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<body>
<h1 class="title toc-ignore">Exact Procedures</h1>
<div id="TOC">
<ul>
<li><a href="#ordinary-poisson-binomial-distribution" id="toc-ordinary-poisson-binomial-distribution">Ordinary Poisson
Binomial Distribution</a>
<ul>
<li><a href="#direct-convolution" id="toc-direct-convolution">Direct
Convolution</a></li>
<li><a href="#divide-conquer-fft-tree-convolution" id="toc-divide-conquer-fft-tree-convolution">Divide & Conquer FFT
Tree Convolution</a></li>
<li><a href="#discrete-fourier-transformation-of-the-characteristic-function" id="toc-discrete-fourier-transformation-of-the-characteristic-function">Discrete
Fourier Transformation of the Characteristic Function</a></li>
<li><a href="#recursive-formula" id="toc-recursive-formula">Recursive
Formula</a></li>
<li><a href="#processing-speed-comparisons" id="toc-processing-speed-comparisons">Processing Speed
Comparisons</a></li>
</ul></li>
<li><a href="#generalized-poisson-binomial-distribution" id="toc-generalized-poisson-binomial-distribution">Generalized Poisson
Binomial Distribution</a>
<ul>
<li><a href="#generalized-direct-convolution" id="toc-generalized-direct-convolution">Generalized Direct
Convolution</a></li>
<li><a href="#generalized-divide-conquer-fft-tree-convolution" id="toc-generalized-divide-conquer-fft-tree-convolution">Generalized
Divide & Conquer FFT Tree Convolution</a></li>
<li><a href="#generalized-discrete-fourier-transformation-of-the-characteristic-function" id="toc-generalized-discrete-fourier-transformation-of-the-characteristic-function">Generalized
Discrete Fourier Transformation of the Characteristic Function</a></li>
<li><a href="#processing-speed-comparisons-1" id="toc-processing-speed-comparisons-1">Processing Speed
Comparisons</a></li>
</ul></li>
</ul>
</div>
<div id="ordinary-poisson-binomial-distribution" class="section level2">
<h2>Ordinary Poisson Binomial Distribution</h2>
<div id="direct-convolution" class="section level3">
<h3>Direct Convolution</h3>
<p>The <em>Direct Convolution</em> (DC) approach is requested with
<code>method = "Convolve"</code>.</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb1-3"><a href="#cb1-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb1-4"><a href="#cb1-4" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Convolve"</span>)</span>
<span id="cb1-6"><a href="#cb1-6" tabindex="-1"></a><span class="co">#> [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26</span></span>
<span id="cb1-7"><a href="#cb1-7" tabindex="-1"></a><span class="co">#> [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19</span></span>
<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a><span class="co">#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13</span></span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a><span class="co">#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08</span></span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a><span class="co">#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05</span></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a><span class="co">#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03</span></span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a><span class="co">#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01</span></span>
<span id="cb1-13"><a href="#cb1-13" tabindex="-1"></a><span class="co">#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02</span></span>
<span id="cb1-14"><a href="#cb1-14" tabindex="-1"></a><span class="co">#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03</span></span>
<span id="cb1-15"><a href="#cb1-15" tabindex="-1"></a><span class="co">#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06</span></span>
<span id="cb1-16"><a href="#cb1-16" tabindex="-1"></a><span class="co">#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10</span></span>
<span id="cb1-17"><a href="#cb1-17" tabindex="-1"></a><span class="co">#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17</span></span>
<span id="cb1-18"><a href="#cb1-18" tabindex="-1"></a><span class="co">#> [61] 9.411166e-19 6.727527e-21</span></span>
<span id="cb1-19"><a href="#cb1-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Convolve"</span>)</span>
<span id="cb1-20"><a href="#cb1-20" tabindex="-1"></a><span class="co">#> [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26</span></span>
<span id="cb1-21"><a href="#cb1-21" tabindex="-1"></a><span class="co">#> [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19</span></span>
<span id="cb1-22"><a href="#cb1-22" tabindex="-1"></a><span class="co">#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13</span></span>
<span id="cb1-23"><a href="#cb1-23" tabindex="-1"></a><span class="co">#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08</span></span>
<span id="cb1-24"><a href="#cb1-24" tabindex="-1"></a><span class="co">#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05</span></span>
<span id="cb1-25"><a href="#cb1-25" tabindex="-1"></a><span class="co">#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02</span></span>
<span id="cb1-26"><a href="#cb1-26" tabindex="-1"></a><span class="co">#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01</span></span>
<span id="cb1-27"><a href="#cb1-27" tabindex="-1"></a><span class="co">#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01</span></span>
<span id="cb1-28"><a href="#cb1-28" tabindex="-1"></a><span class="co">#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01</span></span>
<span id="cb1-29"><a href="#cb1-29" tabindex="-1"></a><span class="co">#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01</span></span>
<span id="cb1-30"><a href="#cb1-30" tabindex="-1"></a><span class="co">#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb1-31"><a href="#cb1-31" tabindex="-1"></a><span class="co">#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb1-32"><a href="#cb1-32" tabindex="-1"></a><span class="co">#> [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
</div>
<div id="divide-conquer-fft-tree-convolution" class="section level3">
<h3>Divide & Conquer FFT Tree Convolution</h3>
<p>The <em>Divide & Conquer FFT Tree Convolution</em> (DC-FFT)
approach is requested with <code>method = "DivideFFT"</code>.</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb2-2"><a href="#cb2-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb2-3"><a href="#cb2-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb2-4"><a href="#cb2-4" tabindex="-1"></a></span>
<span id="cb2-5"><a href="#cb2-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"DivideFFT"</span>)</span>
<span id="cb2-6"><a href="#cb2-6" tabindex="-1"></a><span class="co">#> [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26</span></span>
<span id="cb2-7"><a href="#cb2-7" tabindex="-1"></a><span class="co">#> [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19</span></span>
<span id="cb2-8"><a href="#cb2-8" tabindex="-1"></a><span class="co">#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13</span></span>
<span id="cb2-9"><a href="#cb2-9" tabindex="-1"></a><span class="co">#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08</span></span>
<span id="cb2-10"><a href="#cb2-10" tabindex="-1"></a><span class="co">#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05</span></span>
<span id="cb2-11"><a href="#cb2-11" tabindex="-1"></a><span class="co">#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03</span></span>
<span id="cb2-12"><a href="#cb2-12" tabindex="-1"></a><span class="co">#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01</span></span>
<span id="cb2-13"><a href="#cb2-13" tabindex="-1"></a><span class="co">#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02</span></span>
<span id="cb2-14"><a href="#cb2-14" tabindex="-1"></a><span class="co">#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03</span></span>
<span id="cb2-15"><a href="#cb2-15" tabindex="-1"></a><span class="co">#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06</span></span>
<span id="cb2-16"><a href="#cb2-16" tabindex="-1"></a><span class="co">#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10</span></span>
<span id="cb2-17"><a href="#cb2-17" tabindex="-1"></a><span class="co">#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17</span></span>
<span id="cb2-18"><a href="#cb2-18" tabindex="-1"></a><span class="co">#> [61] 9.411166e-19 6.727527e-21</span></span>
<span id="cb2-19"><a href="#cb2-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"DivideFFT"</span>)</span>
<span id="cb2-20"><a href="#cb2-20" tabindex="-1"></a><span class="co">#> [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26</span></span>
<span id="cb2-21"><a href="#cb2-21" tabindex="-1"></a><span class="co">#> [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19</span></span>
<span id="cb2-22"><a href="#cb2-22" tabindex="-1"></a><span class="co">#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13</span></span>
<span id="cb2-23"><a href="#cb2-23" tabindex="-1"></a><span class="co">#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08</span></span>
<span id="cb2-24"><a href="#cb2-24" tabindex="-1"></a><span class="co">#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05</span></span>
<span id="cb2-25"><a href="#cb2-25" tabindex="-1"></a><span class="co">#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02</span></span>
<span id="cb2-26"><a href="#cb2-26" tabindex="-1"></a><span class="co">#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01</span></span>
<span id="cb2-27"><a href="#cb2-27" tabindex="-1"></a><span class="co">#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01</span></span>
<span id="cb2-28"><a href="#cb2-28" tabindex="-1"></a><span class="co">#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01</span></span>
<span id="cb2-29"><a href="#cb2-29" tabindex="-1"></a><span class="co">#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01</span></span>
<span id="cb2-30"><a href="#cb2-30" tabindex="-1"></a><span class="co">#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb2-31"><a href="#cb2-31" tabindex="-1"></a><span class="co">#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb2-32"><a href="#cb2-32" tabindex="-1"></a><span class="co">#> [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>By design, as proposed by <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &
Brunner (2018)</a>, its results are identical to the DC procedure, if
<span class="math inline">\(n \leq 750\)</span>. Thus, differences can
be observed for larger <span class="math inline">\(n >
750\)</span>:</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb3-2"><a href="#cb3-2" tabindex="-1"></a>pp1 <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">751</span>)</span>
<span id="cb3-3"><a href="#cb3-3" tabindex="-1"></a>pp2 <span class="ot"><-</span> pp1[<span class="dv">1</span><span class="sc">:</span><span class="dv">750</span>]</span>
<span id="cb3-4"><a href="#cb3-4" tabindex="-1"></a></span>
<span id="cb3-5"><a href="#cb3-5" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">abs</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp2, <span class="at">method =</span> <span class="st">"DivideFFT"</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp2, <span class="at">method =</span> <span class="st">"Convolve"</span>)))</span>
<span id="cb3-6"><a href="#cb3-6" tabindex="-1"></a><span class="co">#> [1] 0</span></span>
<span id="cb3-7"><a href="#cb3-7" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">abs</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp1, <span class="at">method =</span> <span class="st">"DivideFFT"</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp1, <span class="at">method =</span> <span class="st">"Convolve"</span>)))</span>
<span id="cb3-8"><a href="#cb3-8" tabindex="-1"></a><span class="co">#> [1] 0</span></span></code></pre></div>
<p>The reason is that the DC-FFT method splits the input
<code>probs</code> vector into as equally sized parts as possible and
computes their distributions separately with the DC approach. The
results of the portions are then convoluted by means of the Fast Fourier
Transformation. As proposed by <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &
Brunner (2018)</a>, no splitting is done for <span class="math inline">\(n \leq 750\)</span>. In addition, the DC-FFT
procedure does not produce probabilities <span class="math inline">\(\leq 5.55e\text{-}17\)</span>, i.e. smaller values
are rounded off to 0, if <span class="math inline">\(n >
750\)</span>, whereas the smallest possible result of the DC algorithm
is <span class="math inline">\(\sim 1e\text{-}323\)</span>. This is most
likely caused by the used FFTW3 library.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb4-2"><a href="#cb4-2" tabindex="-1"></a>pp1 <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">751</span>)</span>
<span id="cb4-3"><a href="#cb4-3" tabindex="-1"></a></span>
<span id="cb4-4"><a href="#cb4-4" tabindex="-1"></a>d1 <span class="ot"><-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp1, <span class="at">method =</span> <span class="st">"DivideFFT"</span>)</span>
<span id="cb4-5"><a href="#cb4-5" tabindex="-1"></a>d2 <span class="ot"><-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp1, <span class="at">method =</span> <span class="st">"Convolve"</span>)</span>
<span id="cb4-6"><a href="#cb4-6" tabindex="-1"></a></span>
<span id="cb4-7"><a href="#cb4-7" tabindex="-1"></a><span class="fu">min</span>(d1[d1 <span class="sc">></span> <span class="dv">0</span>])</span>
<span id="cb4-8"><a href="#cb4-8" tabindex="-1"></a><span class="co">#> [1] 1.635357e-321</span></span>
<span id="cb4-9"><a href="#cb4-9" tabindex="-1"></a><span class="fu">min</span>(d2[d2 <span class="sc">></span> <span class="dv">0</span>])</span>
<span id="cb4-10"><a href="#cb4-10" tabindex="-1"></a><span class="co">#> [1] 1.635357e-321</span></span></code></pre></div>
</div>
<div id="discrete-fourier-transformation-of-the-characteristic-function" class="section level3">
<h3>Discrete Fourier Transformation of the Characteristic Function</h3>
<p>The <em>Discrete Fourier Transformation of the Characteristic
Function</em> (DFT-CF) approach is requested with
<code>method = "Characteristic"</code>.</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb5-2"><a href="#cb5-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb5-3"><a href="#cb5-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb5-4"><a href="#cb5-4" tabindex="-1"></a></span>
<span id="cb5-5"><a href="#cb5-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Characteristic"</span>)</span>
<span id="cb5-6"><a href="#cb5-6" tabindex="-1"></a><span class="co">#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb5-7"><a href="#cb5-7" tabindex="-1"></a><span class="co">#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb5-8"><a href="#cb5-8" tabindex="-1"></a><span class="co">#> [11] 0.000000e+00 2.238353e-16 3.549132e-15 4.829828e-14 5.804377e-13</span></span>
<span id="cb5-9"><a href="#cb5-9" tabindex="-1"></a><span class="co">#> [16] 6.158818e-12 5.784702e-11 4.822438e-10 3.576566e-09 2.364563e-08</span></span>
<span id="cb5-10"><a href="#cb5-10" tabindex="-1"></a><span class="co">#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05</span></span>
<span id="cb5-11"><a href="#cb5-11" tabindex="-1"></a><span class="co">#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03</span></span>
<span id="cb5-12"><a href="#cb5-12" tabindex="-1"></a><span class="co">#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01</span></span>
<span id="cb5-13"><a href="#cb5-13" tabindex="-1"></a><span class="co">#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02</span></span>
<span id="cb5-14"><a href="#cb5-14" tabindex="-1"></a><span class="co">#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03</span></span>
<span id="cb5-15"><a href="#cb5-15" tabindex="-1"></a><span class="co">#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06</span></span>
<span id="cb5-16"><a href="#cb5-16" tabindex="-1"></a><span class="co">#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110923e-10</span></span>
<span id="cb5-17"><a href="#cb5-17" tabindex="-1"></a><span class="co">#> [56] 2.392079e-11 1.468354e-12 6.994931e-14 2.513558e-15 0.000000e+00</span></span>
<span id="cb5-18"><a href="#cb5-18" tabindex="-1"></a><span class="co">#> [61] 0.000000e+00 0.000000e+00</span></span>
<span id="cb5-19"><a href="#cb5-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Characteristic"</span>)</span>
<span id="cb5-20"><a href="#cb5-20" tabindex="-1"></a><span class="co">#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb5-21"><a href="#cb5-21" tabindex="-1"></a><span class="co">#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb5-22"><a href="#cb5-22" tabindex="-1"></a><span class="co">#> [11] 0.000000e+00 2.238353e-16 3.772968e-15 5.207125e-14 6.325089e-13</span></span>
<span id="cb5-23"><a href="#cb5-23" tabindex="-1"></a><span class="co">#> [16] 6.791327e-12 6.463834e-11 5.468822e-10 4.123448e-09 2.776908e-08</span></span>
<span id="cb5-24"><a href="#cb5-24" tabindex="-1"></a><span class="co">#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05</span></span>
<span id="cb5-25"><a href="#cb5-25" tabindex="-1"></a><span class="co">#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02</span></span>
<span id="cb5-26"><a href="#cb5-26" tabindex="-1"></a><span class="co">#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01</span></span>
<span id="cb5-27"><a href="#cb5-27" tabindex="-1"></a><span class="co">#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01</span></span>
<span id="cb5-28"><a href="#cb5-28" tabindex="-1"></a><span class="co">#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01</span></span>
<span id="cb5-29"><a href="#cb5-29" tabindex="-1"></a><span class="co">#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01</span></span>
<span id="cb5-30"><a href="#cb5-30" tabindex="-1"></a><span class="co">#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb5-31"><a href="#cb5-31" tabindex="-1"></a><span class="co">#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb5-32"><a href="#cb5-32" tabindex="-1"></a><span class="co">#> [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>As can be seen, the DFT-CF procedure does not produce probabilities
<span class="math inline">\(\leq 2.22e\text{-}16\)</span>, i.e. smaller
values are rounded off to 0, most likely due to the used FFTW3
library.</p>
</div>
<div id="recursive-formula" class="section level3">
<h3>Recursive Formula</h3>
<p>The <em>Recursive Formula</em> (RF) approach is requested with
<code>method = "Recursive"</code>.</p>
<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb6-2"><a href="#cb6-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb6-3"><a href="#cb6-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb6-4"><a href="#cb6-4" tabindex="-1"></a></span>
<span id="cb6-5"><a href="#cb6-5" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Recursive"</span>)</span>
<span id="cb6-6"><a href="#cb6-6" tabindex="-1"></a><span class="co">#> [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26</span></span>
<span id="cb6-7"><a href="#cb6-7" tabindex="-1"></a><span class="co">#> [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19</span></span>
<span id="cb6-8"><a href="#cb6-8" tabindex="-1"></a><span class="co">#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13</span></span>
<span id="cb6-9"><a href="#cb6-9" tabindex="-1"></a><span class="co">#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08</span></span>
<span id="cb6-10"><a href="#cb6-10" tabindex="-1"></a><span class="co">#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05</span></span>
<span id="cb6-11"><a href="#cb6-11" tabindex="-1"></a><span class="co">#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03</span></span>
<span id="cb6-12"><a href="#cb6-12" tabindex="-1"></a><span class="co">#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01</span></span>
<span id="cb6-13"><a href="#cb6-13" tabindex="-1"></a><span class="co">#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02</span></span>
<span id="cb6-14"><a href="#cb6-14" tabindex="-1"></a><span class="co">#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03</span></span>
<span id="cb6-15"><a href="#cb6-15" tabindex="-1"></a><span class="co">#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06</span></span>
<span id="cb6-16"><a href="#cb6-16" tabindex="-1"></a><span class="co">#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10</span></span>
<span id="cb6-17"><a href="#cb6-17" tabindex="-1"></a><span class="co">#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17</span></span>
<span id="cb6-18"><a href="#cb6-18" tabindex="-1"></a><span class="co">#> [61] 9.411166e-19 6.727527e-21</span></span>
<span id="cb6-19"><a href="#cb6-19" tabindex="-1"></a><span class="fu">ppbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Recursive"</span>)</span>
<span id="cb6-20"><a href="#cb6-20" tabindex="-1"></a><span class="co">#> [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26</span></span>
<span id="cb6-21"><a href="#cb6-21" tabindex="-1"></a><span class="co">#> [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19</span></span>
<span id="cb6-22"><a href="#cb6-22" tabindex="-1"></a><span class="co">#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13</span></span>
<span id="cb6-23"><a href="#cb6-23" tabindex="-1"></a><span class="co">#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08</span></span>
<span id="cb6-24"><a href="#cb6-24" tabindex="-1"></a><span class="co">#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05</span></span>
<span id="cb6-25"><a href="#cb6-25" tabindex="-1"></a><span class="co">#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02</span></span>
<span id="cb6-26"><a href="#cb6-26" tabindex="-1"></a><span class="co">#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01</span></span>
<span id="cb6-27"><a href="#cb6-27" tabindex="-1"></a><span class="co">#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01</span></span>
<span id="cb6-28"><a href="#cb6-28" tabindex="-1"></a><span class="co">#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01</span></span>
<span id="cb6-29"><a href="#cb6-29" tabindex="-1"></a><span class="co">#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01</span></span>
<span id="cb6-30"><a href="#cb6-30" tabindex="-1"></a><span class="co">#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb6-31"><a href="#cb6-31" tabindex="-1"></a><span class="co">#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb6-32"><a href="#cb6-32" tabindex="-1"></a><span class="co">#> [61] 1.000000e+00 1.000000e+00</span></span></code></pre></div>
<p>Obviously, the RF procedure does produce probabilities <span class="math inline">\(\leq 5.55e\text{-}17\)</span>, because it does not
rely on the FFTW3 library. Furthermore, it yields the same results as
the DC method.</p>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb7-2"><a href="#cb7-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">1000</span>)</span>
<span id="cb7-3"><a href="#cb7-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">1000</span>, <span class="cn">TRUE</span>)</span>
<span id="cb7-4"><a href="#cb7-4" tabindex="-1"></a></span>
<span id="cb7-5"><a href="#cb7-5" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">abs</span>(<span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Convolve"</span>) <span class="sc">-</span> <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp, wt, <span class="st">"Recursive"</span>)))</span>
<span id="cb7-6"><a href="#cb7-6" tabindex="-1"></a><span class="co">#> [1] 0</span></span></code></pre></div>
</div>
<div id="processing-speed-comparisons" class="section level3">
<h3>Processing Speed Comparisons</h3>
<p>To assess the performance of the exact procedures, we use the
<code>microbenchmark</code> package. Each algorithm has to calculate the
PMF repeatedly based on random probability vectors. The run times are
then summarized in a table that presents, among other statistics, their
minima, maxima and means. The following results were recorded on an AMD
Ryzen 9 5900X with 64 GiB of RAM and Windows 10 Education (22H2).</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" tabindex="-1"></a><span class="fu">library</span>(microbenchmark)</span>
<span id="cb8-2"><a href="#cb8-2" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb8-3"><a href="#cb8-3" tabindex="-1"></a></span>
<span id="cb8-4"><a href="#cb8-4" tabindex="-1"></a>f1 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">6000</span>), <span class="at">method =</span> <span class="st">"DivideFFT"</span>)</span>
<span id="cb8-5"><a href="#cb8-5" tabindex="-1"></a>f2 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">6000</span>), <span class="at">method =</span> <span class="st">"Convolve"</span>)</span>
<span id="cb8-6"><a href="#cb8-6" tabindex="-1"></a>f3 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">6000</span>), <span class="at">method =</span> <span class="st">"Recursive"</span>)</span>
<span id="cb8-7"><a href="#cb8-7" tabindex="-1"></a>f4 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(<span class="dv">6000</span>), <span class="at">method =</span> <span class="st">"Characteristic"</span>)</span>
<span id="cb8-8"><a href="#cb8-8" tabindex="-1"></a></span>
<span id="cb8-9"><a href="#cb8-9" tabindex="-1"></a><span class="fu">microbenchmark</span>(<span class="fu">f1</span>(), <span class="fu">f2</span>(), <span class="fu">f3</span>(), <span class="fu">f4</span>(), <span class="at">times =</span> <span class="dv">51</span>)</span>
<span id="cb8-10"><a href="#cb8-10" tabindex="-1"></a><span class="co">#> Unit: milliseconds</span></span>
<span id="cb8-11"><a href="#cb8-11" tabindex="-1"></a><span class="co">#> expr min lq mean median uq max neval</span></span>
<span id="cb8-12"><a href="#cb8-12" tabindex="-1"></a><span class="co">#> f1() 20.9010 21.82365 23.03580 22.3124 22.60315 33.6471 51</span></span>
<span id="cb8-13"><a href="#cb8-13" tabindex="-1"></a><span class="co">#> f2() 44.1096 44.98895 45.64250 45.2764 45.94010 53.8505 51</span></span>
<span id="cb8-14"><a href="#cb8-14" tabindex="-1"></a><span class="co">#> f3() 80.0998 80.81575 83.09995 81.6625 82.49565 129.9786 51</span></span>
<span id="cb8-15"><a href="#cb8-15" tabindex="-1"></a><span class="co">#> f4() 198.8036 203.35170 206.31347 205.5394 207.87380 237.2220 51</span></span></code></pre></div>
<p>Clearly, the DC-FFT procedure is the fastest, followed by DC, RF and
DFT-CF methods.</p>
</div>
</div>
<div id="generalized-poisson-binomial-distribution" class="section level2">
<h2>Generalized Poisson Binomial Distribution</h2>
<div id="generalized-direct-convolution" class="section level3">
<h3>Generalized Direct Convolution</h3>
<p>The <em>Generalized Direct Convolution</em> (G-DC) approach is
requested with <code>method = "Convolve"</code>.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb9-2"><a href="#cb9-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb9-3"><a href="#cb9-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb9-4"><a href="#cb9-4" tabindex="-1"></a>va <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb9-5"><a href="#cb9-5" tabindex="-1"></a>vb <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb9-6"><a href="#cb9-6" tabindex="-1"></a></span>
<span id="cb9-7"><a href="#cb9-7" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">"Convolve"</span>)</span>
<span id="cb9-8"><a href="#cb9-8" tabindex="-1"></a><span class="co">#> [1] 1.140600e-31 5.349930e-30 1.164698e-28 1.572037e-27 1.491024e-26</span></span>
<span id="cb9-9"><a href="#cb9-9" tabindex="-1"></a><span class="co">#> [6] 1.077204e-25 6.336147e-25 3.215011e-24 1.466295e-23 6.127671e-23</span></span>
<span id="cb9-10"><a href="#cb9-10" tabindex="-1"></a><span class="co">#> [11] 2.363402e-22 8.484857e-22 2.866109e-21 9.171228e-21 2.788507e-20</span></span>
<span id="cb9-11"><a href="#cb9-11" tabindex="-1"></a><span class="co">#> [16] 8.091940e-20 2.254155e-19 6.051395e-19 1.570129e-18 3.953458e-18</span></span>
<span id="cb9-12"><a href="#cb9-12" tabindex="-1"></a><span class="co">#> [21] 9.696098e-18 2.321913e-17 5.442392e-17 1.251302e-16 2.824507e-16</span></span>
<span id="cb9-13"><a href="#cb9-13" tabindex="-1"></a><span class="co">#> [26] 6.264454e-16 1.366745e-15 2.934598e-15 6.203639e-15 1.292697e-14</span></span>
<span id="cb9-14"><a href="#cb9-14" tabindex="-1"></a><span class="co">#> [31] 2.657759e-14 5.394727e-14 1.081983e-13 2.144873e-13 4.201625e-13</span></span>
<span id="cb9-15"><a href="#cb9-15" tabindex="-1"></a><span class="co">#> [36] 8.135609e-13 1.557745e-12 2.949821e-12 5.527695e-12 1.025815e-11</span></span>
<span id="cb9-16"><a href="#cb9-16" tabindex="-1"></a><span class="co">#> [41] 1.885777e-11 3.434641e-11 6.196981e-11 1.106787e-10 1.956340e-10</span></span>
<span id="cb9-17"><a href="#cb9-17" tabindex="-1"></a><span class="co">#> [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753751e-09 2.972596e-09</span></span>
<span id="cb9-18"><a href="#cb9-18" tabindex="-1"></a><span class="co">#> [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08</span></span>
<span id="cb9-19"><a href="#cb9-19" tabindex="-1"></a><span class="co">#> [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07</span></span>
<span id="cb9-20"><a href="#cb9-20" tabindex="-1"></a><span class="co">#> [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06</span></span>
<span id="cb9-21"><a href="#cb9-21" tabindex="-1"></a><span class="co">#> [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05</span></span>
<span id="cb9-22"><a href="#cb9-22" tabindex="-1"></a><span class="co">#> [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05</span></span>
<span id="cb9-23"><a href="#cb9-23" tabindex="-1"></a><span class="co">#> [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04</span></span>
<span id="cb9-24"><a href="#cb9-24" tabindex="-1"></a><span class="co">#> [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03</span></span>
<span id="cb9-25"><a href="#cb9-25" tabindex="-1"></a><span class="co">#> [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03</span></span>
<span id="cb9-26"><a href="#cb9-26" tabindex="-1"></a><span class="co">#> [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03</span></span>
<span id="cb9-27"><a href="#cb9-27" tabindex="-1"></a><span class="co">#> [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02</span></span>
<span id="cb9-28"><a href="#cb9-28" tabindex="-1"></a><span class="co">#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02</span></span>
<span id="cb9-29"><a href="#cb9-29" tabindex="-1"></a><span class="co">#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02</span></span>
<span id="cb9-30"><a href="#cb9-30" tabindex="-1"></a><span class="co">#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02</span></span>
<span id="cb9-31"><a href="#cb9-31" tabindex="-1"></a><span class="co">#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02</span></span>
<span id="cb9-32"><a href="#cb9-32" tabindex="-1"></a><span class="co">#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02</span></span>
<span id="cb9-33"><a href="#cb9-33" tabindex="-1"></a><span class="co">#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02</span></span>
<span id="cb9-34"><a href="#cb9-34" tabindex="-1"></a><span class="co">#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02</span></span>
<span id="cb9-35"><a href="#cb9-35" tabindex="-1"></a><span class="co">#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03</span></span>
<span id="cb9-36"><a href="#cb9-36" tabindex="-1"></a><span class="co">#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03</span></span>
<span id="cb9-37"><a href="#cb9-37" tabindex="-1"></a><span class="co">#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03</span></span>
<span id="cb9-38"><a href="#cb9-38" tabindex="-1"></a><span class="co">#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04</span></span>
<span id="cb9-39"><a href="#cb9-39" tabindex="-1"></a><span class="co">#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04</span></span>
<span id="cb9-40"><a href="#cb9-40" tabindex="-1"></a><span class="co">#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05</span></span>
<span id="cb9-41"><a href="#cb9-41" tabindex="-1"></a><span class="co">#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05</span></span>
<span id="cb9-42"><a href="#cb9-42" tabindex="-1"></a><span class="co">#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06</span></span>
<span id="cb9-43"><a href="#cb9-43" tabindex="-1"></a><span class="co">#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07</span></span>
<span id="cb9-44"><a href="#cb9-44" tabindex="-1"></a><span class="co">#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08</span></span>
<span id="cb9-45"><a href="#cb9-45" tabindex="-1"></a><span class="co">#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09</span></span>
<span id="cb9-46"><a href="#cb9-46" tabindex="-1"></a><span class="co">#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11</span></span>
<span id="cb9-47"><a href="#cb9-47" tabindex="-1"></a><span class="co">#> [196] 1.676154e-11 7.585978e-12 3.326429e-12 1.407527e-12 5.717370e-13</span></span>
<span id="cb9-48"><a href="#cb9-48" tabindex="-1"></a><span class="co">#> [201] 2.216349e-13 8.149241e-14 2.824954e-14 9.179165e-15 2.780017e-15</span></span>
<span id="cb9-49"><a href="#cb9-49" tabindex="-1"></a><span class="co">#> [206] 7.803525e-16 2.018046e-16 4.775552e-17 1.025798e-17 1.979767e-18</span></span>
<span id="cb9-50"><a href="#cb9-50" tabindex="-1"></a><span class="co">#> [211] 3.386554e-19 5.038594e-20 6.336865e-21 6.424747e-22 4.821385e-23</span></span>
<span id="cb9-51"><a href="#cb9-51" tabindex="-1"></a><span class="co">#> [216] 2.108301e-24</span></span>
<span id="cb9-52"><a href="#cb9-52" tabindex="-1"></a><span class="fu">pgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">"Convolve"</span>)</span>
<span id="cb9-53"><a href="#cb9-53" tabindex="-1"></a><span class="co">#> [1] 1.140600e-31 5.463990e-30 1.219337e-28 1.693971e-27 1.660421e-26</span></span>
<span id="cb9-54"><a href="#cb9-54" tabindex="-1"></a><span class="co">#> [6] 1.243246e-25 7.579393e-25 3.972950e-24 1.863590e-23 7.991261e-23</span></span>
<span id="cb9-55"><a href="#cb9-55" tabindex="-1"></a><span class="co">#> [11] 3.162528e-22 1.164739e-21 4.030847e-21 1.320208e-20 4.108715e-20</span></span>
<span id="cb9-56"><a href="#cb9-56" tabindex="-1"></a><span class="co">#> [16] 1.220065e-19 3.474220e-19 9.525615e-19 2.522691e-18 6.476149e-18</span></span>
<span id="cb9-57"><a href="#cb9-57" tabindex="-1"></a><span class="co">#> [21] 1.617225e-17 3.939138e-17 9.381530e-17 2.189455e-16 5.013962e-16</span></span>
<span id="cb9-58"><a href="#cb9-58" tabindex="-1"></a><span class="co">#> [26] 1.127842e-15 2.494586e-15 5.429184e-15 1.163282e-14 2.455979e-14</span></span>
<span id="cb9-59"><a href="#cb9-59" tabindex="-1"></a><span class="co">#> [31] 5.113739e-14 1.050847e-13 2.132829e-13 4.277703e-13 8.479327e-13</span></span>
<span id="cb9-60"><a href="#cb9-60" tabindex="-1"></a><span class="co">#> [36] 1.661494e-12 3.219239e-12 6.169059e-12 1.169675e-11 2.195491e-11</span></span>
<span id="cb9-61"><a href="#cb9-61" tabindex="-1"></a><span class="co">#> [41] 4.081268e-11 7.515909e-11 1.371289e-10 2.478076e-10 4.434415e-10</span></span>
<span id="cb9-62"><a href="#cb9-62" tabindex="-1"></a><span class="co">#> [46] 7.859810e-10 1.380789e-09 2.406013e-09 4.159763e-09 7.132360e-09</span></span>
<span id="cb9-63"><a href="#cb9-63" tabindex="-1"></a><span class="co">#> [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08</span></span>
<span id="cb9-64"><a href="#cb9-64" tabindex="-1"></a><span class="co">#> [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07</span></span>
<span id="cb9-65"><a href="#cb9-65" tabindex="-1"></a><span class="co">#> [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06</span></span>
<span id="cb9-66"><a href="#cb9-66" tabindex="-1"></a><span class="co">#> [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05</span></span>
<span id="cb9-67"><a href="#cb9-67" tabindex="-1"></a><span class="co">#> [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04</span></span>
<span id="cb9-68"><a href="#cb9-68" tabindex="-1"></a><span class="co">#> [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03</span></span>
<span id="cb9-69"><a href="#cb9-69" tabindex="-1"></a><span class="co">#> [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03</span></span>
<span id="cb9-70"><a href="#cb9-70" tabindex="-1"></a><span class="co">#> [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02</span></span>
<span id="cb9-71"><a href="#cb9-71" tabindex="-1"></a><span class="co">#> [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02</span></span>
<span id="cb9-72"><a href="#cb9-72" tabindex="-1"></a><span class="co">#> [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01</span></span>
<span id="cb9-73"><a href="#cb9-73" tabindex="-1"></a><span class="co">#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01</span></span>
<span id="cb9-74"><a href="#cb9-74" tabindex="-1"></a><span class="co">#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01</span></span>
<span id="cb9-75"><a href="#cb9-75" tabindex="-1"></a><span class="co">#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01</span></span>
<span id="cb9-76"><a href="#cb9-76" tabindex="-1"></a><span class="co">#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01</span></span>
<span id="cb9-77"><a href="#cb9-77" tabindex="-1"></a><span class="co">#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01</span></span>
<span id="cb9-78"><a href="#cb9-78" tabindex="-1"></a><span class="co">#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01</span></span>
<span id="cb9-79"><a href="#cb9-79" tabindex="-1"></a><span class="co">#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01</span></span>
<span id="cb9-80"><a href="#cb9-80" tabindex="-1"></a><span class="co">#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01</span></span>
<span id="cb9-81"><a href="#cb9-81" tabindex="-1"></a><span class="co">#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01</span></span>
<span id="cb9-82"><a href="#cb9-82" tabindex="-1"></a><span class="co">#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01</span></span>
<span id="cb9-83"><a href="#cb9-83" tabindex="-1"></a><span class="co">#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01</span></span>
<span id="cb9-84"><a href="#cb9-84" tabindex="-1"></a><span class="co">#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01</span></span>
<span id="cb9-85"><a href="#cb9-85" tabindex="-1"></a><span class="co">#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01</span></span>
<span id="cb9-86"><a href="#cb9-86" tabindex="-1"></a><span class="co">#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01</span></span>
<span id="cb9-87"><a href="#cb9-87" tabindex="-1"></a><span class="co">#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01</span></span>
<span id="cb9-88"><a href="#cb9-88" tabindex="-1"></a><span class="co">#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01</span></span>
<span id="cb9-89"><a href="#cb9-89" tabindex="-1"></a><span class="co">#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-90"><a href="#cb9-90" tabindex="-1"></a><span class="co">#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-91"><a href="#cb9-91" tabindex="-1"></a><span class="co">#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-92"><a href="#cb9-92" tabindex="-1"></a><span class="co">#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-93"><a href="#cb9-93" tabindex="-1"></a><span class="co">#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-94"><a href="#cb9-94" tabindex="-1"></a><span class="co">#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-95"><a href="#cb9-95" tabindex="-1"></a><span class="co">#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb9-96"><a href="#cb9-96" tabindex="-1"></a><span class="co">#> [216] 1.000000e+00</span></span></code></pre></div>
</div>
<div id="generalized-divide-conquer-fft-tree-convolution" class="section level3">
<h3>Generalized Divide & Conquer FFT Tree Convolution</h3>
<p>The <em>Generalized Divide & Conquer FFT Tree Convolution</em>
(G-DC-FFT) approach is requested with
<code>method = "DivideFFT"</code>.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb10-2"><a href="#cb10-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb10-3"><a href="#cb10-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb10-4"><a href="#cb10-4" tabindex="-1"></a>va <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb10-5"><a href="#cb10-5" tabindex="-1"></a>vb <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb10-6"><a href="#cb10-6" tabindex="-1"></a></span>
<span id="cb10-7"><a href="#cb10-7" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">"DivideFFT"</span>)</span>
<span id="cb10-8"><a href="#cb10-8" tabindex="-1"></a><span class="co">#> [1] 1.140600e-31 5.349930e-30 1.164698e-28 1.572037e-27 1.491024e-26</span></span>
<span id="cb10-9"><a href="#cb10-9" tabindex="-1"></a><span class="co">#> [6] 1.077204e-25 6.336147e-25 3.215011e-24 1.466295e-23 6.127671e-23</span></span>
<span id="cb10-10"><a href="#cb10-10" tabindex="-1"></a><span class="co">#> [11] 2.363402e-22 8.484857e-22 2.866109e-21 9.171228e-21 2.788507e-20</span></span>
<span id="cb10-11"><a href="#cb10-11" tabindex="-1"></a><span class="co">#> [16] 8.091940e-20 2.254155e-19 6.051395e-19 1.570129e-18 3.953458e-18</span></span>
<span id="cb10-12"><a href="#cb10-12" tabindex="-1"></a><span class="co">#> [21] 9.696098e-18 2.321913e-17 5.442392e-17 1.251302e-16 2.824507e-16</span></span>
<span id="cb10-13"><a href="#cb10-13" tabindex="-1"></a><span class="co">#> [26] 6.264454e-16 1.366745e-15 2.934598e-15 6.203639e-15 1.292697e-14</span></span>
<span id="cb10-14"><a href="#cb10-14" tabindex="-1"></a><span class="co">#> [31] 2.657759e-14 5.394727e-14 1.081983e-13 2.144873e-13 4.201625e-13</span></span>
<span id="cb10-15"><a href="#cb10-15" tabindex="-1"></a><span class="co">#> [36] 8.135609e-13 1.557745e-12 2.949821e-12 5.527695e-12 1.025815e-11</span></span>
<span id="cb10-16"><a href="#cb10-16" tabindex="-1"></a><span class="co">#> [41] 1.885777e-11 3.434641e-11 6.196981e-11 1.106787e-10 1.956340e-10</span></span>
<span id="cb10-17"><a href="#cb10-17" tabindex="-1"></a><span class="co">#> [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753751e-09 2.972596e-09</span></span>
<span id="cb10-18"><a href="#cb10-18" tabindex="-1"></a><span class="co">#> [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08</span></span>
<span id="cb10-19"><a href="#cb10-19" tabindex="-1"></a><span class="co">#> [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07</span></span>
<span id="cb10-20"><a href="#cb10-20" tabindex="-1"></a><span class="co">#> [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06</span></span>
<span id="cb10-21"><a href="#cb10-21" tabindex="-1"></a><span class="co">#> [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05</span></span>
<span id="cb10-22"><a href="#cb10-22" tabindex="-1"></a><span class="co">#> [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05</span></span>
<span id="cb10-23"><a href="#cb10-23" tabindex="-1"></a><span class="co">#> [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04</span></span>
<span id="cb10-24"><a href="#cb10-24" tabindex="-1"></a><span class="co">#> [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03</span></span>
<span id="cb10-25"><a href="#cb10-25" tabindex="-1"></a><span class="co">#> [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03</span></span>
<span id="cb10-26"><a href="#cb10-26" tabindex="-1"></a><span class="co">#> [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03</span></span>
<span id="cb10-27"><a href="#cb10-27" tabindex="-1"></a><span class="co">#> [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02</span></span>
<span id="cb10-28"><a href="#cb10-28" tabindex="-1"></a><span class="co">#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02</span></span>
<span id="cb10-29"><a href="#cb10-29" tabindex="-1"></a><span class="co">#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02</span></span>
<span id="cb10-30"><a href="#cb10-30" tabindex="-1"></a><span class="co">#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02</span></span>
<span id="cb10-31"><a href="#cb10-31" tabindex="-1"></a><span class="co">#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02</span></span>
<span id="cb10-32"><a href="#cb10-32" tabindex="-1"></a><span class="co">#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02</span></span>
<span id="cb10-33"><a href="#cb10-33" tabindex="-1"></a><span class="co">#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02</span></span>
<span id="cb10-34"><a href="#cb10-34" tabindex="-1"></a><span class="co">#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02</span></span>
<span id="cb10-35"><a href="#cb10-35" tabindex="-1"></a><span class="co">#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03</span></span>
<span id="cb10-36"><a href="#cb10-36" tabindex="-1"></a><span class="co">#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03</span></span>
<span id="cb10-37"><a href="#cb10-37" tabindex="-1"></a><span class="co">#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03</span></span>
<span id="cb10-38"><a href="#cb10-38" tabindex="-1"></a><span class="co">#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04</span></span>
<span id="cb10-39"><a href="#cb10-39" tabindex="-1"></a><span class="co">#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04</span></span>
<span id="cb10-40"><a href="#cb10-40" tabindex="-1"></a><span class="co">#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05</span></span>
<span id="cb10-41"><a href="#cb10-41" tabindex="-1"></a><span class="co">#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05</span></span>
<span id="cb10-42"><a href="#cb10-42" tabindex="-1"></a><span class="co">#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06</span></span>
<span id="cb10-43"><a href="#cb10-43" tabindex="-1"></a><span class="co">#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07</span></span>
<span id="cb10-44"><a href="#cb10-44" tabindex="-1"></a><span class="co">#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08</span></span>
<span id="cb10-45"><a href="#cb10-45" tabindex="-1"></a><span class="co">#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09</span></span>
<span id="cb10-46"><a href="#cb10-46" tabindex="-1"></a><span class="co">#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11</span></span>
<span id="cb10-47"><a href="#cb10-47" tabindex="-1"></a><span class="co">#> [196] 1.676154e-11 7.585978e-12 3.326429e-12 1.407527e-12 5.717370e-13</span></span>
<span id="cb10-48"><a href="#cb10-48" tabindex="-1"></a><span class="co">#> [201] 2.216349e-13 8.149241e-14 2.824954e-14 9.179165e-15 2.780017e-15</span></span>
<span id="cb10-49"><a href="#cb10-49" tabindex="-1"></a><span class="co">#> [206] 7.803525e-16 2.018046e-16 4.775552e-17 1.025798e-17 1.979767e-18</span></span>
<span id="cb10-50"><a href="#cb10-50" tabindex="-1"></a><span class="co">#> [211] 3.386554e-19 5.038594e-20 6.336865e-21 6.424747e-22 4.821385e-23</span></span>
<span id="cb10-51"><a href="#cb10-51" tabindex="-1"></a><span class="co">#> [216] 2.108301e-24</span></span>
<span id="cb10-52"><a href="#cb10-52" tabindex="-1"></a><span class="fu">pgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">"DivideFFT"</span>)</span>
<span id="cb10-53"><a href="#cb10-53" tabindex="-1"></a><span class="co">#> [1] 1.140600e-31 5.463990e-30 1.219337e-28 1.693971e-27 1.660421e-26</span></span>
<span id="cb10-54"><a href="#cb10-54" tabindex="-1"></a><span class="co">#> [6] 1.243246e-25 7.579393e-25 3.972950e-24 1.863590e-23 7.991261e-23</span></span>
<span id="cb10-55"><a href="#cb10-55" tabindex="-1"></a><span class="co">#> [11] 3.162528e-22 1.164739e-21 4.030847e-21 1.320208e-20 4.108715e-20</span></span>
<span id="cb10-56"><a href="#cb10-56" tabindex="-1"></a><span class="co">#> [16] 1.220065e-19 3.474220e-19 9.525615e-19 2.522691e-18 6.476149e-18</span></span>
<span id="cb10-57"><a href="#cb10-57" tabindex="-1"></a><span class="co">#> [21] 1.617225e-17 3.939138e-17 9.381530e-17 2.189455e-16 5.013962e-16</span></span>
<span id="cb10-58"><a href="#cb10-58" tabindex="-1"></a><span class="co">#> [26] 1.127842e-15 2.494586e-15 5.429184e-15 1.163282e-14 2.455979e-14</span></span>
<span id="cb10-59"><a href="#cb10-59" tabindex="-1"></a><span class="co">#> [31] 5.113739e-14 1.050847e-13 2.132829e-13 4.277703e-13 8.479327e-13</span></span>
<span id="cb10-60"><a href="#cb10-60" tabindex="-1"></a><span class="co">#> [36] 1.661494e-12 3.219239e-12 6.169059e-12 1.169675e-11 2.195491e-11</span></span>
<span id="cb10-61"><a href="#cb10-61" tabindex="-1"></a><span class="co">#> [41] 4.081268e-11 7.515909e-11 1.371289e-10 2.478076e-10 4.434415e-10</span></span>
<span id="cb10-62"><a href="#cb10-62" tabindex="-1"></a><span class="co">#> [46] 7.859810e-10 1.380789e-09 2.406013e-09 4.159763e-09 7.132360e-09</span></span>
<span id="cb10-63"><a href="#cb10-63" tabindex="-1"></a><span class="co">#> [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08</span></span>
<span id="cb10-64"><a href="#cb10-64" tabindex="-1"></a><span class="co">#> [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07</span></span>
<span id="cb10-65"><a href="#cb10-65" tabindex="-1"></a><span class="co">#> [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06</span></span>
<span id="cb10-66"><a href="#cb10-66" tabindex="-1"></a><span class="co">#> [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05</span></span>
<span id="cb10-67"><a href="#cb10-67" tabindex="-1"></a><span class="co">#> [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04</span></span>
<span id="cb10-68"><a href="#cb10-68" tabindex="-1"></a><span class="co">#> [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03</span></span>
<span id="cb10-69"><a href="#cb10-69" tabindex="-1"></a><span class="co">#> [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03</span></span>
<span id="cb10-70"><a href="#cb10-70" tabindex="-1"></a><span class="co">#> [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02</span></span>
<span id="cb10-71"><a href="#cb10-71" tabindex="-1"></a><span class="co">#> [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02</span></span>
<span id="cb10-72"><a href="#cb10-72" tabindex="-1"></a><span class="co">#> [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01</span></span>
<span id="cb10-73"><a href="#cb10-73" tabindex="-1"></a><span class="co">#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01</span></span>
<span id="cb10-74"><a href="#cb10-74" tabindex="-1"></a><span class="co">#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01</span></span>
<span id="cb10-75"><a href="#cb10-75" tabindex="-1"></a><span class="co">#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01</span></span>
<span id="cb10-76"><a href="#cb10-76" tabindex="-1"></a><span class="co">#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01</span></span>
<span id="cb10-77"><a href="#cb10-77" tabindex="-1"></a><span class="co">#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01</span></span>
<span id="cb10-78"><a href="#cb10-78" tabindex="-1"></a><span class="co">#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01</span></span>
<span id="cb10-79"><a href="#cb10-79" tabindex="-1"></a><span class="co">#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01</span></span>
<span id="cb10-80"><a href="#cb10-80" tabindex="-1"></a><span class="co">#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01</span></span>
<span id="cb10-81"><a href="#cb10-81" tabindex="-1"></a><span class="co">#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01</span></span>
<span id="cb10-82"><a href="#cb10-82" tabindex="-1"></a><span class="co">#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01</span></span>
<span id="cb10-83"><a href="#cb10-83" tabindex="-1"></a><span class="co">#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01</span></span>
<span id="cb10-84"><a href="#cb10-84" tabindex="-1"></a><span class="co">#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01</span></span>
<span id="cb10-85"><a href="#cb10-85" tabindex="-1"></a><span class="co">#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01</span></span>
<span id="cb10-86"><a href="#cb10-86" tabindex="-1"></a><span class="co">#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01</span></span>
<span id="cb10-87"><a href="#cb10-87" tabindex="-1"></a><span class="co">#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01</span></span>
<span id="cb10-88"><a href="#cb10-88" tabindex="-1"></a><span class="co">#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01</span></span>
<span id="cb10-89"><a href="#cb10-89" tabindex="-1"></a><span class="co">#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-90"><a href="#cb10-90" tabindex="-1"></a><span class="co">#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-91"><a href="#cb10-91" tabindex="-1"></a><span class="co">#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-92"><a href="#cb10-92" tabindex="-1"></a><span class="co">#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-93"><a href="#cb10-93" tabindex="-1"></a><span class="co">#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-94"><a href="#cb10-94" tabindex="-1"></a><span class="co">#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-95"><a href="#cb10-95" tabindex="-1"></a><span class="co">#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb10-96"><a href="#cb10-96" tabindex="-1"></a><span class="co">#> [216] 1.000000e+00</span></span></code></pre></div>
<p>By design, similar to the ordinary DC-FFT algorithm by <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &
Brunner (2018)</a>, its results are identical to the G-DC procedure, if
<span class="math inline">\(n\)</span> and the number of possible
observed values is small. Thus, differences can be observed for larger
numbers:</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb11-2"><a href="#cb11-2" tabindex="-1"></a>pp1 <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">250</span>)</span>
<span id="cb11-3"><a href="#cb11-3" tabindex="-1"></a>va1 <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">50</span>, <span class="dv">250</span>, <span class="cn">TRUE</span>)</span>
<span id="cb11-4"><a href="#cb11-4" tabindex="-1"></a>vb1 <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">50</span>, <span class="dv">250</span>, <span class="cn">TRUE</span>)</span>
<span id="cb11-5"><a href="#cb11-5" tabindex="-1"></a>pp2 <span class="ot"><-</span> pp1[<span class="dv">1</span><span class="sc">:</span><span class="dv">248</span>]</span>
<span id="cb11-6"><a href="#cb11-6" tabindex="-1"></a>va2 <span class="ot"><-</span> va1[<span class="dv">1</span><span class="sc">:</span><span class="dv">248</span>]</span>
<span id="cb11-7"><a href="#cb11-7" tabindex="-1"></a>vb2 <span class="ot"><-</span> vb1[<span class="dv">1</span><span class="sc">:</span><span class="dv">248</span>]</span>
<span id="cb11-8"><a href="#cb11-8" tabindex="-1"></a></span>
<span id="cb11-9"><a href="#cb11-9" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">abs</span>(<span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp1, va1, vb1, <span class="at">method =</span> <span class="st">"DivideFFT"</span>)</span>
<span id="cb11-10"><a href="#cb11-10" tabindex="-1"></a> <span class="sc">-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp1, va1, vb1, <span class="at">method =</span> <span class="st">"Convolve"</span>)))</span>
<span id="cb11-11"><a href="#cb11-11" tabindex="-1"></a><span class="co">#> [1] 0</span></span>
<span id="cb11-12"><a href="#cb11-12" tabindex="-1"></a></span>
<span id="cb11-13"><a href="#cb11-13" tabindex="-1"></a><span class="fu">sum</span>(<span class="fu">abs</span>(<span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp2, va2, vb2, <span class="at">method =</span> <span class="st">"DivideFFT"</span>)</span>
<span id="cb11-14"><a href="#cb11-14" tabindex="-1"></a> <span class="sc">-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp2, va2, vb2, <span class="at">method =</span> <span class="st">"Convolve"</span>)))</span>
<span id="cb11-15"><a href="#cb11-15" tabindex="-1"></a><span class="co">#> [1] 0</span></span></code></pre></div>
<p>The reason is that the G-DC-FFT method splits the input
<code>probs</code>, <code>val_p</code> and <code>val_q</code> vectors
into parts such that the numbers of possible observations of all parts
are as equally sized as possible. Their distributions are then computed
separately with the G-DC approach. The results of the portions are then
convoluted by means of the Fast Fourier Transformation. For small <span class="math inline">\(n\)</span> and small distribution sizes, no
splitting is needed. In addition, the G-DC-FFT procedure, just like the
DC-FFT method, does not produce probabilities <span class="math inline">\(\leq 5.55e\text{-}17\)</span>, i.e. smaller values
are rounded off to <span class="math inline">\(0\)</span>, if the total
number of possible observations is smaller than <span class="math inline">\(750\)</span>, whereas the smallest possible result
of the DC algorithm is <span class="math inline">\(\sim
1e\text{-}323\)</span>. This is most likely caused by the used FFTW3
library.</p>
<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" tabindex="-1"></a>d1 <span class="ot"><-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp1, va1, vb1, <span class="at">method =</span> <span class="st">"DivideFFT"</span>)</span>
<span id="cb12-2"><a href="#cb12-2" tabindex="-1"></a>d2 <span class="ot"><-</span> <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp1, va1, vb1, <span class="at">method =</span> <span class="st">"Convolve"</span>)</span>
<span id="cb12-3"><a href="#cb12-3" tabindex="-1"></a></span>
<span id="cb12-4"><a href="#cb12-4" tabindex="-1"></a><span class="fu">min</span>(d1[d1 <span class="sc">></span> <span class="dv">0</span>])</span>
<span id="cb12-5"><a href="#cb12-5" tabindex="-1"></a><span class="co">#> [1] 2.839368e-99</span></span>
<span id="cb12-6"><a href="#cb12-6" tabindex="-1"></a><span class="fu">min</span>(d2[d2 <span class="sc">></span> <span class="dv">0</span>])</span>
<span id="cb12-7"><a href="#cb12-7" tabindex="-1"></a><span class="co">#> [1] 2.839368e-99</span></span></code></pre></div>
</div>
<div id="generalized-discrete-fourier-transformation-of-the-characteristic-function" class="section level3">
<h3>Generalized Discrete Fourier Transformation of the Characteristic
Function</h3>
<p>The <em>Generalized Discrete Fourier Transformation of the
Characteristic Function</em> (G-DFT-CF) approach is requested with
<code>method = "Characteristic"</code>.</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb13-2"><a href="#cb13-2" tabindex="-1"></a>pp <span class="ot"><-</span> <span class="fu">runif</span>(<span class="dv">10</span>)</span>
<span id="cb13-3"><a href="#cb13-3" tabindex="-1"></a>wt <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb13-4"><a href="#cb13-4" tabindex="-1"></a>va <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb13-5"><a href="#cb13-5" tabindex="-1"></a>vb <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">10</span>, <span class="dv">10</span>, <span class="cn">TRUE</span>)</span>
<span id="cb13-6"><a href="#cb13-6" tabindex="-1"></a></span>
<span id="cb13-7"><a href="#cb13-7" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">"Characteristic"</span>)</span>
<span id="cb13-8"><a href="#cb13-8" tabindex="-1"></a><span class="co">#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-9"><a href="#cb13-9" tabindex="-1"></a><span class="co">#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-10"><a href="#cb13-10" tabindex="-1"></a><span class="co">#> [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-11"><a href="#cb13-11" tabindex="-1"></a><span class="co">#> [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-12"><a href="#cb13-12" tabindex="-1"></a><span class="co">#> [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.837237e-16</span></span>
<span id="cb13-13"><a href="#cb13-13" tabindex="-1"></a><span class="co">#> [26] 6.250144e-16 1.365163e-15 2.931811e-15 6.199773e-15 1.292382e-14</span></span>
<span id="cb13-14"><a href="#cb13-14" tabindex="-1"></a><span class="co">#> [31] 2.657288e-14 5.394142e-14 1.081912e-13 2.144812e-13 4.201536e-13</span></span>
<span id="cb13-15"><a href="#cb13-15" tabindex="-1"></a><span class="co">#> [36] 8.135511e-13 1.557734e-12 2.949810e-12 5.527683e-12 1.025814e-11</span></span>
<span id="cb13-16"><a href="#cb13-16" tabindex="-1"></a><span class="co">#> [41] 1.885776e-11 3.434640e-11 6.196980e-11 1.106787e-10 1.956340e-10</span></span>
<span id="cb13-17"><a href="#cb13-17" tabindex="-1"></a><span class="co">#> [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753750e-09 2.972596e-09</span></span>
<span id="cb13-18"><a href="#cb13-18" tabindex="-1"></a><span class="co">#> [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08</span></span>
<span id="cb13-19"><a href="#cb13-19" tabindex="-1"></a><span class="co">#> [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07</span></span>
<span id="cb13-20"><a href="#cb13-20" tabindex="-1"></a><span class="co">#> [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06</span></span>
<span id="cb13-21"><a href="#cb13-21" tabindex="-1"></a><span class="co">#> [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05</span></span>
<span id="cb13-22"><a href="#cb13-22" tabindex="-1"></a><span class="co">#> [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05</span></span>
<span id="cb13-23"><a href="#cb13-23" tabindex="-1"></a><span class="co">#> [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04</span></span>
<span id="cb13-24"><a href="#cb13-24" tabindex="-1"></a><span class="co">#> [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03</span></span>
<span id="cb13-25"><a href="#cb13-25" tabindex="-1"></a><span class="co">#> [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03</span></span>
<span id="cb13-26"><a href="#cb13-26" tabindex="-1"></a><span class="co">#> [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03</span></span>
<span id="cb13-27"><a href="#cb13-27" tabindex="-1"></a><span class="co">#> [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02</span></span>
<span id="cb13-28"><a href="#cb13-28" tabindex="-1"></a><span class="co">#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02</span></span>
<span id="cb13-29"><a href="#cb13-29" tabindex="-1"></a><span class="co">#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02</span></span>
<span id="cb13-30"><a href="#cb13-30" tabindex="-1"></a><span class="co">#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02</span></span>
<span id="cb13-31"><a href="#cb13-31" tabindex="-1"></a><span class="co">#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02</span></span>
<span id="cb13-32"><a href="#cb13-32" tabindex="-1"></a><span class="co">#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02</span></span>
<span id="cb13-33"><a href="#cb13-33" tabindex="-1"></a><span class="co">#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02</span></span>
<span id="cb13-34"><a href="#cb13-34" tabindex="-1"></a><span class="co">#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02</span></span>
<span id="cb13-35"><a href="#cb13-35" tabindex="-1"></a><span class="co">#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03</span></span>
<span id="cb13-36"><a href="#cb13-36" tabindex="-1"></a><span class="co">#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03</span></span>
<span id="cb13-37"><a href="#cb13-37" tabindex="-1"></a><span class="co">#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03</span></span>
<span id="cb13-38"><a href="#cb13-38" tabindex="-1"></a><span class="co">#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04</span></span>
<span id="cb13-39"><a href="#cb13-39" tabindex="-1"></a><span class="co">#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04</span></span>
<span id="cb13-40"><a href="#cb13-40" tabindex="-1"></a><span class="co">#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05</span></span>
<span id="cb13-41"><a href="#cb13-41" tabindex="-1"></a><span class="co">#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05</span></span>
<span id="cb13-42"><a href="#cb13-42" tabindex="-1"></a><span class="co">#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06</span></span>
<span id="cb13-43"><a href="#cb13-43" tabindex="-1"></a><span class="co">#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07</span></span>
<span id="cb13-44"><a href="#cb13-44" tabindex="-1"></a><span class="co">#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08</span></span>
<span id="cb13-45"><a href="#cb13-45" tabindex="-1"></a><span class="co">#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09</span></span>
<span id="cb13-46"><a href="#cb13-46" tabindex="-1"></a><span class="co">#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11</span></span>
<span id="cb13-47"><a href="#cb13-47" tabindex="-1"></a><span class="co">#> [196] 1.676155e-11 7.585978e-12 3.326431e-12 1.407528e-12 5.717366e-13</span></span>
<span id="cb13-48"><a href="#cb13-48" tabindex="-1"></a><span class="co">#> [201] 2.216380e-13 8.149294e-14 2.825106e-14 9.182984e-15 2.782753e-15</span></span>
<span id="cb13-49"><a href="#cb13-49" tabindex="-1"></a><span class="co">#> [206] 7.822960e-16 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-50"><a href="#cb13-50" tabindex="-1"></a><span class="co">#> [211] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-51"><a href="#cb13-51" tabindex="-1"></a><span class="co">#> [216] 0.000000e+00</span></span>
<span id="cb13-52"><a href="#cb13-52" tabindex="-1"></a><span class="fu">pgpbinom</span>(<span class="cn">NULL</span>, pp, va, vb, wt, <span class="st">"Characteristic"</span>)</span>
<span id="cb13-53"><a href="#cb13-53" tabindex="-1"></a><span class="co">#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-54"><a href="#cb13-54" tabindex="-1"></a><span class="co">#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-55"><a href="#cb13-55" tabindex="-1"></a><span class="co">#> [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-56"><a href="#cb13-56" tabindex="-1"></a><span class="co">#> [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00</span></span>
<span id="cb13-57"><a href="#cb13-57" tabindex="-1"></a><span class="co">#> [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.837237e-16</span></span>
<span id="cb13-58"><a href="#cb13-58" tabindex="-1"></a><span class="co">#> [26] 9.087381e-16 2.273901e-15 5.205712e-15 1.140549e-14 2.432930e-14</span></span>
<span id="cb13-59"><a href="#cb13-59" tabindex="-1"></a><span class="co">#> [31] 5.090218e-14 1.048436e-13 2.130348e-13 4.275160e-13 8.476697e-13</span></span>
<span id="cb13-60"><a href="#cb13-60" tabindex="-1"></a><span class="co">#> [36] 1.661221e-12 3.218955e-12 6.168765e-12 1.169645e-11 2.195459e-11</span></span>
<span id="cb13-61"><a href="#cb13-61" tabindex="-1"></a><span class="co">#> [41] 4.081235e-11 7.515874e-11 1.371285e-10 2.478072e-10 4.434412e-10</span></span>
<span id="cb13-62"><a href="#cb13-62" tabindex="-1"></a><span class="co">#> [46] 7.859806e-10 1.380788e-09 2.406013e-09 4.159763e-09 7.132359e-09</span></span>
<span id="cb13-63"><a href="#cb13-63" tabindex="-1"></a><span class="co">#> [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08</span></span>
<span id="cb13-64"><a href="#cb13-64" tabindex="-1"></a><span class="co">#> [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07</span></span>
<span id="cb13-65"><a href="#cb13-65" tabindex="-1"></a><span class="co">#> [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06</span></span>
<span id="cb13-66"><a href="#cb13-66" tabindex="-1"></a><span class="co">#> [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05</span></span>
<span id="cb13-67"><a href="#cb13-67" tabindex="-1"></a><span class="co">#> [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04</span></span>
<span id="cb13-68"><a href="#cb13-68" tabindex="-1"></a><span class="co">#> [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03</span></span>
<span id="cb13-69"><a href="#cb13-69" tabindex="-1"></a><span class="co">#> [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03</span></span>
<span id="cb13-70"><a href="#cb13-70" tabindex="-1"></a><span class="co">#> [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02</span></span>
<span id="cb13-71"><a href="#cb13-71" tabindex="-1"></a><span class="co">#> [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02</span></span>
<span id="cb13-72"><a href="#cb13-72" tabindex="-1"></a><span class="co">#> [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01</span></span>
<span id="cb13-73"><a href="#cb13-73" tabindex="-1"></a><span class="co">#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01</span></span>
<span id="cb13-74"><a href="#cb13-74" tabindex="-1"></a><span class="co">#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01</span></span>
<span id="cb13-75"><a href="#cb13-75" tabindex="-1"></a><span class="co">#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01</span></span>
<span id="cb13-76"><a href="#cb13-76" tabindex="-1"></a><span class="co">#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01</span></span>
<span id="cb13-77"><a href="#cb13-77" tabindex="-1"></a><span class="co">#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01</span></span>
<span id="cb13-78"><a href="#cb13-78" tabindex="-1"></a><span class="co">#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01</span></span>
<span id="cb13-79"><a href="#cb13-79" tabindex="-1"></a><span class="co">#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01</span></span>
<span id="cb13-80"><a href="#cb13-80" tabindex="-1"></a><span class="co">#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01</span></span>
<span id="cb13-81"><a href="#cb13-81" tabindex="-1"></a><span class="co">#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01</span></span>
<span id="cb13-82"><a href="#cb13-82" tabindex="-1"></a><span class="co">#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01</span></span>
<span id="cb13-83"><a href="#cb13-83" tabindex="-1"></a><span class="co">#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01</span></span>
<span id="cb13-84"><a href="#cb13-84" tabindex="-1"></a><span class="co">#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01</span></span>
<span id="cb13-85"><a href="#cb13-85" tabindex="-1"></a><span class="co">#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01</span></span>
<span id="cb13-86"><a href="#cb13-86" tabindex="-1"></a><span class="co">#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01</span></span>
<span id="cb13-87"><a href="#cb13-87" tabindex="-1"></a><span class="co">#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01</span></span>
<span id="cb13-88"><a href="#cb13-88" tabindex="-1"></a><span class="co">#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01</span></span>
<span id="cb13-89"><a href="#cb13-89" tabindex="-1"></a><span class="co">#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-90"><a href="#cb13-90" tabindex="-1"></a><span class="co">#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-91"><a href="#cb13-91" tabindex="-1"></a><span class="co">#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-92"><a href="#cb13-92" tabindex="-1"></a><span class="co">#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-93"><a href="#cb13-93" tabindex="-1"></a><span class="co">#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-94"><a href="#cb13-94" tabindex="-1"></a><span class="co">#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-95"><a href="#cb13-95" tabindex="-1"></a><span class="co">#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00</span></span>
<span id="cb13-96"><a href="#cb13-96" tabindex="-1"></a><span class="co">#> [216] 1.000000e+00</span></span></code></pre></div>
<p>As can be seen, the G-DFT-CF procedure does not produce probabilities
<span class="math inline">\(\leq 2.2e\text{-}16\)</span>, i.e. smaller
values are rounded off to 0, most likely due to the used FFTW3
library.</p>
</div>
<div id="processing-speed-comparisons-1" class="section level3">
<h3>Processing Speed Comparisons</h3>
<p>To assess the performance of the exact procedures, we use the
<code>microbenchmark</code> package. Each algorithm has to calculate the
PMF repeatedly based on random probability and value vectors. The run
times are then summarized in a table that presents, among other
statistics, their minima, maxima and means. The following results were
recorded on an AMD Ryzen 9 5900X with 64 GiB of RAM and Windows 10
Education (22H2).</p>
<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" tabindex="-1"></a><span class="fu">library</span>(microbenchmark)</span>
<span id="cb14-2"><a href="#cb14-2" tabindex="-1"></a>n <span class="ot"><-</span> <span class="dv">2500</span></span>
<span id="cb14-3"><a href="#cb14-3" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb14-4"><a href="#cb14-4" tabindex="-1"></a>va <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">50</span>, n, <span class="cn">TRUE</span>)</span>
<span id="cb14-5"><a href="#cb14-5" tabindex="-1"></a>vb <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">50</span>, n, <span class="cn">TRUE</span>)</span>
<span id="cb14-6"><a href="#cb14-6" tabindex="-1"></a></span>
<span id="cb14-7"><a href="#cb14-7" tabindex="-1"></a>f1 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(n), va, vb, <span class="at">method =</span> <span class="st">"DivideFFT"</span>)</span>
<span id="cb14-8"><a href="#cb14-8" tabindex="-1"></a>f2 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(n), va, vb, <span class="at">method =</span> <span class="st">"Convolve"</span>)</span>
<span id="cb14-9"><a href="#cb14-9" tabindex="-1"></a>f3 <span class="ot"><-</span> <span class="cf">function</span>() <span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">runif</span>(n), va, vb, <span class="at">method =</span> <span class="st">"Characteristic"</span>)</span>
<span id="cb14-10"><a href="#cb14-10" tabindex="-1"></a></span>
<span id="cb14-11"><a href="#cb14-11" tabindex="-1"></a><span class="fu">microbenchmark</span>(<span class="fu">f1</span>(), <span class="fu">f2</span>(), <span class="fu">f3</span>(), <span class="at">times =</span> <span class="dv">51</span>)</span>
<span id="cb14-12"><a href="#cb14-12" tabindex="-1"></a><span class="co">#> Unit: milliseconds</span></span>
<span id="cb14-13"><a href="#cb14-13" tabindex="-1"></a><span class="co">#> expr min lq mean median uq max neval</span></span>
<span id="cb14-14"><a href="#cb14-14" tabindex="-1"></a><span class="co">#> f1() 78.0103 80.94265 85.83106 82.3256 84.77965 222.4554 51</span></span>
<span id="cb14-15"><a href="#cb14-15" tabindex="-1"></a><span class="co">#> f2() 185.1651 189.36355 192.53650 191.5913 194.92825 216.8803 51</span></span>
<span id="cb14-16"><a href="#cb14-16" tabindex="-1"></a><span class="co">#> f3() 639.9733 721.50785 739.89051 747.7238 764.83850 801.4566 51</span></span></code></pre></div>
<p>Clearly, the G-DC-FFT procedure is the fastest one. It outperforms
both the G-DC and G-DFT-CF approaches. The latter one needs a lot more
time than the others. Generally, the computational speed advantage of
the G-DC-FFT procedure increases with larger <span class="math inline">\(n\)</span> (and <span class="math inline">\(m\)</span>).</p>
</div>
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