Module: Distribution::NormalBivariate

Defined in:
lib/distribution/normalbivariate.rb

Overview

Calculate pdf and cdf for bivariate normal distribution.

Pdf if easy to calculate, but CDF is not trivial. Several papers describe methods to calculate the integral.

Three methods are implemented on this module:

  • Genz

    Used by default, with improvement to calculate p on rho > 0.95

  • Hull

    Port from a C++ code

  • Jantaravareerat

    Iterative (and slow)

Constant Summary collapse

SIDE =

:nodoc:

0.1
LIMIT =

:nodoc:

5

Class Method Summary collapse

Class Method Details

.cdf(a, b, rho) ⇒ Object

CDF for a given x, y and rho value. Uses Genz algorithm (cdf_genz method).



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# File 'lib/distribution/normalbivariate.rb', line 35

def cdf(a,b,rho)
  cdf_genz(a,b,rho)
end

.cdf_genz(x, y, rho) ⇒ Object

Normal cumulative distribution function (cdf) for a given x, y and rho. Based on 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:



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# File 'lib/distribution/normalbivariate.rb', line 143

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

.cdf_hull(a, b, rho) ⇒ Object

Normal cumulative distribution function (cdf) for a given x, y and rho. Based on Hull (1993, cited by Arne, 2003)

References:



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# File 'lib/distribution/normalbivariate.rb', line 52

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_jantaravareerat(x, y, rho, s1 = 1, s2 = 1) ⇒ Object

CDF. Iterative method by Jantaravareerat (n/d)

Reference:

  • Jantaravareerat, M. & Thomopoulos, N. (n/d). Tables for standard bivariate normal distribution



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# File 'lib/distribution/normalbivariate.rb', line 95

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

.f(x, y, aprime, bprime, rho) ⇒ Object 



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# File 'lib/distribution/normalbivariate.rb', line 27

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

.pdf(x, y, rho, s1 = 1.0, s2 = 1.0) ⇒ Object

Probability density function for a given x, y and rho value.

Source: en.wikipedia.org/wiki/Multivariate_normal_distribution



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# File 'lib/distribution/normalbivariate.rb', line 22

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

.sgn(x) ⇒ Object 



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# File 'lib/distribution/normalbivariate.rb', line 39

def sgn(x)
  if(x>=0)
  1
  else
  -1
  end     
end