radu
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SSMBook
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   State Space Models: A Modern Approach
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   Particle filtering
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   Sequential Monte Carlo
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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">distrax</span>
<span class="k">except</span><span class="p">:</span>
    <span class="o">%</span><span class="k">pip</span> install --upgrade  distrax
    <span class="kn">import</span> <span class="nn">distrax</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{\hid}{\hidden}
\newcommand{\observed}{\vy}
\newcommand{\obs}{\observed}
\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{\ssmDynFn}{f}
\newcommand{\ssmObsFn}{h}\\
%%%
\newcommand{\gauss}{\mathcal{N}}\\\newcommand{\diag}{\mathrm{diag}}\end{aligned}\end{align} \]</div>
<div class="tex2jax_ignore mathjax_ignore section" id="linear-gaussian-ssms">
<span id="sec-lds-intro"></span><h1>Linear Gaussian SSMs<a class="headerlink" href="#linear-gaussian-ssms" title="Permalink to this headline">¶</a></h1>
<p>Consider the state space model in
<a class="reference internal" href="ssm_intro.html#equation-eq-ssm-ar">(2)</a>
where we assume the observations are conditionally iid given the
hidden states and inputs (i.e. there are no auto-regressive dependencies
between the observables).
We can rewrite this model as
a stochastic nonlinear dynamical system (NLDS)
by defining the distribution of the next hidden state
as a deterministic function of the past state
plus random process noise <span class="math notranslate nohighlight">\(\vepsilon_t\)</span></p>
<div class="amsmath math notranslate nohighlight" id="equation-e31147c5-366d-4025-ad02-0d0886a1dffa">
<span class="eqno">(5)<a class="headerlink" href="#equation-e31147c5-366d-4025-ad02-0d0886a1dffa" title="Permalink to this equation">¶</a></span>\[\begin{align}
\hmmhid_t &amp;= \ssmDynFn(\hmmhid_{t-1}, \inputs_t, \vepsilon_t)  
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\vepsilon_t\)</span> is drawn from the distribution such
that the induced distribution
on <span class="math notranslate nohighlight">\(\hmmhid_t\)</span> matches <span class="math notranslate nohighlight">\(p(\hmmhid_t|\hmmhid_{t-1}, \inputs_t)\)</span>.
Similarly we can rewrite the observation distributions
as a deterministic function of the hidden state
plus observation noise <span class="math notranslate nohighlight">\(\veta_t\)</span>:</p>
<div class="amsmath math notranslate nohighlight" id="equation-f582ebd3-bbc8-4436-8a13-41e99905dfbd">
<span class="eqno">(6)<a class="headerlink" href="#equation-f582ebd3-bbc8-4436-8a13-41e99905dfbd" title="Permalink to this equation">¶</a></span>\[\begin{align}
\hmmobs_t &amp;= \ssmObsFn(\hmmhid_{t}, \inputs_t, \veta_t)
\end{align}\]</div>
<p>If we assume additive Gaussian noise,
the model becomes</p>
<div class="amsmath math notranslate nohighlight" id="equation-09c96fd9-5478-4aa7-ac38-43504373febc">
<span class="eqno">(7)<a class="headerlink" href="#equation-09c96fd9-5478-4aa7-ac38-43504373febc" title="Permalink to this equation">¶</a></span>\[\begin{align}
\hmmhid_t &amp;= \ssmDynFn(\hmmhid_{t-1}, \inputs_t) +  \vepsilon_t  \\
\hmmobs_t &amp;= \ssmObsFn(\hmmhid_{t}, \inputs_t) + \veta_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\vepsilon_t \sim \gauss(\vzero,\vQ_t)\)</span>
and <span class="math notranslate nohighlight">\(\veta_t \sim \gauss(\vzero,\vR_t)\)</span>.
We will call these Gaussian SSMs.</p>
<p>If we additionally assume
the transition function <span class="math notranslate nohighlight">\(\ssmDynFn\)</span>
and the observation function <span class="math notranslate nohighlight">\(\ssmObsFn\)</span> are both linear,
then we can rewrite the model as follows:</p>
<div class="amsmath math notranslate nohighlight" id="equation-44914eac-1d98-497d-87e7-1d6b5346a8f2">
<span class="eqno">(8)<a class="headerlink" href="#equation-44914eac-1d98-497d-87e7-1d6b5346a8f2" title="Permalink to this equation">¶</a></span>\[\begin{align}
p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) &amp;= \gauss(\hmmhid_t|\ldsDyn_t \hmmhid_{t-1}
+ \ldsDynIn_t \inputs_t, \vQ_t)
\\
p(\hmmobs_t|\hmmhid_t,\inputs_t) &amp;= \gauss(\hmmobs_t|\ldsObs_t \hmmhid_{t}
+ \ldsObsIn_t \inputs_t, \vR_t)
\end{align}\]</div>
<p>This is called a
linear-Gaussian state space model
(LG-SSM),
or a
linear dynamical system (LDS).
We usually assume the parameters are independent of time, in which case
the model is said to be time-invariant or homogeneous.</p>
<div class="section" id="example-tracking-a-2d-point">
<span id="sec-kalman-tracking"></span><span id="sec-tracking-lds"></span><h2>Example: tracking a 2d point<a class="headerlink" href="#example-tracking-a-2d-point" title="Permalink to this headline">¶</a></h2>
<p>Consider an object moving in <span class="math notranslate nohighlight">\(\real^2\)</span>.
Let the state be
the position and velocity of the object,
<span class="math notranslate nohighlight">\(\vz_t =\begin{pmatrix} u_t &amp; \dot{u}_t &amp; v_t &amp; \dot{v}_t \end{pmatrix}\)</span>.
(We use <span class="math notranslate nohighlight">\(u\)</span> and <span class="math notranslate nohighlight">\(v\)</span> for the two coordinates,
to avoid confusion with the state and observation variables.)
If we use Euler discretization,
the dynamics become</p>
<div class="amsmath math notranslate nohighlight" id="equation-56d91323-8e7f-4b13-8792-ca0ca145de95">
<span class="eqno">(9)<a class="headerlink" href="#equation-56d91323-8e7f-4b13-8792-ca0ca145de95" title="Permalink to this equation">¶</a></span>\[\begin{align}
\underbrace{\begin{pmatrix} u_t\\ \dot{u}_t \\ v_t \\ \dot{v}_t \end{pmatrix}}_{\vz_t}
  = 
\underbrace{
\begin{pmatrix}
1 &amp; 0 &amp; \Delta &amp; 0 \\
0 &amp; 1 &amp; 0 &amp; \Delta\\
0 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1
\end{pmatrix}
}_{\ldsDyn}
\
\underbrace{\begin{pmatrix} u_{t-1} \\ \dot{u}_{t-1} \\ v_{t-1} \\ \dot{v}_{t-1} \end{pmatrix}}_{\vz_{t-1}}
+ \vepsilon_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\vepsilon_t \sim \gauss(\vzero,\vQ)\)</span> is
the process noise.</p>
<p>Let us assume
that the process noise is
a white noise process added to the velocity components
of the state, but not to the location.
(This is known as a random accelerations model.)
We can approximate the resulting process in discrete time by assuming
<span class="math notranslate nohighlight">\(\vQ = \diag(0, q, 0, q)\)</span>.
(See  <span id="id1">[<a class="reference internal" href="../../bib.html#id18" title="Simo Sarkka. Bayesian Filtering and Smoothing. Cambridge University Press, 2013. URL: https://users.aalto.fi/~ssarkka/pub/cup_book_online_20131111.pdf.">Sar13</a>]</span> p60 for a more accurate way
to convert the continuous time process to discrete time.)</p>
<p>Now suppose that at each discrete time point we
observe the location,
corrupted by  Gaussian noise.
Thus the observation model becomes</p>
<div class="amsmath math notranslate nohighlight" id="equation-54905b1b-79c0-47fc-b0cc-f790a1e50550">
<span class="eqno">(10)<a class="headerlink" href="#equation-54905b1b-79c0-47fc-b0cc-f790a1e50550" title="Permalink to this equation">¶</a></span>\[\begin{align}
\underbrace{\begin{pmatrix}  y_{1,t} \\  y_{2,t} \end{pmatrix}}_{\vy_t}
  &amp;=
    \underbrace{
    \begin{pmatrix}
1 &amp; 0 &amp; 0 &amp; 0 \\
0 &amp; 0 &amp; 1 &amp; 0
    \end{pmatrix}
    }_{\ldsObs}
    \
\underbrace{\begin{pmatrix} u_t\\ \dot{u}_t \\ v_t \\ \dot{v}_t \end{pmatrix}}_{\vz_t}    
 + \veta_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\veta_t \sim \gauss(\vzero,\vR)\)</span> is the \keywordDef{observation noise}.
We see that the observation matrix <span class="math notranslate nohighlight">\(\ldsObs\)</span> simply ``extracts’’ the
relevant parts  of the state vector.</p>
<p>Suppose we sample a trajectory and corresponding set
of noisy observations from this model,
<span class="math notranslate nohighlight">\((\vz_{1:T}, \vy_{1:T}) \sim p(\vz,\vy|\vtheta)\)</span>.
(We use diagonal observation noise,
<span class="math notranslate nohighlight">\(\vR = \diag(\sigma_1^2, \sigma_2^2)\)</span>.)
The results are shown below.</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="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">314</span><span class="p">)</span>
<span class="n">timesteps</span> <span class="o">=</span> <span class="mi">15</span>
<span class="n">delta</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">jnp</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="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">delta</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="p">])</span>

<span class="n">C</span> <span class="o">=</span> <span class="n">jnp</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="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
    <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="p">])</span>

<span class="n">state_size</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span>
<span class="n">observation_size</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">C</span><span class="o">.</span><span class="n">shape</span>

<span class="n">Q</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">state_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.001</span>
<span class="n">R</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">observation_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">1.0</span>
<span class="c1"># Prior parameter distribution</span>
<span class="n">mu0</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">8</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">float</span><span class="p">)</span>
<span class="n">Sigma0</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">state_size</span><span class="p">)</span> <span class="o">*</span> <span class="mf">1.0</span>

<span class="kn">from</span> <span class="nn">jsl.lds.kalman_filter</span> <span class="kn">import</span> <span class="n">LDS</span><span class="p">,</span> <span class="n">smooth</span><span class="p">,</span> <span class="nb">filter</span>

<span class="n">lds</span> <span class="o">=</span> <span class="n">LDS</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">C</span><span class="p">,</span> <span class="n">Q</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">mu0</span><span class="p">,</span> <span class="n">Sigma0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">lds</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>LDS(A=DeviceArray([[1., 0., 1., 0.],
             [0., 1., 0., 1.],
             [0., 0., 1., 0.],
             [0., 0., 0., 1.]], dtype=float32), C=DeviceArray([[1, 0, 0, 0],
             [0, 1, 0, 0]], dtype=int32), Q=DeviceArray([[0.001, 0.   , 0.   , 0.   ],
             [0.   , 0.001, 0.   , 0.   ],
             [0.   , 0.   , 0.001, 0.   ],
             [0.   , 0.   , 0.   , 0.001]], dtype=float32), R=DeviceArray([[1., 0.],
             [0., 1.]], dtype=float32), mu=DeviceArray([ 8., 10.,  1.,  0.], dtype=float32), Sigma=DeviceArray([[1., 0., 0., 0.],
             [0., 1., 0., 0.],
             [0., 0., 1., 0.],
             [0., 0., 0., 1.]], dtype=float32), state_offset=None, obs_offset=None, nstates=4, nobs=2)
</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="kn">from</span> <span class="nn">jsl.demos.plot_utils</span> <span class="kn">import</span> <span class="n">plot_ellipse</span>

<span class="k">def</span> <span class="nf">plot_tracking_values</span><span class="p">(</span><span class="n">observed</span><span class="p">,</span> <span class="n">filtered</span><span class="p">,</span> <span class="n">cov_hist</span><span class="p">,</span> <span class="n">signal_label</span><span class="p">,</span> <span class="n">ax</span><span class="p">):</span>
    <span class="n">timesteps</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">observed</span><span class="o">.</span><span class="n">shape</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">observed</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">observed</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span>
            <span class="n">markerfacecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;observed&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="o">*</span><span class="n">filtered</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">signal_label</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:red&quot;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">covn</span> <span class="o">=</span> <span class="n">cov_hist</span><span class="p">[</span><span class="n">t</span><span class="p">][:</span><span class="mi">2</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
        <span class="n">plot_ellipse</span><span class="p">(</span><span class="n">covn</span><span class="p">,</span> <span class="n">filtered</span><span class="p">[</span><span class="n">t</span><span class="p">,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="n">ax</span><span class="p">,</span> <span class="n">n_std</span><span class="o">=</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">plot_center</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;equal&quot;</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
</pre></div>
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<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">z_hist</span><span class="p">,</span> <span class="n">x_hist</span> <span class="o">=</span> <span class="n">lds</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">timesteps</span><span class="p">)</span>

<span class="n">fig_truth</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">axs</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_hist</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">x_hist</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
        <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">markerfacecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">,</span>
        <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span>
        <span class="n">label</span><span class="o">=</span><span class="s2">&quot;observed&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;tab:green&quot;</span><span class="p">)</span>

<span class="n">axs</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z_hist</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">z_hist</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
        <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;truth&quot;</span><span class="p">,</span>
        <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;s&quot;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
<span class="n">axs</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">axs</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s2">&quot;equal&quot;</span><span class="p">)</span>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(7.24486608505249, 23.857812213897706, 8.042076778411865, 11.636079120635987)
</pre></div>
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<img alt="../../_images/lds_7_1.png" src="../../_images/lds_7_1.png" />
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</div>
<p>The main task is to infer the hidden states given the noisy
observations, i.e., <span class="math notranslate nohighlight">\(p(\vz|\vy,\vtheta)\)</span>. We discuss the topic of inference in <a class="reference internal" href="inference.html#sec-inference"><span class="std std-ref">Inferential goals</span></a>.</p>
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