The Poisson Approximation (DC) approach is requested with
method = "Poisson". It is based on a Poisson distribution,
whose parameter is the sum of the probabilities of success.
set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
dpbinom(NULL, pp, wt, "Poisson")
#> [1] 2.263593e-16 8.154460e-15 1.468798e-13 1.763753e-12 1.588454e-11
#> [6] 1.144462e-10 6.871428e-10 3.536273e-09 1.592402e-08 6.373926e-08
#> [11] 2.296169e-07 7.519830e-07 2.257479e-06 6.255718e-06 1.609704e-05
#> [16] 3.865908e-05 8.704191e-05 1.844490e-04 3.691482e-04 6.999128e-04
#> [21] 1.260697e-03 2.162661e-03 3.541299e-03 5.546660e-03 8.325631e-03
#> [26] 1.199704e-02 1.662255e-02 2.217842e-02 2.853445e-02 3.544609e-02
#> [31] 4.256414e-02 4.946284e-02 5.568342e-02 6.078674e-02 6.440607e-02
#> [36] 6.629115e-02 6.633610e-02 6.458699e-02 6.122916e-02 5.655755e-02
#> [41] 5.093630e-02 4.475488e-02 3.838734e-02 3.216003e-02 2.633059e-02
#> [46] 2.107875e-02 1.650760e-02 1.265269e-02 9.495953e-03 6.981348e-03
#> [51] 5.029979e-03 3.552981e-03 2.461424e-03 1.673044e-03 1.116119e-03
#> [56] 7.310458e-04 4.702766e-04 2.972182e-04 1.846053e-04 1.127169e-04
#> [61] 6.767601e-05 9.288901e-05
ppbinom(NULL, pp, wt, "Poisson")
#> [1] 2.263593e-16 8.380820e-15 1.552606e-13 1.919013e-12 1.780355e-11
#> [6] 1.322498e-10 8.193925e-10 4.355666e-09 2.027968e-08 8.401894e-08
#> [11] 3.136359e-07 1.065619e-06 3.323097e-06 9.578815e-06 2.567585e-05
#> [16] 6.433494e-05 1.513768e-04 3.358259e-04 7.049740e-04 1.404887e-03
#> [21] 2.665584e-03 4.828245e-03 8.369543e-03 1.391620e-02 2.224184e-02
#> [26] 3.423887e-02 5.086142e-02 7.303984e-02 1.015743e-01 1.370204e-01
#> [31] 1.795845e-01 2.290474e-01 2.847308e-01 3.455175e-01 4.099236e-01
#> [36] 4.762147e-01 5.425508e-01 6.071378e-01 6.683670e-01 7.249245e-01
#> [41] 7.758608e-01 8.206157e-01 8.590031e-01 8.911631e-01 9.174937e-01
#> [46] 9.385724e-01 9.550800e-01 9.677327e-01 9.772287e-01 9.842100e-01
#> [51] 9.892400e-01 9.927930e-01 9.952544e-01 9.969275e-01 9.980436e-01
#> [56] 9.987746e-01 9.992449e-01 9.995421e-01 9.997267e-01 9.998394e-01
#> [61] 9.999071e-01 1.000000e+00A comparison with exact computation shows that the approximation quality of the PA procedure increases with smaller probabilities of success. The reason is that the Poisson Binomial distribution approaches a Poisson distribution when the probabilities are very small.
set.seed(1)
# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "Poisson")
#> [1] 0.0000150619 0.0001672374 0.0009284471 0.0034362888 0.0095385726
#> [6] 0.0211820073 0.0391985129 0.0621763578 0.0862956727 0.1064633767
#> [11] 0.1182099310 0.1193204840 0.1104046811 0.0942969970 0.0747865595
#> [16] 0.0553587178 0.0384166744 0.0250913815 0.0154776776 0.0090449448
#> [21] 0.0101904160
dpbinom(NULL, pp)
#> [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#> [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "Poisson") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -9.555e-02 1.506e-05 9.437e-03 0.000e+00 2.407e-02 4.379e-02
# U(0, 0.01) random probabilities of success
pp <- runif(20, 0, 0.01)
dpbinom(NULL, pp, method = "Poisson")
#> [1] 9.095763e-01 8.620639e-02 4.085167e-03 1.290592e-04 3.057942e-06
#> [6] 5.796418e-08 9.156063e-10 1.239684e-11 1.468661e-13 1.546605e-15
#> [11] 1.465817e-17 1.262953e-19 9.974852e-22 7.272161e-24 4.923067e-26
#> [16] 3.110605e-28 1.842575e-30 1.027251e-32 5.408845e-35 2.698058e-37
#> [21] 1.284357e-39
dpbinom(NULL, pp)
#> [1] 9.093051e-01 8.672423e-02 3.861917e-03 1.066765e-04 2.048094e-06
#> [6] 2.902198e-08 3.145829e-10 2.667571e-12 1.794592e-14 9.656258e-17
#> [11] 4.170114e-19 1.444465e-21 3.994453e-24 8.738444e-27 1.490372e-29
#> [16] 1.938487e-32 1.859939e-35 1.249654e-38 5.381374e-42 1.245845e-45
#> [21] 9.511846e-50
summary(dpbinom(NULL, pp, method = "Poisson") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -5.178e-04 0.000e+00 0.000e+00 0.000e+00 6.000e-10 2.712e-04The Arithmetic Mean Binomial Approximation (AMBA) approach
is requested with method = "Mean". It is based on a
Binomial distribution, whose parameter is the arithmetic mean of the
probabilities of success.
set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
mean(rep(pp, wt))
#> [1] 0.5905641
dpbinom(NULL, pp, wt, "Mean")
#> [1] 2.204668e-24 1.939788e-22 8.393759e-21 2.381049e-19 4.979863e-18
#> [6] 8.188480e-17 1.102354e-15 1.249300e-14 1.216331e-13 1.033156e-12
#> [11] 7.749086e-12 5.182139e-11 3.114432e-10 1.693217e-09 8.373498e-09
#> [16] 3.784379e-08 1.569327e-07 5.991812e-07 2.112610e-06 6.896287e-06
#> [21] 2.088890e-05 5.882491e-05 1.542694e-04 3.773093e-04 8.616897e-04
#> [26] 1.839474e-03 3.673702e-03 6.868933e-03 1.203071e-02 1.974641e-02
#> [31] 3.038072e-02 4.382068e-02 5.925587e-02 7.510979e-02 8.921887e-02
#> [36] 9.927353e-02 1.034154e-01 1.007871e-01 9.181496e-02 7.810121e-02
#> [41] 6.195859e-02 4.577391e-02 3.143980e-02 2.003761e-02 1.182352e-02
#> [46] 6.442647e-03 3.232269e-03 1.487928e-03 6.259647e-04 2.395401e-04
#> [51] 8.292214e-05 2.579729e-05 7.155695e-06 1.752667e-06 3.745215e-07
#> [56] 6.875325e-08 1.062521e-08 1.344354e-09 1.337294e-10 9.807924e-12
#> [61] 4.715599e-13 1.115034e-14
ppbinom(NULL, pp, wt, "Mean")
#> [1] 2.204668e-24 1.961834e-22 8.589942e-21 2.466948e-19 5.226557e-18
#> [6] 8.711136e-17 1.189465e-15 1.368247e-14 1.353155e-13 1.168472e-12
#> [11] 8.917558e-12 6.073895e-11 3.721822e-10 2.065399e-09 1.043890e-08
#> [16] 4.828268e-08 2.052154e-07 8.043966e-07 2.917007e-06 9.813294e-06
#> [21] 3.070220e-05 8.952711e-05 2.437965e-04 6.211058e-04 1.482796e-03
#> [26] 3.322270e-03 6.995972e-03 1.386490e-02 2.589561e-02 4.564203e-02
#> [31] 7.602274e-02 1.198434e-01 1.790993e-01 2.542091e-01 3.434279e-01
#> [36] 4.427015e-01 5.461169e-01 6.469040e-01 7.387189e-01 8.168201e-01
#> [41] 8.787787e-01 9.245526e-01 9.559924e-01 9.760300e-01 9.878536e-01
#> [46] 9.942962e-01 9.975285e-01 9.990164e-01 9.996424e-01 9.998819e-01
#> [51] 9.999648e-01 9.999906e-01 9.999978e-01 9.999995e-01 9.999999e-01
#> [56] 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00 1.000000e+00
#> [61] 1.000000e+00 1.000000e+00A comparison with exact computation shows that the approximation quality of the AMBA procedure increases when the probabilities of success are closer to each other. The reason is that, although the expectation remains unchanged, the distribution’s variance becomes smaller the less the probabilities differ. Since this variance is minimized by equal probabilities (but still underestimated), the AMBA method is best suited for situations with very similar probabilities of success.
set.seed(1)
# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "Mean")
#> [1] 9.203176e-08 2.297178e-06 2.723611e-05 2.039497e-04 1.081780e-03
#> [6] 4.320318e-03 1.347977e-02 3.364646e-02 6.823695e-02 1.135495e-01
#> [11] 1.558851e-01 1.768638e-01 1.655492e-01 1.271454e-01 7.934094e-02
#> [16] 3.960811e-02 1.544760e-02 4.536271e-03 9.435709e-04 1.239589e-04
#> [21] 7.735255e-06
dpbinom(NULL, pp)
#> [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#> [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "Mean") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -3.801e-02 2.290e-06 6.360e-04 0.000e+00 8.837e-03 1.662e-02
# U(0.3, 0.5) random probabilities of success
pp <- runif(20, 0.3, 0.5)
dpbinom(NULL, pp, method = "Mean")
#> [1] 4.348271e-05 5.672598e-04 3.515127e-03 1.375712e-02 3.813748e-02
#> [6] 7.960444e-02 1.298114e-01 1.693472e-01 1.795010e-01 1.561137e-01
#> [11] 1.120132e-01 6.642197e-02 3.249439e-02 1.304339e-02 4.253984e-03
#> [16] 1.109919e-03 2.262438e-04 3.472347e-05 3.774915e-06 2.591904e-07
#> [21] 8.453263e-09
dpbinom(NULL, pp)
#> [1] 4.015121e-05 5.344728e-04 3.370391e-03 1.338738e-02 3.756479e-02
#> [6] 7.915145e-02 1.299445e-01 1.702071e-01 1.806555e-01 1.569062e-01
#> [11] 1.121277e-01 6.604356e-02 3.200604e-02 1.269255e-02 4.078679e-03
#> [16] 1.045709e-03 2.088926e-04 3.133484e-05 3.320483e-06 2.216332e-07
#> [21] 7.008006e-09
summary(dpbinom(NULL, pp, method = "Mean") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -1.155e-03 1.400e-09 1.735e-05 0.000e+00 3.508e-04 5.727e-04
# U(0.39, 0.41) random probabilities of success
pp <- runif(20, 0.39, 0.41)
dpbinom(NULL, pp, method = "Mean")
#> [1] 3.638616e-05 4.854405e-04 3.076305e-03 1.231262e-02 3.490673e-02
#> [6] 7.451247e-02 1.242621e-01 1.657824e-01 1.797056e-01 1.598344e-01
#> [11] 1.172824e-01 7.112295e-02 3.558286e-02 1.460687e-02 4.871885e-03
#> [16] 1.299951e-03 2.709859e-04 4.253314e-05 4.728746e-06 3.320414e-07
#> [21] 1.107470e-08
dpbinom(NULL, pp)
#> [1] 3.636149e-05 4.851935e-04 3.075192e-03 1.230970e-02 3.490204e-02
#> [6] 7.450845e-02 1.242626e-01 1.657891e-01 1.797153e-01 1.598415e-01
#> [11] 1.172840e-01 7.112011e-02 3.557873e-02 1.460374e-02 4.870251e-03
#> [16] 1.299328e-03 2.708111e-04 4.249771e-05 4.723809e-06 3.316172e-07
#> [21] 1.105772e-08
summary(dpbinom(NULL, pp, method = "Mean") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -9.641e-06 1.700e-11 1.747e-07 0.000e+00 2.844e-06 4.689e-06The Geometric Mean Binomial Approximation (Variant A)
(GMBA-A) approach is requested with method = "GeoMean". It
is based on a Binomial distribution, whose parameter is the geometric
mean of the probabilities of success: $$\hat{p} = \sqrt[n]{p_1 \cdot ... \cdot
p_n}$$
set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
prod(rep(pp, wt))^(1/sum(wt))
#> [1] 0.4669916
dpbinom(NULL, pp, wt, "GeoMean")
#> [1] 2.141782e-17 1.144670e-15 3.008684e-14 5.184208e-13 6.586057e-12
#> [6] 6.578175e-11 5.379195e-10 3.703028e-09 2.189958e-08 1.129911e-07
#> [11] 5.147813e-07 2.091103e-06 7.633772e-06 2.520966e-05 7.572779e-05
#> [16] 2.078916e-04 5.236606e-04 1.214475e-03 2.601021e-03 5.157435e-03
#> [21] 9.489168e-03 1.623184e-02 2.585712e-02 3.841422e-02 5.328923e-02
#> [26] 6.909972e-02 8.382634e-02 9.520502e-02 1.012875e-01 1.009827e-01
#> [31] 9.437363e-02 8.268481e-02 6.791600e-02 5.229152e-02 3.772988e-02
#> [36] 2.550094e-02 1.613623e-02 9.552467e-03 5.285892e-03 2.731219e-03
#> [41] 1.316117e-03 5.906156e-04 2.464113e-04 9.539397e-05 3.419132e-05
#> [46] 1.131690e-05 3.448772e-06 9.643463e-07 2.464308e-07 5.728188e-08
#> [51] 1.204491e-08 2.276152e-09 3.835067e-10 5.705775e-11 7.406038e-12
#> [56] 8.258409e-13 7.752374e-14 5.958061e-15 3.600079e-16 1.603823e-17
#> [61] 4.683928e-19 6.727527e-21
ppbinom(NULL, pp, wt, "GeoMean")
#> [1] 2.141782e-17 1.166088e-15 3.125293e-14 5.496737e-13 7.135731e-12
#> [6] 7.291748e-11 6.108370e-10 4.313865e-09 2.621345e-08 1.392046e-07
#> [11] 6.539859e-07 2.745088e-06 1.037886e-05 3.558852e-05 1.113163e-04
#> [16] 3.192079e-04 8.428685e-04 2.057343e-03 4.658364e-03 9.815799e-03
#> [21] 1.930497e-02 3.553681e-02 6.139393e-02 9.980815e-02 1.530974e-01
#> [26] 2.221971e-01 3.060234e-01 4.012285e-01 5.025160e-01 6.034986e-01
#> [31] 6.978723e-01 7.805571e-01 8.484731e-01 9.007646e-01 9.384945e-01
#> [36] 9.639954e-01 9.801316e-01 9.896841e-01 9.949700e-01 9.977012e-01
#> [41] 9.990173e-01 9.996080e-01 9.998544e-01 9.999498e-01 9.999840e-01
#> [46] 9.999953e-01 9.999987e-01 9.999997e-01 9.999999e-01 1.000000e+00
#> [51] 1.000000e+00 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+00It is known that the geometric mean of the probabilities of success is always smaller than their arithmetic mean. Thus, we get a stochastically smaller binomial distribution. A comparison with exact computation shows that the approximation quality of the GMBA-A procedure increases when the probabilities of success are closer to each other:
set.seed(1)
# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "GeoMean")
#> [1] 4.557123e-06 7.742984e-05 6.249130e-04 3.185359e-03 1.150098e-02
#> [6] 3.126602e-02 6.640491e-02 1.128282e-01 1.557610e-01 1.764351e-01
#> [11] 1.648790e-01 1.273387e-01 8.113517e-02 4.241734e-02 1.801777e-02
#> [16] 6.122779e-03 1.625497e-03 3.249263e-04 4.600672e-05 4.114199e-06
#> [21] 1.747603e-07
dpbinom(NULL, pp)
#> [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#> [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "GeoMean") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.11151 -0.01493 0.00000 0.00000 0.01140 0.10279
# U(0.4, 0.6) random probabilities of success
pp <- runif(20, 0.4, 0.6)
dpbinom(NULL, pp, method = "GeoMean")
#> [1] 1.317886e-06 2.551200e-05 2.345875e-04 1.362363e-03 5.604265e-03
#> [6] 1.735823e-02 4.200318e-02 8.131092e-02 1.278907e-01 1.650496e-01
#> [11] 1.757292e-01 1.546280e-01 1.122499e-01 6.686047e-02 3.235759e-02
#> [16] 1.252775e-02 3.789307e-03 8.629936e-04 1.392173e-04 1.418425e-05
#> [21] 6.864565e-07
dpbinom(NULL, pp)
#> [1] 1.046635e-06 2.098187e-05 1.993006e-04 1.192678e-03 5.043114e-03
#> [6] 1.601621e-02 3.964022e-02 7.829406e-02 1.253351e-01 1.642218e-01
#> [11] 1.770816e-01 1.574210e-01 1.151700e-01 6.896627e-02 3.347297e-02
#> [16] 1.296524e-02 3.913788e-03 8.873960e-04 1.421738e-04 1.435144e-05
#> [21] 6.864565e-07
summary(dpbinom(NULL, pp, method = "GeoMean") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.0029201 -0.0004375 0.0000000 0.0000000 0.0005612 0.0030169
# U(0.49, 0.51) random probabilities of success
pp <- runif(20, 0.49, 0.51)
dpbinom(NULL, pp, method = "GeoMean")
#> [1] 9.491177e-07 1.899145e-05 1.805052e-04 1.083550e-03 4.607292e-03
#> [6] 1.475040e-02 3.689366e-02 7.382266e-02 1.200193e-01 1.601024e-01
#> [11] 1.761970e-01 1.602558e-01 1.202494e-01 7.403508e-02 3.703527e-02
#> [16] 1.482120e-02 4.633845e-03 1.090839e-03 1.818935e-04 1.915586e-05
#> [21] 9.582517e-07
dpbinom(NULL, pp)
#> [1] 9.472606e-07 1.895984e-05 1.802539e-04 1.082315e-03 4.603107e-03
#> [6] 1.474011e-02 3.687497e-02 7.379784e-02 1.199969e-01 1.600932e-01
#> [11] 1.762060e-01 1.602781e-01 1.202742e-01 7.405383e-02 3.704562e-02
#> [16] 1.482542e-02 4.635093e-03 1.091093e-03 1.819256e-04 1.915775e-05
#> [21] 9.582517e-07
summary(dpbinom(NULL, pp, method = "GeoMean") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -2.485e-05 -4.219e-06 0.000e+00 0.000e+00 4.185e-06 2.482e-05The Geometric Mean Binomial Approximation (Variant B)
(GMBA-B) approach is requested with
method = "GeoMeanCounter". It is based on a Binomial
distribution, whose parameter is 1 minus the geometric mean of the
probabilities of failure: $$\hat{p} = 1 - \sqrt[n]{(1 - p_1) \cdot ... \cdot
(1 - p_n)}$$
set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
1 - prod(1 - rep(pp, wt))^(1/sum(wt))
#> [1] 0.7275426
dpbinom(NULL, pp, wt, "GeoMeanCounter")
#> [1] 3.574462e-35 5.822379e-33 4.664248e-31 2.449471e-29 9.484189e-28
#> [6] 2.887121e-26 7.195512e-25 1.509685e-23 2.721134e-22 4.279009e-21
#> [11] 5.941642e-20 7.356037e-19 8.184508e-18 8.237686e-17 7.541858e-16
#> [16] 6.310225e-15 4.844429e-14 3.424255e-13 2.235148e-12 1.350769e-11
#> [21] 7.574609e-11 3.948978e-10 1.917264e-09 8.681177e-09 3.670379e-08
#> [26] 1.450549e-07 5.363170e-07 1.856461e-06 6.019586e-06 1.829121e-05
#> [31] 5.209921e-05 1.391205e-04 3.482749e-04 8.172712e-04 1.797236e-03
#> [36] 3.702208e-03 7.139892e-03 1.288219e-02 2.172588e-02 3.421374e-02
#> [41] 5.024851e-02 6.872559e-02 8.738947e-02 1.031108e-01 1.126377e-01
#> [46] 1.136267e-01 1.055364e-01 8.994057e-02 7.004907e-02 4.962603e-02
#> [51] 3.180393e-02 1.831737e-02 9.406320e-03 4.265268e-03 1.687339e-03
#> [56] 5.734528e-04 1.640669e-04 3.843049e-05 7.077304e-06 9.609416e-07
#> [61] 8.553338e-08 3.744258e-09
ppbinom(NULL, pp, wt, "GeoMeanCounter")
#> [1] 3.574462e-35 5.858123e-33 4.722829e-31 2.496699e-29 9.733859e-28
#> [6] 2.984460e-26 7.493958e-25 1.584624e-23 2.879597e-22 4.566969e-21
#> [11] 6.398339e-20 7.995871e-19 8.984095e-18 9.136095e-17 8.455467e-16
#> [16] 7.155772e-15 5.560007e-14 3.980256e-13 2.633173e-12 1.614086e-11
#> [21] 9.188695e-11 4.867847e-10 2.404049e-09 1.108523e-08 4.778901e-08
#> [26] 1.928440e-07 7.291610e-07 2.585622e-06 8.605207e-06 2.689642e-05
#> [31] 7.899562e-05 2.181161e-04 5.663910e-04 1.383662e-03 3.180899e-03
#> [36] 6.883107e-03 1.402300e-02 2.690519e-02 4.863107e-02 8.284481e-02
#> [41] 1.330933e-01 2.018189e-01 2.892084e-01 3.923192e-01 5.049569e-01
#> [46] 6.185836e-01 7.241200e-01 8.140606e-01 8.841097e-01 9.337357e-01
#> [51] 9.655396e-01 9.838570e-01 9.932633e-01 9.975286e-01 9.992159e-01
#> [56] 9.997894e-01 9.999534e-01 9.999919e-01 9.999989e-01 9.999999e-01
#> [61] 1.000000e+00 1.000000e+00It is known that the geometric mean of the probabilities of failure is always smaller than their arithmetic mean. As a result, 1 minus the geometric mean is larger than 1 minus the arithmetic mean. Thus, we get a stochastically larger binomial distribution. A comparison with exact computation shows that the approximation quality of the GMBA-B procedure again increases when the probabilities of success are closer to each other:
set.seed(1)
# U(0, 1) random probabilities of success
pp <- runif(20)
dpbinom(NULL, pp, method = "GeoMeanCounter")
#> [1] 4.401037e-11 2.019854e-09 4.403304e-08 6.062685e-07 5.912743e-06
#> [6] 4.341843e-05 2.490859e-04 1.143179e-03 4.262876e-03 1.304297e-02
#> [11] 3.292337e-02 6.868258e-02 1.182069e-01 1.669263e-01 1.915269e-01
#> [16] 1.758024e-01 1.260695e-01 6.807004e-02 2.603394e-02 6.288561e-03
#> [21] 7.215333e-04
dpbinom(NULL, pp)
#> [1] 4.401037e-11 7.873212e-09 3.624610e-07 7.952504e-06 1.014602e-04
#> [6] 8.311558e-04 4.642470e-03 1.838525e-02 5.297347e-02 1.129135e-01
#> [11] 1.798080e-01 2.148719e-01 1.926468e-01 1.289706e-01 6.384266e-02
#> [16] 2.299142e-02 5.871700e-03 1.021142e-03 1.129421e-04 6.977021e-06
#> [21] 1.747603e-07
summary(dpbinom(NULL, pp, method = "GeoMeanCounter") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -1.469e-01 -1.724e-02 -3.200e-07 0.000e+00 2.592e-02 1.528e-01
# U(0.4, 0.6) random probabilities of success
pp <- runif(20, 0.4, 0.6)
dpbinom(NULL, pp, method = "GeoMeanCounter")
#> [1] 1.046635e-06 2.073844e-05 1.951870e-04 1.160254e-03 4.885321e-03
#> [6] 1.548796e-02 3.836059e-02 7.600922e-02 1.223688e-01 1.616443e-01
#> [11] 1.761588e-01 1.586582e-01 1.178895e-01 7.187414e-02 3.560358e-02
#> [16] 1.410928e-02 4.368234e-03 1.018282e-03 1.681387e-04 1.753458e-05
#> [21] 8.685930e-07
dpbinom(NULL, pp)
#> [1] 1.046635e-06 2.098187e-05 1.993006e-04 1.192678e-03 5.043114e-03
#> [6] 1.601621e-02 3.964022e-02 7.829406e-02 1.253351e-01 1.642218e-01
#> [11] 1.770816e-01 1.574210e-01 1.151700e-01 6.896627e-02 3.347297e-02
#> [16] 1.296524e-02 3.913788e-03 8.873960e-04 1.421738e-04 1.435144e-05
#> [21] 6.864565e-07
summary(dpbinom(NULL, pp, method = "GeoMeanCounter") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.0029663 -0.0005283 0.0000000 0.0000000 0.0004544 0.0029079
# U(0.49, 0.51) random probabilities of success
pp <- runif(20, 0.49, 0.51)
dpbinom(NULL, pp, method = "GeoMeanCounter")
#> [1] 9.472606e-07 1.895800e-05 1.802225e-04 1.082065e-03 4.601880e-03
#> [6] 1.473596e-02 3.686475e-02 7.377926e-02 1.199722e-01 1.600709e-01
#> [11] 1.761969e-01 1.602871e-01 1.202964e-01 7.407854e-02 3.706427e-02
#> [16] 1.483571e-02 4.639289e-03 1.092334e-03 1.821786e-04 1.918963e-05
#> [21] 9.601293e-07
dpbinom(NULL, pp)
#> [1] 9.472606e-07 1.895984e-05 1.802539e-04 1.082315e-03 4.603107e-03
#> [6] 1.474011e-02 3.687497e-02 7.379784e-02 1.199969e-01 1.600932e-01
#> [11] 1.762060e-01 1.602781e-01 1.202742e-01 7.405383e-02 3.704562e-02
#> [16] 1.482542e-02 4.635093e-03 1.091093e-03 1.819256e-04 1.915775e-05
#> [21] 9.582517e-07
summary(dpbinom(NULL, pp, method = "GeoMeanCounter") - dpbinom(NULL, pp))
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -2.467e-05 -4.159e-06 0.000e+00 0.000e+00 4.196e-06 2.470e-05The Normal Approximation (NA) approach is requested with
method = "Normal". It is based on a Normal distribution,
whose parameters are derived from the theoretical mean and variance of
the input probabilities of success.
set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
dpbinom(NULL, pp, wt, "Normal")
#> [1] 2.552770e-32 1.207834e-30 5.219650e-29 2.022022e-27 7.021785e-26
#> [6] 2.185917e-24 6.100302e-23 1.526188e-21 3.423032e-20 6.882841e-19
#> [11] 1.240755e-17 2.005270e-16 2.905604e-15 3.774712e-14 4.396661e-13
#> [16] 4.591569e-12 4.299381e-11 3.609645e-10 2.717342e-09 1.834224e-08
#> [21] 1.110185e-07 6.025326e-07 2.932337e-06 1.279682e-05 5.007841e-05
#> [26] 1.757379e-04 5.530339e-04 1.560683e-03 3.949650e-03 8.963710e-03
#> [31] 1.824341e-02 3.329786e-02 5.450317e-02 8.000636e-02 1.053238e-01
#> [36] 1.243451e-01 1.316535e-01 1.250080e-01 1.064497e-01 8.129267e-02
#> [41] 5.567468e-02 3.419491e-02 1.883477e-02 9.303614e-03 4.121280e-03
#> [46] 1.637186e-03 5.832371e-04 1.863241e-04 5.337829e-05 1.371282e-05
#> [51] 3.159002e-06 6.525712e-07 1.208800e-07 2.007813e-08 2.990389e-09
#> [56] 3.993563e-10 4.782059e-11 5.134327e-12 4.942641e-13 4.266130e-14
#> [61] 3.301422e-15 2.441468e-16
ppbinom(NULL, pp, wt, "Normal")
#> [1] 2.552770e-32 1.233362e-30 5.342987e-29 2.075452e-27 7.229330e-26
#> [6] 2.258210e-24 6.326123e-23 1.589449e-21 3.581977e-20 7.241039e-19
#> [11] 1.313165e-17 2.136587e-16 3.119262e-15 4.086639e-14 4.805325e-13
#> [16] 5.072102e-12 4.806591e-11 4.090305e-10 3.126373e-09 2.146861e-08
#> [21] 1.324871e-07 7.350197e-07 3.667357e-06 1.646417e-05 6.654258e-05
#> [26] 2.422805e-04 7.953144e-04 2.355997e-03 6.305647e-03 1.526936e-02
#> [31] 3.351276e-02 6.681062e-02 1.213138e-01 2.013201e-01 3.066439e-01
#> [36] 4.309891e-01 5.626426e-01 6.876506e-01 7.941003e-01 8.753930e-01
#> [41] 9.310676e-01 9.652625e-01 9.840973e-01 9.934009e-01 9.975222e-01
#> [46] 9.991594e-01 9.997426e-01 9.999290e-01 9.999823e-01 9.999960e-01
#> [51] 9.999992e-01 9.999999e-01 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+00A comparison with exact computation shows that the approximation quality of the NA procedure increases with larger numbers of probabilities of success:
set.seed(1)
# 10 random probabilities of success
pp <- runif(10)
dpn <- dpbinom(NULL, pp, method = "Normal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.0053305 -0.0010422 0.0005271 0.0000000 0.0016579 0.0026553
# 1000 random probabilities of success
pp <- runif(1000)
dpn <- dpbinom(NULL, pp, method = "Normal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -8.412e-06 0.000e+00 0.000e+00 0.000e+00 0.000e+00 3.815e-06
# 100000 random probabilities of success
pp <- runif(100000)
dpn <- dpbinom(NULL, pp, method = "Normal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -4.484e-09 0.000e+00 8.990e-13 0.000e+00 4.919e-10 2.734e-09The Refined Normal Approximation (RNA) approach is requested
with method = "RefinedNormal". It is based on a Normal
distribution, whose parameters are derived from the theoretical mean,
variance and skewness of the input probabilities of success.
set.seed(1)
pp <- runif(10)
wt <- sample(1:10, 10, TRUE)
dpbinom(NULL, pp, wt, "RefinedNormal")
#> [1] 2.579548e-31 1.128297e-29 4.507210e-28 1.611452e-26 5.156486e-25
#> [6] 1.476806e-23 3.785627e-22 8.685911e-21 1.783953e-19 3.280039e-18
#> [11] 5.399492e-17 7.959230e-16 1.050796e-14 1.242802e-13 1.317210e-12
#> [16] 1.251531e-11 1.066498e-10 8.155390e-10 5.599786e-09 3.455053e-08
#> [21] 1.917106e-07 9.574753e-07 4.308224e-06 1.748069e-05 6.401569e-05
#> [26] 2.117447e-04 6.329842e-04 1.710740e-03 4.180480e-03 9.234968e-03
#> [31] 1.843341e-02 3.322175e-02 5.401115e-02 7.912655e-02 1.043358e-01
#> [36] 1.236782e-01 1.316360e-01 1.256489e-01 1.074322e-01 8.218619e-02
#> [41] 5.618825e-02 3.428872e-02 1.865323e-02 9.032795e-03 3.886960e-03
#> [46] 1.483178e-03 5.004545e-04 1.487517e-04 3.873113e-05 8.757189e-06
#> [51] 1.693868e-06 2.722346e-07 3.388544e-08 2.218356e-09 0.000000e+00
#> [56] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [61] 0.000000e+00 0.000000e+00
ppbinom(NULL, pp, wt, "RefinedNormal")
#> [1] 2.579548e-31 1.154092e-29 4.622620e-28 1.657678e-26 5.322254e-25
#> [6] 1.530028e-23 3.938629e-22 9.079774e-21 1.874750e-19 3.467514e-18
#> [11] 5.746244e-17 8.533855e-16 1.136134e-14 1.356415e-13 1.452852e-12
#> [16] 1.396817e-11 1.206179e-10 9.361569e-10 6.535943e-09 4.108647e-08
#> [21] 2.327971e-07 1.190272e-06 5.498496e-06 2.297918e-05 8.699487e-05
#> [26] 2.987396e-04 9.317238e-04 2.642463e-03 6.822944e-03 1.605791e-02
#> [31] 3.449132e-02 6.771307e-02 1.217242e-01 2.008508e-01 3.051866e-01
#> [36] 4.288648e-01 5.605008e-01 6.861497e-01 7.935820e-01 8.757682e-01
#> [41] 9.319564e-01 9.662451e-01 9.848984e-01 9.939312e-01 9.978181e-01
#> [46] 9.993013e-01 9.998018e-01 9.999505e-01 9.999892e-01 9.999980e-01
#> [51] 9.999997e-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+00A comparison with exact computation shows that the approximation quality of the RNA procedure increases with larger numbers of probabilities of success:
set.seed(1)
# 10 random probabilities of success
pp <- runif(10)
dpn <- dpbinom(NULL, pp, method = "RefinedNormal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.0039538 -0.0006920 0.0003543 0.0000000 0.0017167 0.0023597
# 1000 random probabilities of success
pp <- runif(1000)
dpn <- dpbinom(NULL, pp, method = "RefinedNormal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -2.974e-06 0.000e+00 0.000e+00 0.000e+00 0.000e+00 2.270e-06
# 100000 random probabilities of success
pp <- runif(100000)
dpn <- dpbinom(NULL, pp, method = "RefinedNormal")
dpd <- dpbinom(NULL, pp)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -3.126e-09 0.000e+00 6.337e-13 0.000e+00 4.632e-10 2.293e-09To assess the performance of the approximation 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(4000), method = "Normal")
f2 <- function() dpbinom(NULL, runif(4000), method = "Poisson")
f3 <- function() dpbinom(NULL, runif(4000), method = "RefinedNormal")
f4 <- function() dpbinom(NULL, runif(4000), method = "Mean")
f5 <- function() dpbinom(NULL, runif(4000), method = "GeoMean")
f6 <- function() dpbinom(NULL, runif(4000), method = "GeoMeanCounter")
f7 <- function() dpbinom(NULL, runif(4000), method = "DivideFFT")
microbenchmark(f1(), f2(), f3(), f4(), f5(), f6(), f7(), times = 51)
#> Unit: microseconds
#> expr min lq mean median uq max neval
#> f1() 652.669 657.2025 685.9348 667.397 676.6845 1462.973 51
#> f2() 871.899 879.8740 906.2233 887.207 894.0305 1763.104 51
#> f3() 880.826 886.6520 1025.2515 890.684 903.4730 2840.977 51
#> f4() 664.481 670.6580 696.5130 674.901 687.0380 1484.974 51
#> f5() 693.436 699.4520 731.9917 706.570 716.6785 1784.304 51
#> f6() 687.454 695.8245 722.3078 698.565 711.8945 1551.078 51
#> f7() 27003.457 27089.6475 27310.0641 27131.696 27208.6295 29223.664 51Clearly, the NA procedure is the fastest, followed by the PA and RNA methods. The next fastest algorithms are AMBA, GMBA-A and GMBA-B. They exhibit almost equal mean execution speed, with the AMBA algorithm being slightly faster. All of the approximation procedures outperform the fastest exact approach, DC-FFT, by far.
The Generalized Normal Approximation (G-NA) approach is
requested with method = "Normal". It is based on a Normal
distribution, whose parameters are derived from the theoretical mean and
variance of the input probabilities of success (see Introduction.
set.seed(2)
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, "Normal")
#> [1] 5.607923e-34 8.868899e-34 2.266907e-33 5.759009e-33 1.454159e-32
#> [6] 3.649437e-32 9.103112e-32 2.256856e-31 5.561194e-31 1.362016e-30
#> [11] 3.315478e-30 8.021587e-30 1.928965e-29 4.610400e-29 1.095224e-28
#> [16] 2.585931e-28 6.068497e-28 1.415453e-27 3.281403e-27 7.560907e-27
#> [21] 1.731562e-26 3.941418e-26 8.916960e-26 2.005077e-25 4.481212e-25
#> [26] 9.954281e-25 2.197730e-24 4.822684e-24 1.051849e-23 2.280173e-23
#> [31] 4.912836e-23 1.052075e-22 2.239296e-22 4.737247e-22 9.960718e-22
#> [36] 2.081639e-21 4.323844e-21 8.926573e-21 1.831680e-20 3.735634e-20
#> [41] 7.572323e-20 1.525612e-19 3.054984e-19 6.080284e-19 1.202787e-18
#> [46] 2.364851e-18 4.621350e-18 8.976023e-18 1.732802e-17 3.324790e-17
#> [51] 6.340586e-17 1.201834e-16 2.264174e-16 4.239603e-16 7.890246e-16
#> [56] 1.459506e-15 2.683313e-15 4.903282e-15 8.905378e-15 1.607563e-14
#> [61] 2.884254e-14 5.143387e-14 9.116221e-14 1.605945e-13 2.811877e-13
#> [66] 4.893417e-13 8.464047e-13 1.455104e-12 2.486337e-12 4.222561e-12
#> [71] 7.127579e-12 1.195799e-11 1.993996e-11 3.304764e-11 5.443857e-11
#> [76] 8.912982e-11 1.450405e-10 2.345880e-10 3.771137e-10 6.025440e-10
#> [81] 9.568753e-10 1.510330e-09 2.369401e-09 3.694497e-09 5.725614e-09
#> [86] 8.819398e-09 1.350224e-08 2.054578e-08 3.107347e-08 4.670967e-08
#> [91] 6.978689e-08 1.036313e-07 1.529531e-07 2.243755e-07 3.271469e-07
#> [96] 4.740893e-07 6.828536e-07 9.775638e-07 1.390954e-06 1.967117e-06
#> [101] 2.765018e-06 3.862920e-06 5.363935e-06 7.402890e-06 1.015475e-05
#> [106] 1.384482e-05 1.876097e-05 2.526814e-05 3.382528e-05 4.500488e-05
#> [111] 5.951520e-05 7.822512e-05 1.021915e-04 1.326884e-04 1.712386e-04
#> [116] 2.196444e-04 2.800198e-04 3.548195e-04 4.468649e-04 5.593647e-04
#> [121] 6.959275e-04 8.605635e-04 1.057674e-03 1.292025e-03 1.568701e-03
#> [126] 1.893038e-03 2.270537e-03 2.706749e-03 3.207136e-03 3.776912e-03
#> [131] 4.420856e-03 5.143112e-03 5.946968e-03 6.834635e-03 7.807017e-03
#> [136] 8.863494e-03 1.000172e-02 1.121747e-02 1.250446e-02 1.385431e-02
#> [141] 1.525651e-02 1.669842e-02 1.816543e-02 1.964112e-02 2.110749e-02
#> [146] 2.254536e-02 2.393468e-02 2.525505e-02 2.648616e-02 2.760831e-02
#> [151] 2.860294e-02 2.945314e-02 3.014411e-02 3.066363e-02 3.100235e-02
#> [156] 3.115414e-02 3.111624e-02 3.088932e-02 3.047753e-02 2.988830e-02
#> [161] 2.913216e-02 2.822242e-02 2.717477e-02 2.600684e-02 2.473770e-02
#> [166] 2.338736e-02 2.197622e-02 2.052462e-02 1.905228e-02 1.757799e-02
#> [171] 1.611912e-02 1.469141e-02 1.330871e-02 1.198280e-02 1.072335e-02
#> [176] 9.537908e-03 8.431904e-03 7.408807e-03 6.470249e-03 5.616215e-03
#> [181] 4.845254e-03 4.154698e-03 3.540890e-03 2.999407e-03 2.525274e-03
#> [186] 2.113156e-03 1.757538e-03 1.452874e-03 1.193717e-03 9.748208e-04
#> [191] 7.912218e-04 6.382955e-04 5.117942e-04 4.078674e-04 3.230671e-04
#> [196] 2.543411e-04 1.990171e-04 1.547798e-04 1.196432e-04 9.192046e-05
#> [201] 7.019178e-05 5.327340e-05 4.018691e-05 3.013068e-05 2.245346e-05
#> [206] 1.663059e-05 1.224284e-05 8.957907e-06 6.514501e-06 1.614725e-05
pgpbinom(NULL, pp, va, vb, wt, "Normal")
#> [1] 5.607923e-34 1.447682e-33 3.714589e-33 9.473598e-33 2.401518e-32
#> [6] 6.050955e-32 1.515407e-31 3.772263e-31 9.333457e-31 2.295361e-30
#> [11] 5.610840e-30 1.363243e-29 3.292208e-29 7.902608e-29 1.885484e-28
#> [16] 4.471416e-28 1.053991e-27 2.469444e-27 5.750847e-27 1.331175e-26
#> [21] 3.062738e-26 7.004156e-26 1.592112e-25 3.597189e-25 8.078401e-25
#> [26] 1.803268e-24 4.000998e-24 8.823682e-24 1.934217e-23 4.214390e-23
#> [31] 9.127226e-23 1.964798e-22 4.204093e-22 8.941340e-22 1.890206e-21
#> [36] 3.971844e-21 8.295689e-21 1.722226e-20 3.553906e-20 7.289540e-20
#> [41] 1.486186e-19 3.011798e-19 6.066782e-19 1.214707e-18 2.417494e-18
#> [46] 4.782345e-18 9.403695e-18 1.837972e-17 3.570774e-17 6.895564e-17
#> [51] 1.323615e-16 2.525449e-16 4.789624e-16 9.029227e-16 1.691947e-15
#> [56] 3.151453e-15 5.834767e-15 1.073805e-14 1.964343e-14 3.571905e-14
#> [61] 6.456159e-14 1.159955e-13 2.071577e-13 3.677521e-13 6.489399e-13
#> [66] 1.138282e-12 1.984686e-12 3.439790e-12 5.926127e-12 1.014869e-11
#> [71] 1.727627e-11 2.923425e-11 4.917421e-11 8.222186e-11 1.366604e-10
#> [76] 2.257903e-10 3.708308e-10 6.054188e-10 9.825325e-10 1.585076e-09
#> [81] 2.541952e-09 4.052282e-09 6.421683e-09 1.011618e-08 1.584179e-08
#> [86] 2.466119e-08 3.816343e-08 5.870922e-08 8.978268e-08 1.364924e-07
#> [91] 2.062792e-07 3.099106e-07 4.628636e-07 6.872392e-07 1.014386e-06
#> [96] 1.488475e-06 2.171329e-06 3.148893e-06 4.539847e-06 6.506964e-06
#> [101] 9.271982e-06 1.313490e-05 1.849884e-05 2.590173e-05 3.605648e-05
#> [106] 4.990129e-05 6.866226e-05 9.393040e-05 1.277557e-04 1.727606e-04
#> [111] 2.322758e-04 3.105009e-04 4.126924e-04 5.453808e-04 7.166194e-04
#> [116] 9.362638e-04 1.216284e-03 1.571103e-03 2.017968e-03 2.577333e-03
#> [121] 3.273260e-03 4.133824e-03 5.191498e-03 6.483523e-03 8.052224e-03
#> [126] 9.945263e-03 1.221580e-02 1.492255e-02 1.812968e-02 2.190660e-02
#> [131] 2.632745e-02 3.147056e-02 3.741753e-02 4.425217e-02 5.205918e-02
#> [136] 6.092268e-02 7.092440e-02 8.214187e-02 9.464633e-02 1.085006e-01
#> [141] 1.237572e-01 1.404556e-01 1.586210e-01 1.782621e-01 1.993696e-01
#> [146] 2.219150e-01 2.458497e-01 2.711047e-01 2.975909e-01 3.251992e-01
#> [151] 3.538021e-01 3.832553e-01 4.133994e-01 4.440630e-01 4.750653e-01
#> [156] 5.062195e-01 5.373357e-01 5.682250e-01 5.987026e-01 6.285909e-01
#> [161] 6.577230e-01 6.859454e-01 7.131202e-01 7.391271e-01 7.638648e-01
#> [166] 7.872521e-01 8.092283e-01 8.297529e-01 8.488052e-01 8.663832e-01
#> [171] 8.825023e-01 8.971938e-01 9.105025e-01 9.224853e-01 9.332086e-01
#> [176] 9.427465e-01 9.511784e-01 9.585872e-01 9.650575e-01 9.706737e-01
#> [181] 9.755189e-01 9.796736e-01 9.832145e-01 9.862139e-01 9.887392e-01
#> [186] 9.908524e-01 9.926099e-01 9.940628e-01 9.952565e-01 9.962313e-01
#> [191] 9.970225e-01 9.976608e-01 9.981726e-01 9.985805e-01 9.989036e-01
#> [196] 9.991579e-01 9.993569e-01 9.995117e-01 9.996314e-01 9.997233e-01
#> [201] 9.997935e-01 9.998467e-01 9.998869e-01 9.999171e-01 9.999395e-01
#> [206] 9.999561e-01 9.999684e-01 9.999773e-01 9.999839e-01 1.000000e+00A comparison with exact computation shows that the approximation quality of the NA procedure increases with larger numbers of probabilities of success:
set.seed(2)
# 10 random probabilities of success
pp <- runif(10)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "Normal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.0346309 -0.0042919 0.0001378 0.0000000 0.0038447 0.0317044
# 100 random probabilities of success
pp <- runif(100)
va <- sample(0:100, 100, TRUE)
vb <- sample(0:100, 100, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "Normal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -3.006e-05 -1.126e-09 0.000e+00 0.000e+00 1.854e-09 2.967e-05
# 1000 random probabilities of success
pp <- runif(1000)
va <- sample(0:1000, 1000, TRUE)
vb <- sample(0:1000, 1000, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "Normal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -3.152e-08 0.000e+00 3.060e-12 0.000e+00 8.992e-10 3.707e-08The Generalized Refined Normal Approximation (G-RNA)
approach is requested with method = "RefinedNormal". It is
based on a Normal distribution, whose parameters are derived from the
theoretical mean, variance and skewness of the input probabilities of
success.
set.seed(2)
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, "RefinedNormal")
#> [1] 5.100768e-32 7.816039e-32 1.959106e-31 4.880045e-31 1.208047e-30
#> [6] 2.971921e-30 7.265798e-30 1.765311e-29 4.262362e-29 1.022751e-28
#> [11] 2.438814e-28 5.779315e-28 1.361012e-27 3.185186e-27 7.407878e-27
#> [16] 1.712136e-26 3.932484e-26 8.975930e-26 2.035985e-25 4.589352e-25
#> [21] 1.028037e-24 2.288476e-24 5.062470e-24 1.112900e-23 2.431235e-23
#> [26] 5.278047e-23 1.138660e-22 2.441116e-22 5.200621e-22 1.101015e-21
#> [31] 2.316333e-21 4.842591e-21 1.006056e-20 2.076983e-20 4.260973e-20
#> [36] 8.686571e-20 1.759748e-19 3.542530e-19 7.086575e-19 1.408697e-18
#> [41] 2.782630e-18 5.461965e-18 1.065359e-17 2.064884e-17 3.976912e-17
#> [46] 7.611065e-17 1.447413e-16 2.735176e-16 5.135966e-16 9.582999e-16
#> [51] 1.776730e-15 3.273256e-15 5.992053e-15 1.089949e-14 1.970017e-14
#> [56] 3.538058e-14 6.313772e-14 1.119541e-13 1.972495e-13 3.453144e-13
#> [61] 6.006676e-13 1.038179e-12 1.782897e-12 3.042246e-12 5.157913e-12
#> [66] 8.688860e-12 1.454315e-11 2.418568e-11 3.996319e-11 6.560867e-11
#> [71] 1.070186e-10 1.734408e-10 2.792769e-10 4.467944e-10 7.101774e-10
#> [76] 1.121527e-09 1.759679e-09 2.743061e-09 4.248282e-09 6.536785e-09
#> [81] 9.992759e-09 1.517660e-08 2.289965e-08 3.432780e-08 5.112383e-08
#> [86] 7.564129e-08 1.111860e-07 1.623661e-07 2.355550e-07 3.394997e-07
#> [91] 4.861107e-07 6.914779e-07 9.771650e-07 1.371840e-06 1.913307e-06
#> [96] 2.651012e-06 3.649099e-06 4.990081e-06 6.779222e-06 9.149662e-06
#> [101] 1.226837e-05 1.634294e-05 2.162919e-05 2.843967e-05 3.715276e-05
#> [106] 4.822249e-05 6.218875e-05 7.968764e-05 1.014618e-04 1.283702e-04
#> [111] 1.613972e-04 2.016606e-04 2.504176e-04 3.090698e-04 3.791651e-04
#> [116] 4.623982e-04 5.606082e-04 6.757744e-04 8.100102e-04 9.655553e-04
#> [121] 1.144767e-03 1.350110e-03 1.584150e-03 1.849543e-03 2.149024e-03
#> [126] 2.485405e-03 2.861561e-03 3.280420e-03 3.744950e-03 4.258135e-03
#> [131] 4.822941e-03 5.442277e-03 6.118927e-03 6.855467e-03 7.654163e-03
#> [136] 8.516833e-03 9.444692e-03 1.043817e-02 1.149671e-02 1.261856e-02
#> [141] 1.380053e-02 1.503782e-02 1.632377e-02 1.764978e-02 1.900514e-02
#> [146] 2.037702e-02 2.175055e-02 2.310888e-02 2.443348e-02 2.570445e-02
#> [151] 2.690096e-02 2.800177e-02 2.898579e-02 2.983278e-02 3.052397e-02
#> [156] 3.104271e-02 3.137515e-02 3.151071e-02 3.144261e-02 3.116818e-02
#> [161] 3.068902e-02 3.001109e-02 2.914456e-02 2.810352e-02 2.690563e-02
#> [166] 2.557147e-02 2.412399e-02 2.258773e-02 2.098813e-02 1.935073e-02
#> [171] 1.770044e-02 1.606093e-02 1.445398e-02 1.289904e-02 1.141287e-02
#> [176] 1.000927e-02 8.699011e-03 7.489773e-03 6.386301e-03 5.390581e-03
#> [181] 4.502114e-03 3.718233e-03 3.034469e-03 2.444914e-03 1.942594e-03
#> [186] 1.519822e-03 1.168521e-03 8.805066e-04 6.477360e-04 4.625001e-04
#> [191] 2.621189e-04 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [196] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [201] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [206] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
pgpbinom(NULL, pp, va, vb, wt, "RefinedNormal")
#> [1] 5.100768e-32 1.291681e-31 3.250786e-31 8.130831e-31 2.021130e-30
#> [6] 4.993051e-30 1.225885e-29 2.991196e-29 7.253558e-29 1.748106e-28
#> [11] 4.186920e-28 9.966236e-28 2.357636e-27 5.542822e-27 1.295070e-26
#> [16] 3.007206e-26 6.939690e-26 1.591562e-25 3.627547e-25 8.216899e-25
#> [21] 1.849727e-24 4.138203e-24 9.200673e-24 2.032968e-23 4.464203e-23
#> [26] 9.742250e-23 2.112885e-22 4.554002e-22 9.754623e-22 2.076477e-21
#> [31] 4.392810e-21 9.235402e-21 1.929596e-20 4.006579e-20 8.267552e-20
#> [36] 1.695412e-19 3.455160e-19 6.997690e-19 1.408427e-18 2.817123e-18
#> [41] 5.599754e-18 1.106172e-17 2.171531e-17 4.236415e-17 8.213328e-17
#> [46] 1.582439e-16 3.029852e-16 5.765028e-16 1.090099e-15 2.048399e-15
#> [51] 3.825129e-15 7.098385e-15 1.309044e-14 2.398993e-14 4.369010e-14
#> [56] 7.907068e-14 1.422084e-13 2.541625e-13 4.514120e-13 7.967264e-13
#> [61] 1.397394e-12 2.435573e-12 4.218470e-12 7.260717e-12 1.241863e-11
#> [66] 2.110749e-11 3.565064e-11 5.983632e-11 9.979950e-11 1.654082e-10
#> [71] 2.724267e-10 4.458675e-10 7.251445e-10 1.171939e-09 1.882116e-09
#> [76] 3.003643e-09 4.763322e-09 7.506383e-09 1.175466e-08 1.829145e-08
#> [81] 2.828421e-08 4.346081e-08 6.636046e-08 1.006883e-07 1.518121e-07
#> [86] 2.274534e-07 3.386394e-07 5.010055e-07 7.365605e-07 1.076060e-06
#> [91] 1.562171e-06 2.253649e-06 3.230814e-06 4.602653e-06 6.515960e-06
#> [96] 9.166972e-06 1.281607e-05 1.780615e-05 2.458537e-05 3.373504e-05
#> [101] 4.600341e-05 6.234634e-05 8.397554e-05 1.124152e-04 1.495680e-04
#> [106] 1.977905e-04 2.599792e-04 3.396668e-04 4.411286e-04 5.694988e-04
#> [111] 7.308960e-04 9.325566e-04 1.182974e-03 1.492044e-03 1.871209e-03
#> [116] 2.333607e-03 2.894215e-03 3.569990e-03 4.380000e-03 5.345555e-03
#> [121] 6.490322e-03 7.840432e-03 9.424583e-03 1.127413e-02 1.342315e-02
#> [126] 1.590855e-02 1.877011e-02 2.205053e-02 2.579549e-02 3.005362e-02
#> [131] 3.487656e-02 4.031884e-02 4.643777e-02 5.329323e-02 6.094740e-02
#> [136] 6.946423e-02 7.890892e-02 8.934709e-02 1.008438e-01 1.134624e-01
#> [141] 1.272629e-01 1.423007e-01 1.586245e-01 1.762743e-01 1.952794e-01
#> [146] 2.156564e-01 2.374070e-01 2.605159e-01 2.849493e-01 3.106538e-01
#> [151] 3.375548e-01 3.655565e-01 3.945423e-01 4.243751e-01 4.548991e-01
#> [156] 4.859418e-01 5.173169e-01 5.488276e-01 5.802702e-01 6.114384e-01
#> [161] 6.421274e-01 6.721385e-01 7.012831e-01 7.293866e-01 7.562922e-01
#> [166] 7.818637e-01 8.059877e-01 8.285754e-01 8.495636e-01 8.689143e-01
#> [171] 8.866147e-01 9.026757e-01 9.171296e-01 9.300287e-01 9.414415e-01
#> [176] 9.514508e-01 9.601498e-01 9.676396e-01 9.740259e-01 9.794165e-01
#> [181] 9.839186e-01 9.876368e-01 9.906713e-01 9.931162e-01 9.950588e-01
#> [186] 9.965786e-01 9.977471e-01 9.986276e-01 9.992754e-01 9.997379e-01
#> [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+00A comparison with exact computation shows that the approximation quality of the RNA procedure increases with larger numbers of probabilities of success:
set.seed(2)
# 10 random probabilities of success
pp <- runif(10)
va <- sample(0:10, 10, TRUE)
vb <- sample(0:10, 10, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "RefinedNormal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -3.045e-02 -4.084e-03 1.727e-04 1.179e-05 4.324e-03 3.161e-02
# 100 random probabilities of success
pp <- runif(100)
va <- sample(0:100, 100, TRUE)
vb <- sample(0:100, 100, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "RefinedNormal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -8.831e-06 0.000e+00 1.000e-12 9.000e-12 3.642e-07 1.333e-05
# 1000 random probabilities of success
pp <- runif(1000)
va <- sample(0:1000, 1000, TRUE)
vb <- sample(0:1000, 1000, TRUE)
dpn <- dgpbinom(NULL, pp, va, vb, method = "RefinedNormal")
dpd <- dgpbinom(NULL, pp, va, vb)
idx <- which(dpn != 0 & dpd != 0)
summary((dpn - dpd)[idx])
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -1.980e-08 0.000e+00 4.960e-12 0.000e+00 1.561e-09 3.197e-08To assess the performance of the approximation 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)
n <- 1500
set.seed(2)
va <- sample(1:50, n, TRUE)
vb <- sample(1:50, n, TRUE)
f1 <- function() dgpbinom(NULL, runif(n), va, vb, method = "Normal")
f2 <- function() dgpbinom(NULL, runif(n), va, vb, method = "RefinedNormal")
f3 <- function() dgpbinom(NULL, runif(n), va, vb, method = "DivideFFT")
microbenchmark(f1(), f2(), f3(), times = 51)
#> Unit: milliseconds
#> expr min lq mean median uq max neval
#> f1() 5.030929 5.061941 5.365583 5.131487 5.194379 7.063646 51
#> f2() 6.047488 6.105090 6.252433 6.163283 6.222370 8.071970 51
#> f3() 234.748486 235.069610 236.136893 235.513685 236.567683 245.275040 51Clearly, the G-NA procedure is the fastest, followed by the G-RNA method. Both are hugely faster than G-DC-FFT.