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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 :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 and b<=0 and 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=sum*(Math::sqrt(1.0-rho**2).quo(Math::PI))
return sum
elsif(a*b*rho<=0.0)
#puts "ruta 2"
if(a<=0 and b>=0 and rho>=0)
return Distribution::Normal.cdf(a) - cdf(a,-b,-rho)
elsif (a>=0.0 and b<=0.0 and rho>=0)
return Distribution::Normal.cdf(b) - cdf(-a,b,-rho)
elsif (a>=0.0 and b>=0.0 and 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
raise "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 and y>LIMIT
return 0 if x<-LIMIT or 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 {|i|
w[i][1]=data[(i-1)*2]
x[i][1]=data[(i-1)*2+1]
}
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 {|i|
w[i][2]=data[(i-1)*2]
x[i][2]=data[(i-1)*2+1]
}
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 {|i|
w[i][3]=data[(i-1)*2]
x[i][3]=data[(i-1)*2+1]
}
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 = bvn + w[i][ng] * Math::exp( ( sn*hk-hs ).quo( 1-sn*sn ) )
end # do
end # do
bvn = bvn*asr.quo( 2*twopi )
end # if
bvn = 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 = 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 = 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 = 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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