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<title>Chapter 23. Backgrounders</title>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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@@ -51,7 +50,9 @@
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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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
@@ -59,7 +60,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Chapter 4. Mathematical Constants</title>
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@@ -44,7 +43,9 @@
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -52,7 +53,8 @@
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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@@ -52,7 +51,9 @@
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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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
@@ -60,7 +61,8 @@
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>
<link rel="stylesheet" href="math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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@@ -207,7 +206,9 @@
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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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
@@ -215,7 +216,8 @@
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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@@ -35,7 +34,9 @@
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -43,7 +44,8 @@
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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@@ -31,7 +30,9 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -39,7 +40,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Chapter 18. Integer Utilities (Greatest Common Divisor and Least Common Multiple)</title>
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@@ -36,7 +35,9 @@
continue to exist and redirect to the moved headers.
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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
@@ -44,7 +45,8 @@
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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<html>
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@@ -23,7 +22,7 @@
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<div><h1 class="title">
<a name="math_toolkit"></a>Math Toolkit 4.1.1</h1></div>
<a name="math_toolkit"></a>Math Toolkit 4.1.0</h1></div>
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<span class="firstname">Nikhar</span> <span class="surname">Agrawal</span>
@@ -134,7 +133,10 @@ 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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@@ -37,7 +36,9 @@
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<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
@@ -45,7 +46,8 @@
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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@@ -2824,7 +2822,6 @@
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<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/empirical_cdf.html" title="Empirical Cumulative Distribution Function"><span class="index-entry-level-1">Empirical Cumulative Distribution Function</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/number_series/fibonacci_numbers.html" title="Fibonacci Numbers"><span class="index-entry-level-1">Fibonacci Numbers</span></a></p></li>
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/diff0.html" title="Lanczos Smoothing Derivatives"><span class="index-entry-level-1">Lanczos Smoothing Derivatives</span></a></p></li>
@@ -4014,7 +4011,9 @@
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -4022,7 +4021,8 @@
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
@@ -475,7 +476,8 @@
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
@@ -394,7 +395,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -382,7 +383,8 @@
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<p><span class="index-entry-level-0">Chebyshev Polynomials</span></p>
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@@ -2487,7 +2477,6 @@
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@@ -7865,7 +7852,6 @@
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@@ -10310,7 +10296,9 @@
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@@ -10318,7 +10306,8 @@
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@@ -55,7 +56,8 @@
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@@ -42,7 +43,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -42,7 +43,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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
@@ -43,7 +44,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -124,7 +125,8 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -116,7 +117,8 @@
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doc/html/math_toolkit/airy/airy_root.html
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Alpha</a>.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -469,7 +470,8 @@ airy_ai_zeros:
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -115,7 +116,8 @@
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
@@ -116,7 +117,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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of the Anderson-Darling test statistic agrees with Mathematica.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -199,7 +200,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -178,7 +177,9 @@
implemented in there too.)
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -186,7 +187,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -48,7 +47,9 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -56,7 +57,8 @@
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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@@ -48,7 +47,9 @@
<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/asinh.png"></span>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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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
@@ -56,7 +57,8 @@
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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@@ -4,11 +4,10 @@
<title>atan</title>
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@@ -48,7 +47,9 @@
<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/atan.png"></span>
</p>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -56,7 +57,8 @@
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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@@ -4,11 +4,10 @@
<title>atanh</title>
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@@ -48,7 +47,9 @@
<span class="bold"><strong>Formula: </strong></span> <span class="inlinemediaobject"><img src="../../images/atanh.png"></span>
</p>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -56,7 +57,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -4,11 +4,10 @@
<title>Automatic Differentiation</title>
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@@ -400,7 +399,9 @@ put gamma = 0.00199852
manual</a>.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -408,7 +409,8 @@ put gamma = 0.00199852
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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@@ -4,11 +4,10 @@
<title>The Effect of a Poor Initial Guess</title>
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@@ -758,7 +757,9 @@
when using derivatives.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -766,7 +767,8 @@
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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@@ -4,11 +4,10 @@
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@@ -109,7 +108,9 @@
and for <span class="emphasis"><em>z<sub>0</sub> &gt; 0</em></span> the steps are actually divergent.
</p>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -117,7 +118,8 @@
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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@@ -269,7 +268,9 @@ Abscissa value that yields a potential of 3 = 0.604728
Root was found in 10 iterations.
</pre>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -277,7 +278,8 @@ Root was found in 10 iterations.
Distributed under the Boost Software License, Version 1.0. (See accompanying
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doc/html/math_toolkit/bessel.html
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<title>Bessel Functions</title>
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@@ -41,7 +40,9 @@
the Bessel Functions</a></span></dt>
</dl></div>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -49,7 +50,8 @@
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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doc/html/math_toolkit/bessel/bessel_derivatives.html
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@@ -4,11 +4,10 @@
<title>Derivatives of the Bessel Functions</title>
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@@ -1437,7 +1436,9 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -1445,7 +1446,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1368,7 +1367,9 @@ are also computed by recursions (involving gamma functions), but
recurrence.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -1376,7 +1377,8 @@ are also computed by recursions (involving gamma functions), but
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -216,7 +215,9 @@ is also known as the spherical
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@@ -224,7 +225,8 @@ is also known as the spherical
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -4,11 +4,10 @@
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@@ -756,7 +755,9 @@
Alpha</a>.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -764,7 +765,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -1181,7 +1180,9 @@ are also computed by recursions (involving gamma functions), but
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@@ -242,7 +243,8 @@
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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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@@ -133,7 +134,8 @@
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@@ -146,7 +147,8 @@
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@@ -658,7 +659,8 @@
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@@ -154,7 +155,8 @@
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ISO Standard</a>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -470,7 +471,8 @@
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@@ -312,7 +311,9 @@ The population of the United States surpassed 100 million on the 11th month of 1
Found in 3 iterations
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@@ -320,7 +321,8 @@ Found in 3 iterations
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differentiable, but not three or four times differentiable.
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@@ -92,7 +93,8 @@
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of data by spline functions.</em></span> Diss. City, University of London, 1975.
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<span class="special">}</span> <span class="comment">// Namespaces</span>
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<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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@@ -4727,7 +4728,8 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -111,7 +112,8 @@
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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in Boost.Math.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -198,7 +199,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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the problem size gets larger.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -253,7 +254,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -141,7 +140,9 @@ The residual is -1.56586e-14, and the expected residual is 4.64155e-14. This is
estimate.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -149,7 +150,8 @@ 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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58
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<title>Daubechies Wavelets and Scaling Functions</title>
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@@ -184,54 +183,6 @@
<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>
<a name="math_toolkit.daubechies.h1"></a>
<span class="phrase"><a name="math_toolkit.daubechies.references"></a></span><a class="link" href="daubechies.html#math_toolkit.daubechies.references">References</a>
@@ -251,7 +202,9 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -259,7 +212,8 @@
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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@@ -188,7 +189,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Numerical Differentiation</title>
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -355,7 +356,8 @@
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>Lanczos Smoothing Derivatives</title>
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@@ -231,7 +230,9 @@
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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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
@@ -239,7 +240,8 @@
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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@@ -110,7 +109,9 @@
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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
@@ -118,7 +119,8 @@
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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Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -103,7 +104,8 @@
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of normally distributed weights to meet a specification.
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@@ -126,7 +127,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -98,7 +99,8 @@
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@@ -637,7 +638,8 @@
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@@ -357,7 +358,8 @@
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@@ -602,7 +601,9 @@ from presumed-known mean and variance.
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@@ -610,7 +611,8 @@ from presumed-known mean and variance.
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@@ -697,7 +696,7 @@
<p>
Using the relation:
</p>
<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>
<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>
<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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@@ -886,7 +885,9 @@
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@@ -894,7 +895,8 @@
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
@@ -300,7 +301,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -395,7 +394,9 @@ independent, normally distributed random
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -403,7 +404,8 @@ independent, normally distributed random
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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@@ -186,7 +185,9 @@
is itself a fairly expensive operation.
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@@ -194,7 +195,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -306,13 +305,15 @@
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&amp;gbpv=0" target="_top">Extreme
<a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&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.
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -320,7 +321,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -66,7 +65,7 @@
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<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&amp;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&gbpv=0" target="_top">Extreme
Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
Nadarajah</a>.
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@@ -315,7 +314,9 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -323,7 +324,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -427,7 +428,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -358,7 +359,8 @@
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
@@ -835,7 +836,8 @@
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
@@ -1461,7 +1462,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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<title>Hypergeometric Distribution</title>
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@@ -42,11 +41,11 @@
<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="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="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="comment">// Accessors:</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="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="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>
@@ -105,25 +104,25 @@
<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="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 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>
<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="identifier">uint64_t</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<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>
<p>
Returns the total number of objects <span class="emphasis"><em>N</em></span>.
</p>
<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 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>
<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="identifier">uint64_t</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="keyword">unsigned</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
@@ -159,8 +158,8 @@
<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 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>
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>
is raised if the random variable is outside this range, or is not an integral
value.
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@@ -344,7 +343,9 @@
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@@ -352,7 +353,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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@@ -465,7 +466,8 @@
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@@ -351,7 +350,9 @@
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
@@ -359,7 +360,8 @@
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
@@ -438,7 +439,8 @@
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
@@ -342,7 +343,8 @@
Distributed under the Boost Software License, Version 1.0. (See accompanying
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