# Module: Distribution::BivariateNormal::Ruby_

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

SIDE =

:nodoc:

`0.1`
LIMIT =

:nodoc:

`5`

## Class Method Summary collapse

• CDF for a given x, y and rho value.

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

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

• CDF.

• .partial_derivative_cdf_x(x, y, rho) ⇒ Object (also: 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.

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

## Class Method Details

### .cdf(a, b, rho) ⇒ Object

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

 ``` 37 38 39``` ```# 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:

 ``` 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267``` ```# 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:

 ``` 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90``` ```# 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
 ``` 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123``` ```# 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

 ``` 29 30 31 32``` ```# 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) ⇒ ObjectAlso 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
 ``` 17 18 19``` ```# 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.

 ``` 24 25 26 27``` ```# 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

 ``` 41 42 43 44 45 46 47``` ```# File 'lib/distribution/bivariatenormal/ruby.rb', line 41 def sgn(x) if (x >= 0) 1 else -1 end end```