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  94.    State Space Models: A Modern Approach
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  111.      What are State Space Models?
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  126.      Nonlinear Gaussian SSMs
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  131.      States estimation (inference)
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  143.    Hidden Markov Models
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  153.      HMM filtering (forwards algorithm)
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  158.      HMM smoothing (forwards-backwards algorithm)
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  168.      Parallel HMM  smoothing
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  173.      Forwards-filtering backwards-sampling algorithm
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  180.    Linear-Gaussian SSMs
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  190.      Kalman filtering
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  205.      Forwards-filtering backwards sampling
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  222.      Extended Kalman filtering
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  227.      Extended Kalman smoother
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  232.      Parallel extended Kalman smoothing
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  249.      Unscented filtering
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  254.      Unscented smoothing
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    261. 

  261.    Quadrature and cubature methods
    262. 

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  266.    Posterior linearization
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  271.    Assumed Density Filtering
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  276.    Variational inference
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  281.    Particle filtering
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  286.    Sequential Monte Carlo
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  291.    Offline parameter estimation (learning)
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    450. 

  450.    Example: inference in the casino HMM
    451. 

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    468. 

  468.                 <h1>States estimation (inference)</h1>
    469. 

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  472.                         
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    474. 

  474.                             <h2> Contents </h2>
    475. 

  475.                         </div>
    476. 

  476.                         <nav aria-label="Page">
    477. 

  477.                             <ul class="visible nav section-nav flex-column">
    478. 

  478.  <li class="toc-h2 nav-item toc-entry">
    479. 

  479.   <a class="reference internal nav-link" href="#example-inference-in-the-casino-hmm">
    480. 

  480.    Example: inference in the casino HMM
    481. 

  481.   </a>
    482. 

  482.  </li>
    483. 

  483.  <li class="toc-h2 nav-item toc-entry">
    484. 

  484.   <a class="reference internal nav-link" href="#example-inference-in-the-tracking-lg-ssm">
    485. 

  485.    Example: inference in the tracking LG-SSM
    486. 

  486.   </a>
    487. 

  487.  </li>
    488. 

  488. </ul>
    489. 

  489. 

  490.                         </nav>
    491. 

  491.                     </div>
    492. 

  492.                 </div>
    493. 

  493.             </div>
    494. 

  494.             
    495. 

  495.               <div>
    496. 

  496.                 
    497. 

  497.   <div class="cell docutils container">
    498. 

  498. <div class="cell_input docutils container">
    499. 

  499. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">{</span>
    500. 

  500.     <span class="s2">&quot;tags&quot;</span><span class="p">:</span> <span class="p">[</span>
    501. 

  501.         <span class="s2">&quot;hide-cell&quot;</span>
    502. 

  502.     <span class="p">]</span>
    503. 

  503. <span class="p">}</span>
    504. 

  504. 

  505. 

  506. <span class="c1">### Import standard libraries</span>
    507. 

  507. 

  508. <span class="kn">import</span> <span class="nn">abc</span>
    509. 

  509. <span class="kn">from</span> <span class="nn">dataclasses</span> <span class="kn">import</span> <span class="n">dataclass</span>
    510. 

  510. <span class="kn">import</span> <span class="nn">functools</span>
    511. 

  511. <span class="kn">import</span> <span class="nn">itertools</span>
    512. 

  512. 

  513. <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>
    514. 

  514. 

  515. <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
    516. 

  516. <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
    517. 

  517. 

  518. 

  519. <span class="kn">import</span> <span class="nn">jax</span>
    520. 

  520. <span class="kn">import</span> <span class="nn">jax.numpy</span> <span class="k">as</span> <span class="nn">jnp</span>
    521. 

  521. <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>
    522. 

  522. <span class="kn">from</span> <span class="nn">jax.scipy.special</span> <span class="kn">import</span> <span class="n">logit</span>
    523. 

  523. <span class="kn">from</span> <span class="nn">jax.nn</span> <span class="kn">import</span> <span class="n">softmax</span>
    524. 

  524. <span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">partial</span>
    525. 

  525. <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>
    526. 

  526. 

  527. <span class="kn">import</span> <span class="nn">jsl</span>
    528. 

  528. <span class="kn">import</span> <span class="nn">ssm_jax</span>
    529. 

  529. </pre></div>
    530. 

  530. </div>
    531. 

  531. </div>
    532. 

  532. </div>
    533. 

  533. <div class="tex2jax_ignore mathjax_ignore section" id="states-estimation-inference">
    534. 

  534. <span id="sec-inference"></span><h1>States estimation (inference)<a class="headerlink" href="#states-estimation-inference" title="Permalink to this headline">¶</a></h1>
    535. 

  535. <p>Given the sequence of observations, and a known model,
    536. 

  536. one of the main tasks with SSMs
    537. 

  537. to perform posterior inference,
    538. 

  538. about the hidden states; this is also called
    539. 

  539. state estimation.
    540. 

  540. At each time step <span class="math notranslate nohighlight">\(t\)</span>,
    541. 

  541. there are multiple forms of posterior we may be interested in computing,
    542. 

  542. including the following:</p>
    543. 

  543. <ul class="simple">
    544. 

  544. <li><p>the filtering distribution
    545. 

  545. <span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:t})\)</span></p></li>
    546. 

  546. <li><p>the smoothing distribution
    547. 

  547. <span class="math notranslate nohighlight">\(p(\hidden_t|\obs_{1:T})\)</span> (note that this conditions on future data <span class="math notranslate nohighlight">\(T&gt;t\)</span>)</p></li>
    548. 

  548. <li><p>the fixed-lag smoothing distribution
    549. 

  549. <span class="math notranslate nohighlight">\(p(\hidden_{t-\ell}|\obs_{1:t})\)</span> (note that this
    550. 

  550. infers <span class="math notranslate nohighlight">\(\ell\)</span> steps in the past given data up to the present).</p></li>
    551. 

  551. </ul>
    552. 

  552. <p>We may also want to compute the
    553. 

  553. predictive distribution <span class="math notranslate nohighlight">\(h\)</span> steps into the future:</p>
    554. 

  554. <div class="amsmath math notranslate nohighlight" id="equation-1f208dad-d4e8-412e-aed2-aa723b3f8bbf">
    555. 

  555. <span class="eqno">(15)<a class="headerlink" href="#equation-1f208dad-d4e8-412e-aed2-aa723b3f8bbf" title="Permalink to this equation">¶</a></span>\[\begin{align}
    556. 

  556. p(\obs_{t+h}|\obs_{1:t})
    557. 

  557. = \sum_{\hidden_{t+h}} p(\obs_{t+h}|\hidden_{t+h}) p(\hidden_{t+h}|\obs_{1:t})
    558. 

  558. \end{align}\]</div>
    559. 

  559. <p>where the hidden state predictive distribution is</p>
    560. 

  560. <div class="amsmath math notranslate nohighlight" id="equation-aeb53008-8650-4419-932d-1f31ed6427f1">
    561. 

  561. <span class="eqno">(16)<a class="headerlink" href="#equation-aeb53008-8650-4419-932d-1f31ed6427f1" title="Permalink to this equation">¶</a></span>\[\begin{align}
    562. 

  562. p(\hidden_{t+h}|\obs_{1:t})
    563. 

  563. &amp;= \sum_{\hidden_{t:t+h-1}}
    564. 

  564.  p(\hidden_t|\obs_{1:t}) 
    565. 

  565.  p(\hidden_{t+1}|\hidden_{t})
    566. 

  566.  p(\hidden_{t+2}|\hidden_{t+1})
    567. 

  567. \cdots
    568. 

  568.  p(\hidden_{t+h}|\hidden_{t+h-1})
    569. 

  569. \end{align}\]</div>
    570. 

  570. <p>See
    571. 

  571. <a class="reference internal" href="#fig-dbn-inference"><span class="std std-numref">Fig. 5</span></a> for a summary of these distributions.</p>
    572. 

  572. <div class="figure align-default" id="fig-dbn-inference">
    573. 

  573. <a class="reference internal image-reference" href="../../_images/inference-problems-tikz.png"><img alt="../../_images/inference-problems-tikz.png" src="../../_images/inference-problems-tikz.png" style="width: 352.2px; height: 269.4px;" /></a>
    574. 

  574. <p class="caption"><span class="caption-number">Fig. 5 </span><span class="caption-text">Illustration of the different kinds of inference in an SSM.
    575. 

  575. The main kinds of inference for state-space models.
    576. 

  576. The shaded region is the interval for which we have data.
    577. 

  577. The arrow represents the time step at which we want to perform inference.
    578. 

  578. <span class="math notranslate nohighlight">\(t\)</span> is the current time,  <span class="math notranslate nohighlight">\(T\)</span> is the sequence length,
    579. 

  579. <span class="math notranslate nohighlight">\(\ell\)</span> is the lag and <span class="math notranslate nohighlight">\(h\)</span> is the prediction horizon.</span><a class="headerlink" href="#fig-dbn-inference" title="Permalink to this image">¶</a></p>
    580. 

  580. </div>
    581. 

  581. <p>In addition  to comuting posterior marginals,
    582. 

  582. we may want to compute the most probable hidden sequence,
    583. 

  583. i.e., the joint MAP estimate</p>
    584. 

  584. <div class="math notranslate nohighlight">
    585. 

  585. \[\arg \max_{\hidden_{1:T}} p(\hidden_{1:T}|\obs_{1:T})\]</div>
    586. 

  586. <p>or sample sequences from the posterior</p>
    587. 

  587. <div class="math notranslate nohighlight">
    588. 

  588. \[\hidden_{1:T} \sim p(\hidden_{1:T}|\obs_{1:T})\]</div>
    589. 

  589. <p>Algorithms for all these task are discussed in the following chapters,
    590. 

  590. since the details depend on the form of the SSM.</p>
    591. 

  591. <div class="section" id="example-inference-in-the-casino-hmm">
    592. 

  592. <span id="sec-casino-inference"></span><h2>Example: inference in the casino HMM<a class="headerlink" href="#example-inference-in-the-casino-hmm" title="Permalink to this headline">¶</a></h2>
    593. 

  593. <p>We now illustrate filtering, smoothing and MAP decoding applied
    594. 

  594. to the casino HMM from <span class="xref std std-ref">sec:casino</span> and <span class="xref myst"></span>.</p>
    595. 

  595. <div class="cell docutils container">
    596. 

  596. <div class="cell_input docutils container">
    597. 

  597. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># state transition matrix</span>
    598. 

  598. <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>
    599. 

  599.     <span class="p">[</span><span class="mf">0.95</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">],</span>
    600. 

  600.     <span class="p">[</span><span class="mf">0.10</span><span class="p">,</span> <span class="mf">0.90</span><span class="p">]</span>
    601. 

  601. <span class="p">])</span>
    602. 

  602. 

  603. <span class="c1"># observation matrix</span>
    604. 

  604. <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>
    605. 

  605.     <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>
    606. 

  606.     <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>
    607. 

  607. <span class="p">])</span>
    608. 

  608. 

  609. <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>
    610. 

  610. 

  611. <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>
    612. 

  612. 

  613. <span class="kn">import</span> <span class="nn">distrax</span>
    614. 

  614. <span class="kn">from</span> <span class="nn">distrax</span> <span class="kn">import</span> <span class="n">HMM</span>
    615. 

  615. 

  616. 

  617. <span class="n">hmm</span> <span class="o">=</span> <span class="n">HMM</span><span class="p">(</span><span class="n">trans_dist</span><span class="o">=</span><span class="n">distrax</span><span class="o">.</span><span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="o">=</span><span class="n">A</span><span class="p">),</span>
    618. 

  618.             <span class="n">init_dist</span><span class="o">=</span><span class="n">distrax</span><span class="o">.</span><span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="o">=</span><span class="n">pi</span><span class="p">),</span>
    619. 

  619.             <span class="n">obs_dist</span><span class="o">=</span><span class="n">distrax</span><span class="o">.</span><span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="o">=</span><span class="n">B</span><span class="p">))</span>
    620. 

  620. 

  621. 

  622. <span class="n">seed</span> <span class="o">=</span> <span class="mi">314</span>
    623. 

  623. <span class="n">n_samples</span> <span class="o">=</span> <span class="mi">300</span>
    624. 

  624. <span class="n">z_hist</span><span class="p">,</span> <span class="n">x_hist</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="n">PRNGKey</span><span class="p">(</span><span class="n">seed</span><span class="p">),</span> <span class="n">seq_len</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>
    625. 

  625. </pre></div>
    626. 

  626. </div>
    627. 

  627. </div>
    628. 

  628. <div class="cell_output docutils container">
    629. 

  629. <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.)
    630. 

  630. </pre></div>
    631. 

  631. </div>
    632. 

  632. </div>
    633. 

  633. </div>
    634. 

  634. <div class="cell docutils container">
    635. 

  635. <div class="cell_input docutils container">
    636. 

  636. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Call inference engine</span>
    637. 

  637. 

  638. <span class="n">filtered_dist</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">smoothed_dist</span><span class="p">,</span> <span class="n">loglik</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">forward_backward</span><span class="p">(</span><span class="n">x_hist</span><span class="p">)</span>
    639. 

  639. <span class="n">map_path</span> <span class="o">=</span> <span class="n">hmm</span><span class="o">.</span><span class="n">viterbi</span><span class="p">(</span><span class="n">x_hist</span><span class="p">)</span>
    640. 

  640. </pre></div>
    641. 

  641. </div>
    642. 

  642. </div>
    643. 

  643. <div class="cell_output docutils container">
    644. 

  644. <div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/anaconda3/lib/python3.8/site-packages/jax/_src/numpy/lax_numpy.py:4457: UserWarning: Explicitly requested dtype &lt;class &#39;jax.numpy.int64&#39;&gt; requested in astype is not available, and will be truncated to dtype int32. To enable more dtypes, set the jax_enable_x64 configuration option or the JAX_ENABLE_X64 shell environment variable. See https://github.com/google/jax#current-gotchas for more.
    645. 

  645.   lax_internal._check_user_dtype_supported(dtype, &quot;astype&quot;)
    646. 

  646. </pre></div>
    647. 

  647. </div>
    648. 

  648. </div>
    649. 

  649. </div>
    650. 

  650. <div class="cell docutils container">
    651. 

  651. <div class="cell_input docutils container">
    652. 

  652. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Find the span of timesteps that the    simulated systems turns to be in state 1</span>
    653. 

  653. <span class="k">def</span> <span class="nf">find_dishonest_intervals</span><span class="p">(</span><span class="n">z_hist</span><span class="p">):</span>
    654. 

  654.     <span class="n">spans</span> <span class="o">=</span> <span class="p">[]</span>
    655. 

  655.     <span class="n">x_init</span> <span class="o">=</span> <span class="mi">0</span>
    656. 

  656.     <span class="k">for</span> <span class="n">t</span><span class="p">,</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">z_hist</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]):</span>
    657. 

  657.         <span class="k">if</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
    658. 

  658.             <span class="n">x_end</span> <span class="o">=</span> <span class="n">t</span>
    659. 

  659.             <span class="n">spans</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">x_init</span><span class="p">,</span> <span class="n">x_end</span><span class="p">))</span>
    660. 

  660.         <span class="k">elif</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">z_hist</span><span class="p">[</span><span class="n">t</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
    661. 

  661.             <span class="n">x_init</span> <span class="o">=</span> <span class="n">t</span> <span class="o">+</span> <span class="mi">1</span>
    662. 

  662.     <span class="k">return</span> <span class="n">spans</span>
    663. 

  663. </pre></div>
    664. 

  664. </div>
    665. 

  665. </div>
    666. 

  666. </div>
    667. 

  667. <div class="cell docutils container">
    668. 

  668. <div class="cell_input docutils container">
    669. 

  669. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot posterior</span>
    670. 

  670. <span class="k">def</span> <span class="nf">plot_inference</span><span class="p">(</span><span class="n">inference_values</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">state</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">map_estimate</span><span class="o">=</span><span class="kc">False</span><span class="p">):</span>
    671. 

  671.     <span class="n">n_samples</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">inference_values</span><span class="p">)</span>
    672. 

  672.     <span class="n">xspan</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_samples</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
    673. 

  673.     <span class="n">spans</span> <span class="o">=</span> <span class="n">find_dishonest_intervals</span><span class="p">(</span><span class="n">z_hist</span><span class="p">)</span>
    674. 

  674.     <span class="k">if</span> <span class="n">map_estimate</span><span class="p">:</span>
    675. 

  675.         <span class="n">ax</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">xspan</span><span class="p">,</span> <span class="n">inference_values</span><span class="p">,</span> <span class="n">where</span><span class="o">=</span><span class="s2">&quot;post&quot;</span><span class="p">)</span>
    676. 

  676.     <span class="k">else</span><span class="p">:</span>
    677. 

  677.         <span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xspan</span><span class="p">,</span> <span class="n">inference_values</span><span class="p">[:,</span> <span class="n">state</span><span class="p">])</span>
    678. 

  678. 

  679.     <span class="k">for</span> <span class="n">span</span> <span class="ow">in</span> <span class="n">spans</span><span class="p">:</span>
    680. 

  680.         <span class="n">ax</span><span class="o">.</span><span class="n">axvspan</span><span class="p">(</span><span class="o">*</span><span class="n">span</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s2">&quot;tab:gray&quot;</span><span class="p">,</span> <span class="n">edgecolor</span><span class="o">=</span><span class="s2">&quot;none&quot;</span><span class="p">)</span>
    681. 

  681.     <span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>
    682. 

  682.     <span class="c1"># ax.set_ylim(0, 1)</span>
    683. 

  683.     <span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">)</span>
    684. 

  684.     <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Observation number&quot;</span><span class="p">)</span>
    685. 

  685. </pre></div>
    686. 

  686. </div>
    687. 

  687. </div>
    688. 

  688. </div>
    689. 

  689. <div class="cell docutils container">
    690. 

  690. <div class="cell_input docutils container">
    691. 

  691. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span> <span class="c1"># Filtering</span>
    692. 

  692. <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>
    693. 

  693. <span class="n">plot_inference</span><span class="p">(</span><span class="n">filtered_dist</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">)</span>
    694. 

  694. <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;p(loaded)&quot;</span><span class="p">)</span>
    695. 

  695. <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Filtered&quot;</span><span class="p">)</span>
    696. 

  696.  
    697. 

  697. 

  698.   
    699. 

  699. </pre></div>
    700. 

  700. </div>
    701. 

  701. </div>
    702. 

  702. <div class="cell_output docutils container">
    703. 

  703. <div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Filtered&#39;)
    704. 

  704. </pre></div>
    705. 

  705. </div>
    706. 

  706. <img alt="../../_images/inference_7_1.png" src="../../_images/inference_7_1.png" />
    707. 

  707. </div>
    708. 

  708. </div>
    709. 

  709. <div class="cell docutils container">
    710. 

  710. <div class="cell_input docutils container">
    711. 

  711. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Smoothing</span>
    712. 

  712. <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>
    713. 

  713. <span class="n">plot_inference</span><span class="p">(</span><span class="n">smoothed_dist</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">)</span>
    714. 

  714. <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;p(loaded)&quot;</span><span class="p">)</span>
    715. 

  715. <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Smoothed&quot;</span><span class="p">)</span>
    716. 

  716. 

  717. 

  718.  
    719. 

  719. </pre></div>
    720. 

  720. </div>
    721. 

  721. </div>
    722. 

  722. <div class="cell_output docutils container">
    723. 

  723. <div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Smoothed&#39;)
    724. 

  724. </pre></div>
    725. 

  725. </div>
    726. 

  726. <img alt="../../_images/inference_8_1.png" src="../../_images/inference_8_1.png" />
    727. 

  727. </div>
    728. 

  728. </div>
    729. 

  729. <div class="cell docutils container">
    730. 

  730. <div class="cell_input docutils container">
    731. 

  731. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># MAP estimation</span>
    732. 

  732. <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>
    733. 

  733. <span class="n">plot_inference</span><span class="p">(</span><span class="n">map_path</span><span class="p">,</span> <span class="n">z_hist</span><span class="p">,</span> <span class="n">ax</span><span class="p">,</span> <span class="n">map_estimate</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
    734. 

  734. <span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;MAP state&quot;</span><span class="p">)</span>
    735. 

  735. <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Viterbi&quot;</span><span class="p">)</span>
    736. 

  736. </pre></div>
    737. 

  737. </div>
    738. 

  738. </div>
    739. 

  739. <div class="cell_output docutils container">
    740. 

  740. <div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0.5, 1.0, &#39;Viterbi&#39;)
    741. 

  741. </pre></div>
    742. 

  742. </div>
    743. 

  743. <img alt="../../_images/inference_9_1.png" src="../../_images/inference_9_1.png" />
    744. 

  744. </div>
    745. 

  745. </div>
    746. 

  746. <div class="cell docutils container">
    747. 

  747. <div class="cell_input docutils container">
    748. 

  748. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># TODO: posterior samples</span>
    749. 

  749. </pre></div>
    750. 

  750. </div>
    751. 

  751. </div>
    752. 

  752. </div>
    753. 

  753. </div>
    754. 

  754. <div class="section" id="example-inference-in-the-tracking-lg-ssm">
    755. 

  755. <h2>Example: inference in the tracking LG-SSM<a class="headerlink" href="#example-inference-in-the-tracking-lg-ssm" title="Permalink to this headline">¶</a></h2>
    756. 

  756. <p>We now illustrate filtering, smoothing and MAP decoding applied
    757. 

  757. to the 2d tracking HMM from <a class="reference internal" href="lds.html#sec-tracking-lds"><span class="std std-ref">Example: tracking a 2d point</span></a>.</p>
    758. 

  758. <div class="cell docutils container">
    759. 

  759. <div class="cell_input docutils container">
    760. 

  760. <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>
    761. 

  761. <span class="n">timesteps</span> <span class="o">=</span> <span class="mi">15</span>
    762. 

  762. <span class="n">delta</span> <span class="o">=</span> <span class="mf">1.0</span>
    763. 

  763. <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>
    764. 

  764.     <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>
    765. 

  765.     <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>
    766. 

  766.     <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>
    767. 

  767.     <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>
    768. 

  768. <span class="p">])</span>
    769. 

  769. 

  770. <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>
    771. 

  771.     <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>
    772. 

  772.     <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>
    773. 

  773. <span class="p">])</span>
    774. 

  774. 

  775. <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>
    776. 

  776. <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>
    777. 

  777. 

  778. <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>
    779. 

  779. <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>
    780. 

  780. <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>
    781. 

  781. <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>
    782. 

  782. 

  783. <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>
    784. 

  784. 

  785. <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>
    786. 

  786. 

  787. 

  788. <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>
    789. 

  789. </pre></div>
    790. 

  790. </div>
    791. 

  791. </div>
    792. 

  792. </div>
    793. 

  793. <div class="cell docutils container">
    794. 

  794. <div class="cell_input docutils container">
    795. 

  795. <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>
    796. 

  796. 

  797. <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>
    798. 

  798.     <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>
    799. 

  799.     <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>
    800. 

  800.             <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>
    801. 

  801.     <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>
    802. 

  802.     <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>
    803. 

  803.         <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>
    804. 

  804.         <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>
    805. 

  805.     <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>
    806. 

  806.     <span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
    807. 

  807. </pre></div>
    808. 

  808. </div>
    809. 

  809. </div>
    810. 

  810. </div>
    811. 

  811. <div class="cell docutils container">
    812. 

  812. <div class="cell_input docutils container">
    813. 

  813. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Filtering</span>
    814. 

  814. 

  815. <span class="n">mu_hist</span><span class="p">,</span> <span class="n">Sigma_hist</span><span class="p">,</span> <span class="n">mu_cond_hist</span><span class="p">,</span> <span class="n">Sigma_cond_hist</span> <span class="o">=</span> <span class="nb">filter</span><span class="p">(</span><span class="n">lds</span><span class="p">,</span> <span class="n">x_hist</span><span class="p">)</span>
    816. 

  816. <span class="n">l2_filter</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">z_hist</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">mu_hist</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
    817. 

  817. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;L2-filter: </span><span class="si">{</span><span class="n">l2_filter</span><span class="si">:</span><span class="s2">0.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
    818. 

  818. <span class="n">fig_filtered</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>
    819. 

  819. <span class="n">plot_tracking_values</span><span class="p">(</span><span class="n">x_hist</span><span class="p">,</span> <span class="n">mu_hist</span><span class="p">,</span> <span class="n">Sigma_hist</span><span class="p">,</span> <span class="s2">&quot;filtered&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">)</span>
    820. 

  820. </pre></div>
    821. 

  821. </div>
    822. 

  822. </div>
    823. 

  823. <div class="cell_output docutils container">
    824. 

  824. <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L2-filter: 3.2481
    825. 

  825. </pre></div>
    826. 

  826. </div>
    827. 

  827. <img alt="../../_images/inference_14_1.png" src="../../_images/inference_14_1.png" />
    828. 

  828. </div>
    829. 

  829. </div>
    830. 

  830. <div class="cell docutils container">
    831. 

  831. <div class="cell_input docutils container">
    832. 

  832. <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Smoothing</span>
    833. 

  833. 

  834. <span class="n">mu_hist_smooth</span><span class="p">,</span> <span class="n">Sigma_hist_smooth</span> <span class="o">=</span> <span class="n">smooth</span><span class="p">(</span><span class="n">lds</span><span class="p">,</span> <span class="n">mu_hist</span><span class="p">,</span> <span class="n">Sigma_hist</span><span class="p">,</span> <span class="n">mu_cond_hist</span><span class="p">,</span> <span class="n">Sigma_cond_hist</span><span class="p">)</span>
    835. 

  835. <span class="n">l2_smooth</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">z_hist</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">mu_hist_smooth</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">],</span> <span class="mi">2</span><span class="p">)</span>
    836. 

  836. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;L2-smooth: </span><span class="si">{</span><span class="n">l2_smooth</span><span class="si">:</span><span class="s2">0.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
    837. 

  837. <span class="n">fig_smoothed</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>
    838. 

  838. <span class="n">plot_tracking_values</span><span class="p">(</span><span class="n">x_hist</span><span class="p">,</span> <span class="n">mu_hist_smooth</span><span class="p">,</span> <span class="n">Sigma_hist_smooth</span><span class="p">,</span> <span class="s2">&quot;smoothed&quot;</span><span class="p">,</span> <span class="n">axs</span><span class="p">)</span>
    839. 

  839. </pre></div>
    840. 

  840. </div>
    841. 

  841. </div>
    842. 

  842. <div class="cell_output docutils container">
    843. 

  843. <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>L2-smooth: 2.0450
    844. 

  844. </pre></div>
    845. 

  845. </div>
    846. 

  846. <img alt="../../_images/inference_15_1.png" src="../../_images/inference_15_1.png" />
    847. 

  847. </div>
    848. 

  848. </div>
    849. 

  849. </div>
    850. 

  850. </div>
    851. 

  851. 

  852.     <script type="text/x-thebe-config">
    853. 

  853.     {
    854. 

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    855. 

  855.         binderOptions: {
    856. 

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    857. 

  857.             ref: "master",
    858. 

  858.         },
    859. 

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    860. 

  860.             theme: "abcdef",
    861. 

  861.             mode: "python"
    862. 

  862.         },
    863. 

  863.         kernelOptions: {
    864. 

  864.             kernelName: "python3",
    865. 

  865.             path: "./chapters/ssm"
    866. 

  866.         },
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    880. 

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