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   State Space Models: A Modern Approach
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  <div class="tex2jax_ignore mathjax_ignore section" id="scratchpad">
<span id="ch-intro"></span><h1>Scratchpad<a class="headerlink" href="#scratchpad" title="Permalink to this headline">¶</a></h1>
<p>In this chapter, we do blah.
Specifically</p>
<ul class="simple">
<li><p>foo</p></li>
<li><p>bar.</p></li>
<li><p>baz</p></li>
</ul>
<p>For more details, see
<a class="reference internal" href="hmm/hmm_index.html#ch-hmm"><span class="std std-ref">Hidden Markov Models</span></a> and  <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>.</p>
<div class="section" id="python">
<h2>Python<a class="headerlink" href="#python" title="Permalink to this headline">¶</a></h2>
<p>We’re now ready to start coding.</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="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">rcParams</span><span class="p">,</span> <span class="n">cycler</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="n">plt</span><span class="o">.</span><span class="n">ion</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Fixing random state for reproducibility</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">19680801</span><span class="p">)</span>

<span class="n">N</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</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">100</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span> <span class="o">+</span> <span class="n">ii</span> <span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">)]</span>
<span class="n">data</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="n">data</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="n">cmap</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">coolwarm</span>
<span class="n">rcParams</span><span class="p">[</span><span class="s1">&#39;axes.prop_cycle&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="n">cycler</span><span class="p">(</span><span class="n">color</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</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="n">N</span><span class="p">)))</span>


<span class="kn">from</span> <span class="nn">matplotlib.lines</span> <span class="kn">import</span> <span class="n">Line2D</span>
<span class="n">custom_lines</span> <span class="o">=</span> <span class="p">[</span><span class="n">Line2D</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="n">color</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="mf">0.</span><span class="p">),</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
                <span class="n">Line2D</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="n">color</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="mf">.5</span><span class="p">),</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">),</span>
                <span class="n">Line2D</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="n">color</span><span class="o">=</span><span class="n">cmap</span><span class="p">(</span><span class="mf">1.</span><span class="p">),</span> <span class="n">lw</span><span class="o">=</span><span class="mi">4</span><span class="p">)]</span>

<span class="n">fig</span><span class="p">,</span> <span class="n">ax</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">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">lines</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">custom_lines</span><span class="p">,</span> <span class="p">[</span><span class="s1">&#39;Cold&#39;</span><span class="p">,</span> <span class="s1">&#39;Medium&#39;</span><span class="p">,</span> <span class="s1">&#39;Hot&#39;</span><span class="p">]);</span>
</pre></div>
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<img alt="../_images/scratch_2_0.png" src="../_images/scratch_2_0.png" />
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></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="nb">print</span><span class="p">(</span><span class="n">jax</span><span class="o">.</span><span class="n">devices</span><span class="p">())</span>
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<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>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[CpuDevice(id=0)]
</pre></div>
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</div>
</div>
<div class="section" id="images">
<h2>Images<a class="headerlink" href="#images" title="Permalink to this headline">¶</a></h2>
<!---
![](https://myst-parser.readthedocs.io/en/latest/_static/logo-wide.svg)


<img src="https://github.com/probml/probml-notebooks/blob/main/images/cat_dog.jpg"
style="height:200">
-->
<div class="figure align-default" id="cat-dog">
<a class="reference internal image-reference" href="../_images/cat_dog.jpg"><img alt="../_images/cat_dog.jpg" src="../_images/cat_dog.jpg" style="width: 320.0px; height: 240.0px;" /></a>
<p class="caption"><span class="caption-number">Fig. 1 </span><span class="caption-text">A photo of a cat and a dog.</span><a class="headerlink" href="#cat-dog" title="Permalink to this image">¶</a></p>
</div>
<div class="figure align-default" id="cat-dog2">
<a class="reference internal image-reference" href="../_images/cat_dog.jpg"><img alt="../_images/cat_dog.jpg" src="../_images/cat_dog.jpg" style="height: 300px;" /></a>
<p class="caption"><span class="caption-number">Fig. 2 </span><span class="caption-text">Another photo of a cat and a dog.</span><a class="headerlink" href="#cat-dog2" title="Permalink to this image">¶</a></p>
</div>
<p>In <a class="reference internal" href="#cat-dog"><span class="std std-numref">Fig. 1</span></a>
we show catdog.
In <a class="reference internal" href="#cat-dog2"><span class="std std-numref">Figure 2</span></a> we show catdog2, its twin.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>I am a useful note!</p>
</div>
</div>
<div class="section" id="math">
<h2>Math<a class="headerlink" href="#math" title="Permalink to this headline">¶</a></h2>
<p>Here is <span class="math notranslate nohighlight">\(\N=10\)</span> and blah. <span class="math notranslate nohighlight">\(\floor{42.3}= 42\)</span>.</p>
<p>We have <span class="math notranslate nohighlight">\(E= mc^2\)</span>, and also</p>
<div class="math notranslate nohighlight" id="equation-foo">
<span class="eqno">(1)<a class="headerlink" href="#equation-foo" title="Permalink to this equation">¶</a></span>\[a x^2 + bx+ c = 0\]</div>
<p>From <a class="reference internal" href="#equation-foo">(1)</a>, it follows that</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align}
0 &amp;= a x^2 + bx+ c \\
0 &amp;= a x^2 + bx+ c 
\end{align}
\end{split}\]</div>
</div>
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