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<h1 class="title">Information theory: HW #2 solution</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#orgheadline1">1. Markov chain entropy</a></li>
<li><a href="#orgheadline2">2. Huffman codes</a></li>
</ul>
</div>
</div>
<div id="outline-container-orgheadline1" class="outline-2">
<h2 id="orgheadline1"><span class="section-number-2">1</span> Markov chain entropy</h2>
<div class="outline-text-2" id="text-1">
<p>
First, short remark I've already given in the previous homework: I'm
putting some haskell code here just for fun/more detailed
explanation/proof-of-work. Please skip it if it's too borkng.
</p>
<p>
Let's consider the following matrix:
</p>
\begin{align*}
E =
\begin{pmatrix}
1/4 & 0 & 3/4\\
0 & 1/4 & 3/4\\
1/4 & 1/2 & 1/4
\end{pmatrix}
\end{align*}
<p>
First task is to find such \(p\) that \(p = pH\). This can be done, in
fact, in two ways. First one is solving the following equation:
</p>
\begin{align*}
\begin{pmatrix}x&y&z\end{pmatrix}=
\begin{pmatrix}x&y&z\end{pmatrix}*E
\end{align*}
<p>
Indeed, this equation system has a solution (skipped): \(P = (1/6, 1/3,
1/2)\). But it can be though obtained in another way. Let's suppose \(E\)
is ergodic, then accordingly to ergodic theorem <sup><a id="fnr.1" class="footref" href="#fn.1">1</a></sup> \(E^n\) converges
to \(A\), where each row of it is equal to \(P\). This can be easily
seen in practice:
</p>
<div class="org-src-container">
<pre class="src src-haskell"><span style="color: #868B44;">--</span><span style="color: #333333;"> </span><span style="color: #868B44;">|</span><span style="color: #333333;"> </span><span style="color: #868B44;">This</span><span style="color: #333333;"> </span><span style="color: #868B44;">function</span><span style="color: #333333;"> </span><span style="color: #868B44;">takes</span><span style="color: #333333;"> </span><span style="color: #868B44;">matrix</span><span style="color: #333333;"> </span><span style="color: #868B44;">and</span><span style="color: #333333;"> </span><span style="color: #868B44;">raises</span><span style="color: #333333;"> </span><span style="color: #868B44;">it</span><span style="color: #333333;"> </span><span style="color: #868B44;">to</span><span style="color: #333333;"> </span><span style="color: #868B44;">the</span><span style="color: #333333;"> </span><span style="color: #868B44;">power</span><span style="color: #333333;"> </span><span style="color: #868B44;">n.</span>
<span style="color: #868B44;">--</span><span style="color: #333333;"> </span><span style="color: #868B44;">(!*!)</span><span style="color: #333333;"> </span><span style="color: #868B44;">is</span><span style="color: #333333;"> </span><span style="color: #868B44;">matrix</span><span style="color: #333333;"> </span><span style="color: #868B44;">multiplication.</span>
(<span style="color: #8AAFFA;">!*^</span>)<span style="color: #333333;"> </span><span style="color: #A197BF;">::</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Matrix</span><span style="color: #333333;"> </span><span style="color: #A197BF;">-></span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Int</span><span style="color: #333333;"> </span><span style="color: #A197BF;">-></span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Matrix</span>
(<span style="color: #8AAFFA;">!*^</span>)<span style="color: #333333;"> </span>m<span style="color: #333333;"> </span>0<span style="color: #333333;"> </span><span style="color: #A197BF;">=</span><span style="color: #333333;"> </span>m
(<span style="color: #8AAFFA;">!*^</span>)<span style="color: #333333;"> </span>m<span style="color: #333333;"> </span>i<span style="color: #333333;"> </span><span style="color: #A197BF;">=</span><span style="color: #333333;"> </span>m<span style="color: #333333;"> </span><span style="color: #A197BF;">!*!</span><span style="color: #333333;"> </span>(m<span style="color: #333333;"> </span><span style="color: #A197BF;">!*^</span><span style="color: #333333;"> </span>(i<span style="color: #A197BF;">-</span>1))
<span style="color: #637579;">--</span><span style="color: #333333;"> </span><span style="color: #637579;">Type</span><span style="color: #333333;"> </span><span style="color: #637579;">aliases</span><span style="color: #333333;"> </span><span style="color: #637579;">for</span><span style="color: #333333;"> </span><span style="color: #637579;">vector</span><span style="color: #333333;"> </span><span style="color: #637579;">and</span><span style="color: #333333;"> </span><span style="color: #637579;">matrix</span>
<span style="color: #FFA070;">type</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Vector</span><span style="color: #333333;"> </span><span style="color: #A197BF;">=</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">V.Vector</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Double</span>
<span style="color: #FFA070;">type</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Matrix</span><span style="color: #333333;"> </span><span style="color: #A197BF;">=</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">V.Vector</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Vector</span>
<span style="color: #868B44;">--</span><span style="color: #333333;"> </span><span style="color: #868B44;">|</span><span style="color: #333333;"> </span><span style="color: #868B44;">Calculate</span><span style="color: #333333;"> </span><span style="color: #868B44;">the</span><span style="color: #333333;"> </span><span style="color: #868B44;">entropy</span>
<span style="color: #8AAFFA;">entropy</span><span style="color: #333333;"> </span><span style="color: #A197BF;">::</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Vector</span><span style="color: #333333;"> </span><span style="color: #A197BF;">-></span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Double</span>
<span style="color: #8AAFFA;">entropy</span><span style="color: #333333;"> </span><span style="color: #A197BF;">=</span><span style="color: #333333;"> </span>sum<span style="color: #333333;"> </span><span style="color: #A197BF;">.</span><span style="color: #333333;"> </span>V.map<span style="color: #333333;"> </span>(<span style="color: #A197BF;">\</span>p<span style="color: #333333;"> </span><span style="color: #A197BF;">-></span><span style="color: #333333;"> </span><span style="color: #A197BF;">-</span><span style="color: #333333;"> </span>p<span style="color: #333333;"> </span><span style="color: #A197BF;">*</span><span style="color: #333333;"> </span>log2<span style="color: #333333;"> </span>p)
<span style="color: #868B44;">--</span><span style="color: #333333;"> </span><span style="color: #868B44;">|</span><span style="color: #333333;"> </span><span style="color: #868B44;">Given</span><span style="color: #333333;"> </span><span style="color: #868B44;">matrix</span>
<span style="color: #8AAFFA;">matrix</span><span style="color: #333333;"> </span><span style="color: #A197BF;">::</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Matrix</span>
<span style="color: #8AAFFA;">matrix</span><span style="color: #333333;"> </span><span style="color: #A197BF;">=</span>
<span style="color: #333333;"> </span>V.fromList<span style="color: #333333;"> </span><span style="color: #A197BF;">$</span><span style="color: #333333;"> </span>map<span style="color: #333333;"> </span>V.fromList<span style="color: #333333;"> </span><span style="color: #A197BF;">$</span>
<span style="color: #333333;"> </span>[[1<span style="color: #A197BF;">/</span>4,<span style="color: #333333;"> </span>0,<span style="color: #333333;"> </span>3<span style="color: #A197BF;">/</span>4],
<span style="color: #333333;"> </span>[<span style="color: #333333;"> </span>0,<span style="color: #333333;"> </span>1<span style="color: #A197BF;">/</span>4,<span style="color: #333333;"> </span>3<span style="color: #A197BF;">/</span>4],
<span style="color: #333333;"> </span>[1<span style="color: #A197BF;">/</span>4,<span style="color: #333333;"> </span>1<span style="color: #A197BF;">/</span>2,<span style="color: #333333;"> </span>1<span style="color: #A197BF;">/</span>4]]
<span style="color: #868B44;">--</span><span style="color: #333333;"> </span><span style="color: #868B44;">|</span><span style="color: #333333;"> </span><span style="color: #868B44;">Solution</span><span style="color: #333333;"> </span><span style="color: #868B44;">vector</span><span style="color: #333333;"> </span><span style="color: #868B44;">calculated</span><span style="color: #333333;"> </span><span style="color: #868B44;">from</span><span style="color: #333333;"> </span><span style="color: #868B44;">raising</span><span style="color: #333333;"> </span><span style="color: #868B44;">matrix</span><span style="color: #333333;"> </span><span style="color: #868B44;">E</span><span style="color: #333333;"> </span><span style="color: #868B44;">to</span><span style="color: #333333;"> </span><span style="color: #868B44;">the</span><span style="color: #333333;"> </span><span style="color: #868B44;">power</span>
<span style="color: #868B44;">--</span><span style="color: #333333;"> </span><span style="color: #868B44;">5000,</span><span style="color: #333333;"> </span><span style="color: #868B44;">takes</span><span style="color: #333333;"> </span><span style="color: #868B44;">0'th</span><span style="color: #333333;"> </span><span style="color: #868B44;">row.</span>
<span style="color: #8AAFFA;">p</span><span style="color: #333333;"> </span><span style="color: #A197BF;">::</span><span style="color: #333333;"> </span><span style="color: #EF5C5F;">Vector</span>
<span style="color: #8AAFFA;">p</span><span style="color: #333333;"> </span><span style="color: #A197BF;">=</span><span style="color: #333333;"> </span>(matrix<span style="color: #333333;"> </span><span style="color: #A197BF;">!*^</span><span style="color: #333333;"> </span>5000)<span style="color: #333333;"> </span><span style="color: #A197BF;">!</span><span style="color: #333333;"> </span>0
</pre>
</div>
<p>
If printed, value of <code>P</code> is
<code>[0.16666666666666663,0.3333333333333333,0.5000000000000001]</code>, that
seems pretty much to convert to the exact solution.
</p>
<p>
The entropy of the random source with given probabilities:
\(H = 1.459147\) (calculated as <code>entropy p</code>).
</p>
<p>
Now let's calculate \(H(X|X^∞)\):
</p>
\begin{align*}
H(X|X^{∞}) = H(X|X) = - ∑_i P_i ∑_j P_{ij} log(P_ij)
\end{align*}
<p>
\(P_ij\) is exactly \(E\). Proof of the first equality can be found in
the course textbook (1.7, example 1.7.2). The value calculated is
\(H(X|X^∞) = 1.155639\). Now we are able to compute \(H_n(X)\) using the
formula in the end of chapter 1.7 of the textbook:
</p>
\begin{align*}
H_n(X) = H(X|X^n) + \frac{s}{n}(H_s(X) - H(X|X^s))
\end{align*}
<p>
We'll use a fact that \(H(X|X^n)\) equals to \(H(X|X)\) and already
computed. Plus the markov's chain is simple, so \(s = 1\). Thus
formula looks like this:
</p>
\begin{align*}
H_n(X) = H(X|X) + \frac{1}{n}(H(X) - H(X|X))
= 1.1556 + \frac{0.303}{n}
\end{align*}
<p>
And so \(H_2 = 1.1556 + 0.303/2 = 1.3071\).
</p>
</div>
</div>
<div id="outline-container-orgheadline2" class="outline-2">
<h2 id="orgheadline2"><span class="section-number-2">2</span> Huffman codes</h2>
<div class="outline-text-2" id="text-2">
<p>
Given three words \(a, b, c\) with probabilities \(\frac{1}{6},
\frac{1}{3}, \frac{1}{2}\), we can assign the following codes to
them:
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Word</th>
<th scope="col" class="org-right">Probability</th>
<th scope="col" class="org-right">Code</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">a</td>
<td class="org-right">0.166</td>
<td class="org-right">01</td>
</tr>
<tr>
<td class="org-left">b</td>
<td class="org-right">0.333</td>
<td class="org-right">00</td>
</tr>
<tr>
<td class="org-left">c</td>
<td class="org-right">0.5</td>
<td class="org-right">1</td>
</tr>
</tbody>
</table>
<p>
Then it's easy to see that average amount of bits required to encode
one word is \(2*0.166+2*0.333+1*0.5 = 1.498\), which is more than
\(H(X)\). Let's then consider blocks of size 2 and calculate \(P(XY) =
P(X)*P(X|Y)\), where second probability is from \(E\):
</p>
<table border="2" cellspacing="0" cellpadding="6" rules="all" frame="border">
<colgroup>
<col class="org-left" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left">Word</th>
<th scope="col" class="org-right">P(X)</th>
<th scope="col" class="org-right">P(X!Y)</th>
<th scope="col" class="org-right">P(XY)</th>
<th scope="col" class="org-right">Code</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left">bc</td>
<td class="org-right">0.333</td>
<td class="org-right">0.75</td>
<td class="org-right">0.250</td>
<td class="org-right">00</td>
</tr>
<tr>
<td class="org-left">cb</td>
<td class="org-right">0.5</td>
<td class="org-right">0.5</td>
<td class="org-right">0.250</td>
<td class="org-right">01</td>
</tr>
<tr>
<td class="org-left">ac</td>
<td class="org-right">0.166</td>
<td class="org-right">0.75</td>
<td class="org-right">0.125</td>
<td class="org-right">101</td>
</tr>
<tr>
<td class="org-left">ca</td>
<td class="org-right">0.5</td>
<td class="org-right">0.25</td>
<td class="org-right">0.125</td>
<td class="org-right">110</td>
</tr>
<tr>
<td class="org-left">cc</td>
<td class="org-right">0.5</td>
<td class="org-right">0.25</td>
<td class="org-right">0.125</td>
<td class="org-right">111</td>
</tr>
<tr>
<td class="org-left">bb</td>
<td class="org-right">0.333</td>
<td class="org-right">0.25</td>
<td class="org-right">0.083</td>
<td class="org-right">1001</td>
</tr>
<tr>
<td class="org-left">aa</td>
<td class="org-right">0.166</td>
<td class="org-right">0.25</td>
<td class="org-right">0.042</td>
<td class="org-right">10001</td>
</tr>
<tr>
<td class="org-left">ab</td>
<td class="org-right">0.166</td>
<td class="org-right">0</td>
<td class="org-right">0</td>
<td class="org-right">100000</td>
</tr>
<tr>
<td class="org-left">ba</td>
<td class="org-right">0.333</td>
<td class="org-right">0</td>
<td class="org-right">0</td>
<td class="org-right">100001</td>
</tr>
</tbody>
</table>
<p>
And then average bit amount per word is:
</p>
\begin{align*}\frac{0.250*2*2 + 0.125*3*3 + 0.083*4+0.042*5}{2} = \frac{2.667}{2} = 1.334\end{align*}
<p>
, which appears to be more than \(1.498\) that we got in the previous
attempt. So encoding info in 2 char blocks is more efficient. I
won't proceed with bigger block sizes because it doesn't provide
further academic experience.
</p>
</div>
</div>
<div id="footnotes">
<h2 class="footnotes">Footnotes: </h2>
<div id="text-footnotes">
<div class="footdef"><sup><a id="fn.1" class="footnum" href="#fnr.1">1</a></sup> <div class="footpara"><p class="footpara">
<a href="http://neerc.ifmo.ru/wiki/index.php?title=%D0%AD%D1%80%D0%B3%D0%BE%D0%B4%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D1%86%D0%B5%D0%BF%D1%8C">http://neerc.ifmo.ru/wiki/index.php?title=%D0%AD%D1%80%D0%B3%D0%BE%D0%B4%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D1%86%D0%B5%D0%BF%D1%8C</a>
</p></div></div>
</div>
</div></div>
<div id="postamble" class="status">
<p class="author">Author: Volkhov Mikhail, M3338</p>
<p class="date">Created: 2016-11-04 Fri 14:18</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>
</html>