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   State Space Models: A Modern Approach
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                <h1>Nonlinear Gaussian SSMs</h1>
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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="nonlinear-gaussian-ssms">
<span id="sec-nlds-intro"></span><h1>Nonlinear Gaussian SSMs<a class="headerlink" href="#nonlinear-gaussian-ssms" title="Permalink to this headline">¶</a></h1>
<p>In this section, we consider SSMs in which the dynamics and/or observation models are nonlinear,
but the process noise and observation noise are Gaussian.
That is,</p>
<div class="amsmath math notranslate nohighlight" id="equation-48c705c3-8479-4828-87b8-fd179fc23bc4">
<span class="eqno">(11)<a class="headerlink" href="#equation-48c705c3-8479-4828-87b8-fd179fc23bc4" 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>.
This is a very widely used model class. We give some examples below.</p>
<div class="section" id="example-tracking-a-1d-pendulum">
<span id="sec-pendulum"></span><h2>Example: tracking a 1d pendulum<a class="headerlink" href="#example-tracking-a-1d-pendulum" title="Permalink to this headline">¶</a></h2>
<div class="figure align-default" id="fig-pendulum">
<a class="reference internal image-reference" href="../../_images/pendulum.png"><img alt="../../_images/pendulum.png" src="../../_images/pendulum.png" style="width: 132.5px; height: 147.5px;" /></a>
<p class="caption"><span class="caption-number">Fig. 6 </span><span class="caption-text">Illustration of a pendulum swinging.
<span class="math notranslate nohighlight">\(g\)</span> is the force of gravity,
<span class="math notranslate nohighlight">\(w(t)\)</span> is a random external force,
and <span class="math notranslate nohighlight">\(\alpha\)</span> is the angle wrt the vertical.
Based on <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> fig 3.10.</span><a class="headerlink" href="#fig-pendulum" title="Permalink to this image">¶</a></p>
</div>
<p>Consider a simple pendulum of unit mass and length swinging from
a fixed attachment, as in
<code class="xref std std-numref docutils literal notranslate"><span class="pre">Figure</span> <span class="pre">%s</span></code>.
Such an object is in principle entirely deterministic in its behavior.
However, in the real world, there are often unknown forces at work
(e.g., air turbulence, friction).
We will model these by a continuous time random Gaussian noise process <span class="math notranslate nohighlight">\(w(t)\)</span>.
This gives rise to the following differential equation:</p>
<div class="amsmath math notranslate nohighlight" id="equation-ce7cd5b9-026f-47f8-b6eb-aeb189d67e25">
<span class="eqno">(12)<a class="headerlink" href="#equation-ce7cd5b9-026f-47f8-b6eb-aeb189d67e25" title="Permalink to this equation">¶</a></span>\[\begin{align}
\frac{d^2 \alpha}{d t^2}
= -g \sin(\alpha) + w(t)
\end{align}\]</div>
<p>We can write this as a nonlinear SSM by defining the state to be
<span class="math notranslate nohighlight">\(z_1(t) = \alpha(t)\)</span> and <span class="math notranslate nohighlight">\(z_2(t) = d\alpha(t)/dt\)</span>.
Thus</p>
<div class="amsmath math notranslate nohighlight" id="equation-fdafb753-eac8-4deb-9ab4-5408727f28f7">
<span class="eqno">(13)<a class="headerlink" href="#equation-fdafb753-eac8-4deb-9ab4-5408727f28f7" title="Permalink to this equation">¶</a></span>\[\begin{align}
\frac{d \vz}{dt}
= \begin{pmatrix} z_2 \\ -g \sin(z_1) \end{pmatrix}
+ \begin{pmatrix} 0 \\ 1 \end{pmatrix} w(t)
\end{align}\]</div>
<p>If we discretize this step size <span class="math notranslate nohighlight">\(\Delta\)</span>,
we get the following
formulation <span id="id2">[<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> p74:</p>
<div class="amsmath math notranslate nohighlight" id="equation-b07b597e-169c-4cc9-a403-ff0a210604e8">
<span class="eqno">(14)<a class="headerlink" href="#equation-b07b597e-169c-4cc9-a403-ff0a210604e8" title="Permalink to this equation">¶</a></span>\[\begin{align}
\underbrace{
  \begin{pmatrix} z_{1,t} \\ z_{2,t} \end{pmatrix}
  }_{\hmmhid_t}
=
\underbrace{
  \begin{pmatrix} z_{1,t-1} + z_{2,t-1} \Delta  \\
    z_{2,t-1} -g \sin(z_{1,t-1}) \Delta  \end{pmatrix}
  }_{\vf(\hmmhid_{t-1})}
+\vq_{t-1}
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(\vq_{t-1} \sim \gauss(\vzero,\vQ)\)</span> with</p>
<div class="amsmath math notranslate nohighlight" id="equation-45e5300e-4cc7-4e93-900a-875db0283ab1">
<span class="eqno">(15)<a class="headerlink" href="#equation-45e5300e-4cc7-4e93-900a-875db0283ab1" title="Permalink to this equation">¶</a></span>\[\begin{align}
\vQ = q^c \begin{pmatrix}
  \frac{\Delta^3}{3} &amp;   \frac{\Delta^2}{2} \\
  \frac{\Delta^2}{2} &amp; \Delta
  \end{pmatrix}
  \end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(q^c\)</span> is the spectral density (continuous time variance)
of the continuous-time noise process.</p>
<p>If we observe the angular position, we
get the linear observation model</p>
<div class="amsmath math notranslate nohighlight" id="equation-adb0a338-37f5-41ae-93d3-b2f9a1a35350">
<span class="eqno">(16)<a class="headerlink" href="#equation-adb0a338-37f5-41ae-93d3-b2f9a1a35350" title="Permalink to this equation">¶</a></span>\[\begin{align}
y_t = \alpha_t + r_t =  h(\hmmhid_t) + r_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(h(\hmmhid_t) = z_{1,t}\)</span>
and <span class="math notranslate nohighlight">\(r_t\)</span> is the observation noise.
If we only observe  the horizontal position,
we get the nonlinear observation model</p>
<div class="amsmath math notranslate nohighlight" id="equation-94cc9e21-b503-4b95-b6cd-7a6bffccae45">
<span class="eqno">(17)<a class="headerlink" href="#equation-94cc9e21-b503-4b95-b6cd-7a6bffccae45" title="Permalink to this equation">¶</a></span>\[\begin{align}
y_t = \sin(\alpha_t) + r_t =  h(\hmmhid_t) + r_t
\end{align}\]</div>
<p>where <span class="math notranslate nohighlight">\(h(\hmmhid_t) = \sin(z_{1,t})\)</span>.</p>
</div>
</div>

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