Exact Procedures

Ordinary Poisson Binomial Distribution

Direct Convolution

The Direct Convolution (DC) approach is requested with method = "Convolve".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

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

Divide & Conquer FFT Tree Convolution

The Divide & Conquer FFT Tree Convolution (DC-FFT) approach is requested with method = "DivideFFT".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

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 ≤ 750. Thus, differences can be observed for larger n > 750:

set.seed(1)
pp1 <- runif(751)
pp2 <- pp1[1:750]

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 ≤ 750. In addition, the DC-FFT procedure does not produce probabilities ≤ 5.55e-17, i.e. smaller values are rounded off to 0, if n > 750, whereas the smallest possible result of the DC algorithm is ∼ 1e-323. This is most likely caused by the used FFTW3 library.

set.seed(1)
pp1 <- runif(751)

d1 <- dpbinom(NULL, pp1, method = "DivideFFT")
d2 <- dpbinom(NULL, pp1, method = "Convolve")

min(d1[d1 > 0])
#> [1] 1.635357e-321
min(d2[d2 > 0])
#> [1] 1.635357e-321

Discrete Fourier Transformation of the Characteristic Function

The Discrete Fourier Transformation of the Characteristic Function (DFT-CF) approach is requested with method = "Characteristic".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

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 ≤ 2.22e-16, i.e. smaller values are rounded off to 0, most likely due to the used FFTW3 library.

Recursive Formula

The Recursive Formula (RF) approach is requested with method = "Recursive".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)

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 ≤ 5.55e-17, because it does not rely on the FFTW3 library. Furthermore, it yields the same results as the DC method.

set.seed(1)
pp <- runif(1000)
wt <- sample(1:10, 1000, TRUE)

sum(abs(dpbinom(NULL, pp, wt, "Convolve") - dpbinom(NULL, pp, wt, "Recursive")))
#> [1] 0

Processing Speed Comparisons

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 9 5900X with 64 GiB of RAM and Windows 10 Education (22H2).

library(microbenchmark)
set.seed(1)

f1 <- function() dpbinom(NULL, runif(6000), method = "DivideFFT")
f2 <- function() dpbinom(NULL, runif(6000), method = "Convolve")
f3 <- function() dpbinom(NULL, runif(6000), method = "Recursive")
f4 <- function() dpbinom(NULL, runif(6000), method = "Characteristic")

microbenchmark(f1(), f2(), f3(), f4(), times = 51)
#> Unit: milliseconds
#>  expr       min        lq      mean   median        uq       max neval
#>  f1()  55.33225  55.48742  55.85213  55.5676  55.74928  59.60287    51
#>  f2() 126.77123 127.29649 127.86562 127.6447 128.04243 132.08101    51
#>  f3() 167.96266 168.48935 169.02589 168.6778 169.31518 171.60450    51
#>  f4()  91.62972  91.74005  92.68508  93.1054  93.20073  94.17543    51

Clearly, the DC-FFT procedure is the fastest, followed by DC, RF and DFT-CF methods.

Generalized Poisson Binomial Distribution

Generalized Direct Convolution

The Generalized Direct Convolution (G-DC) approach is requested with method = "Convolve".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)

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

Generalized Divide & Conquer FFT Tree Convolution

The Generalized Divide & Conquer FFT Tree Convolution (G-DC-FFT) approach is requested with method = "DivideFFT".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)

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)
pp1 <- runif(250)
va1 <- sample(0:50, 250, TRUE)
vb1 <- sample(0:50, 250, TRUE)
pp2 <- pp1[1:248]
va2 <- va1[1:248]
vb2 <- vb1[1:248]

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 ≤ 5.55e-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 ∼ 1e-323. This is most likely caused by the used FFTW3 library.

d1 <- dgpbinom(NULL, pp1, va1, vb1, method = "DivideFFT")
d2 <- dgpbinom(NULL, pp1, va1, vb1, method = "Convolve")

min(d1[d1 > 0])
#> [1] 2.839368e-99
min(d2[d2 > 0])
#> [1] 2.839368e-99

Generalized Discrete Fourier Transformation of the Characteristic Function

The Generalized Discrete Fourier Transformation of the Characteristic Function (G-DFT-CF) approach is requested with method = "Characteristic".

set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)

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 ≤ 2.2e-16, i.e. smaller values are rounded off to 0, most likely due to the used FFTW3 library.

Processing Speed Comparisons

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 9 5900X with 64 GiB of RAM and Windows 10 Education (22H2).

library(microbenchmark)
n <- 2500
set.seed(1)
va <- sample(1:50, n, TRUE)
vb <- sample(1:50, n, TRUE)

f1 <- function() dgpbinom(NULL, runif(n), va, vb, method = "DivideFFT")
f2 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Convolve")
f3 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Characteristic")

microbenchmark(f1(), f2(), f3(), times = 51)
#> Unit: milliseconds
#>  expr      min       lq     mean   median       uq      max neval
#>  f1() 326.4028 326.8275 331.2380 327.2105 328.7831 506.4497    51
#>  f2() 776.7544 780.1921 782.0670 781.7714 784.0173 788.9935    51
#>  f3() 613.5195 621.4151 624.7435 623.1188 625.0551 655.8785    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).

- Ordinary Poisson Binomial Distribution
- Generalized Poisson Binomial Distribution