The Direct Convolution (DC) approach is requested with method = "Convolve"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt
dpbinom(NULL, pp, wt, "Convolve")
#> [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26
#> [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19
#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13
#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10
#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17
#> [61] 9.411166e-19 6.727527e-21
ppbinom(NULL, pp, wt, "Convolve")
#> [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26
#> [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19
#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13
#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00
The Divide & Conquer FFT Tree Convolution (DC-FFT) approach is requested with method = "DivideFFT"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt
dpbinom(NULL, pp, wt, "DivideFFT")
#> [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26
#> [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19
#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13
#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10
#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17
#> [61] 9.411166e-19 6.727527e-21
ppbinom(NULL, pp, wt, "DivideFFT")
#> [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26
#> [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19
#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13
#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00
By design, as proposed by Biscarri, Zhao & Brunner (2018), its results are identical to the DC procedure, if \(n \leq 750\). Thus, differences can be observed for larger \(n > 750\):
set.seed(1)
<- runif(751)
pp1 <- pp1[1:750]
pp2
sum(abs(dpbinom(NULL, pp2, method = "DivideFFT") - dpbinom(NULL, pp2, method = "Convolve")))
#> [1] 0
sum(abs(dpbinom(NULL, pp1, method = "DivideFFT") - dpbinom(NULL, pp1, method = "Convolve")))
#> [1] 0
The reason is that the DC-FFT method splits the input probs
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 Biscarri, Zhao & Brunner (2018), no splitting is done for \(n \leq 750\). In addition, the DC-FFT procedure does not produce probabilities \(\leq 5.55e\text{-}17\), i.e. smaller values are rounded off to 0, if \(n > 750\), whereas the smallest possible result of the DC algorithm is \(\sim 1e\text{-}323\). This is most likely caused by the used FFTW3 library.
set.seed(1)
<- runif(751)
pp1
<- dpbinom(NULL, pp1, method = "DivideFFT")
d1 <- dpbinom(NULL, pp1, method = "Convolve")
d2
min(d1[d1 > 0])
#> [1] 1.635357e-321
min(d2[d2 > 0])
#> [1] 1.635357e-321
The Discrete Fourier Transformation of the Characteristic Function (DFT-CF) approach is requested with method = "Characteristic"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt
dpbinom(NULL, pp, wt, "Characteristic")
#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [11] 0.000000e+00 2.238353e-16 3.549132e-15 4.829828e-14 5.804377e-13
#> [16] 6.158818e-12 5.784702e-11 4.822438e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110923e-10
#> [56] 2.392079e-11 1.468354e-12 6.994931e-14 2.513558e-15 0.000000e+00
#> [61] 0.000000e+00 0.000000e+00
ppbinom(NULL, pp, wt, "Characteristic")
#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [11] 0.000000e+00 2.238353e-16 3.772968e-15 5.207125e-14 6.325089e-13
#> [16] 6.791327e-12 6.463834e-11 5.468822e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00
As can be seen, the DFT-CF procedure does not produce probabilities \(\leq 2.22e\text{-}16\), i.e. smaller values are rounded off to 0, most likely due to the used FFTW3 library.
The Recursive Formula (RF) approach is requested with method = "Recursive"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt
dpbinom(NULL, pp, wt, "Recursive")
#> [1] 3.574462e-35 1.120280e-32 1.685184e-30 1.620524e-28 1.119523e-26
#> [6] 5.920060e-25 2.493263e-23 8.591850e-22 2.470125e-20 6.011429e-19
#> [11] 1.252345e-17 2.253115e-16 3.525477e-15 4.825171e-14 5.803728e-13
#> [16] 6.158735e-12 5.784692e-11 4.822437e-10 3.576566e-09 2.364563e-08
#> [21] 1.395965e-07 7.370448e-07 3.484836e-06 1.477208e-05 5.619632e-05
#> [26] 1.920240e-04 5.897928e-04 1.629272e-03 4.049768e-03 9.060183e-03
#> [31] 1.824629e-02 3.307754e-02 5.396724e-02 7.921491e-02 1.045505e-01
#> [36] 1.239854e-01 1.319896e-01 1.259938e-01 1.077029e-01 8.232174e-02
#> [41] 5.616422e-02 3.413623e-02 1.844304e-02 8.835890e-03 3.743554e-03
#> [46] 1.398320e-03 4.589049e-04 1.318064e-04 3.298425e-05 7.154649e-06
#> [51] 1.337083e-06 2.137543e-07 2.898296e-08 3.298587e-09 3.110922e-10
#> [56] 2.392070e-11 1.468267e-12 6.991155e-14 2.478218e-15 6.130807e-17
#> [61] 9.411166e-19 6.727527e-21
ppbinom(NULL, pp, wt, "Recursive")
#> [1] 3.574462e-35 1.123854e-32 1.696423e-30 1.637488e-28 1.135898e-26
#> [6] 6.033650e-25 2.553600e-23 8.847210e-22 2.558597e-20 6.267289e-19
#> [11] 1.315018e-17 2.384617e-16 3.763939e-15 5.201565e-14 6.323884e-13
#> [16] 6.791123e-12 6.463805e-11 5.468818e-10 4.123448e-09 2.776908e-08
#> [21] 1.673656e-07 9.044104e-07 4.389247e-06 1.916133e-05 7.535765e-05
#> [26] 2.673817e-04 8.571745e-04 2.486446e-03 6.536215e-03 1.559640e-02
#> [31] 3.384269e-02 6.692022e-02 1.208875e-01 2.001024e-01 3.046529e-01
#> [36] 4.286383e-01 5.606280e-01 6.866217e-01 7.943246e-01 8.766463e-01
#> [41] 9.328105e-01 9.669468e-01 9.853898e-01 9.942257e-01 9.979692e-01
#> [46] 9.993676e-01 9.998265e-01 9.999583e-01 9.999913e-01 9.999984e-01
#> [51] 9.999998e-01 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00
Obviously, the RF procedure does produce probabilities \(\leq 5.55e\text{-}17\), because it does not rely on the FFTW3 library. Furthermore, it yields the same results as the DC method.
set.seed(1)
<- runif(1000)
pp <- sample(1:10, 1000, TRUE)
wt
sum(abs(dpbinom(NULL, pp, wt, "Convolve") - dpbinom(NULL, pp, wt, "Recursive")))
#> [1] 0
To assess the performance of the exact procedures, we use the microbenchmark
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 7 1800X with 32 GiB of RAM and Windows 10 Education (20H2).
library(microbenchmark)
set.seed(1)
<- function() dpbinom(NULL, runif(6000), method = "DivideFFT")
f1 <- function() dpbinom(NULL, runif(6000), method = "Convolve")
f2 <- function() dpbinom(NULL, runif(6000), method = "Recursive")
f3 <- function() dpbinom(NULL, runif(6000), method = "Characteristic")
f4
microbenchmark(f1(), f2(), f3(), f4(), times = 51)
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> f1() 12.9458 13.63805 14.21733 13.9593 14.34765 19.3790 51
#> f2() 26.0156 26.63650 27.70315 27.5269 28.03865 30.7545 51
#> f3() 48.5651 49.50555 51.04236 50.6247 51.54655 57.3016 51
#> f4() 135.8826 138.92250 141.05593 140.6084 142.84260 150.7202 51
Clearly, the DC-FFT procedure is the fastest, followed by DC, RF and DFT-CF methods.
The Generalized Direct Convolution (G-DC) approach is requested with method = "Convolve"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt <- sample(0:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb
dgpbinom(NULL, pp, va, vb, wt, "Convolve")
#> [1] 1.140600e-31 5.349930e-30 1.164698e-28 1.572037e-27 1.491024e-26
#> [6] 1.077204e-25 6.336147e-25 3.215011e-24 1.466295e-23 6.127671e-23
#> [11] 2.363402e-22 8.484857e-22 2.866109e-21 9.171228e-21 2.788507e-20
#> [16] 8.091940e-20 2.254155e-19 6.051395e-19 1.570129e-18 3.953458e-18
#> [21] 9.696098e-18 2.321913e-17 5.442392e-17 1.251302e-16 2.824507e-16
#> [26] 6.264454e-16 1.366745e-15 2.934598e-15 6.203639e-15 1.292697e-14
#> [31] 2.657759e-14 5.394727e-14 1.081983e-13 2.144873e-13 4.201625e-13
#> [36] 8.135609e-13 1.557745e-12 2.949821e-12 5.527695e-12 1.025815e-11
#> [41] 1.885777e-11 3.434641e-11 6.196981e-11 1.106787e-10 1.956340e-10
#> [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753751e-09 2.972596e-09
#> [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08
#> [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07
#> [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06
#> [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05
#> [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05
#> [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04
#> [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03
#> [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03
#> [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03
#> [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02
#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02
#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02
#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02
#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02
#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02
#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02
#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02
#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03
#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03
#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03
#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04
#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04
#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05
#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05
#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06
#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07
#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08
#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09
#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11
#> [196] 1.676154e-11 7.585978e-12 3.326429e-12 1.407527e-12 5.717370e-13
#> [201] 2.216349e-13 8.149241e-14 2.824954e-14 9.179165e-15 2.780017e-15
#> [206] 7.803525e-16 2.018046e-16 4.775552e-17 1.025798e-17 1.979767e-18
#> [211] 3.386554e-19 5.038594e-20 6.336865e-21 6.424747e-22 4.821385e-23
#> [216] 2.108301e-24
pgpbinom(NULL, pp, va, vb, wt, "Convolve")
#> [1] 1.140600e-31 5.463990e-30 1.219337e-28 1.693971e-27 1.660421e-26
#> [6] 1.243246e-25 7.579393e-25 3.972950e-24 1.863590e-23 7.991261e-23
#> [11] 3.162528e-22 1.164739e-21 4.030847e-21 1.320208e-20 4.108715e-20
#> [16] 1.220065e-19 3.474220e-19 9.525615e-19 2.522691e-18 6.476149e-18
#> [21] 1.617225e-17 3.939138e-17 9.381530e-17 2.189455e-16 5.013962e-16
#> [26] 1.127842e-15 2.494586e-15 5.429184e-15 1.163282e-14 2.455979e-14
#> [31] 5.113739e-14 1.050847e-13 2.132829e-13 4.277703e-13 8.479327e-13
#> [36] 1.661494e-12 3.219239e-12 6.169059e-12 1.169675e-11 2.195491e-11
#> [41] 4.081268e-11 7.515909e-11 1.371289e-10 2.478076e-10 4.434415e-10
#> [46] 7.859810e-10 1.380789e-09 2.406013e-09 4.159763e-09 7.132360e-09
#> [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08
#> [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07
#> [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06
#> [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05
#> [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04
#> [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03
#> [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03
#> [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02
#> [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02
#> [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01
#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01
#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01
#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01
#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01
#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01
#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01
#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01
#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01
#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01
#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01
#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01
#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01
#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01
#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01
#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01
#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01
#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00
#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [216] 1.000000e+00
The Generalized Divide & Conquer FFT Tree Convolution (G-DC-FFT) approach is requested with method = "DivideFFT"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt <- sample(0:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb
dgpbinom(NULL, pp, va, vb, wt, "DivideFFT")
#> [1] 1.140600e-31 5.349930e-30 1.164698e-28 1.572037e-27 1.491024e-26
#> [6] 1.077204e-25 6.336147e-25 3.215011e-24 1.466295e-23 6.127671e-23
#> [11] 2.363402e-22 8.484857e-22 2.866109e-21 9.171228e-21 2.788507e-20
#> [16] 8.091940e-20 2.254155e-19 6.051395e-19 1.570129e-18 3.953458e-18
#> [21] 9.696098e-18 2.321913e-17 5.442392e-17 1.251302e-16 2.824507e-16
#> [26] 6.264454e-16 1.366745e-15 2.934598e-15 6.203639e-15 1.292697e-14
#> [31] 2.657759e-14 5.394727e-14 1.081983e-13 2.144873e-13 4.201625e-13
#> [36] 8.135609e-13 1.557745e-12 2.949821e-12 5.527695e-12 1.025815e-11
#> [41] 1.885777e-11 3.434641e-11 6.196981e-11 1.106787e-10 1.956340e-10
#> [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753751e-09 2.972596e-09
#> [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08
#> [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07
#> [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06
#> [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05
#> [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05
#> [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04
#> [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03
#> [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03
#> [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03
#> [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02
#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02
#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02
#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02
#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02
#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02
#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02
#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02
#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03
#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03
#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03
#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04
#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04
#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05
#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05
#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06
#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07
#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08
#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09
#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11
#> [196] 1.676154e-11 7.585978e-12 3.326429e-12 1.407527e-12 5.717370e-13
#> [201] 2.216349e-13 8.149241e-14 2.824954e-14 9.179165e-15 2.780017e-15
#> [206] 7.803525e-16 2.018046e-16 4.775552e-17 1.025798e-17 1.979767e-18
#> [211] 3.386554e-19 5.038594e-20 6.336865e-21 6.424747e-22 4.821385e-23
#> [216] 2.108301e-24
pgpbinom(NULL, pp, va, vb, wt, "DivideFFT")
#> [1] 1.140600e-31 5.463990e-30 1.219337e-28 1.693971e-27 1.660421e-26
#> [6] 1.243246e-25 7.579393e-25 3.972950e-24 1.863590e-23 7.991261e-23
#> [11] 3.162528e-22 1.164739e-21 4.030847e-21 1.320208e-20 4.108715e-20
#> [16] 1.220065e-19 3.474220e-19 9.525615e-19 2.522691e-18 6.476149e-18
#> [21] 1.617225e-17 3.939138e-17 9.381530e-17 2.189455e-16 5.013962e-16
#> [26] 1.127842e-15 2.494586e-15 5.429184e-15 1.163282e-14 2.455979e-14
#> [31] 5.113739e-14 1.050847e-13 2.132829e-13 4.277703e-13 8.479327e-13
#> [36] 1.661494e-12 3.219239e-12 6.169059e-12 1.169675e-11 2.195491e-11
#> [41] 4.081268e-11 7.515909e-11 1.371289e-10 2.478076e-10 4.434415e-10
#> [46] 7.859810e-10 1.380789e-09 2.406013e-09 4.159763e-09 7.132360e-09
#> [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08
#> [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07
#> [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06
#> [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05
#> [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04
#> [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03
#> [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03
#> [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02
#> [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02
#> [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01
#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01
#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01
#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01
#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01
#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01
#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01
#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01
#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01
#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01
#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01
#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01
#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01
#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01
#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01
#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01
#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01
#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00
#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [216] 1.000000e+00
By design, similar to the ordinary DC-FFT algorithm by Biscarri, Zhao & Brunner (2018), its results are identical to the G-DC procedure, if \(n\) and the number of possible observed values is small. Thus, differences can be observed for larger numbers:
set.seed(1)
<- runif(250)
pp1 <- sample(0:50, 250, TRUE)
va1 <- sample(0:50, 250, TRUE)
vb1 <- pp1[1:248]
pp2 <- va1[1:248]
va2 <- vb1[1:248]
vb2
sum(abs(dgpbinom(NULL, pp1, va1, vb1, method = "DivideFFT")
- dgpbinom(NULL, pp1, va1, vb1, method = "Convolve")))
#> [1] 0
sum(abs(dgpbinom(NULL, pp2, va2, vb2, method = "DivideFFT")
- dgpbinom(NULL, pp2, va2, vb2, method = "Convolve")))
#> [1] 0
The reason is that the G-DC-FFT method splits the input probs
, val_p
and val_q
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 \(n\) 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 \(\leq 5.55e\text{-}17\), i.e. smaller values are rounded off to \(0\), if the total number of possible observations is smaller than \(750\), whereas the smallest possible result of the DC algorithm is \(\sim 1e\text{-}323\). This is most likely caused by the used FFTW3 library.
<- dgpbinom(NULL, pp1, va1, vb1, method = "DivideFFT")
d1 <- dgpbinom(NULL, pp1, va1, vb1, method = "Convolve")
d2
min(d1[d1 > 0])
#> [1] 2.839368e-99
min(d2[d2 > 0])
#> [1] 2.839368e-99
The Generalized Discrete Fourier Transformation of the Characteristic Function (G-DFT-CF) approach is requested with method = "Characteristic"
.
set.seed(1)
<- runif(10)
pp <- sample(1:10, 10, TRUE)
wt <- sample(0:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb
dgpbinom(NULL, pp, va, vb, wt, "Characteristic")
#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.837237e-16
#> [26] 6.270062e-16 1.364746e-15 2.935666e-15 6.201829e-15 1.292176e-14
#> [31] 2.657237e-14 5.394193e-14 1.081902e-13 2.144802e-13 4.201557e-13
#> [36] 8.135509e-13 1.557735e-12 2.949809e-12 5.527683e-12 1.025814e-11
#> [41] 1.885776e-11 3.434640e-11 6.196980e-11 1.106787e-10 1.956340e-10
#> [46] 3.425394e-10 5.948077e-10 1.025224e-09 1.753750e-09 2.972596e-09
#> [51] 4.985314e-09 8.275458e-09 1.362195e-08 2.227979e-08 3.622799e-08
#> [56] 5.845270e-08 9.332219e-08 1.473012e-07 2.302797e-07 3.576650e-07
#> [61] 5.529336e-07 8.496291e-07 1.292864e-06 1.943382e-06 2.888042e-06
#> [66] 4.257944e-06 6.248675e-06 9.128095e-06 1.322640e-05 1.893515e-05
#> [71] 2.675612e-05 3.741507e-05 5.199255e-05 7.194684e-05 9.895330e-05
#> [76] 1.347017e-04 1.809349e-04 2.399008e-04 3.150314e-04 4.112231e-04
#> [81] 5.341537e-04 6.888863e-04 8.788234e-04 1.106198e-03 1.374340e-03
#> [86] 1.690272e-03 2.065290e-03 2.511885e-03 3.037800e-03 3.641214e-03
#> [91] 4.311837e-03 5.039293e-03 5.824625e-03 6.686091e-03 7.651765e-03
#> [96] 8.740859e-03 9.945159e-03 1.122411e-02 1.252016e-02 1.378863e-02
#> [101] 1.502576e-02 1.627450e-02 1.759663e-02 1.902489e-02 2.052786e-02
#> [106] 2.201243e-02 2.336424e-02 2.450429e-02 2.543095e-02 2.622065e-02
#> [111] 2.697857e-02 2.776636e-02 2.855637e-02 2.924236e-02 2.969655e-02
#> [116] 2.983772e-02 2.967384e-02 2.929746e-02 2.883252e-02 2.836282e-02
#> [121] 2.788971e-02 2.734351e-02 2.663438e-02 2.570794e-02 2.457639e-02
#> [126] 2.331289e-02 2.201380e-02 2.075053e-02 1.954176e-02 1.836001e-02
#> [131] 1.716200e-02 1.592047e-02 1.464084e-02 1.335803e-02 1.211826e-02
#> [136] 1.095708e-02 9.886542e-03 8.897658e-03 7.972694e-03 7.098018e-03
#> [141] 6.270583e-03 5.496952e-03 4.787457e-03 4.149442e-03 3.583427e-03
#> [146] 3.083701e-03 2.641746e-03 2.249767e-03 1.902455e-03 1.596805e-03
#> [151] 1.330879e-03 1.102475e-03 9.084265e-04 7.447312e-04 6.071616e-04
#> [156] 4.918629e-04 3.956251e-04 3.158260e-04 2.502339e-04 1.968330e-04
#> [161] 1.537458e-04 1.192445e-04 9.179821e-05 7.010494e-05 5.308547e-05
#> [166] 3.984854e-05 2.965115e-05 2.187013e-05 1.598631e-05 1.157497e-05
#> [171] 8.295941e-06 5.881266e-06 4.121776e-06 2.854642e-06 1.953341e-06
#> [176] 1.320224e-06 8.809465e-07 5.799307e-07 3.763587e-07 2.406488e-07
#> [181] 1.515662e-07 9.401686e-08 5.742327e-08 3.451481e-08 2.039831e-08
#> [186] 1.184350e-08 6.751380e-09 3.777327e-09 2.073644e-09 1.116337e-09
#> [191] 5.887148e-10 3.036829e-10 1.529887e-10 7.516829e-11 3.598151e-11
#> [196] 1.676154e-11 7.585978e-12 3.326430e-12 1.407529e-12 5.717381e-13
#> [201] 2.216360e-13 8.149551e-14 2.825209e-14 9.182470e-15 2.781725e-15
#> [206] 7.813323e-16 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [211] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [216] 0.000000e+00
pgpbinom(NULL, pp, va, vb, wt, "Characteristic")
#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.837237e-16
#> [26] 9.107298e-16 2.275475e-15 5.211141e-15 1.141297e-14 2.433473e-14
#> [31] 5.090710e-14 1.048490e-13 2.130392e-13 4.275194e-13 8.476751e-13
#> [36] 1.661226e-12 3.218961e-12 6.168770e-12 1.169645e-11 2.195459e-11
#> [41] 4.081235e-11 7.515875e-11 1.371285e-10 2.478072e-10 4.434412e-10
#> [46] 7.859806e-10 1.380788e-09 2.406013e-09 4.159763e-09 7.132359e-09
#> [51] 1.211767e-08 2.039313e-08 3.401508e-08 5.629487e-08 9.252285e-08
#> [56] 1.509756e-07 2.442977e-07 3.915989e-07 6.218786e-07 9.795436e-07
#> [61] 1.532477e-06 2.382106e-06 3.674970e-06 5.618352e-06 8.506394e-06
#> [66] 1.276434e-05 1.901301e-05 2.814111e-05 4.136751e-05 6.030266e-05
#> [71] 8.705877e-05 1.244738e-04 1.764664e-04 2.484132e-04 3.473665e-04
#> [76] 4.820683e-04 6.630032e-04 9.029039e-04 1.217935e-03 1.629158e-03
#> [81] 2.163312e-03 2.852198e-03 3.731022e-03 4.837220e-03 6.211560e-03
#> [86] 7.901832e-03 9.967122e-03 1.247901e-02 1.551681e-02 1.915802e-02
#> [91] 2.346986e-02 2.850915e-02 3.433378e-02 4.101987e-02 4.867163e-02
#> [96] 5.741249e-02 6.735765e-02 7.858176e-02 9.110192e-02 1.048906e-01
#> [101] 1.199163e-01 1.361908e-01 1.537874e-01 1.728123e-01 1.933402e-01
#> [106] 2.153526e-01 2.387169e-01 2.632211e-01 2.886521e-01 3.148727e-01
#> [111] 3.418513e-01 3.696177e-01 3.981740e-01 4.274164e-01 4.571130e-01
#> [116] 4.869507e-01 5.166245e-01 5.459220e-01 5.747545e-01 6.031173e-01
#> [121] 6.310070e-01 6.583505e-01 6.849849e-01 7.106929e-01 7.352692e-01
#> [126] 7.585821e-01 7.805959e-01 8.013465e-01 8.208882e-01 8.392482e-01
#> [131] 8.564102e-01 8.723307e-01 8.869715e-01 9.003296e-01 9.124478e-01
#> [136] 9.234049e-01 9.332914e-01 9.421891e-01 9.501618e-01 9.572598e-01
#> [141] 9.635304e-01 9.690273e-01 9.738148e-01 9.779642e-01 9.815477e-01
#> [146] 9.846314e-01 9.872731e-01 9.895229e-01 9.914253e-01 9.930221e-01
#> [151] 9.943530e-01 9.954555e-01 9.963639e-01 9.971087e-01 9.977158e-01
#> [156] 9.982077e-01 9.986033e-01 9.989191e-01 9.991694e-01 9.993662e-01
#> [161] 9.995199e-01 9.996392e-01 9.997310e-01 9.998011e-01 9.998542e-01
#> [166] 9.998940e-01 9.999237e-01 9.999455e-01 9.999615e-01 9.999731e-01
#> [171] 9.999814e-01 9.999873e-01 9.999914e-01 9.999943e-01 9.999962e-01
#> [176] 9.999975e-01 9.999984e-01 9.999990e-01 9.999994e-01 9.999996e-01
#> [181] 9.999998e-01 9.999999e-01 9.999999e-01 1.000000e+00 1.000000e+00
#> [186] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [191] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [196] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [201] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [206] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [211] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [216] 1.000000e+00
As can be seen, the G-DFT-CF procedure does not produce probabilities \(\leq 2.2e\text{-}16\), i.e. smaller values are rounded off to 0, most likely due to the used FFTW3 library.
To assess the performance of the exact procedures, we use the microbenchmark
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 7 1800X with 32 GiB of RAM and Windows 10 Education (20H2).
library(microbenchmark)
<- 2500
n set.seed(1)
<- sample(1:50, n, TRUE)
va <- sample(1:50, n, TRUE)
vb
<- function() dgpbinom(NULL, runif(n), va, vb, method = "DivideFFT")
f1 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Convolve")
f2 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Characteristic")
f3
microbenchmark(f1(), f2(), f3(), times = 51)
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> f1() 75.7508 78.2083 83.64722 79.8600 84.76995 200.4473 51
#> f2() 108.6950 111.3314 114.78413 113.5170 117.47335 126.8446 51
#> f3() 580.2455 646.8719 665.34329 663.1248 680.58515 758.0827 51
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 \(n\) (and \(m\)).