/************************************************************************* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier Contributors: * Sergey Bochkanov (ALGLIB project). Translation from C to pseudocode. See subroutines comments for additional copyrights. >>> SOURCE LICENSE >>> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation (www.fsf.org); either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses >>> END OF LICENSE >>> *************************************************************************/ using System; namespace alglib { public class fdistr { /************************************************************************* F distribution Returns the area from zero to x under the F density function (also known as Snedcor's density or the variance ratio density). This is the density of x = (u1/df1)/(u2/df2), where u1 and u2 are random variables having Chi square distributions with df1 and df2 degrees of freedom, respectively. The incomplete beta integral is used, according to the formula P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). The arguments a and b are greater than zero, and x is nonnegative. ACCURACY: Tested at random points (a,b,x). x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 0,100 100000 9.8e-15 1.7e-15 IEEE 1,5 0,100 100000 6.5e-15 3.5e-16 IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12 IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double fdistribution(int a, int b, double x) { double result = 0; double w = 0; System.Diagnostics.Debug.Assert(a>=1 & b>=1 & (double)(x)>=(double)(0), "Domain error in FDistribution"); w = a*x; w = w/(b+w); result = ibetaf.incompletebeta(0.5*a, 0.5*b, w); return result; } /************************************************************************* Complemented F distribution Returns the area from x to infinity under the F density function (also known as Snedcor's density or the variance ratio density). inf. - 1 | | a-1 b-1 1-P(x) = ------ | t (1-t) dt B(a,b) | | - x The incomplete beta integral is used, according to the formula P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). ACCURACY: Tested at random points (a,b,x) in the indicated intervals. x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 1,100 100000 3.7e-14 5.9e-16 IEEE 1,5 1,100 100000 8.0e-15 1.6e-15 IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13 IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double fcdistribution(int a, int b, double x) { double result = 0; double w = 0; System.Diagnostics.Debug.Assert(a>=1 & b>=1 & (double)(x)>=(double)(0), "Domain error in FCDistribution"); w = b/(b+a*x); result = ibetaf.incompletebeta(0.5*b, 0.5*a, w); return result; } /************************************************************************* Inverse of complemented F distribution Finds the F density argument x such that the integral from x to infinity of the F density is equal to the given probability p. This is accomplished using the inverse beta integral function and the relations z = incbi( df2/2, df1/2, p ) x = df2 (1-z) / (df1 z). Note: the following relations hold for the inverse of the uncomplemented F distribution: z = incbi( df1/2, df2/2, p ) x = df2 z / (df1 (1-z)). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between .001 and 1: IEEE 1,100 100000 8.3e-15 4.7e-16 IEEE 1,10000 100000 2.1e-11 1.4e-13 For p between 10^-6 and 10^-3: IEEE 1,100 50000 1.3e-12 8.4e-15 IEEE 1,10000 50000 3.0e-12 4.8e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invfdistribution(int a, int b, double y) { double result = 0; double w = 0; System.Diagnostics.Debug.Assert(a>=1 & b>=1 & (double)(y)>(double)(0) & (double)(y)<=(double)(1), "Domain error in InvFDistribution"); // // Compute probability for x = 0.5 // w = ibetaf.incompletebeta(0.5*b, 0.5*a, 0.5); // // If that is greater than y, then the solution w < .5 // Otherwise, solve at 1-y to remove cancellation in (b - b*w) // if( (double)(w)>(double)(y) | (double)(y)<(double)(0.001) ) { w = ibetaf.invincompletebeta(0.5*b, 0.5*a, y); result = (b-b*w)/(a*w); } else { w = ibetaf.invincompletebeta(0.5*a, 0.5*b, 1.0-y); result = b*w/(a*(1.0-w)); } return result; } } }