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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.signal_statistics"></a><a class="link" href="signal_statistics.html" title="Signal Statistics">Signal Statistics</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.signal_statistics.h0"></a>
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<span class="phrase"><a name="math_toolkit.signal_statistics.synopsis"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">signal_statistics</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">Container</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">sample_absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">Container</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">sample_absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">hoyer_sparsity</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">hoyer_sparsity</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">oracle_snr</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">signal</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">noisy_signal</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">oracle_snr_db</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">signal</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">noisy_signal</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">m2m4_snr_estimator</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(*</span><span class="identifier">first</span><span class="special">)</span> <span class="identifier">estimated_signal_kurtosis</span><span class="special">=</span><span class="number">1</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(*</span><span class="identifier">first</span><span class="special">)</span> <span class="identifier">estimated_noise_kurtosis</span><span class="special">=</span><span class="number">3</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">m2m4_snr_estimator</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">noisy_signal</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">estimated_signal_kurtosis</span><span class="special">=</span><span class="number">1</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">estimate_noise_kurtosis</span><span class="special">=</span><span class="number">3</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">m2m4_snr_estimator_db</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(*</span><span class="identifier">first</span><span class="special">)</span> <span class="identifier">estimated_signal_kurtosis</span><span class="special">=</span><span class="number">1</span><span class="special">,</span> <span class="keyword">decltype</span><span class="special">(*</span><span class="identifier">first</span><span class="special">)</span> <span class="identifier">estimated_noise_kurtosis</span><span class="special">=</span><span class="number">3</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">m2m4_snr_estimator_db</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">noisy_signal</span><span class="special">,</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">estimated_signal_kurtosis</span><span class="special">=</span><span class="number">1</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">estimate_noise_kurtosis</span><span class="special">=</span><span class="number">3</span><span class="special">);</span>
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<span class="special">}</span>
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</pre>
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<h4>
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<a name="math_toolkit.signal_statistics.h1"></a>
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<span class="phrase"><a name="math_toolkit.signal_statistics.description"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.description">Description</a>
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</h4>
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<p>
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The file <code class="computeroutput"><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">signal_statistics</span><span class="special">.</span><span class="identifier">hpp</span></code> is a
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set of facilities for computing quantities commonly used in signal analysis.
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</p>
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<p>
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Our examples use <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span></code> to
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hold the data, but this not required. In general, you can store your data in
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an Eigen array, and Armadillo vector, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>,
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and for many of the routines, a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">forward_list</span></code>.
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These routines are usable in float, double, long double, and Boost.Multiprecision
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precision, as well as their complex extensions whenever the computation is
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well-defined.
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</p>
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<h4>
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<a name="math_toolkit.signal_statistics.h2"></a>
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<span class="phrase"><a name="math_toolkit.signal_statistics.absolute_gini_coefficient"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.absolute_gini_coefficient">Absolute
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Gini Coefficient</a>
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</h4>
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<p>
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The Gini coefficient, first used to measure wealth inequality, is also one
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of the best measures of the sparsity of an expansion in a basis. A sparse expansion
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has most of its norm concentrated in just a few coefficients, making the connection
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with wealth inequality obvious. See <a href="https://arxiv.org/pdf/0811.4706.pdf" target="_top">Hurley
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and Rickard</a> for details. However, for measuring sparsity, the phase
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of the numbers is irrelevant, so we provide the <code class="computeroutput"><span class="identifier">absolute_gini_coefficient</span></code>:
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</p>
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<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">sample_absolute_gini_coefficient</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">absolute_gini_coefficient</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">>></span> <span class="identifier">v</span><span class="special">{{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">}};</span>
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<span class="keyword">double</span> <span class="identifier">abs_gini</span> <span class="special">=</span> <span class="identifier">sample_absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
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<span class="comment">// now abs_gini = 1; maximally unequal</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">>></span> <span class="identifier">w</span><span class="special">{{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">},</span> <span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">},</span> <span class="special">{</span><span class="number">0</span><span class="special">,-</span><span class="number">1</span><span class="special">},</span> <span class="special">{-</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">}};</span>
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<span class="identifier">abs_gini</span> <span class="special">=</span> <span class="identifier">absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">w</span><span class="special">);</span>
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<span class="comment">// now abs_gini = 0; every element of the vector has equal magnitude</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">u</span><span class="special">{-</span><span class="number">1</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">};</span>
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<span class="identifier">abs_gini</span> <span class="special">=</span> <span class="identifier">absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">u</span><span class="special">);</span>
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<span class="comment">// now abs_gini = 0</span>
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<span class="comment">// Alternative call useful for computing over subset of the input:</span>
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<span class="identifier">abs_gini</span> <span class="special">=</span> <span class="identifier">absolute_gini_coefficient</span><span class="special">(</span><span class="identifier">u</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">u</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">+</span> <span class="number">1</span><span class="special">);</span>
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</pre>
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<p>
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The sample Gini coefficient returns unity for a vector which has only one nonzero
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coefficient. The population Gini coefficient of a vector with one non-zero
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element is dependent on the length of the input.
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</p>
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<p>
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The sample Gini coefficient lacks one desirable property of the population
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Gini coefficient, namely that "cloning" a vector has the same Gini
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coefficient; though cloning holds to very high accuracy with the sample Gini
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coefficient and can easily be recovered by a rescaling.
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</p>
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<p>
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If sorting the input data is too much expense for a sparsity measure (is it
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going to be perfect anyway?), consider calculating the Hoyer sparsity instead.
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</p>
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<h4>
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<a name="math_toolkit.signal_statistics.h3"></a>
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<span class="phrase"><a name="math_toolkit.signal_statistics.hoyer_sparsity"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.hoyer_sparsity">Hoyer
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Sparsity</a>
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</h4>
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<p>
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The Hoyer sparsity measures a normalized ratio of the ℓ<sup>1</sup> and ℓ<sup>2</sup> norms.
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As the name suggests, it is used to measure the sparsity of an expansion in
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some basis.
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</p>
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<p>
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The Hoyer sparsity computes (√<span class="emphasis"><em>N</em></span> - ℓ<sup>1</sup>(v)/ℓ<sup>2</sup>(v))/(√N
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-1). For details, see <a href="http://www.jmlr.org/papers/volume5/hoyer04a/hoyer04a.pdf" target="_top">Hoyer</a>
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as well as <a href="https://arxiv.org/pdf/0811.4706.pdf" target="_top">Hurley and Rickard</a>.
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</p>
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<p>
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A few special cases will serve to clarify the intended use: If <span class="emphasis"><em>v</em></span>
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has only one nonzero coefficient, the Hoyer sparsity attains its maxima of
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1. If the coefficients of <span class="emphasis"><em>v</em></span> all have the same magnitude,
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then the Hoyer sparsity attains its minima of zero. If the elements of <span class="emphasis"><em>v</em></span>
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are uniformly distributed on an interval [0, <span class="emphasis"><em>b</em></span>], then
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the Hoyer sparsity is approximately 0.133.
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</p>
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<p>
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Usage:
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</p>
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<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
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<span class="identifier">Real</span> <span class="identifier">hs</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">hoyer_sparsity</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
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<span class="comment">// hs = 1</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">Real</span><span class="special">></span> <span class="identifier">v</span><span class="special">{</span><span class="number">1</span><span class="special">,-</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
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<span class="identifier">Real</span> <span class="identifier">hs</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">hoyer_sparsity</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
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<span class="comment">// hs = 0</span>
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</pre>
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<p>
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The container must be forward iterable and the contents are not modified. Accepts
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real, complex, and integer inputs. If the input is an integral type, the output
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is a double precision float.
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</p>
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<h4>
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<a name="math_toolkit.signal_statistics.h4"></a>
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<span class="phrase"><a name="math_toolkit.signal_statistics.oracle_signal_to_noise_ratio"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.oracle_signal_to_noise_ratio">Oracle
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Signal-to-noise ratio</a>
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</h4>
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<p>
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The function <code class="computeroutput"><span class="identifier">oracle_snr</span></code> computes
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the ratio ‖ <span class="emphasis"><em>s</em></span> ‖<sub>2</sub><sup>2</sup> / ‖ <span class="emphasis"><em>s</em></span>
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- <span class="emphasis"><em>x</em></span> ‖<sub>2</sub><sup>2</sup>, where <span class="emphasis"><em>s</em></span> is signal
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and <span class="emphasis"><em>x</em></span> is a noisy signal. The function <code class="computeroutput"><span class="identifier">oracle_snr_db</span></code>
|
|
computes 10<code class="computeroutput"><span class="identifier">log</span></code><sub>10</sub>(‖
|
|
<span class="emphasis"><em>s</em></span> ‖<sup>2</sup> / ‖ <span class="emphasis"><em>s</em></span> - <span class="emphasis"><em>x</em></span>
|
|
‖<sup>2</sup>). The functions are so named because in general, one does not know
|
|
how to decompose a real signal <span class="emphasis"><em>x</em></span> into <span class="emphasis"><em>s</em></span>
|
|
+ <span class="emphasis"><em>w</em></span> and as such <span class="emphasis"><em>s</em></span> is regarded as
|
|
oracle information. Hence this function is mainly useful for unit testing other
|
|
SNR estimators.
|
|
</p>
|
|
<p>
|
|
Usage:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">signal</span><span class="special">(</span><span class="number">500</span><span class="special">,</span> <span class="number">3.2</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">noisy_signal</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
|
|
<span class="comment">// fill 'noisy_signal' signal + noise</span>
|
|
<span class="keyword">double</span> <span class="identifier">snr_db</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">oracle_snr_db</span><span class="special">(</span><span class="identifier">signal</span><span class="special">,</span> <span class="identifier">noisy_signal</span><span class="special">);</span>
|
|
<span class="keyword">double</span> <span class="identifier">snr</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">oracle_snr</span><span class="special">(</span><span class="identifier">signal</span><span class="special">,</span> <span class="identifier">noisy_signal</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
The input can be real, complex, or integral. Integral inputs produce double
|
|
precision floating point outputs. The input data is not modified and must satisfy
|
|
the requirements of a <code class="computeroutput"><span class="identifier">RandomAccessContainer</span></code>.
|
|
</p>
|
|
<h4>
|
|
<a name="math_toolkit.signal_statistics.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.signal_statistics.m_sub_2_m_sub_4_snr_estimation"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.m_sub_2_m_sub_4_snr_estimation"><span class="emphasis"><em>M</em></span><sub>2</sub><span class="emphasis"><em>M</em></span><sub>4</sub> SNR
|
|
Estimation</a>
|
|
</h4>
|
|
<p>
|
|
Estimates the SNR of a noisy signal via the <span class="emphasis"><em>M</em></span><sub>2</sub><span class="emphasis"><em>M</em></span><sub>4</sub> method.
|
|
See <a href="https://doi.org/10.1109/26.871393" target="_top">Pauluzzi and N.C. Beaulieu</a>
|
|
and <a href="https://doi.org/10.1109/ISIT.1994.394869" target="_top">Matzner and Englberger</a>
|
|
for details.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">noisy_signal</span><span class="special">(</span><span class="number">512</span><span class="special">);</span>
|
|
<span class="comment">// fill noisy_signal with data contaminated by Gaussian white noise:</span>
|
|
<span class="keyword">double</span> <span class="identifier">est_snr_db</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">m2m4_snr_estimator_db</span><span class="special">(</span><span class="identifier">noisy_signal</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
The <span class="emphasis"><em>M</em></span><sub>2</sub><span class="emphasis"><em>M</em></span><sub>4</sub> SNR estimator is an "in-service"
|
|
estimator, meaning that the estimate is made using the noisy, data-bearing
|
|
signal, and does not require a background estimate. This estimator has been
|
|
found to be work best between roughly -3 and 15db, tending to overestimate
|
|
the noise below -3db, and underestimate the noise above 15db. See <a href="https://www.mdpi.com/2078-2489/8/3/75/pdf" target="_top">Xue
|
|
et al</a> for details.
|
|
</p>
|
|
<p>
|
|
The <span class="emphasis"><em>M</em></span><sub>2</sub><span class="emphasis"><em>M</em></span><sub>4</sub> SNR estimator, by default,
|
|
assumes that the kurtosis of the signal is 1 and the kurtosis of the noise
|
|
is 3, the latter corresponding to Gaussian noise. These parameters, however,
|
|
can be overridden:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">noisy_signal</span><span class="special">(</span><span class="number">512</span><span class="special">);</span>
|
|
<span class="comment">// fill noisy_signal with the data:</span>
|
|
<span class="keyword">double</span> <span class="identifier">signal_kurtosis</span> <span class="special">=</span> <span class="number">1.5</span><span class="special">;</span>
|
|
<span class="comment">// Noise is assumed to follow Laplace distribution, which has kurtosis of 6:</span>
|
|
<span class="keyword">double</span> <span class="identifier">noise_kurtosis</span> <span class="special">=</span> <span class="number">6</span><span class="special">;</span>
|
|
<span class="keyword">double</span> <span class="identifier">est_snr</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">m2m4_snr_estimator_db</span><span class="special">(</span><span class="identifier">noisy_signal</span><span class="special">,</span> <span class="identifier">signal_kurtosis</span><span class="special">,</span> <span class="identifier">noise_kurtosis</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
Now, technically the method is a "blind SNR estimator", meaning that
|
|
the no <span class="emphasis"><em>a-priori</em></span> information about the signal is required
|
|
to use the method. However, the performance of the method is <span class="emphasis"><em>vastly</em></span>
|
|
better if you can come up with a better estimate of the signal and noise kurtosis.
|
|
How can we do this? Suppose we know that the SNR is much greater than 1. Then
|
|
we can estimate the signal kurtosis simply by using the noisy signal kurtosis.
|
|
If the SNR is much less than one, this method breaks down as the noisy signal
|
|
kurtosis will tend to the noise kurtosis-though in this limit we have an excellent
|
|
estimator of the noise kurtosis! In addition, if you have a model of what your
|
|
signal should look like, you can precompute the signal kurtosis. For example,
|
|
sinusoids have a kurtosis of 1.5. See <a href="http://www.jcomputers.us/vol8/jcp0808-21.pdf" target="_top">here</a>
|
|
for a study which uses estimates of this sort to improve the performance of
|
|
the <span class="emphasis"><em>M</em></span><sub>2</sub><span class="emphasis"><em>M</em></span><sub>4</sub> estimator.
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Nota bene</em></span>: The traditional definition of SNR is <span class="emphasis"><em>not</em></span>
|
|
mean invariant. By this we mean that if a constant is added to every sample
|
|
of a signal, the SNR is changed. For example, adding DC bias to a signal changes
|
|
its SNR. For most use cases, this is really not what you intend; for example
|
|
a signal consisting of zeros plus Gaussian noise has an SNR of zero, whereas
|
|
a signal with a constant DC bias and random Gaussian noise might have a very
|
|
large SNR.
|
|
</p>
|
|
<p>
|
|
The <span class="emphasis"><em>M</em></span><sub>2</sub><span class="emphasis"><em>M</em></span><sub>4</sub> SNR estimator is computed
|
|
from mean-invariant quantities, and hence it should really be compared to the
|
|
mean-invariant SNR.
|
|
</p>
|
|
<p>
|
|
<span class="emphasis"><em>Nota bene</em></span>: This computation requires the solution of a
|
|
system of quadratic equations involving the noise kurtosis, the signal kurtosis,
|
|
and the second and fourth moments of the data. There is no guarantee that a
|
|
solution of this system exists for all value of these parameters, in fact nonexistence
|
|
can easily be demonstrated for certain data. If there is no solution to the
|
|
system, then failure is communicated by returning NaNs. This happens distressingly
|
|
often; if a user is aware of any blind SNR estimators which do not suffer from
|
|
this drawback, please open a github ticket and let us know.
|
|
</p>
|
|
<p>
|
|
The author has not managed to fully characterize the conditions under which
|
|
a real solution with <span class="emphasis"><em>S > 0</em></span> and <span class="emphasis"><em>N >0</em></span>
|
|
exists. However, a very intuitive example demonstrates why nonexistence can
|
|
occur. Suppose the signal and noise kurtosis are equal. Then the method has
|
|
no way to distinguish between the signal and the noise, and the solution is
|
|
non-unique.
|
|
</p>
|
|
<h4>
|
|
<a name="math_toolkit.signal_statistics.h6"></a>
|
|
<span class="phrase"><a name="math_toolkit.signal_statistics.references"></a></span><a class="link" href="signal_statistics.html#math_toolkit.signal_statistics.references">References</a>
|
|
</h4>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
Mallat, Stephane. <span class="emphasis"><em>A wavelet tour of signal processing: the sparse
|
|
way.</em></span> Academic press, 2008.
|
|
</li>
|
|
<li class="listitem">
|
|
Hurley, Niall, and Scott Rickard. <span class="emphasis"><em>Comparing measures of sparsity.</em></span>
|
|
IEEE Transactions on Information Theory 55.10 (2009): 4723-4741.
|
|
</li>
|
|
<li class="listitem">
|
|
Jensen, Arne, and Anders la Cour-Harbo. <span class="emphasis"><em>Ripples in mathematics:
|
|
the discrete wavelet transform.</em></span> Springer Science & Business
|
|
Media, 2001.
|
|
</li>
|
|
<li class="listitem">
|
|
D. R. Pauluzzi and N. C. Beaulieu, <span class="emphasis"><em>A comparison of SNR estimation
|
|
techniques for the AWGN channel,</em></span> IEEE Trans. Communications,
|
|
Vol. 48, No. 10, pp. 1681-1691, 2000.
|
|
</li>
|
|
<li class="listitem">
|
|
Hoyer, Patrik O. <span class="emphasis"><em>Non-negative matrix factorization with sparseness
|
|
constraints.</em></span>, Journal of machine learning research 5.Nov (2004):
|
|
1457-1469.
|
|
</li>
|
|
</ul></div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
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