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() 13.4042 13.65425 13.96337 13.77990 13.91975 17.1088 100 a
#> f2() 20.6392 20.93765 21.15733 21.10240 21.32325 22.6958 100 b
#> f3() 27.5966 27.97090 28.28354 28.17785 28.43690 31.1628 100 c
#> f4() 65.7154 66.30750 66.72891 66.55035 66.91000 70.2184 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() 56.8388 57.76210 63.44748 58.63540 60.26615 338.7246 100 a
#> f2() 91.1376 91.98105 93.34477 93.04655 94.50995 97.0263 100 b
#> f3() 509.4249 517.44770 521.62463 519.51470 522.53045 614.7971 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 \(m\)).