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 8.720e-13 0.000e+00 4.914e-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 6.200e-13 0.000e+00 4.616e-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 7 1800X with 32 GiB of RAM and Windows 10 Education (20H2).
library(microbenchmark)
set.seed(1)
f1 <- function() dpbinom(NULL, runif(4000), method = "Normal")
f2 <- function() dpbinom(NULL, runif(4000), method = "RefinedNormal")
f3 <- function() dpbinom(NULL, runif(4000), method = "Poisson")
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 cld
#> f1() 598.7 645.85 741.4000 653.9 675.75 2962.2 51 a
#> f2() 758.2 805.00 919.8961 825.1 857.15 3110.2 51 ab
#> f3() 914.9 972.15 1073.6471 996.9 1012.05 3302.7 51 b
#> f4() 1326.1 1417.20 1465.6569 1428.2 1456.25 2635.0 51 c
#> f5() 1476.1 1578.15 1622.6667 1589.8 1607.25 2985.4 51 c
#> f6() 1464.9 1571.10 1678.3020 1589.6 1631.40 3940.1 51 c
#> f7() 5762.7 6064.85 6281.2549 6120.8 6185.15 8623.5 51 dClearly, the NA procedure is the fastest, followed by the RNA and PA 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 8.991e-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 7 1800X with 32 GiB of RAM and Windows 10 Education (20H2).
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 cld
#> f1() 5.4328 5.5898 5.875559 5.7124 5.85060 8.1924 51 a
#> f2() 6.7147 7.0995 9.602563 7.2103 7.48995 106.0331 51 a
#> f3() 37.9302 38.6177 39.652061 39.1261 39.57830 57.4351 51 bClearly, the G-NA procedure is the fastest, followed by the G-RNA method. Both are hugely faster than G-DC-FFT.