span> <span class="nn">abc</span>
<span class="kn">from</span> <span class="nn">dataclasses</span> <span class="kn">import</span> <span class="n">dataclass</span>
<span class="kn">import</span> <span class="nn">functools</span>
<span class="kn">import</span> <span class="nn">itertools</span>

<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Any</span><span class="p">,</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">NamedTuple</span><span class="p">,</span> <span class="n">Optional</span><span class="p">,</span> <span class="n">Union</span><span class="p">,</span> <span class="n">Tuple</span>

<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>


<span class="kn">import</span> <span class="nn">jax</span>
<span class="kn">import</span> <span class="nn">jax.numpy</span> <span class="k">as</span> <span class="nn">jnp</span>
<span class="kn">from</span> <span class="nn">jax</span> <span class="kn">import</span> <span class="n">lax</span><span class="p">,</span> <span class="n">vmap</span><span class="p">,</span> <span class="n">jit</span><span class="p">,</span> <span class="n">grad</span>
<span class="kn">from</span> <span class="nn">jax.scipy.special</span> <span class="kn">import</span> <span class="n">logit</span>
<span class="kn">from</span> <span class="nn">jax.nn</span> <span class="kn">import</span> <span class="n">softmax</span>
<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
<span class="kn">from</span> <span class="nn">jax.random</span> <span class="kn">import</span> <span class="n">PRNGKey</span><span class="p">,</span> <span class="n">split</span>

<span class="kn">import</span> <span class="nn">inspect</span>
<span class="kn">import</span> <span class="nn">inspect</span> <span class="k">as</span> <span class="nn">py_inspect</span>
<span class="kn">import</span> <span class="nn">rich</span>

<span class="kn">from</span> <span class="nn">rich</span> <span class="kn">import</span> <span class="n">inspect</span> <span class="k">as</span> <span class="n">r_inspect</span>
<span class="kn">from</span> <span class="nn">rich</span> <span class="kn">import</span> <span class="nb">print</span> <span class="k">as</span> <span class="n">r_print</span>

<span class="k">def</span> <span class="nf">print_source</span><span class="p">(</span><span class="n">fname</span><span class="p">):</span>
    <span class="n">r_print</span><span class="p">(</span><span class="n">py_inspect</span><span class="o">.</span><span class="n">getsource</span><span class="p">(</span><span class="n">fname</span><span class="p">))</span>
</pre></div>
</div>
</div>
</div>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}\newcommand\floor[1]{\lfloor#1\rfloor}\\\newcommand{\real}{\mathbb{R}}\\% Numbers
\newcommand{\vzero}{\boldsymbol{0}}
\newcommand{\vone}{\boldsymbol{1}}\\% Greek https://www.latex-tutorial.com/symbols/greek-alphabet/
\newcommand{\valpha}{\boldsymbol{\alpha}}
\newcommand{\vbeta}{\boldsymbol{\beta}}
\newcommand{\vchi}{\boldsymbol{\chi}}
\newcommand{\vdelta}{\boldsymbol{\delta}}
\newcommand{\vDelta}{\boldsymbol{\Delta}}
\newcommand{\vepsilon}{\boldsymbol{\epsilon}}
\newcommand{\vzeta}{\boldsymbol{\zeta}}
\newcommand{\vXi}{\boldsymbol{\Xi}}
\newcommand{\vell}{\boldsymbol{\ell}}
\newcommand{\veta}{\boldsymbol{\eta}}
%\newcommand{\vEta}{\boldsymbol{\Eta}}
\newcommand{\vgamma}{\boldsymbol{\gamma}}
\newcommand{\vGamma}{\boldsymbol{\Gamma}}
\newcommand{\vmu}{\boldsymbol{\mu}}
\newcommand{\vmut}{\boldsymbol{\tilde{\mu}}}
\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vkappa}{\boldsymbol{\kappa}}
\newcommand{\vlambda}{\boldsymbol{\lambda}}
\newcommand{\vLambda}{\boldsymbol{\Lambda}}
\newcommand{\vLambdaBar}{\overline{\vLambda}}
%\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vomega}{\boldsymbol{\omega}}
\newcommand{\vOmega}{\boldsymbol{\Omega}}
\newcommand{\vphi}{\boldsymbol{\phi}}
\newcommand{\vvarphi}{\boldsymbol{\varphi}}
\newcommand{\vPhi}{\boldsymbol{\Phi}}
\newcommand{\vpi}{\boldsymbol{\pi}}
\newcommand{\vPi}{\boldsymbol{\Pi}}
\newcommand{\vpsi}{\boldsymbol{\psi}}
\newcommand{\vPsi}{\boldsymbol{\Psi}}
\newcommand{\vrho}{\boldsymbol{\rho}}
\newcommand{\vtheta}{\boldsymbol{\theta}}
\newcommand{\vthetat}{\boldsymbol{\tilde{\theta}}}
\newcommand{\vTheta}{\boldsymbol{\Theta}}
\newcommand{\vsigma}{\boldsymbol{\sigma}}
\newcommand{\vSigma}{\boldsymbol{\Sigma}}
\newcommand{\vSigmat}{\boldsymbol{\tilde{\Sigma}}}
\newcommand{\vsigmoid}{\vsigma}
\newcommand{\vtau}{\boldsymbol{\tau}}
\newcommand{\vxi}{\boldsymbol{\xi}}\\
% Lower Roman (Vectors)
\newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vBt}{\mathbf{\tilde{B}}}
\newcommand{\vc}{\mathbf{c}}
\newcommand{\vct}{\mathbf{\tilde{c}}}
\newcommand{\vd}{\mathbf{d}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vf}{\mathbf{f}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\vh}{\mathbf{h}}
%\newcommand{\myvh}{\mathbf{h}}
\newcommand{\vi}{\mathbf{i}}
\newcommand{\vj}{\mathbf{j}}
\newcommand{\vk}{\mathbf{k}}
\newcommand{\vl}{\mathbf{l}}
\newcommand{\vm}{\mathbf{m}}
\newcommand{\vn}{\mathbf{n}}
\newcommand{\vo}{\mathbf{o}}
\newcommand{\vp}{\mathbf{p}}
\newcommand{\vq}{\mathbf{q}}
\newcommand{\vr}{\mathbf{r}}
\newcommand{\vs}{\mathbf{s}}
\newcommand{\vt}{\mathbf{t}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vws}{\vw_s}
\newcommand{\vwt}{\mathbf{\tilde{w}}}
\newcommand{\vWt}{\mathbf{\tilde{W}}}
\newcommand{\vwh}{\hat{\vw}}
\newcommand{\vx}{\mathbf{x}}
%\newcommand{\vx}{\mathbf{x}}
\newcommand{\vxt}{\mathbf{\tilde{x}}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vyt}{\mathbf{\tilde{y}}}
\newcommand{\vz}{\mathbf{z}}
%\newcommand{\vzt}{\mathbf{\tilde{z}}}\\
% Upper Roman (Matrices)
\newcommand{\vA}{\mathbf{A}}
\newcommand{\vB}{\mathbf{B}}
\newcommand{\vC}{\mathbf{C}}
\newcommand{\vD}{\mathbf{D}}
\newcommand{\vE}{\mathbf{E}}
\newcommand{\vF}{\mathbf{F}}
\newcommand{\vG}{\mathbf{G}}
\newcommand{\vH}{\mathbf{H}}
\newcommand{\vI}{\mathbf{I}}
\newcommand{\vJ}{\mathbf{J}}
\newcommand{\vK}{\mathbf{K}}
\newcommand{\vL}{\mathbf{L}}
\newcommand{\vM}{\mathbf{M}}
\newcommand{\vMt}{\mathbf{\tilde{M}}}
\newcommand{\vN}{\mathbf{N}}
\newcommand{\vO}{\mathbf{O}}
\newcommand{\vP}{\mathbf{P}}
\newcommand{\vQ}{\mathbf{Q}}
\newcommand{\vR}{\mathbf{R}}
\newcommand{\vS}{\mathbf{S}}
\newcommand{\vT}{\mathbf{T}}
\newcommand{\vU}{\mathbf{U}}
\newcommand{\vV}{\mathbf{V}}
\newcommand{\vW}{\mathbf{W}}
\newcommand{\vX}{\mathbf{X}}
%\newcommand{\vXs}{\vX_{\vs}}
\newcommand{\vXs}{\vX_{s}}
\newcommand{\vXt}{\mathbf{\tilde{X}}}
\newcommand{\vY}{\mathbf{Y}}
\newcommand{\vZ}{\mathbf{Z}}
\newcommand{\vZt}{\mathbf{\tilde{Z}}}
\newcommand{\vzt}{\mathbf{\tilde{z}}}\\
%%%%
\newcommand{\hidden}{\vz}
\newcommand{\obs}{\vy}
\newcommand{\inputs}{\vu}
\newcommand{\input}{\inputs}\\\newcommand{\hmmTrans}{\vA}
\newcommand{\hmmObs}{\vB}
\newcommand{\hmmInit}{\vpi}
\newcommand{\hmmhid}{\hidden}
\newcommand{\hmmobs}{\obs}\\\newcommand{\ldsDyn}{\vA}
\newcommand{\ldsObs}{\vC}
\newcommand{\ldsDynIn}{\vB}
\newcommand{\ldsObsIn}{\vD}
\newcommand{\ldsDynNoise}{\vQ}
\newcommand{\ldsObsNoise}{\vR}\\\newcommand{\ssmDyn}{f}
\newcommand{\ssmObs}{h}\end{aligned}\end{align} \]</div>
<div class="tex2jax_ignore mathjax_ignore section" id="what-are-state-space-models">
<span id="sec-ssm-intro"></span><h1>What are State Space Models?<a class="headerlink" href="#what-are-state-space-models" title="Permalink to this headline">¶</a></h1>
<p>A state space model or SSM
is a partially observed Markov model,
in which the hidden state,  <span class="math notranslate nohighlight">\(\hidden_t\)</span>,
evolves over time according to a Markov process,
possibly conditional on external inputs or controls <span class="math notranslate nohighlight">\(\input_t\)</span>,
and each hidden state generates some
observations <span class="math notranslate nohighlight">\(\obs_t\)</span> at each time step.
(In this book, we mostly focus on discrete time systems,
although  we consider the continuous-time case in  XXX.)
We get to see the observations, but not the hidden state.
Our main goal is to infer the hidden state given the observations.
However, we can also use the model to predict future observations,
by first predicting future hidden states, and then predicting
what observations they might generate.
By using  a hidden state <span class="math notranslate nohighlight">\(\hidden_t\)</span>
to represent the past observations, <span class="math notranslate nohighlight">\(\obs_{1:t-1}\)</span>,
the  model can have ``infinite’’ memory,
unlike a standard Markov model.</p>
<p>Formally we can define an SSM
as the following joint distribution:</p>
<div class="math notranslate nohighlight" id="equation-ssmfull">
<span class="eqno">(2)<a class="headerlink" href="#equation-ssmfull" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T}|\inputs_{1:T})
 = \left[ p(\hmmhid_1|\inputs_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1}) \right]\]</div>
<p>where <span class="math notranslate nohighlight">\(p(\hmmhid_t|\hmmhid_{t-1},\inputs_t)\)</span> is the
transition model,
<span class="math notranslate nohighlight">\(p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1})\)</span> is the
observation model,
and <span class="math notranslate nohighlight">\(\inputs_{t}\)</span> is an optional input or action.
See <a class="reference internal" href="#ssm-ar"><span class="std std-numref">Figure 3</span></a>
for an illustration of the corresponding graphical model.</p>
<div class="figure align-default" id="ssm-ar">
<a class="reference internal image-reference" href="../../_images/SSM-AR-inputs.png"><img alt="../../_images/SSM-AR-inputs.png" src="../../_images/SSM-AR-inputs.png" style="width: 152.0px; height: 165.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 3 </span><span class="caption-text">Illustration of an SSM as a graphical model.</span><a class="headerlink" href="#ssm-ar" title="Permalink to this image">¶</a></p>
</div>
<p>We often consider a simpler setting in which there
are no external inputs,
and the observations are conditionally independent of each other
(rather than having Markovian dependencies) given the hidden state.
In this case the joint simplifies to</p>
<div class="math notranslate nohighlight" id="equation-ssmsimplified">
<span class="eqno">(3)<a class="headerlink" href="#equation-ssmsimplified" title="Permalink to this equation">¶</a></span>\[p(\hmmobs_{1:T},\hmmhid_{1:T})
 = \left[ p(\hmmhid_1) \prod_{t=2}^{T}
 p(\hmmhid_t|\hmmhid_{t-1}) \right]
 \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t \right]\]</div>
<p>See <a class="reference internal" href="#ssm-simplified"><span class="std std-numref">Figure 4</span></a>
for an illustration of the corresponding graphical model.
Compare <a class="reference internal" href="#equation-ssmfull">(2)</a> and <a class="reference internal" href="#equation-ssmsimplified">(3)</a>.</p>
<div class="figure align-default" id="ssm-simplified">
<a class="reference internal image-reference" href="../../_images/SSM-simplified.png"><img alt="../../_images/SSM-simplified.png" src="../../_images/SSM-simplified.png" style="width: 136.0px; height: 98.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 4 </span><span class="caption-text">Illustration of a simplified SSM.</span><a class="headerlink" href="#ssm-simplified" title="Permalink to this image">¶</a></p>
</div>
</div>
<div class="tex2jax_ignore mathjax_ignore section" id="hidden-markov-models">
<span id="sec-hmm-intro"></span><h1>Hidden Markov Models<a class="headerlink" href="#hidden-markov-models" title="Permalink to this headline">¶</a></h1>
<p>In this section, we discuss the
hidden Markov model or HMM,
which is a state space model in which the hidden states
are discrete, so <span class="math notranslate nohighlight">\(\hmmhid_t \in \{1,\ldots, K\}\)</span>.
The observations may be discrete,
<span class="math notranslate nohighlight">\(\hmmobs_t \in \{1,\ldots, C\}\)</span>,
or continuous,
<span class="math notranslate nohighlight">\(\hmmobs_t \in \real^D\)</span>,
or some combination,
as we illustrate below.
More details can be found in e.g.,
<span id="id1">[<a class="reference internal" href="../../bib.html#id34" title="O. Cappe, E. Moulines, and T. Ryden. Inference in Hidden Markov Models. Springer, 2005.">CMR05</a>, <a class="reference internal" href="../../bib.html#id33" title="A. Fraser. Hidden Markov Models and Dynamical Systems. SIAM Press, 2008.">Fra08</a>, <a class="reference internal" href="../../bib.html#id32" title="L. R. Rabiner. A tutorial on Hidden Markov Models and selected applications in speech recognition. Proc. of the IEEE, 77(2):257–286, 1989.">Rab89</a>]</span>.
For an interactive introduction,
see <a class="reference external" href="https://nipunbatra.github.io/hmm/">https://nipunbatra.github.io/hmm/</a>.</p>
<div class="section" id="example-casino-hmm">
<h2>Example: Casino HMM<a class="headerlink" href="#example-casino-hmm" title="Permalink to this headline">¶</a></h2>
<p>To illustrate HMMs with categorical observation model,
we consider the “Ocassionally dishonest casino” model from <span id="id2">[<a class="reference internal" href="../../bib.html#id3" title="R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1998.">DEKM98</a>]</span>.
There are 2 hidden states, representing whether the dice being used in the casino is fair or loaded.
Each state defines a distribution over the 6 possible observations.</p>
<p>The transition model is denoted by</p>
<div class="math notranslate nohighlight">
\[p(z_t=j|z_{t-1}=i) = \hmmTrans_{ij}\]</div>
<p>Here the <span class="math notranslate nohighlight">\(i\)</span>’th row of <span class="math notranslate nohighlight">\(\vA\)</span> corresponds to the outgoing distribution from state <span class="math notranslate nohighlight">\(i\)</span>.
This is  a row stochastic matrix,
meaning each row sums to one.
We can visualize
the non-zero entries in the transition matrix by creating a state transition diagram,
as shown in <a class="reference internal" href="#casino-fig"><span class="std std-ref">Illustration of the casino HMM.</span></a>.</p>
<div class="figure align-default" id="casino-fig">
<a class="reference internal image-reference" href="../../_images/casino.png"><img alt="../../_images/casino.png" src="../../_images/casino.png" style="width: 208.5px; height: 142.5px;" /></a>
<p class="caption"><span class="caption-number">Fig. 5 </span><span class="caption-text">Illustration of the casino HMM.</span><a class="headerlink" href="#casino-fig" title="Permalink to this image">¶</a></p>
</div>
<p>The  observation model
<span class="math notranslate nohighlight">\(p(\obs_t|\hiddden_t=j)\)</span> has the form</p>
<div class="math notranslate nohighlight">
\[p(\obs_t=k|\hidden_t=j) = \hmmObs_{jk} \]</div>
<p>This is represented by the histograms associated with each
state in <a class="reference internal" href="#casino-fig"><span class="std std-ref">Illustration of the casino HMM.</span></a>.</p>
<p>Finally,
the initial state distribution is denoted by</p>
<div class="math notranslate nohighlight">
\[p(z_1=j) = \hmmInit_j\]</div>
<p>Collectively we denote all the parameters by <span class="math notranslate nohighlight">\(\vtheta=(\hmmTrans, \hmmObs, \hmmInit)\)</span>.</p>
<p>Now let us implement this model code.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># state transition matrix</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
    <span class="p">[</span><span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">],</span>
    <span class="p">[</span><span class="mf">0.10</span><span class="p">,</span> <span class="mf">0.90</span><span class="p">]</span>
<span class="p">])</span>

<span class="c1"># observation matrix</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span>
    <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">],</span> <span class="c1"># fair die</span>
    <span class="p">[</span><span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="o">/</span><span class="mi">10</span><span class="p">]</span> <span class="c1"># loaded die</span>
<span class="p">])</span>

<span class="n">pi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">])</span>

<span class="p">(</span><span class="n">nstates</span><span class="p">,</span> <span class="n">nobs</span><span class="p">)</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">shape</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
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