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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.diff0"></a><a class="link" href="diff0.html" title="Lanczos Smoothing Derivatives">Lanczos Smoothing Derivatives</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.diff0.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.diff0.synopsis"></a></span><a class="link" href="diff0.html#math_toolkit.diff0.synopsis">Synopsis</a>
  32     </h4>
  33 <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">differentiation</span><span class="special">/</span><span class="identifier">lanczos_smoothing</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34
  35 <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">differentiation</span> <span class="special">{</span>
  36
  37     <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">order</span><span class="special">=</span><span class="number">1</span><span class="special">&gt;</span>
  38     <span class="keyword">class</span> <span class="identifier">discrete_lanczos_derivative</span> <span class="special">{</span>
  39     <span class="keyword">public</span><span class="special">:</span>
  40         <span class="identifier">discrete_lanczos_derivative</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">spacing</span><span class="special">,</span>
  41                                     <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">18</span><span class="special">,</span>
  42                                     <span class="identifier">size_t</span> <span class="identifier">approximation_order</span> <span class="special">=</span> <span class="number">3</span><span class="special">);</span>
  43
  44         <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  45         <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  46
  47         <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  48         <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&amp;</span> <span class="identifier">dvdt</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  49
  50         <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  51         <span class="identifier">RandomAccessContainer</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  52
  53         <span class="identifier">Real</span> <span class="identifier">get_spacing</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
  54     <span class="special">};</span>
  55
  56 <span class="special">}</span> <span class="comment">// namespaces</span>
  57 </pre>
  58 <h4>
  59 <a name="math_toolkit.diff0.h1"></a>
  60       <span class="phrase"><a name="math_toolkit.diff0.description"></a></span><a class="link" href="diff0.html#math_toolkit.diff0.description">Description</a>
  61     </h4>
  62 <p>
  63       The <code class="computeroutput"><span class="identifier">discrete_lanczos_derivative</span></code>
  64       class calculates a finite-difference approximation to the derivative of a noisy
  65       sequence of equally-spaced values <span class="emphasis"><em>v</em></span>. A basic usage is
  66     </p>
  67 <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">v</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
  68 <span class="comment">// fill v with noisy data.</span>
  69 <span class="keyword">double</span> <span class="identifier">spacing</span> <span class="special">=</span> <span class="number">0.001</span><span class="special">;</span>
  70 <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">differentiation</span><span class="special">::</span><span class="identifier">discrete_lanczos_derivative</span><span class="special">;</span>
  71 <span class="keyword">auto</span> <span class="identifier">lanczos</span> <span class="special">=</span> <span class="identifier">discrete_lanczos_derivative</span><span class="special">(</span><span class="identifier">spacing</span><span class="special">);</span>
  72 <span class="comment">// Compute dvdt at index 30:</span>
  73 <span class="keyword">double</span> <span class="identifier">dvdt30</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="number">30</span><span class="special">);</span>
  74 <span class="comment">// Compute derivative of entire vector:</span>
  75 <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">dvdt</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
  76 </pre>
  77 <p>
  78       Noise-suppressing second derivatives can also be computed. The syntax is as
  79       follows:
  80     </p>
  81 <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">v</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
  82 <span class="comment">// fill v with noisy data.</span>
  83 <span class="keyword">auto</span> <span class="identifier">lanczos</span> <span class="special">=</span> <span class="identifier">lanczos_derivative</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;(</span><span class="identifier">spacing</span><span class="special">);</span>
  84 <span class="comment">// evaluate second derivative at a point:</span>
  85 <span class="keyword">double</span> <span class="identifier">d2vdt2</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="number">18</span><span class="special">);</span>
  86 <span class="comment">// evaluate second derivative of entire vector:</span>
  87 <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">d2vdt2</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
  88 </pre>
  89 <p>
  90       For memory conscious programmers, you can reuse the memory space for the derivative
  91       as follows:
  92     </p>
  93 <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">v</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
  94 <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">dvdt</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
  95 <span class="comment">// . . . define spacing, create and instance of discrete_lanczos_derivative</span>
  96
  97 <span class="comment">// populate dvdt, perhaps in a loop:</span>
  98 <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">dvdt</span><span class="special">);</span>
  99 </pre>
 100 <p>
 101       If the data has variance &#963;<sup>2</sup>, then the variance of the computed derivative
 102       is roughly &#963;<sup>2</sup><span class="emphasis"><em>p</em></span><sup>3</sup> <span class="emphasis"><em>n</em></span><sup>-3</sup> &#916;
 103       <span class="emphasis"><em>t</em></span><sup>-2</sup>, i.e., it increases cubically with the approximation
 104       order <span class="emphasis"><em>p</em></span>, linearly with the data variance, and decreases
 105       at the cube of the filter length <span class="emphasis"><em>n</em></span>. In addition, we must
 106       not forget the discretization error which is <span class="emphasis"><em>O</em></span>(&#916;
 107       <span class="emphasis"><em>t</em></span><sup><span class="emphasis"><em>p</em></span></sup>). You can play around with the
 108       approximation order <span class="emphasis"><em>p</em></span> and the filter length <span class="emphasis"><em>n</em></span>:
 109     </p>
 110 <pre class="programlisting"><span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
 111 <span class="identifier">size_t</span> <span class="identifier">p</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>
 112 <span class="keyword">auto</span> <span class="identifier">lanczos</span> <span class="special">=</span> <span class="identifier">lanczos_derivative</span><span class="special">(</span><span class="identifier">spacing</span><span class="special">,</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">p</span><span class="special">);</span>
 113 <span class="keyword">double</span> <span class="identifier">dvdt</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">i</span><span class="special">);</span>
 114 </pre>
 115 <p>
 116       If <span class="emphasis"><em>p=2n</em></span>, then the discrete Lanczos derivative is not smoothing:
 117       It reduces to the standard <span class="emphasis"><em>2n+1</em></span>-point finite-difference
 118       formula. For <span class="emphasis"><em>p&gt;2n</em></span>, an assertion is hit as the filter
 119       is undefined.
 120     </p>
 121 <p>
 122       In our tests with AWGN, we have found the error decreases monotonically with
 123       <span class="emphasis"><em>n</em></span>, as is expected from the theory discussed above. So
 124       the choice of <span class="emphasis"><em>n</em></span> is simple: As high as possible given your
 125       speed requirements (larger <span class="emphasis"><em>n</em></span> implies a longer filter and
 126       hence more compute), balanced against the danger of overfitting and averaging
 127       over non-stationarity.
 128     </p>
 129 <p>
 130       The choice of approximation order <span class="emphasis"><em>p</em></span> for a given <span class="emphasis"><em>n</em></span>
 131       is more difficult. If your signal is believed to be a polynomial, it does not
 132       make sense to set <span class="emphasis"><em>p</em></span> to larger than the polynomial degree-
 133       though it may be sensible to take <span class="emphasis"><em>p</em></span> less than this.
 134     </p>
 135 <p>
 136       For a sinusoidal signal contaminated with AWGN, we ran a few tests showing
 137       that for SNR = 1, p = n/8 gave the best results, for SNR = 10, p = n/7 was
 138       the best, and for SNR = 100, p = n/6 was the most reasonable choice. For SNR
 139       = 0.1, the method appears to be useless. The user is urged to use these results
 140       with caution: they have no theoretical backing and are extrapolated from a
 141       single case.
 142     </p>
 143 <p>
 144       The filters are (regrettably) computed at runtime-the vast number of combinations
 145       of approximation order and filter length makes the number of filters that must
 146       be stored excessive for compile-time data. The constructor call computes the
 147       filters. Since each filter has length <span class="emphasis"><em>2n+1</em></span> and there are
 148       <span class="emphasis"><em>n</em></span> filters, whose element each consist of <span class="emphasis"><em>p</em></span>
 149       summands, the complexity of the constructor call is O(<span class="emphasis"><em>n</em></span><sup>2</sup><span class="emphasis"><em>p</em></span>).
 150       This is not cheap-though for most cases small <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>n</em></span>
 151       not too large (&lt; 20) is desired. However, for concreteness, on the author's
 152       2.7GHz Intel laptop CPU, the <span class="emphasis"><em>n=16</em></span>, <span class="emphasis"><em>p=3</em></span>
 153       filter takes 9 microseconds to compute. This is far from negligible, and as
 154       such the filters may be used with multiple data, and even shared between threads:
 155     </p>
 156 <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">v1</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
 157 <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">v2</span><span class="special">(</span><span class="number">500</span><span class="special">);</span>
 158 <span class="comment">// fill v1, v2 with noisy data.</span>
 159 <span class="keyword">auto</span> <span class="identifier">lanczos</span> <span class="special">=</span> <span class="identifier">lanczos_derivative</span><span class="special">(</span><span class="identifier">spacing</span><span class="special">);</span>
 160 <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">dv1dt</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v1</span><span class="special">);</span> <span class="comment">// threadsafe</span>
 161 <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">dv2dt</span> <span class="special">=</span> <span class="identifier">lanczos</span><span class="special">(</span><span class="identifier">v2</span><span class="special">);</span> <span class="comment">// threadsafe</span>
 162 <span class="comment">// need to use a different spacing?</span>
 163 <span class="identifier">lanczos</span><span class="special">.</span><span class="identifier">reset_spacing</span><span class="special">(</span><span class="number">0.02</span><span class="special">);</span> <span class="comment">// not threadsafe</span>
 164 </pre>
 165 <p>
 166       The implementation follows <a href="https://doi.org/10.1080/00207160.2012.666348" target="_top">McDevitt,
 167       2012</a>, who vastly expanded the ideas of Lanczos to create a very general
 168       framework for numerically differentiating noisy equispaced data.
 169     </p>
 170 <h4>
 171 <a name="math_toolkit.diff0.h2"></a>
 172       <span class="phrase"><a name="math_toolkit.diff0.example"></a></span><a class="link" href="diff0.html#math_toolkit.diff0.example">Example</a>
 173     </h4>
 174 <p>
 175       We have extracted some data from the <a href="https://www.gw-openscience.org/data/" target="_top">LIGO
 176       signal</a> and differentiated it using the (<span class="emphasis"><em>n</em></span>, <span class="emphasis"><em>p</em></span>)
 177       = (60, 4) Lanczos smoothing derivative, as well as using the (<span class="emphasis"><em>n</em></span>,
 178       <span class="emphasis"><em>p</em></span>) = (4, 8) (nonsmoothing) derivative.
 179     </p>
 180 <div class="blockquote"><blockquote class="blockquote"><p>
 181         <span class="inlinemediaobject"><img src="../../graphs/ligo_derivative.svg" align="middle"></span>
 182
 183       </p></blockquote></div>
 184 <p>
 185       The original data is in orange, the smoothing derivative in blue, and the non-smoothing
 186       standard finite difference formula is in gray. (Each time series has been rescaled
 187       to fit in the same graph.) We can see that the smoothing derivative tracks
 188       the increase and decrease in the trend well, whereas the standard finite difference
 189       formula produces nonsense and amplifies noise.
 190     </p>
 191 <h4>
 192 <a name="math_toolkit.diff0.h3"></a>
 193       <span class="phrase"><a name="math_toolkit.diff0.caveats"></a></span><a class="link" href="diff0.html#math_toolkit.diff0.caveats">Caveats</a>
 194     </h4>
 195 <p>
 196       The computation of the filters is ill-conditioned for large <span class="emphasis"><em>p</em></span>.
 197       In order to mitigate this problem, we have computed the filters in higher precision
 198       and cast the results to the desired type. However, this simply pushes the problem
 199       to larger <span class="emphasis"><em>p</em></span>. In practice, this is not a problem, as large
 200       <span class="emphasis"><em>p</em></span> corresponds to less powerful denoising, but keep it
 201       in mind.
 202     </p>
 203 <p>
 204       In addition, the <code class="computeroutput"><span class="special">-</span><span class="identifier">ffast</span><span class="special">-</span><span class="identifier">math</span></code> flag
 205       has a very large effect on the speed of these functions. In our benchmarks,
 206       they were 50% faster with the flag enabled, which is much larger than the usual
 207       performance increases we see by turning on this flag. Hence, if the default
 208       performance is not sufficient, this flag is available, though it comes with
 209       its own problems.
 210     </p>
 211 <p>
 212       This requires C++17 <code class="computeroutput"><span class="keyword">if</span> <span class="keyword">constexpr</span></code>.
 213     </p>
 214 <h4>
 215 <a name="math_toolkit.diff0.h4"></a>
 216       <span class="phrase"><a name="math_toolkit.diff0.references"></a></span><a class="link" href="diff0.html#math_toolkit.diff0.references">References</a>
 217     </h4>
 218 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 219 <li class="listitem">
 220           Corless, Robert M., and Nicolas Fillion. <span class="emphasis"><em>A graduate introduction
 221           to numerical methods.</em></span> AMC 10 (2013): 12.
 222         </li>
 223 <li class="listitem">
 224           Lanczos, Cornelius. <span class="emphasis"><em>Applied analysis.</em></span> Courier Corporation,
 225           1988.
 226         </li>
 227 <li class="listitem">
 228           Timothy J. McDevitt (2012): <span class="emphasis"><em>Discrete Lanczos derivatives of noisy
 229           data</em></span>, International Journal of Computer Mathematics, 89:7, 916-931
 230         </li>
 231 </ul></div>
 232 </div>
 233 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 234 <td align="left"></td>
 235 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 236       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 237       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 238       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 239       Daryle Walker and Xiaogang Zhang<p>
 240         Distributed under the Boost Software License, Version 1.0. (See accompanying
 241         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>)
 242       </p>
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