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
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 \(n \leq 750\). 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 \(n \leq 750\). 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.
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 \(\leq 2.22e\text{-}16\), 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)
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 \(\leq 5.55e\text{-}17\), because it does not rely on the FFTW3 library. Furthermore, it yields the same results as the DC method.
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)
f1 <- function() dpbinom(NULL, runif(5000), method = "DivideFFT")
f2 <- function() dpbinom(NULL, runif(5000), method = "Convolve")
f3 <- function() dpbinom(NULL, runif(5000), method = "Recursive")
f4 <- function() dpbinom(NULL, runif(5000), method = "Characteristic")
microbenchmark(f1(), f2(), f3(), f4())
#> Unit: milliseconds
#> expr min lq mean median uq max neval cld
#> f1() 13.7388 13.98920 14.24036 14.07430 14.18050 18.6597 100 a
#> f2() 20.8997 21.40085 21.51780 21.52930 21.61505 22.7991 100 b
#> f3() 28.3662 28.80610 29.00823 28.96065 29.07945 31.8566 100 c
#> f4() 66.7758 67.37795 67.66081 67.63415 67.92170 69.4124 100 d
Clearly, the DC-FFT procedure is the fastest, followed by the DC method, which needs roughly 1.5 times as much time here. RF and DFT-CF are almost two and five times, respectively, slower than DC-FFT, respectively (although RF has some advantages in precision, as stated before).
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
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 \(\leq 5.55e\text{-}17\), 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.
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 \(\leq 2.2e\text{-}16\), 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)
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())
#> Unit: milliseconds
#> expr min lq mean median uq max neval cld
#> f1() 55.4850 58.28000 63.65059 58.79150 60.58415 333.5652 100 a
#> f2() 89.4355 93.52265 95.50789 94.32915 95.95510 195.0174 100 b
#> f3() 511.4210 524.34290 528.88073 528.69490 532.80085 563.9598 100 c
Clearly, the G-DC-FFT procedure is the fastest one. If \(n\) and \(m\) are large enough, it outperforms both the G-DC and G-DFT-CF approaches. The latter one needs almost six times as much time as G-DC. Generally, the computational speed advantage of the G-DC-FFT procedure increases with larger \(n\) (and \(m\)).