Module: Distribution::BivariateNormal::Ruby_

Defined in:
lib/distribution/bivariatenormal/ruby.rb

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/bivariatenormal/ruby.rb', line 37

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. 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:



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# File 'lib/distribution/bivariatenormal/ruby.rb', line 145

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/bivariatenormal/ruby.rb', line 54

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/bivariatenormal/ruby.rb', line 97

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/bivariatenormal/ruby.rb', line 29

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

.partial_derivative_cdf_x(x, y, rho) ⇒ Object Also known as: pd_cdf_x

Return the partial derivative of cdf over x, with y and rho constant Reference:

  • Tallis, 1962, p.346, cited by Olsson, 1979



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# File 'lib/distribution/bivariatenormal/ruby.rb', line 17

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

.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/bivariatenormal/ruby.rb', line 24

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/bivariatenormal/ruby.rb', line 41

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