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 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

.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 && 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_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 && 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

.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: http://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