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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.linear_regression"></a><a class="link" href="linear_regression.html" title="Linear Regression">Linear Regression</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.linear_regression.h0"></a>
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<span class="phrase"><a name="math_toolkit.linear_regression.synopsis"></a></span><a class="link" href="linear_regression.html#math_toolkit.linear_regression.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">linear_regression</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">simple_ordinary_least_squares</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">,</span>
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<span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">y</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">tuple</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">simple_ordinary_least_squares_with_R_squared</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">x</span><span class="special">,</span>
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<span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">y</span><span class="special">);</span>
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<span class="special">}}}</span>
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</pre>
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<h4>
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<a name="math_toolkit.linear_regression.h1"></a>
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<span class="phrase"><a name="math_toolkit.linear_regression.background"></a></span><a class="link" href="linear_regression.html#math_toolkit.linear_regression.background">Background</a>
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</h4>
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<p>
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A simple ordinary least squares finds the numbers <span class="emphasis"><em>c</em></span><sub>0</sub> and
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<span class="emphasis"><em>c</em></span><sub>1</sub> which minimizes the merit function
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/simple_ordinary_least_squares_merit.svg" width="246" height="54"></object></span>
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</p>
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<p>
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The predictive model generated from the minima of this functional is <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>)
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= <span class="emphasis"><em>c</em></span><sub>0</sub> + <span class="emphasis"><em>c</em></span><sub>1</sub> <span class="emphasis"><em>x</em></span>.
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</p>
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<p>
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It turns out that numerically, computing the numbers <span class="emphasis"><em>c</em></span><sub>0</sub> and
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<span class="emphasis"><em>c</em></span><sub>1</sub> is not <span class="emphasis"><em>quite</em></span> trivial. See <a href="https://www.johndcook.com/blog/2008/10/20/comparing-two-ways-to-fit-a-line-to-data/" target="_top">here</a>
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for an explanation of some ways linear regression can go wrong. A better method
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of computing the model parameters uses one-pass, numerically stable methods
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to compute means, variances, and covariances, and then assembles the parameters
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from these.
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</p>
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<p>
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An example usage of the simple linear regression is given below:
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</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
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<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
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<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">linear_regression</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">simple_ordinary_least_squares</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span> <span class="number">3</span><span class="special">,</span> <span class="number">7</span><span class="special">,</span> <span class="number">12</span><span class="special">};</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">y</span><span class="special">{</span><span class="number">8</span><span class="special">,</span><span class="number">13</span><span class="special">,</span> <span class="number">26</span><span class="special">,</span> <span class="number">35</span><span class="special">};</span>
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<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">c0</span><span class="special">,</span> <span class="identifier">c1</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">simple_ordinary_least_squares</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="identifier">y</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"f(x) = "</span> <span class="special"><<</span> <span class="identifier">c0</span> <span class="special"><<</span> <span class="string">" + "</span> <span class="special"><<</span> <span class="identifier">c1</span> <span class="special"><<</span> <span class="string">"*x"</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"f(2) = "</span> <span class="special"><<</span> <span class="identifier">c0</span> <span class="special">+</span> <span class="identifier">c1</span><span class="special">*</span><span class="number">2</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="comment">// Output:</span>
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<span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="number">6.0742</span> <span class="special">+</span> <span class="number">2.50883</span><span class="special">*</span><span class="identifier">x</span>
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<span class="identifier">f</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">=</span> <span class="number">11.0919</span>
|
||
</pre>
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<p>
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||
If, in addition, you wish to assess how appropriate linear regression is for
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your data, you can calculate <span class="emphasis"><em>R</em></span><sup>2</sup> as well, via
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</p>
|
||
<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">c0</span><span class="special">,</span> <span class="identifier">c1</span><span class="special">,</span> <span class="identifier">R2</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">simple_ordinary_least_squares_with_R_squared</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="identifier">y</span><span class="special">);</span>
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||
</pre>
|
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<p>
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||
It seems a number of definitions exist for <span class="emphasis"><em>R</em></span><sup>2</sup>; we use
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||
</p>
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||
<p>
|
||
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/r_squared_definition.svg" width="259" height="54"></object></span>
|
||
</p>
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<p>
|
||
The fit is good if <span class="emphasis"><em>R</em></span><sup>2</sup> is close to 1.
|
||
</p>
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<h4>
|
||
<a name="math_toolkit.linear_regression.h2"></a>
|
||
<span class="phrase"><a name="math_toolkit.linear_regression.performance"></a></span><a class="link" href="linear_regression.html#math_toolkit.linear_regression.performance">Performance</a>
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||
</h4>
|
||
<p>
|
||
There are two cases: When you want to compute <span class="emphasis"><em>R</em></span><sup>2</sup>, and
|
||
when you don't want to simultaneously compute <span class="emphasis"><em>R</em></span><sup>2</sup>, although
|
||
the cost of computing <span class="emphasis"><em>R</em></span><sup>2</sup> is not high:
|
||
</p>
|
||
<pre class="programlisting"><span class="special">-----------------------------------------------------------------------------------------------------------------</span>
|
||
<span class="identifier">Benchmark</span> <span class="identifier">Time</span> <span class="identifier">Bytes</span><span class="special">/</span><span class="identifier">second</span>
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||
<span class="special">-----------------------------------------------------------------------------------------------------------------</span>
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||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8</span> <span class="number">51.9</span> <span class="identifier">ns</span> <span class="number">588.476</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16</span> <span class="number">112</span> <span class="identifier">ns</span> <span class="number">546.087</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32</span> <span class="number">277</span> <span class="identifier">ns</span> <span class="number">440.823</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">64</span> <span class="number">608</span> <span class="identifier">ns</span> <span class="number">401.659</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">128</span> <span class="number">1276</span> <span class="identifier">ns</span> <span class="number">382.622</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">256</span> <span class="number">2606</span> <span class="identifier">ns</span> <span class="number">374.748</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">512</span> <span class="number">5266</span> <span class="identifier">ns</span> <span class="number">370.868</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1024</span> <span class="number">10601</span> <span class="identifier">ns</span> <span class="number">368.466</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2048</span> <span class="number">21243</span> <span class="identifier">ns</span> <span class="number">367.775</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4096</span> <span class="number">42502</span> <span class="identifier">ns</span> <span class="number">367.631</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8192</span> <span class="number">85239</span> <span class="identifier">ns</span> <span class="number">366.618</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16384</span> <span class="number">169736</span> <span class="identifier">ns</span> <span class="number">368.22</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32768</span> <span class="number">340220</span> <span class="identifier">ns</span> <span class="number">367.409</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">65536</span> <span class="number">678907</span> <span class="identifier">ns</span> <span class="number">368.247</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">131072</span> <span class="number">1357145</span> <span class="identifier">ns</span> <span class="number">368.422</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">262144</span> <span class="number">2720207</span> <span class="identifier">ns</span> <span class="number">367.635</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">524288</span> <span class="number">5430141</span> <span class="identifier">ns</span> <span class="number">368.332</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">10896708</span> <span class="identifier">ns</span> <span class="number">367.095</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">21797935</span> <span class="identifier">ns</span> <span class="number">367.047</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">43723059</span> <span class="identifier">ns</span> <span class="number">365.944</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">87229180</span> <span class="identifier">ns</span> <span class="number">366.864</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">174988864</span> <span class="identifier">ns</span> <span class="number">365.74</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">10.42</span> <span class="identifier">N</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8</span> <span class="number">52.4</span> <span class="identifier">ns</span> <span class="number">1.13779</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16</span> <span class="number">122</span> <span class="identifier">ns</span> <span class="number">1002.14</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32</span> <span class="number">307</span> <span class="identifier">ns</span> <span class="number">795.253</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">64</span> <span class="number">685</span> <span class="identifier">ns</span> <span class="number">712.628</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">128</span> <span class="number">1445</span> <span class="identifier">ns</span> <span class="number">675.805</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">2966</span> <span class="identifier">ns</span> <span class="number">658.488</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">6062</span> <span class="identifier">ns</span> <span class="number">644.35</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">12166</span> <span class="identifier">ns</span> <span class="number">642.173</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">24336</span> <span class="identifier">ns</span> <span class="number">642.064</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">48862</span> <span class="identifier">ns</span> <span class="number">639.567</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">98008</span> <span class="identifier">ns</span> <span class="number">637.708</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">196394</span> <span class="identifier">ns</span> <span class="number">636.481</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">392810</span> <span class="identifier">ns</span> <span class="number">636.434</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">783903</span> <span class="identifier">ns</span> <span class="number">637.859</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">1556741</span> <span class="identifier">ns</span> <span class="number">642.378</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">3121184</span> <span class="identifier">ns</span> <span class="number">640.792</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">6265681</span> <span class="identifier">ns</span> <span class="number">638.404</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">12421627</span> <span class="identifier">ns</span> <span class="number">644.076</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">24907611</span> <span class="identifier">ns</span> <span class="number">642.417</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">49773317</span> <span class="identifier">ns</span> <span class="number">642.934</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">100034750</span> <span class="identifier">ns</span> <span class="number">639.79</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">199477349</span> <span class="identifier">ns</span> <span class="number">641.685</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">11.90</span> <span class="identifier">N</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquares</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">0</span> <span class="special">%</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8</span> <span class="number">69.2</span> <span class="identifier">ns</span> <span class="number">441.314</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16</span> <span class="number">147</span> <span class="identifier">ns</span> <span class="number">415.939</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32</span> <span class="number">356</span> <span class="identifier">ns</span> <span class="number">342.815</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">64</span> <span class="number">736</span> <span class="identifier">ns</span> <span class="number">331.749</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">128</span> <span class="number">1494</span> <span class="identifier">ns</span> <span class="number">326.765</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">256</span> <span class="number">3161</span> <span class="identifier">ns</span> <span class="number">308.909</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">512</span> <span class="number">6343</span> <span class="identifier">ns</span> <span class="number">307.94</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1024</span> <span class="number">12707</span> <span class="identifier">ns</span> <span class="number">307.409</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2048</span> <span class="number">25390</span> <span class="identifier">ns</span> <span class="number">307.699</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4096</span> <span class="number">50820</span> <span class="identifier">ns</span> <span class="number">307.455</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8192</span> <span class="number">101738</span> <span class="identifier">ns</span> <span class="number">307.161</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16384</span> <span class="number">203033</span> <span class="identifier">ns</span> <span class="number">307.835</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">32768</span> <span class="number">403366</span> <span class="identifier">ns</span> <span class="number">309.894</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">65536</span> <span class="number">767080</span> <span class="identifier">ns</span> <span class="number">325.911</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">131072</span> <span class="number">1515010</span> <span class="identifier">ns</span> <span class="number">330.034</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">262144</span> <span class="number">2965539</span> <span class="identifier">ns</span> <span class="number">337.21</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">524288</span> <span class="number">5774329</span> <span class="identifier">ns</span> <span class="number">346.372</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">11384267</span> <span class="identifier">ns</span> <span class="number">351.371</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">22899097</span> <span class="identifier">ns</span> <span class="number">349.406</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">45923903</span> <span class="identifier">ns</span> <span class="number">348.423</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">92138186</span> <span class="identifier">ns</span> <span class="number">347.306</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">183263471</span> <span class="identifier">ns</span> <span class="number">349.226</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">10.94</span> <span class="identifier">N</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">float</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">1</span> <span class="special">%</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8</span> <span class="number">68.7</span> <span class="identifier">ns</span> <span class="number">887.806</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16</span> <span class="number">166</span> <span class="identifier">ns</span> <span class="number">734.816</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32</span> <span class="number">385</span> <span class="identifier">ns</span> <span class="number">633.918</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">64</span> <span class="number">812</span> <span class="identifier">ns</span> <span class="number">601.394</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">128</span> <span class="number">1774</span> <span class="identifier">ns</span> <span class="number">550.424</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">3554</span> <span class="identifier">ns</span> <span class="number">549.562</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">7151</span> <span class="identifier">ns</span> <span class="number">546.25</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">14335</span> <span class="identifier">ns</span> <span class="number">545.006</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">28608</span> <span class="identifier">ns</span> <span class="number">546.163</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">57228</span> <span class="identifier">ns</span> <span class="number">546.067</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">113732</span> <span class="identifier">ns</span> <span class="number">549.537</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">227686</span> <span class="identifier">ns</span> <span class="number">549.004</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">453989</span> <span class="identifier">ns</span> <span class="number">550.668</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">901696</span> <span class="identifier">ns</span> <span class="number">554.517</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">1771910</span> <span class="identifier">ns</span> <span class="number">564.365</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">3430961</span> <span class="identifier">ns</span> <span class="number">582.933</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">6751237</span> <span class="identifier">ns</span> <span class="number">592.511</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">13544819</span> <span class="identifier">ns</span> <span class="number">590.639</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2097152</span> <span class="number">27331142</span> <span class="identifier">ns</span> <span class="number">585.422</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4194304</span> <span class="number">54944987</span> <span class="identifier">ns</span> <span class="number">582.425</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8388608</span> <span class="number">109574257</span> <span class="identifier">ns</span> <span class="number">584.087</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16777216</span> <span class="number">221449209</span> <span class="identifier">ns</span> <span class="number">578.003</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">13.17</span> <span class="identifier">N</span>
|
||
<span class="identifier">BMSimpleOrdinaryLeastSquaresWRSquared</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">1</span> <span class="special">%</span>
|
||
</pre>
|
||
</div>
|
||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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