EKB This is an implementation of the incomplete Beta function Ix(a, b), defined as:
/ x
1 | a-1 b-1
Ix(a,b) = ------- | dt t (1 - t)
B(a, b) |
/ 0where B(a,b) is the Beta function. The incomplete Beta function is the cumulative probability function for the Beta distribution. This code has been tested as part of the tests run on the Beta distribution. Here are the results of the tests, where the expected results were calculated using R:
Call: ::beta::cdf-beta 2.1 3.0 0.2
Error: PASSED
Result: 0.16220409270890568
Expected: 0.16220409275804 ± 1.0e-9
Call: ::beta::cdf-beta 4.2 17.3 0.5
Error: PASSED
Result: 0.998630771122991
Expected: 0.998630771123192 ± 1.0e-9
Call: ::beta::cdf-beta 500 375 0.7
Error: PASSED
Result: 1.0
Expected: 1.0 ± 1.0e-9
Call: ::beta::cdf-beta 250 760 0.2
Error: PASSED
Result: 0.0001252342758440994
Expected: 0.000125234318666948 ± 1.0e-9
Call: ::beta::cdf-beta 43.2 19.7 0.6
Error: PASSED
Result: 0.07288812943897244
Expected: 0.0728881294218269 ± 1.0e-9
Call: ::beta::cdf-beta 500 640 0.3
Error: PASSED
Result: 0.0
Expected: 2.99872547567313e-23 ± 1.0e-9
Call: ::beta::cdf-beta 400 640 0.3
Error: PASSED
Result: 3.040258600428558e-9
Expected: 3.07056696205524e-09 ± 1.0e-9
Call: ::beta::cdf-beta 0.1 30 0.1
Error: PASSED
Result: 0.9986410086716231
Expected: 0.998641008671625 ± 1.0e-9
Call: ::beta::cdf-beta 0.01 0.03 0.9
Error: PASSED
Result: 0.7658650057031781
Expected: 0.765865005703006 ± 1.0e-9
Call: ::beta::cdf-beta 2 3 0.9999
Error: PASSED
Result: 0.9999999999960003
Expected: 0.999999999996 ± 1.0e-9
Call: ::beta::cdf-beta 249.9999 759.99999 0.2
Error: PASSED
Result: 0.00012523703106570583
Expected: 0.000125237075575121 ± 1.0e-9
Time for ::beta::cdf-beta 2 3 0.9999:
81.78 microseconds per iteration
Time for ::beta::cdf-beta 250 760 0.2:
455.21 microseconds per iteration
Time for ::beta::cdf-beta 249.9999 759.99999 0.2:
3827.91 microseconds per iterationThe time for the last example is not good. it would take 3 seconds to run 1000 values, so performance at larger values of a & b does not appear to be very good (unless b is an integer, in which case there's a shortcut). More efficient algorithms are available, such as the one used in DCDFLIB (http://people.scs.fsu.edu/~burkardt/f_src/dcdflib/dcdflib.html ).
Here's the code:
package require math
namespace import ::math::ln_Gamma
namespace import ::math::Beta
#
# Implement the incomplete beta function Ix(a, b)
#
proc incompleteBeta {a b x {tol 1.0e-9}} {
if {$x < 0.0 || $x > 1.0} {
error "Value out of range in incomplete Beta function: x = $x, not in \[0, 1\]"
}
if {$a <= 0.0} {
error "Value out of range in incomplete Beta function: a = $a, must be > 0"
}
if {$b <= 0.0} {
error "Value out of range in incomplete Beta function: b = $b, must be > 0"
}
if {$x < $tol} {
return 0.0
}
if {$x > 1.0 - $tol} {
return 1.0
}
# Adjust to get into regime where convergence is faster
if {$a >= $b && $x < 0.9} {
return [incBeta_series $a $b $x $tol]
} else {
set z [incBeta_series $b $a [expr {1.0 - $x}] $tol]
return [expr {1.0 - $z}]
}
}
#####################################################
#
# Series expansion for Ix(a, b)
#
# Abramowitz & Stegun formula 26.5.4
#
# This series has terms
#
# [B(a+1,n)/B(a+b,n)] * x^n
#
# This is guaranteed to converge only if
# b < 1. Use the recurrence formula 26.5.15
# from A&S to bring b below 1. Recurrence is:
#
# Ix(a,b) = (G(a+b)/(G(a+1)G(b))) * x^a * (1-x)^(b-1) +
# Ix(a+1, b-1)
#
# Also, use B(a,b) = G(a)G(b)/G(a+b) to
# rewrite coeff as
#
# Cn = [G(a+1)/G(a+b)] * G(a+b+n)/G(a+n+1)
#
#####################################################
proc incBeta_series {a b x tol} {
# a+b is invariant under recurrence formula
set aplusb [expr {$a + $b}]
set lnGapb [ln_Gamma $aplusb]
# Calculate for convenience -- these don't change
set lnx [expr {log($x)}]
set ln1mx [expr {log(1.0 - $x)}]
set z [expr {$x / (1.0 - $x)}]
set retval 0.0
# Pack everything in one big exp, since individual terms can over/underflow
set factor [expr {exp($lnGapb - [ln_Gamma [expr {$a + 1}]] - [ln_Gamma $b] + \
$a * $lnx + ($b - 1) * $ln1mx)}]
while {$b > 1} {
set retval [expr {$retval + $factor}]
set a [expr {$a + 1}]
set b [expr {$b - 1}]
set factor [expr {$factor * $z * double($b)/double($a)}]
}
set pref_num [expr {pow($x, $a) * pow(1.0 - $x, $b)}]
set pref_denom [expr {$a * [Beta $a $b]}]
set pref [expr {$pref_num/$pref_denom}]
set adjtol [expr {$tol * $pref}]
if {$adjtol == 0.0} {
# Is pref so small that it is evaluated at exactly zero?
set sum 0.0
} elseif {[string is integer $b] && $b == 1} {
# In the case of integer b, this reduces to 1 + x + x^2 + ... = 1/(1-x)
# From routine above, b must be 1, but check
set sum [expr {1.0/(1.0 - $x)}]
} else {
set term 1.0
set z 1.0
set sum 1.0
set n 0
set factor 1.0
while 1 {
set z [expr {$z * $x}]
set abn [expr {$a + $n + $b}]
set an1 [expr {$a + $n + 1}]
set factor [expr {$factor * double($abn)/double($an1)}]
incr n
set nexterm [expr {$z * $factor}]
if {abs($nexterm - $term) < $adjtol} {
break
}
set term $nexterm
set sum [expr {$sum + $term}]
}
}
# Add previous cumulative sum to the series expansion
set retval [expr {$retval + $pref * $sum}]
# Because of imprecision in underlying Tcl calculations, may fall out of bounds
if {$retval < 0.0} {
set retval 0.0
} elseif {$retval > 1.0} {
set retval 1.0
}
return $retval
}