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<title>Chapter 23. Backgrounders</title>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -60,8 +59,7 @@
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>)
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<title>Chapter 4. Mathematical Constants</title>
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -53,8 +52,7 @@
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>)
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<title>Chapter 3. Specified-width floating-point typedefs</title>
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -61,8 +60,7 @@
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>)
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<title>Chapter 5. Statistical Distributions and Functions</title>
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@@ -206,9 +207,7 @@
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -216,8 +215,7 @@
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>)
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<title>Chapter 9. TR1 and C99 external "C" Functions</title>
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@@ -34,9 +35,7 @@
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -44,8 +43,7 @@
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>)
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<title>Chapter 14. Filters</title>
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -40,8 +39,7 @@
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>)
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<title>Chapter 18. Integer Utilities (Greatest Common Divisor and Least Common Multiple)</title>
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@@ -35,9 +36,7 @@
continue to exist and redirect to the moved headers.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -45,8 +44,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<span class="firstname">Nikhar</span> <span class="surname">Agrawal</span>
@@ -133,10 +134,7 @@ This manual is also available in <a href="http://sourceforge.net/projects/boost/
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<title>Chapter 25. Indexes</title>
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -46,8 +45,7 @@
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>)
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@@ -24,7 +25,7 @@
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@@ -2822,6 +2824,7 @@
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@@ -4011,9 +4014,7 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -4021,8 +4022,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -476,8 +475,7 @@
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@@ -395,8 +394,7 @@
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@@ -383,8 +382,7 @@
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@@ -2477,6 +2487,7 @@
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@@ -7852,6 +7865,7 @@
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@@ -10296,9 +10310,7 @@
</li></ul></div></dd>
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@@ -10306,8 +10318,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Chapter 19. Internal Details: Series, Rationals and Continued Fractions, Testing, and Development Tools</title>
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@@ -66,8 +65,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -60,8 +59,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Chapter 15. Complex Number Functions</title>
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Report on C++ Library Extensions</a>.
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@@ -56,8 +55,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Acknowledgements</title>
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section. Thank you to all who contributed to the discution about this library.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -43,8 +42,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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section. Thank you to all who contributed to the discussion about this library.
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@@ -43,8 +42,7 @@
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>)
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<title>acos</title>
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<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/acos.png"></span>
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@@ -57,8 +56,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/acosh.png"></span>
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@@ -57,8 +56,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -44,8 +43,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -125,8 +124,7 @@
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@@ -117,8 +116,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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Alpha</a>.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -470,8 +469,7 @@ airy_ai_zeros:
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -116,8 +115,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -117,8 +116,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -190,9 +191,7 @@
of the Anderson-Darling test statistic agrees with Mathematica.
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@@ -200,8 +199,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -177,9 +178,7 @@
implemented in there too.)
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -187,8 +186,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/asin.png"></span>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -57,8 +56,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/asinh.png"></span>
</p>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -57,8 +56,7 @@
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>)
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<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/atan.png"></span>
</p>
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -57,8 +56,7 @@
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>)
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<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/atanh.png"></span>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -57,8 +56,7 @@
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>)
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@@ -399,9 +400,7 @@ put gamma = 0.00199852
manual</a>.
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@@ -409,8 +408,7 @@ put gamma = 0.00199852
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -757,9 +758,7 @@
when using derivatives.
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@@ -767,8 +766,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -108,9 +109,7 @@
and for <span class="emphasis"><em>z<sub>0</sub> &gt; 0</em></span> the steps are actually divergent.
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@@ -118,8 +117,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -268,9 +269,7 @@ Abscissa value that yields a potential of 3 = 0.604728
Root was found in 10 iterations.
</pre>
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@@ -278,8 +277,7 @@ Root was found in 10 iterations.
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -40,9 +41,7 @@
the Bessel Functions</a></span></dt>
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@@ -50,8 +49,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1436,9 +1437,7 @@
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@@ -1446,8 +1445,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1367,9 +1368,7 @@ are also computed by recursions (involving gamma functions), but
recurrence.
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@@ -1377,8 +1376,7 @@ are also computed by recursions (involving gamma functions), but
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -215,9 +216,7 @@ is also known as the spherical
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@@ -225,8 +224,7 @@ is also known as the spherical
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@@ -755,9 +756,7 @@
Alpha</a>.
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@@ -765,8 +764,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1180,9 +1181,7 @@ are also computed by recursions (involving gamma functions), but
two values and f<sub>ν</sub>, the Wronskian yields I<sub>ν</sub>(x) directly.
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@@ -1190,8 +1189,7 @@ are also computed by recursions (involving gamma functions), but
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main definition as x → 0.
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@@ -243,8 +242,7 @@
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<span class="identifier">Bilinear</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span> <span class="number">13.2</span> <span class="identifier">ns</span>
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@@ -147,8 +146,7 @@
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@@ -155,8 +154,7 @@
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ISO Standard</a>
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@@ -471,8 +470,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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Found in 3 iterations
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@@ -321,8 +320,7 @@ Found in 3 iterations
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differentiable, but not three or four times differentiable.
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@@ -93,8 +92,7 @@
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of data by spline functions.</em></span> Diss. City, University of London, 1975.
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@@ -158,8 +157,7 @@
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.chatterjee_correlation"></a><a class="link" href="chatterjee_correlation.html" title="Chatterjee Correlation">Chatterjee Correlation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.chatterjee_correlation.h0"></a>
<span class="phrase"><a name="math_toolkit.chatterjee_correlation.synopsis"></a></span><a class="link" href="chatterjee_correlation.html#math_toolkit.chatterjee_correlation.synopsis">Synopsis</a>
</h4>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">chatterjee_correlation</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<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>
<span class="identifier">C</span><span class="special">++</span><span class="number">17</span><span class="special">:</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">ExecutionPolicy</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">&gt;</span>
<span class="keyword">auto</span> <span class="identifier">chatterjee_correlation</span><span class="special">(</span><span class="identifier">ExecutionPolicy</span><span class="special">&amp;&amp;</span> <span class="identifier">exec</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Container</span><span class="special">&amp;</span> <span class="identifier">u</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Container</span><span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
<span class="identifier">C</span><span class="special">++</span><span class="number">11</span><span class="special">:</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">&gt;</span>
<span class="keyword">auto</span> <span class="identifier">chatterjee_correlation</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Container</span><span class="special">&amp;</span> <span class="identifier">u</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Container</span><span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
<span class="special">}</span>
</pre>
<h4>
<a name="math_toolkit.chatterjee_correlation.h1"></a>
<span class="phrase"><a name="math_toolkit.chatterjee_correlation.description"></a></span><a class="link" href="chatterjee_correlation.html#math_toolkit.chatterjee_correlation.description">Description</a>
</h4>
<p>
The classical correlation coefficients like the Pearson's correlation are useful
primarily for distinguishing when one dataset depends linearly on another.
However, Pearson's correlation coefficient has a known weakness in that when
the dependent variable has an obvious functional relationship with the independent
variable, the value of the correlation coefficient can take on any value. As
Chatterjee says:
</p>
<p>
&gt; Ideally, one would like a coefficient that approaches its maximum value
if and only if one variable looks more and more like a noiseless function of
the other, just as Pearson correlation is close to its maximum value if and
only if one variable is close to being a noiseless linear function of the other.
</p>
<p>
This is the problem Chatterjee's coefficient solves. Let X and Y be random
variables, where Y is not constant, and let (X_i, Y_i) be samples from this
distribution. Rearrange these samples so that X_(0) &lt; X_{(1)} &lt; ... X_{(n-1)}
and create (R(X_{(i)}), R(Y_{(i)})). The Chatterjee correlation is then given
by
</p>
<p>
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../equations/chatterjee_correlation.svg"></object></span>
</p>
<p>
In the limit of an infinite amount of i.i.d data, the statistic lies in [0,
1]. However, if the data is not infinite, the statistic may be negative. If
X and Y are independent, the value is zero, and if Y is a measurable function
of X, then the statistic is unity. The complexity is O(n log n).
</p>
<p>
An example is given below:
</p>
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">X</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">Y</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
<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">chatterjee_correlation</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">coeff</span> <span class="special">=</span> <span class="identifier">chatterjee_correlation</span><span class="special">(</span><span class="identifier">X</span><span class="special">,</span> <span class="identifier">Y</span><span class="special">);</span>
</pre>
<p>
The implementation follows <a href="https://arxiv.org/pdf/1909.10140.pdf" target="_top">Chatterjee's
paper</a>.
</p>
<p>
<span class="emphasis"><em>Nota bene:</em></span> If the input is an integer type the output
will be a double precision type.
</p>
<h4>
<a name="math_toolkit.chatterjee_correlation.h2"></a>
<span class="phrase"><a name="math_toolkit.chatterjee_correlation.invariants"></a></span><a class="link" href="chatterjee_correlation.html#math_toolkit.chatterjee_correlation.invariants">Invariants</a>
</h4>
<p>
The function expects at least two samples, a non-constant vector Y, and the
same number of X's as Y's. If Y is constant, the result is a quiet NaN. The
data set must be sorted by X values. If there are ties in the values of X,
then the statistic is random due to the random breaking of ties. Of course,
random numbers are not used internally, but the result is not guaranteed to
be identical on different systems.
</p>
<h4>
<a name="math_toolkit.chatterjee_correlation.h3"></a>
<span class="phrase"><a name="math_toolkit.chatterjee_correlation.references"></a></span><a class="link" href="chatterjee_correlation.html#math_toolkit.chatterjee_correlation.references">References</a>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
Chatterjee, Sourav. "A new coefficient of correlation." Journal
of the American Statistical Association 116.536 (2021): 2009-2022.
</li></ul></div>
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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>)
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@@ -3202,8 +3201,7 @@
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@@ -4728,8 +4727,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -674,8 +673,7 @@
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@@ -45,8 +44,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -54,8 +53,7 @@
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@@ -176,8 +175,7 @@
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@@ -374,8 +373,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1765,8 +1764,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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Some physical constants may be available in Boost.Units.
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@@ -130,8 +129,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -64,8 +63,7 @@
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@@ -78,8 +77,7 @@
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test file</a>).
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@@ -112,8 +111,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -199,8 +198,7 @@
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@@ -244,9 +245,7 @@
the problem size gets larger.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -254,8 +253,7 @@
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@@ -140,9 +141,7 @@ The residual is -1.56586e-14, and the expected residual is 4.64155e-14. This is
estimate.
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@@ -150,8 +149,7 @@ The residual is -1.56586e-14, and the expected residual is 4.64155e-14. This is
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Daubechies Wavelets and Scaling Functions</title>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/daubechies_8_scaling.svg"></object></span> The 8 vanishing
moment scaling function.
</p>
<p>
Boost.Math also provides numerical evaluation of the Fourier transform of these
functions. This is useful in sparse recovery problems where the measurements
are taken in the Fourier basis. The usage is exhibited below:
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">fourier_transform_daubechies_scaling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<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">fourier_transform_daubechies_scaling</span><span class="special">;</span>
<span class="comment">// Evaluate the Fourier transform of the 4-vanishing moment Daubechies scaling function at ω=1.8:</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span> <span class="identifier">hat_phi</span> <span class="special">=</span> <span class="identifier">fourier_transform_daubechies_scaling</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">,</span> <span class="number">4</span><span class="special">&gt;(</span><span class="number">1.8f</span><span class="special">);</span>
</pre>
<p>
The Fourier transform convention is unitary with the sign of the imaginary
unit being given in Daubechies Ten Lectures. In particular, this means that
<code class="computeroutput"><span class="identifier">fourier_transform_daubechies_scaling</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">,</span>
<span class="identifier">p</span><span class="special">&gt;(</span><span class="number">0.0</span><span class="special">)</span></code> returns
1/sqrt(2π).
</p>
<p>
The implementation computes an infinite product of trigonometric polynomials
as can be found from recursive application of the identity 𝓕[φ](ω) = m(ω/2)𝓕[φ](ω/2).
This is neither particularly fast nor accurate, but there appears to be no
literature on this extremely useful topic, and hence the naive method must
suffice.
</p>
<p>
<span class="inlinemediaobject"><img src="../../graphs/fourier_transform_daubechies.png"></span>
</p>
<p>
A benchmark can be found in <code class="computeroutput"><span class="identifier">reporting</span><span class="special">/</span><span class="identifier">performance</span><span class="special">/</span><span class="identifier">fourier_transform_daubechies_performance</span><span class="special">.</span><span class="identifier">cpp</span></code>; the
results on a ~2021 M1 Macbook pro are presented below:
</p>
<p>
Run on (10 X 24.1212 MHz CPU s) CPU Caches: L1 Data 64 KiB (x10) L1 Instruction
128 KiB (x10) L2 Unified 4096 KiB (x5) Load Average: 1.33, 1.52, 1.62 -----------------------------------------------------------
Benchmark Time -----------------------------------------------------------
FourierTransformDaubechiesScaling&lt;double, 1&gt; 70.3 ns FourierTransformDaubechiesScaling&lt;double,
2&gt; 330 ns FourierTransformDaubechiesScaling&lt;double, 3&gt; 335 ns FourierTransformDaubechiesScaling&lt;double,
4&gt; 364 ns FourierTransformDaubechiesScaling&lt;double, 5&gt; 386 ns FourierTransformDaubechiesScaling&lt;double,
6&gt; 436 ns FourierTransformDaubechiesScaling&lt;double, 7&gt; 447 ns FourierTransformDaubechiesScaling&lt;double,
8&gt; 473 ns FourierTransformDaubechiesScaling&lt;double, 9&gt; 503 ns FourierTransformDaubechiesScaling&lt;double,
10&gt; 554 ns
</p>
<p>
Due to the low accuracy of this method, <code class="computeroutput"><span class="keyword">float</span></code>
precision is arg-promoted to <code class="computeroutput"><span class="keyword">double</span></code>,
and hence takes just as long as <code class="computeroutput"><span class="keyword">double</span></code>
precision to execute.
</p>
<h4>
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@@ -202,9 +251,7 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -212,8 +259,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -189,8 +188,7 @@
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@@ -356,8 +355,7 @@
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@@ -240,8 +239,7 @@
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@@ -119,8 +118,7 @@
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@@ -388,8 +387,7 @@
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@@ -104,8 +103,7 @@
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@@ -117,9 +118,7 @@
of normally distributed weights to meet a specification.
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@@ -127,8 +126,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -99,8 +98,7 @@
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@@ -638,8 +637,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -358,8 +357,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -601,9 +602,7 @@ from presumed-known mean and variance.
MathWorld</a>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -611,8 +610,7 @@ from presumed-known mean and variance.
Distributed under the Boost Software License, Version 1.0. (See accompanying
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Using the relation:
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<pre xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" class="table-programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">I</span><span class="special">[</span><span class="identifier">sub</span> <span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">](</span><span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">k</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span>
<pre class="table-programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">I</span><span class="special">[</span><span class="identifier">sub</span> <span class="number">1</span><span class="special">-</span><span class="identifier">p</span><span class="special">](</span><span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">k</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="identifier">I</span><span class="special">[</span><span class="identifier">sub</span> <span class="identifier">p</span><span class="special">](</span><span class="identifier">k</span> <span class="special">+</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">)</span>
<span class="special">=</span> <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a><span class="special">(</span><span class="identifier">k</span> <span class="special">+</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">n</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span></pre>
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@@ -885,9 +886,7 @@
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@@ -895,8 +894,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -301,8 +300,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -394,9 +395,7 @@ independent, normally distributed random
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -404,8 +403,7 @@ independent, normally distributed random
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -185,9 +186,7 @@
is itself a fairly expensive operation.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -195,8 +194,7 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -305,15 +306,13 @@
Distributions</a>.)
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
<a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&gbpv=0" target="_top">Extreme
<a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&amp;gbpv=0" target="_top">Extreme
Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
Nadarajah</a> discuss the relationship of the types of extreme
value distributions.
</li></ul></div>
</div>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -321,8 +320,7 @@
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>)
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<title>Extreme Value Distribution</title>
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@@ -65,7 +66,7 @@
</p>
<p>
The relationship of the types of extreme value distributions, of which
this is but one, is discussed by <a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&gbpv=0" target="_top">Extreme
this is but one, is discussed by <a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&amp;gbpv=0" target="_top">Extreme
Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
Nadarajah</a>.
</p>
@@ -314,9 +315,7 @@
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -324,8 +323,7 @@
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>)
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<title>F Distribution</title>
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@@ -418,9 +419,7 @@
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -428,8 +427,7 @@
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>)
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<title>Gamma (and Erlang) Distribution</title>
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@@ -349,9 +350,7 @@
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -359,8 +358,7 @@
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>)
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<title>Geometric Distribution</title>
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@@ -826,9 +827,7 @@
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -836,8 +835,7 @@
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>)
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@@ -1452,9 +1453,7 @@
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -1462,8 +1461,7 @@
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>)
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<title>Hypergeometric Distribution</title>
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@@ -41,11 +42,11 @@
<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
<span class="comment">// Construct:</span>
<span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span> <span class="comment">// r=defective/failures/success, n=trials/draws, N=total population.</span>
<span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="identifier">uint64_t</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">uint64_t</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">uint64_t</span> <span class="identifier">N</span><span class="special">);</span> <span class="comment">// r=defective/failures/success, n=trials/draws, N=total population.</span>
<span class="comment">// Accessors:</span>
<span class="keyword">unsigned</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="keyword">unsigned</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="keyword">unsigned</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">uint64_t</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">uint64_t</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="identifier">uint64_t</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="keyword">typedef</span> <span class="identifier">hypergeometric_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">hypergeometric</span><span class="special">;</span>
@@ -104,25 +105,25 @@
<span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.member_functions"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.member_functions">Member
Functions</a>
</h5>
<pre class="programlisting"><span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span>
<pre class="programlisting"><span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="identifier">uint64_t</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">uint64_t</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">uint64_t</span> <span class="identifier">N</span><span class="special">);</span>
</pre>
<p>
Constructs a hypergeometric distribution with a population of <span class="emphasis"><em>N</em></span>
objects, of which <span class="emphasis"><em>r</em></span> are defective, and from which
<span class="emphasis"><em>n</em></span> are sampled.
</p>
<pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<pre class="programlisting"><span class="identifier">uint64_t</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the total number of objects <span class="emphasis"><em>N</em></span>.
</p>
<pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<pre class="programlisting"><span class="identifier">uint64_t</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the number of objects <span class="emphasis"><em>r</em></span> in population <span class="emphasis"><em>N</em></span>
which are defective.
</p>
<pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<pre class="programlisting"><span class="identifier">uint64_t</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the number of objects <span class="emphasis"><em>n</em></span> which are sampled
@@ -158,8 +159,8 @@
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is the unsigned integers in the range
[max(0, n + r - N), min(n, r)]. A <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
The domain of the random variable are the 64-bit unsigned integers in the
range [max(0, n + r - N), min(n, r)]. A <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
is raised if the random variable is outside this range, or is not an integral
value.
</p>
@@ -343,9 +344,7 @@
</p></blockquote></div>
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -353,8 +352,7 @@
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>)
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<title>Inverse Chi Squared Distribution</title>
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@@ -456,9 +457,7 @@
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -466,8 +465,7 @@
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>)
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@@ -350,9 +351,7 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -360,8 +359,7 @@
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>)
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@@ -439,8 +438,7 @@
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@@ -343,8 +342,7 @@
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