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<h1 class="title">Wiener process regression in LazyPPL</h1>
</header>
<details class="code-details">
<summary>
Extensions and imports for this Literate Haskell file
</summary>
<div class="sourceCode" id="cb1"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">module</span> <span class="dt">WienerDemo</span> <span class="kw">where</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">LazyPPL</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Distr</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.List</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.Map</span> (empty,lookup,insert,size,keys)</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Data.IORef</span></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">System.IO.Unsafe</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="kw">import</span> <span class="dt">Graphics.Matplotlib</span> <span class="kw">hiding</span> (density)</span></code></pre></div>
</details>
<p><br></p>
We can define a random function <code
class="sourceCode haskell"><span class="ot">wiener ::</span> <span class="dt">Prob</span>(<span class="dt">Double</span> <span class="ot">-></span> <span class="dt">Double</span>)</code>
which describes a Wiener process. This requires an infinite number of
random choices, first because it is defined for arbitrarily large
<code>x</code>, but also it needs an infinite number of random choices
in each finite interval, because it is no-where differentiable. This is
all dealt with using laziness. We can’t plot this unbounded,
undifferentiable function precisely, but when we plot it with a fixed
resolution and viewport, the necessary finite random choices are
triggered. <img src="images/wiener-prior.svg" />
<details class="code-details">
<summary>
(Plotting code)
</summary>
<div class="sourceCode" id="cb2"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>plotWienerPrior <span class="ot">=</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">do</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a> fws <span class="ot"><-</span> mh <span class="dv">1</span> <span class="op">$</span> sample wiener</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> xys <span class="ot">=</span> <span class="fu">map</span> (\f <span class="ot">-></span> <span class="fu">map</span> (\x <span class="ot">-></span> (x,f x)) [<span class="dv">0</span>,<span class="fl">0.1</span><span class="op">..</span><span class="dv">6</span>]) <span class="op">$</span> <span class="fu">map</span> <span class="fu">fst</span> <span class="op">$</span> <span class="fu">take</span> <span class="dv">100</span> <span class="op">$</span> fws</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> plotCoords <span class="st">"images/wiener-prior.svg"</span> [] xys (<span class="op">-</span><span class="dv">7</span>) <span class="dv">7</span> <span class="fl">0.2</span></span></code></pre></div>
</details>
<br> We will use this random function as a prior for Bayesian
regression, as in <a href="Regression.html">the other regression
examples</a>. Here is our example data set:
<div class="sourceCode" id="cb3"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="ot">dataset ::</span> [(<span class="dt">Double</span>, <span class="dt">Double</span>)]</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>dataset <span class="ot">=</span> [(<span class="dv">0</span>,<span class="fl">0.6</span>), (<span class="dv">1</span>, <span class="fl">0.7</span>), (<span class="dv">2</span>,<span class="fl">1.2</span>), (<span class="dv">3</span>,<span class="fl">3.2</span>), (<span class="dv">4</span>,<span class="fl">6.8</span>), (<span class="dv">5</span>, <span class="fl">8.2</span>), (<span class="dv">6</span>,<span class="fl">8.4</span>)]</span></code></pre></div>
And here is our model where we combine a Wiener function <code>g</code>
plus a random start point <code>a</code>. (Note that we are treating
this <code>g</code> as a function like any other. And we could also have
built this model with the second-order <code>regress</code> function
from <a href="Regression.html">the other regression examples</a>.)
<div class="sourceCode" id="cb4"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="ot">example ::</span> <span class="dt">Meas</span> (<span class="dt">Double</span> <span class="ot">-></span> <span class="dt">Double</span>)</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>example <span class="ot">=</span> <span class="kw">do</span> g <span class="ot"><-</span> sample wiener</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> a <span class="ot"><-</span> sample <span class="op">$</span> normal <span class="dv">0</span> <span class="dv">3</span></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> f x <span class="ot">=</span> a <span class="op">+</span> <span class="dv">2</span> <span class="op">*</span> (g x)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">mapM</span> (\(x,y) <span class="ot">-></span> score <span class="op">$</span> normalPdf (f x) <span class="fl">0.3</span> y) dataset</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span> f</span></code></pre></div>
We can now sample from the unnormalized distribution, using
Metropolis-Hastings. Because of laziness, the values of the functions
will be sampled at different times, some only when we come to plot the
functions.
<div class="sourceCode" id="cb5"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>plotWienerRegression <span class="ot">=</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a> <span class="kw">do</span></span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a> fws <span class="ot"><-</span> mh <span class="fl">0.1</span> example</span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> xys <span class="ot">=</span> <span class="fu">map</span> (\f <span class="ot">-></span> <span class="fu">map</span> (\x <span class="ot">-></span> (x,f x)) [<span class="dv">0</span>,<span class="fl">0.1</span><span class="op">..</span><span class="dv">6</span>]) <span class="op">$</span> <span class="fu">map</span> <span class="fu">fst</span> <span class="op">$</span> <span class="fu">take</span> <span class="dv">100</span> <span class="op">$</span> every <span class="dv">1000</span> <span class="op">$</span> <span class="fu">drop</span> <span class="dv">10000</span> <span class="op">$</span> fws</span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a> plotCoords <span class="st">"images/wiener-reg.svg"</span> dataset xys (<span class="op">-</span><span class="dv">2</span>) <span class="dv">10</span> <span class="fl">0.1</span></span></code></pre></div>
<p><img src="images/wiener-reg.svg" /></p>
The Wiener function itself is defined by using a Brownian bridge and a
hidden memo table. Although it uses hidden state, it is still safe,
i.e. statistically commutative and discardable.
<details class="code-details">
<summary>
Definition of Wiener function
</summary>
<div class="sourceCode" id="cb6"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="ot">wiener ::</span> <span class="dt">Prob</span> (<span class="dt">Double</span> <span class="ot">-></span> <span class="dt">Double</span>)</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>wiener <span class="ot">=</span> <span class="dt">Prob</span> <span class="op">$</span> \(<span class="dt">Tree</span> r gs) <span class="ot">-></span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a> unsafePerformIO <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a> ref <span class="ot"><-</span> newIORef Data.Map.empty</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a> modifyIORef' ref (Data.Map.insert <span class="dv">0</span> <span class="dv">0</span>)</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span> <span class="op">$</span> \x <span class="ot">-></span> unsafePerformIO <span class="op">$</span> <span class="kw">do</span></span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a> table <span class="ot"><-</span> readIORef ref</span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a> <span class="kw">case</span> Data.Map.lookup x table <span class="kw">of</span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a> <span class="dt">Just</span> y <span class="ot">-></span> <span class="kw">do</span> {<span class="fu">return</span> y}</span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a> <span class="dt">Nothing</span> <span class="ot">-></span> <span class="kw">do</span> <span class="kw">let</span> lower <span class="ot">=</span> <span class="kw">do</span> {l <span class="ot"><-</span> findMaxLower x (keys table) ;</span>
<span id="cb6-11"><a href="#cb6-11" aria-hidden="true" tabindex="-1"></a> v <span class="ot"><-</span> Data.Map.lookup l table ; <span class="fu">return</span> (l,v) }</span>
<span id="cb6-12"><a href="#cb6-12" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> upper <span class="ot">=</span> <span class="kw">do</span> {u <span class="ot"><-</span> find (<span class="op">></span> x) (keys table) ;</span>
<span id="cb6-13"><a href="#cb6-13" aria-hidden="true" tabindex="-1"></a> v <span class="ot"><-</span> Data.Map.lookup u table ; <span class="fu">return</span> (u,v) }</span>
<span id="cb6-14"><a href="#cb6-14" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> m <span class="ot">=</span> bridge lower x upper</span>
<span id="cb6-15"><a href="#cb6-15" aria-hidden="true" tabindex="-1"></a> <span class="kw">let</span> y <span class="ot">=</span> runProb m (gs <span class="op">!!</span> (<span class="dv">1</span> <span class="op">+</span> size table))</span>
<span id="cb6-16"><a href="#cb6-16" aria-hidden="true" tabindex="-1"></a> modifyIORef' ref (Data.Map.insert x y)</span>
<span id="cb6-17"><a href="#cb6-17" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span> y</span>
<span id="cb6-18"><a href="#cb6-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-19"><a href="#cb6-19" aria-hidden="true" tabindex="-1"></a><span class="ot">bridge ::</span> <span class="dt">Maybe</span> (<span class="dt">Double</span>,<span class="dt">Double</span>) <span class="ot">-></span> <span class="dt">Double</span> <span class="ot">-></span> <span class="dt">Maybe</span> (<span class="dt">Double</span>,<span class="dt">Double</span>) <span class="ot">-></span> <span class="dt">Prob</span> <span class="dt">Double</span></span>
<span id="cb6-20"><a href="#cb6-20" aria-hidden="true" tabindex="-1"></a><span class="co">-- not needed since the table is always initialized with (0, 0)</span></span>
<span id="cb6-21"><a href="#cb6-21" aria-hidden="true" tabindex="-1"></a><span class="co">-- bridge Nothing y Nothing = if y==0 then return 0 else normal 0 (sqrt y) </span></span>
<span id="cb6-22"><a href="#cb6-22" aria-hidden="true" tabindex="-1"></a>bridge (<span class="dt">Just</span> (x,x')) y <span class="dt">Nothing</span> <span class="ot">=</span> normal x' (<span class="fu">sqrt</span> (y<span class="op">-</span>x))</span>
<span id="cb6-23"><a href="#cb6-23" aria-hidden="true" tabindex="-1"></a>bridge <span class="dt">Nothing</span> y (<span class="dt">Just</span> (z,z')) <span class="ot">=</span> normal z' (<span class="fu">sqrt</span> (z<span class="op">-</span>y))</span>
<span id="cb6-24"><a href="#cb6-24" aria-hidden="true" tabindex="-1"></a>bridge (<span class="dt">Just</span> (x,x')) y (<span class="dt">Just</span> (z,z')) <span class="ot">=</span> normal (x' <span class="op">+</span> ((y<span class="op">-</span>x)<span class="op">*</span>(z'<span class="op">-</span>x')<span class="op">/</span>(z<span class="op">-</span>x))) (<span class="fu">sqrt</span> ((z<span class="op">-</span>y)<span class="op">*</span>(y<span class="op">-</span>x)<span class="op">/</span>(z<span class="op">-</span>x)))</span>
<span id="cb6-25"><a href="#cb6-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb6-26"><a href="#cb6-26" aria-hidden="true" tabindex="-1"></a><span class="ot">findMaxLower ::</span> <span class="dt">Double</span> <span class="ot">-></span> [<span class="dt">Double</span>] <span class="ot">-></span> <span class="dt">Maybe</span> <span class="dt">Double</span> </span>
<span id="cb6-27"><a href="#cb6-27" aria-hidden="true" tabindex="-1"></a>findMaxLower d [] <span class="ot">=</span> <span class="dt">Nothing</span></span>
<span id="cb6-28"><a href="#cb6-28" aria-hidden="true" tabindex="-1"></a>findMaxLower d (x<span class="op">:</span>xs) <span class="ot">=</span> <span class="kw">let</span> y <span class="ot">=</span> findMaxLower d xs <span class="kw">in</span></span>
<span id="cb6-29"><a href="#cb6-29" aria-hidden="true" tabindex="-1"></a> <span class="kw">case</span> y <span class="kw">of</span> </span>
<span id="cb6-30"><a href="#cb6-30" aria-hidden="true" tabindex="-1"></a> <span class="dt">Nothing</span> <span class="ot">-></span> <span class="kw">if</span> x <span class="op"><</span> d <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Nothing</span> </span>
<span id="cb6-31"><a href="#cb6-31" aria-hidden="true" tabindex="-1"></a> <span class="dt">Just</span> m <span class="ot">-></span> <span class="kw">do</span> </span>
<span id="cb6-32"><a href="#cb6-32" aria-hidden="true" tabindex="-1"></a> <span class="kw">if</span> x <span class="op">></span> m <span class="op">&&</span> x <span class="op"><</span> d <span class="kw">then</span> <span class="dt">Just</span> x <span class="kw">else</span> <span class="dt">Just</span> m </span></code></pre></div>
</details>
<details class="code-details">
<summary>
Graphing routines
</summary>
<div class="sourceCode" id="cb7"><pre
class="sourceCode haskell literate"><code class="sourceCode haskell"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="ot">plotCoords ::</span> <span class="dt">String</span> <span class="ot">-></span> [(<span class="dt">Double</span>,<span class="dt">Double</span>)] <span class="ot">-></span> [[(<span class="dt">Double</span>,<span class="dt">Double</span>)]] <span class="ot">-></span> <span class="dt">Double</span> <span class="ot">-></span> <span class="dt">Double</span> <span class="ot">-></span> <span class="dt">Double</span> <span class="ot">-></span> <span class="dt">IO</span> ()</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>plotCoords filename dataset xyss ymin ymax alpha <span class="ot">=</span> </span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a> <span class="kw">do</span> <span class="fu">putStrLn</span> <span class="op">$</span> <span class="st">"Plotting "</span> <span class="op">++</span> filename <span class="op">++</span> <span class="st">"..."</span></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a> file filename <span class="op">$</span> <span class="fu">foldl</span> (\a xys <span class="ot">-></span> a <span class="op">%</span> plot (<span class="fu">map</span> <span class="fu">fst</span> xys) (<span class="fu">map</span> <span class="fu">snd</span> xys) <span class="op">@@</span> [o1 <span class="st">"go-"</span>, o2 <span class="st">"linewidth"</span> (<span class="fl">0.5</span><span class="ot"> ::</span> <span class="dt">Double</span>), o2 <span class="st">"alpha"</span> alpha, o2 <span class="st">"ms"</span> (<span class="dv">0</span><span class="ot"> ::</span> <span class="dt">Int</span>)]) (scatter (<span class="fu">map</span> <span class="fu">fst</span> dataset) (<span class="fu">map</span> <span class="fu">snd</span> dataset) <span class="op">@@</span> [o2 <span class="st">"c"</span> <span class="st">"black"</span>] <span class="op">%</span> xlim (<span class="dv">0</span><span class="ot"> ::</span> <span class="dt">Int</span>) (<span class="dv">6</span><span class="ot"> ::</span> <span class="dt">Int</span>) <span class="op">%</span> ylim ymin ymax) xyss</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">putStrLn</span> <span class="st">"Done."</span></span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span> ()</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></a><span class="ot">main ::</span> <span class="dt">IO</span> ()</span>
<span id="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></a>main <span class="ot">=</span> <span class="kw">do</span> { plotWienerPrior ; plotWienerRegression } </span></code></pre></div>
</details>
<br>
<hr>
<small><i> Generated by Pandoc from Literate Haskell. Full source on <a href="https://github.com/lazyppl-team/lazyppl"><i class="fa-brands fa-github"></i></a>.</i></small>
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