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: 0.16220409276889514
Expected: 0.16220409275804 ± 1.0e-9
Call: ::beta::cdf-beta 4.2 17.3 0.5
Error: PASSED
Result: 0.9986307712891124
Expected: 0.998630771123192 ± 1.0e-9
Call: ::beta::cdf-beta 500 375 0.7
Error: PASSED
Result: 0.9999999999999996
Expected: 1.0 ± 1.0e-9
Call: ::beta::cdf-beta 250 760 0.2
Error: PASSED
Result: 0.00012523431867230727
Expected: 0.000125234318666948 ± 1.0e-9
Call: ::beta::cdf-beta 43.2 19.7 0.6
Error: PASSED
Result: 0.07288812920336896
Expected: 0.0728881294218269 ± 1.0e-9
Call: ::beta::cdf-beta 500 640 0.3
Error: PASSED
Result: 2.998725475754239e-23
Expected: 2.99872547567313e-23 ± 1.0e-9
Call: ::beta::cdf-beta 400 640 0.3
Error: PASSED
Result: 3.0705669621484744e-9
Expected: 3.07056696205524e-09 ± 1.0e-9
Call: ::beta::cdf-beta 0.1 30 0.1
Error: PASSED
Result: 0.9986410087106836
Expected: 0.998641008671625 ± 1.0e-9
Call: ::beta::cdf-beta 0.01 0.03 0.9
Error: PASSED
Result: 0.7658650057150924
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: 0.00012523707558061254
Expected: 0.000125237075575121 ± 1.0e-9
Average time/1000 iterations for ::beta::cdf-beta 2 3 0.9999:
76.469 microseconds per iteration
Average time/1000 iterations for ::beta::cdf-beta 250 760 0.47:
1276.926 microseconds per iteration
Average time/1000 iterations for ::beta::cdf-beta 249.9999 759.99999 0.47:
1515.179 microseconds per iteration
Average time/1000 iterations for ::beta::cdf-beta 2.1 3.2 0.7:
165.804 microseconds per iteration
Average time/1000 iterations for ::beta::cdf-beta 4.3 9.2 0.3:
193.342 microseconds per iteration
Average time/1000 iterations for ::beta::cdf-beta 2.5 1.9 0.1:
148.551 microseconds per iteration
Average time/1000 iterations for ::beta::cdf-beta 5 7 0.3:
84.338 microseconds per iterationThe running times are average times over 1000 runs (using time). The results are OK, but not great (compare to a nearly constant rate of about 70 microseconds/iteration for the current implementation of the complete Beta function in tcllib). I've shortened them from significantly longer values in earlier tests. However, it may make sense to use more efficient algorithms, 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 - $tol} {
return 1.0
}
# Adjust to get into regime where convergence is faster
if {$x <= 0.5} {
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}]
if {[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 - $x)}]
} else {
set term 1.0
set z 1.0
set sum 1.0
set n 0
if {$pref > 1} {
set adjtol [expr {$tol * $pref}]
} else {
set adjtol $tol
}
set factor 1.0
while 1 {
set z [expr {$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
}