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module Distribution
#
# Ruby version implements three methods on this module:
# * Genz:: Used by default, with improvement to calculate p on rho > 0.95
# * Hull:: Port from a C++ code
# * Jantaravareerat:: Iterative (slow and buggy)
#
module BivariateNormal
module Ruby_
class << self
SIDE = 0.1 # :nodoc:
LIMIT = 5 # :nodoc:
# Return the partial derivative of cdf over x, with y and rho constant
# Reference:
# * Tallis, 1962, p.346, cited by Olsson, 1979
def partial_derivative_cdf_x(x, y, rho)
Distribution::Normal.pdf(x) * Distribution::Normal.cdf((y - rho * x).quo(Math.sqrt(1 - rho**2)))
end
alias_method :pd_cdf_x, :partial_derivative_cdf_x
# Probability density function for a given x, y and rho value.
#
# Source: http://en.wikipedia.org/wiki/Multivariate_normal_distribution
def pdf(x, y, rho, s1 = 1.0, s2 = 1.0)
1.quo(2 * Math::PI * s1 * s2 * Math.sqrt(1 - rho**2)) * (Math.exp(-(1.quo(2 * (1 - rho**2))) *
((x**2.quo(s1)) + (y**2.quo(s2)) - (2 * rho * x * y).quo(s1 * s2))))
end
def f(x, y, aprime, bprime, rho)
r = aprime * (2 * x - aprime) + bprime * (2 * y - bprime) + 2 * rho * (x - aprime) * (y - bprime)
Math.exp(r)
end
# CDF for a given x, y and rho value.
# Uses Genz algorithm (cdf_genz method).
#
def cdf(a, b, rho)
cdf_genz(a, b, rho)
end
def sgn(x)
if (x >= 0)
1
else
-1
end
end
# Normal cumulative distribution function (cdf) for a given x, y and rho.
# Based on Hull (1993, cited by Arne, 2003)
#
# References:
# * Arne, B.(2003). Financial Numerical Recipes in C ++. Available on http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node23.html
def cdf_hull(a, b, rho)
# puts "a:#{a} - b:#{b} - rho:#{rho}"
if a <= 0 && b <= 0 && rho <= 0
# puts "ruta 1"
aprime = a.quo(Math.sqrt(2.0 * (1.0 - rho**2)))
bprime = b.quo(Math.sqrt(2.0 * (1.0 - rho**2)))
aa = [0.3253030, 0.4211071, 0.1334425, 0.006374323]
bb = [0.1337764, 0.6243247, 1.3425378, 2.2626645]
sum = 0
4.times do |i|
4.times do |j|
sum += aa[i] * aa[j] * f(bb[i], bb[j], aprime, bprime, rho)
end
end
sum *= (Math.sqrt(1.0 - rho**2).quo(Math::PI))
return sum
elsif (a * b * rho <= 0.0)
# puts "ruta 2"
if a <= 0 && b >= 0 && rho >= 0
return Distribution::Normal.cdf(a) - cdf(a, -b, -rho)
elsif a >= 0.0 && b <= 0.0 && rho >= 0
return Distribution::Normal.cdf(b) - cdf(-a, b, -rho)
elsif a >= 0.0 && b >= 0.0 && rho <= 0
return Distribution::Normal.cdf(a) + Distribution::Normal.cdf(b) - 1.0 + cdf(-a, -b, rho)
end
elsif (a * b * rho >= 0.0)
# puts "ruta 3"
denum = Math.sqrt(a**2 - 2 * rho * a * b + b**2)
rho1 = ((rho * a - b) * sgn(a)).quo(denum)
rho2 = ((rho * b - a) * sgn(b)).quo(denum)
delta = (1.0 - sgn(a) * sgn(b)).quo(4)
# puts "#{rho1} - #{rho2}"
return cdf(a, 0.0, rho1) + cdf(b, 0.0, rho2) - delta
end
fail "Should'nt be here! #{a} - #{b} #{rho}"
end
# CDF. Iterative method by Jantaravareerat (n/d)
#
# Reference:
# * Jantaravareerat, M. & Thomopoulos, N. (n/d). Tables for standard bivariate normal distribution
def cdf_jantaravareerat(x, y, rho, s1 = 1, s2 = 1)
# Special cases
return 1 if x > LIMIT && y > LIMIT
return 0 if x < -LIMIT || y < -LIMIT
return Distribution::Normal.cdf(y) if x > LIMIT
return Distribution::Normal.cdf(x) if y > LIMIT
# puts "x:#{x} - y:#{y}"
x = -LIMIT if x < -LIMIT
x = LIMIT if x > LIMIT
y = -LIMIT if y < -LIMIT
y = LIMIT if y > LIMIT
x_squares = ((LIMIT + x) / SIDE).to_i
y_squares = ((LIMIT + y) / SIDE).to_i
sum = 0
x_squares.times do |i|
y_squares.times do |j|
z1 = -LIMIT + (i + 1) * SIDE
z2 = -LIMIT + (j + 1) * SIDE
# puts " #{z1}-#{z2}"
h = (pdf(z1, z2, rho, s1, s2) + pdf(z1 - SIDE, z2, rho, s1, s2) + pdf(z1, z2 - SIDE, rho, s1, s2) + pdf(z1 - SIDE, z2 - SIDE, rho, s1, s2)).quo(4)
sum += (SIDE**2) * h # area
end
end
sum
end
# Normal cumulative distribution function (cdf) for a given x, y and rho.
# Ported from Fortran code by Alan Genz
#
# Original documentation
# DOUBLE PRECISION FUNCTION BVND( DH, DK, R )
# A function for computing bivariate normal probabilities.
#
# Alan Genz
# Department of Mathematics
# Washington State University
# Pullman, WA 99164-3113
# Email : alangenz_AT_wsu.edu
#
# This function is based on the method described by
# Drezner, Z and G.O. Wesolowsky, (1989),
# On the computation of the bivariate normal integral,
# Journal of Statist. Comput. Simul. 35, pp. 101-107,
# with major modifications for double precision, and for |R| close to 1.
#
# Original location:
# * http://www.math.wsu.edu/faculty/genz/software/fort77/tvpack.f
def cdf_genz(x, y, rho)
dh = -x
dk = -y
r = rho
twopi = 6.283185307179586
w = 11.times.collect { [nil] * 4 }
x = 11.times.collect { [nil] * 4 }
data = [
0.1713244923791705E+00, -0.9324695142031522E+00,
0.3607615730481384E+00, -0.6612093864662647E+00,
0.4679139345726904E+00, -0.2386191860831970E+00]
(1..3).each do|i|
w[i][1] = data[(i - 1) * 2]
x[i][1] = data[(i - 1) * 2 + 1]
end
data = [
0.4717533638651177E-01, -0.9815606342467191E+00,
0.1069393259953183E+00, -0.9041172563704750E+00,
0.1600783285433464E+00, -0.7699026741943050E+00,
0.2031674267230659E+00, -0.5873179542866171E+00,
0.2334925365383547E+00, -0.3678314989981802E+00,
0.2491470458134029E+00, -0.1252334085114692E+00]
(1..6).each do|i|
w[i][2] = data[(i - 1) * 2]
x[i][2] = data[(i - 1) * 2 + 1]
end
data = [
0.1761400713915212E-01, -0.9931285991850949E+00,
0.4060142980038694E-01, -0.9639719272779138E+00,
0.6267204833410906E-01, -0.9122344282513259E+00,
0.8327674157670475E-01, -0.8391169718222188E+00,
0.1019301198172404E+00, -0.7463319064601508E+00,
0.1181945319615184E+00, -0.6360536807265150E+00,
0.1316886384491766E+00, -0.5108670019508271E+00,
0.1420961093183821E+00, -0.3737060887154196E+00,
0.1491729864726037E+00, -0.2277858511416451E+00,
0.1527533871307259E+00, -0.7652652113349733E-01]
(1..10).each do|i|
w[i][3] = data[(i - 1) * 2]
x[i][3] = data[(i - 1) * 2 + 1]
end
if r.abs < 0.3
ng = 1
lg = 3
elsif r.abs < 0.75
ng = 2
lg = 6
else
ng = 3
lg = 10
end
h = dh
k = dk
hk = h * k
bvn = 0
if r.abs < 0.925
if r.abs > 0
hs = (h * h + k * k).quo(2)
asr = Math.asin(r)
(1..lg).each do |i|
[-1, 1].each do |is|
sn = Math.sin(asr * (is * x[i][ng] + 1).quo(2))
bvn += w[i][ng] * Math.exp((sn * hk - hs).quo(1 - sn * sn))
end # do
end # do
bvn *= asr.quo(2 * twopi)
end # if
bvn += Distribution::Normal.cdf(-h) * Distribution::Normal.cdf(-k)
else # r.abs
if r < 0
k = -k
hk = -hk
end
if r.abs < 1
as = (1 - r) * (1 + r)
a = Math.sqrt(as)
bs = (h - k)**2
c = (4 - hk).quo(8)
d = (12 - hk).quo(16)
asr = -(bs.quo(as) + hk).quo(2)
if asr > -100
bvn = a * Math.exp(asr) * (1 - c * (bs - as) * (1 - d * bs.quo(5)).quo(3) + c * d * as * as.quo(5))
end
if -hk < 100
b = Math.sqrt(bs)
bvn -= Math.exp(-hk.quo(2)) * Math.sqrt(twopi) * Distribution::Normal.cdf(-b.quo(a)) * b *
(1 - c * bs * (1 - d * bs.quo(5)).quo(3))
end
a = a.quo(2)
(1..lg).each do |i|
[-1, 1].each do |is|
xs = (a * (is * x[i][ng] + 1))**2
rs = Math.sqrt(1 - xs)
asr = -(bs / xs + hk).quo(2)
if asr > -100
bvn += a * w[i][ng] * Math.exp(asr) *
(Math.exp(-hk * (1 - rs).quo(2 * (1 + rs))) .quo(rs) - (1 + c * xs * (1 + d * xs)))
end
end
end
bvn = -bvn / twopi
end
if r > 0
bvn += Distribution::Normal.cdf(-[h, k].max)
else
bvn = -bvn
if k > h
bvn = bvn + Distribution::Normal.cdf(k) - Distribution::Normal.cdf(h)
end
end
end
bvn
end
private :f, :sgn
end
end
end
end
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