// NewBinomial returns a Binomial distribution.
func NewBinomial(n int, p float64, src rand.Source) *Binomial {
if float64(n)*math.Min(p, 1.0-p) <= 0 {
panic("Illegal argument for Binomial distribution: n*p <= 0")
}
binomial := &Binomial{N: n, P: p}
binomial.randomGenerator = rand.New(src)
binomial.n_last = -1
binomial.n_prev = -1
binomial.p_last = -1.0
binomial.p_prev = -1.0
binomial.log_p = math.Log(binomial.P)
binomial.log_q = math.Log(1.0 - binomial.P)
binomial.log_n = specfunc.LogFactorial(binomial.N)
return binomial
}
// Pdf returns the probability distribution function.
func (poisson Poisson) Pdf(k int) (p float64) {
p = math.Exp(float64(k)*math.Log(poisson.Mean) - specfunc.LogFactorial(k) - poisson.Mean)
return
}
// Helper function
func (poisson Poisson) f(k int, l_nu, c_pm float64) float64 {
return math.Exp(float64(k)*l_nu - specfunc.LogFactorial(k) - c_pm)
}
// Generate poisson random number
// Implementation: High performance implementation.
// Patchwork Rejection/Inversion method.
// This is a port of Poisson.java from the colt java library, which is based upon
// H. Zechner (1994): Efficient sampling from continuous and discrete unimodal distributions,
// Doctoral Dissertation, 156 pp., Technical University Graz, Austria.
// Also see
// Stadlober E., H. Zechner (1999), The patchwork rejection method for sampling from unimodal distributions,
// to appear in ACM Transactions on Modelling and Simulation.
func (poisson *Poisson) random() (k int) {
/******************************************************************
* *
* Poisson Distribution - Patchwork Rejection/Inversion *
* *
******************************************************************
* *
* For parameter my < 10 Tabulated Inversion is applied. *
* For my >= 10 Patchwork Rejection is employed: *
* The area below the histogram function f(x) is rearranged in *
* its body by certain point reflections. Within a large center *
* interval variates are sampled efficiently by rejection from *
* uniform hats. Rectangular immediate acceptance regions speed *
* up the generation. The remaining tails are covered by *
* exponential functions. *
* *
*****************************************************************/
my := poisson.Mean
if my < SWITCH_MEAN { // CASE B: Inversion- start new table and calculate p0
if my != poisson.my_old {
poisson.my_old = my
poisson.llll = 0
poisson.p = math.Exp(-my)
poisson.q = poisson.p
poisson.p0 = poisson.p
}
if my > 1.0 {
poisson.m = int(my)
} else {
poisson.m = 1
}
for {
u := poisson.randGenerator.Float64()
if u <= poisson.p0 {
return k
}
if poisson.llll != 0 {
if u > 0.458 {
if poisson.llll < poisson.m {
k = poisson.llll
} else {
k = poisson.m
}
} else {
k = 1
}
for ; k <= poisson.llll; k++ {
if u <= poisson.pp[k] {
return k
}
}
if poisson.llll == 35 {
continue
}
}
for k = poisson.llll + 1; k <= 35; k++ {
poisson.p *= my / float64(k)
poisson.q += poisson.p
poisson.pp[k] = poisson.q
if u <= poisson.q {
poisson.llll = k
return k
}
}
poisson.llll = 35
}
} else {
var (
Dk, X, Y int
Ds, U, V, W float64
)
poisson.m = int(my)
if my != poisson.my_last {
poisson.my_last = my
// approximate deviation of reflection points k2, k4 from my - 1/2
Ds = math.Sqrt(my + 0.25)
// mode m, reflection points k2 and k4, and points k1 and k5, which
// delimit the centre region of h(x)
poisson.k2 = int(math.Ceil(my - 0.5 - Ds))
poisson.k4 = int(my - 0.5 + Ds)
poisson.k1 = int(poisson.k2 + poisson.k2 - poisson.m + 1.0)
poisson.k5 = int(poisson.k4 + poisson.k4 - poisson.m)
// range width of the critical left and right centre region
poisson.dl = float64(poisson.k2 - poisson.k1)
poisson.dr = float64(poisson.k5 - poisson.k4)
// recurrence constants r(k) = p(k)/p(k-1) at k = k1, k2, k4+1, k5+1
poisson.r1 = my / float64(poisson.k1)
poisson.r2 = my / float64(poisson.k2)
poisson.r4 = my / float64(poisson.k4+1)
poisson.r5 = my / float64(poisson.k5+1)
// reciprocal values of the scale parameters of expon. tail envelopes
poisson.ll = math.Log(poisson.r1) // expon. tail left
poisson.lr = -math.Log(poisson.r5) // expon. tail right
// Poisson constants, necessary for computing function values f(k)
poisson.l_my = math.Log(my)
poisson.c_pm = float64(poisson.m)*poisson.l_my - specfunc.LogFactorial(poisson.m)
// function values f(k) = p(k)/p(m) at k = k2, k4, k1, k5
poisson.f2 = poisson.f(poisson.k2, poisson.l_my, poisson.c_pm)
poisson.f4 = poisson.f(poisson.k4, poisson.l_my, poisson.c_pm)
poisson.f1 = poisson.f(poisson.k1, poisson.l_my, poisson.c_pm)
poisson.f5 = poisson.f(poisson.k5, poisson.l_my, poisson.c_pm)
// area of the two centre and the two exponential tail regions
// area of the two immediate acceptance regions between k2, k4
poisson.p1 = poisson.f2 * (poisson.dl + 1.0) // immed. left
poisson.p2 = poisson.f2*poisson.dl + poisson.p1 // centre left
poisson.p3 = poisson.f4*(poisson.dr+1.0) + poisson.p2 // immed. right
poisson.p4 = poisson.f4*poisson.dr + poisson.p3 // centre right
poisson.p5 = poisson.f1/poisson.ll + poisson.p4 // expon. tail left
poisson.p6 = poisson.f5/poisson.lr + poisson.p5 // expon. tail right
} // end set-up
for {
// generate uniform number U -- U(0, p6)
// case distinction corresponding to U
U = poisson.randGenerator.Float64() * poisson.p6
if U < poisson.p2 {
// immediate acceptance region R2 = [k2, m) *[0, f2), X = k2, ... m -1
V = U - poisson.p1
if V < 0.0 {
k = poisson.k2 + int(U/poisson.f2)
return
}
// immediate acceptance region R1 = [k1, k2)*[0, f1), X = k1, ... k2-1
W = V / poisson.dl
if W < poisson.f1 {
k = poisson.k1 + int(V/poisson.f1)
return
}
// computation of candidate X < k2, and its counterpart Y > k2
// either squeeze-acceptance of X or acceptance-rejection of Y
Dk = int(poisson.dl*poisson.randGenerator.Float64()) + 1
if W <= poisson.f2-float64(Dk)*(poisson.f2-poisson.f2/poisson.r2) {
k = poisson.k2 - Dk
return
}
V = poisson.f2 + poisson.f2 - W
if V < 1.0 {
Y = poisson.k2 + Dk
if V < poisson.f2+float64(Dk)*(1.0-poisson.f2)/(poisson.dl+1.0) {
k = Y
return
}
if V <= poisson.f(Y, poisson.l_my, poisson.c_pm) {
k = Y
return
}
}
X = poisson.k2 - Dk
} else if U < poisson.p4 { // centre right
// immediate acceptance region R3 = [m, k4+1)*[0, f4), X = m, ... k4
V = U - poisson.p3
if V < 0.0 {
k = poisson.k4 - int((U-poisson.p2)/poisson.f4)
return
}
// immediate acceptance region R4 = [k4+1, k5+1)*[0, f5)
W = V / poisson.dr
if W < poisson.f5 {
k = poisson.k5 - int(V/poisson.f5)
return
}
// computation of candidate X > k4, and its counterpart Y < k4
// either squeeze-acceptance of X or acceptance-rejection of Y
Dk = int(poisson.dr*poisson.randGenerator.Float64()) + 1
if W <= poisson.f4-float64(Dk)*(poisson.f4-poisson.f4*poisson.f4) {
k = poisson.k4 + Dk
return
}
V = poisson.f4 + poisson.f4 - W
if V < 1.0 {
Y = poisson.k4 - Dk
if V <= poisson.f4+float64(Dk)*(1.0-poisson.f4)/poisson.dr {
k = Y
return
}
if V <= poisson.f(Y, poisson.l_my, poisson.c_pm) {
k = Y
return
}
}
X = poisson.k4 + Dk
} else {
W = poisson.randGenerator.Float64()
if U < poisson.p5 {
Dk = int(1.0 - math.Log(W)/poisson.ll)
X = poisson.k1 - Dk
if X < 0 {
continue
}
W *= (U - poisson.p4) * poisson.ll
if W <= poisson.f1-float64(Dk)*(poisson.f1-poisson.f1/poisson.r1) {
k = X
return
}
} else {
Dk = int(1.0 - math.Log(W)/poisson.lr)
X = poisson.k5 + Dk
W *= (U - poisson.p5) * poisson.lr
if W <= poisson.f5-float64(Dk)*(poisson.f5-poisson.f5*poisson.f5) {
k = X
return
}
}
}
// acceptance-rejection test of candidate X from the original area
// test, whether W <= f(k), with W = U*h(x) and U -- U(0, 1)
// log f(X) = (X - m)*log(my) - log X! + log m!
if math.Log(W) <= float64(X)*poisson.l_my-specfunc.LogFactorial(X)-poisson.c_pm {
k = X
return
}
}
}
return
}
// Pdf returns the probability distribution function.
func (binomial Binomial) Pdf(k int) (p float64) {
r := binomial.N - k
p = math.Exp(binomial.log_n - specfunc.LogFactorial(k) - specfunc.LogFactorial(r) + binomial.log_p*float64(k) + binomial.log_q*float64(r))
return
}
