package weka.core;
/**
* Class implementing some distributions, tests, etc. The code is mostly adapted from the CERN
* Jet Java libraries:
*
* Copyright 2001 University of Waikato
* Copyright 1999 CERN - European Organization for Nuclear Research.
* Permission to use, copy, modify, distribute and sell this software and its documentation for
* any purpose is hereby granted without fee, provided that the above copyright notice appear
* in all copies and that both that copyright notice and this permission notice appear in
* supporting documentation.
* CERN and the University of Waikato make no representations about the suitability of this
* software for any purpose. It is provided "as is" without expressed or implied warranty.
*
* @author peter.gedeck@pharma.Novartis.com
* @author wolfgang.hoschek@cern.ch
* @author Eibe Frank (eibe@cs.waikato.ac.nz)
* @author Richard Kirkby (rkirkby@cs.waikato.ac.nz)
* @version $Revision: 1.9 $
*/
public class Statistics {
/** Some constants */
protected static final double MACHEP = 1.11022302462515654042E-16;
protected static final double MAXLOG = 7.09782712893383996732E2;
protected static final double MINLOG = -7.451332191019412076235E2;
protected static final double MAXGAM = 171.624376956302725;
protected static final double SQTPI = 2.50662827463100050242E0;
protected static final double SQRTH = 7.07106781186547524401E-1;
protected static final double LOGPI = 1.14472988584940017414;
protected static final double big = 4.503599627370496e15;
protected static final double biginv = 2.22044604925031308085e-16;
/*************************************************
* COEFFICIENTS FOR METHOD normalInverse() *
*************************************************/
/* approximation for 0 <= |y - 0.5| <= 3/8 */
protected static final double P0[] = {
-5.99633501014107895267E1,
9.80010754185999661536E1,
-5.66762857469070293439E1,
1.39312609387279679503E1,
-1.23916583867381258016E0,
};
protected static final double Q0[] = {
/* 1.00000000000000000000E0,*/
1.95448858338141759834E0,
4.67627912898881538453E0,
8.63602421390890590575E1,
-2.25462687854119370527E2,
2.00260212380060660359E2,
-8.20372256168333339912E1,
1.59056225126211695515E1,
-1.18331621121330003142E0,
};
/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8
* i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
*/
protected static final double P1[] = {
4.05544892305962419923E0,
3.15251094599893866154E1,
5.71628192246421288162E1,
4.40805073893200834700E1,
1.46849561928858024014E1,
2.18663306850790267539E0,
-1.40256079171354495875E-1,
-3.50424626827848203418E-2,
-8.57456785154685413611E-4,
};
protected static final double Q1[] = {
/* 1.00000000000000000000E0,*/
1.57799883256466749731E1,
4.53907635128879210584E1,
4.13172038254672030440E1,
1.50425385692907503408E1,
2.50464946208309415979E0,
-1.42182922854787788574E-1,
-3.80806407691578277194E-2,
-9.33259480895457427372E-4,
};
/* Approximation for interval z = sqrt(-2 log y ) between 8 and 64
* i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
*/
protected static final double P2[] = {
3.23774891776946035970E0,
6.91522889068984211695E0,
3.93881025292474443415E0,
1.33303460815807542389E0,
2.01485389549179081538E-1,
1.23716634817820021358E-2,
3.01581553508235416007E-4,
2.65806974686737550832E-6,
6.23974539184983293730E-9,
};
protected static final double Q2[] = {
/* 1.00000000000000000000E0,*/
6.02427039364742014255E0,
3.67983563856160859403E0,
1.37702099489081330271E0,
2.16236993594496635890E-1,
1.34204006088543189037E-2,
3.28014464682127739104E-4,
2.89247864745380683936E-6,
6.79019408009981274425E-9,
};
/**
* Computes standard error for observed values of a binomial
* random variable.
*
* @param p the probability of success
* @param n the size of the sample
* @return the standard error
*/
public static double binomialStandardError(double p, int n) {
if (n == 0) {
return 0;
}
return Math.sqrt((p*(1-p))/(double) n);
}
/**
* Returns chi-squared probability for given value and degrees
* of freedom. (The probability that the chi-squared variate
* will be greater than x for the given degrees of freedom.)
*
* @param x the value
* @param v the number of degrees of freedom
* @return the chi-squared probability
*/
public static double chiSquaredProbability(double x, double v) {
if( x < 0.0 || v < 1.0 ) return 0.0;
return incompleteGammaComplement( v/2.0, x/2.0 );
}
/**
* Computes probability of F-ratio.
*
* @param F the F-ratio
* @param df1 the first number of degrees of freedom
* @param df2 the second number of degrees of freedom
* @return the probability of the F-ratio.
*/
public static double FProbability(double F, int df1, int df2) {
return incompleteBeta( df2/2.0, df1/2.0, df2/(df2+df1*F) );
}
/**
* Returns the area under the Normal (Gaussian) probability density
* function, integrated from minus infinity to <tt>x</tt>
* (assumes mean is zero, variance is one).
* <pre>
* x
* -
* 1 | | 2
* normal(x) = --------- | exp( - t /2 ) dt
* sqrt(2pi) | |
* -
* -inf.
*
* = ( 1 + erf(z) ) / 2
* = erfc(z) / 2
* </pre>
* where <tt>z = x/sqrt(2)</tt>.
* Computation is via the functions <tt>errorFunction</tt> and <tt>errorFunctionComplement</tt>.
*
* @param a the z-value
* @return the probability of the z value according to the normal pdf
*/
public static double normalProbability(double a) {
double x, y, z;
x = a * SQRTH;
z = Math.abs(x);
if( z < SQRTH ) y = 0.5 + 0.5 * errorFunction(x);
else {
y = 0.5 * errorFunctionComplemented(z);
if( x > 0 ) y = 1.0 - y;
}
return y;
}
/**
* Returns the value, <tt>x</tt>, for which the area under the
* Normal (Gaussian) probability density function (integrated from
* minus infinity to <tt>x</tt>) is equal to the argument <tt>y</tt>
* (assumes mean is zero, variance is one).
* <p>
* For small arguments <tt>0 < y < exp(-2)</tt>, the program computes
* <tt>z = sqrt( -2.0 * log(y) )</tt>; then the approximation is
* <tt>x = z - log(z)/z - (1/z) P(1/z) / Q(1/z)</tt>.
* There are two rational functions P/Q, one for <tt>0 < y < exp(-32)</tt>
* and the other for <tt>y</tt> up to <tt>exp(-2)</tt>.
* For larger arguments,
* <tt>w = y - 0.5</tt>, and <tt>x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2))</tt>.
*
* @param y0 the area under the normal pdf
* @return the z-value
*/
public static double normalInverse(double y0) {
double x, y, z, y2, x0, x1;
int code;
final double s2pi = Math.sqrt(2.0*Math.PI);
if( y0 <= 0.0 ) throw new IllegalArgumentException();
if( y0 >= 1.0 ) throw new IllegalArgumentException();
code = 1;
y = y0;
if( y > (1.0 - 0.13533528323661269189) ) { /* 0.135... = exp(-2) */
y = 1.0 - y;
code = 0;
}
if( y > 0.13533528323661269189 ) {
y = y - 0.5;
y2 = y * y;
x = y + y * (y2 * polevl( y2, P0, 4)/p1evl( y2, Q0, 8 ));
x = x * s2pi;
return(x);
}
x = Math.sqrt( -2.0 * Math.log(y) );
x0 = x - Math.log(x)/x;
z = 1.0/x;
if( x < 8.0 ) /* y > exp(-32) = 1.2664165549e-14 */
x1 = z * polevl( z, P1, 8 )/p1evl( z, Q1, 8 );
else
x1 = z * polevl( z, P2, 8 )/p1evl( z, Q2, 8 );
x = x0 - x1;
if( code != 0 )
x = -x;
return( x );
}
/**
* Returns natural logarithm of gamma function.
*
* @param x the value
* @return natural logarithm of gamma function
*/
public static double lnGamma(double x) {
double p, q, w, z;
double A[] = {
8.11614167470508450300E-4,
-5.95061904284301438324E-4,
7.93650340457716943945E-4,
-2.77777777730099687205E-3,
8.33333333333331927722E-2
};
double B[] = {
-1.37825152569120859100E3,
-3.88016315134637840924E4,
-3.31612992738871184744E5,
-1.16237097492762307383E6,
-1.72173700820839662146E6,
-8.53555664245765465627E5
};
double C[] = {
/* 1.00000000000000000000E0, */
-3.51815701436523470549E2,
-1.70642106651881159223E4,
-2.20528590553854454839E5,
-1.13933444367982507207E6,
-2.53252307177582951285E6,
-2.01889141433532773231E6
};
if( x < -34.0 ) {
q = -x;
w = lnGamma(q);
p = Math.floor(q);
if( p == q ) throw new ArithmeticException("lnGamma: Overflow");
z = q - p;
if( z > 0.5 ) {
p += 1.0;
z = p - q;
}
z = q * Math.sin( Math.PI * z );
if( z == 0.0 ) throw new
ArithmeticException("lnGamma: Overflow");
z = LOGPI - Math.log( z ) - w;
return z;
}
if( x < 13.0 ) {
z = 1.0;
while( x >= 3.0 ) {
x -= 1.0;
z *= x;
}
while( x < 2.0 ) {
if( x == 0.0 ) throw new
ArithmeticException("lnGamma: Overflow");
z /= x;
x += 1.0;
}
if( z < 0.0 ) z = -z;
if( x == 2.0 ) return Math.log(z);
x -= 2.0;
p = x * polevl( x, B, 5 ) / p1evl( x, C, 6);
return( Math.log(z) + p );
}
if( x > 2.556348e305 ) throw new ArithmeticException("lnGamma: Overflow");
q = ( x - 0.5 ) * Math.log(x) - x + 0.91893853320467274178;
if( x > 1.0e8 ) return( q );
p = 1.0/(x*x);
if( x >= 1000.0 )
q += (( 7.9365079365079365079365e-4 * p
- 2.7777777777777777777778e-3) *p
+ 0.0833333333333333333333) / x;
else
q += polevl( p, A, 4 ) / x;
return q;
}
/**
* Returns the error function of the normal distribution.
* The integral is
* <pre>
* x
* -
* 2 | | 2
* erf(x) = -------- | exp( - t ) dt.
* sqrt(pi) | |
* -
* 0
* </pre>
* <b>Implementation:</b>
* For <tt>0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2)</tt>; otherwise
* <tt>erf(x) = 1 - erfc(x)</tt>.
* <p>
* Code adapted from the <A HREF="http://www.sci.usq.edu.au/staff/leighb/graph/Top.html">
* Java 2D Graph Package 2.4</A>,
* which in turn is a port from the
* <A HREF="http://people.ne.mediaone.net/moshier/index.html#Cephes">Cephes 2.2</A>
* Math Library (C).
*
* @param a the argument to the function.
*/
static double errorFunction(double x) {
double y, z;
final double T[] = {
9.60497373987051638749E0,
9.00260197203842689217E1,
2.23200534594684319226E3,
7.00332514112805075473E3,
5.55923013010394962768E4
};
final double U[] = {
//1.00000000000000000000E0,
3.35617141647503099647E1,
5.21357949780152679795E2,
4.59432382970980127987E3,
2.26290000613890934246E4,
4.92673942608635921086E4
};
if( Math.abs(x) > 1.0 ) return( 1.0 - errorFunctionComplemented(x) );
z = x * x;
y = x * polevl( z, T, 4 ) / p1evl( z, U, 5 );
return y;
}
/**
* Returns the complementary Error function of the normal distribution.
* <pre>
* 1 - erf(x) =
*
* inf.
* -
* 2 | | 2
* erfc(x) = -------- | exp( - t ) dt
* sqrt(pi) | |
* -
* x
* </pre>
* <b>Implementation:</b>
* For small x, <tt>erfc(x) = 1 - erf(x)</tt>; otherwise rational
* approximations are computed.
* <p>
* Code adapted from the <A HREF="http://www.sci.usq.edu.au/staff/leighb/graph/Top.html">
* Java 2D Graph Package 2.4</A>,
* which in turn is a port from the
* <A HREF="http://people.ne.mediaone.net/moshier/index.html#Cephes">Cephes 2.2</A>
* Math Library (C).
*
* @param a the argument to the function.
*/
static double errorFunctionComplemented(double a) {
double x,y,z,p,q;
double P[] = {
2.46196981473530512524E-10,
5.64189564831068821977E-1,
7.46321056442269912687E0,
4.86371970985681366614E1,
1.96520832956077098242E2,
5.26445194995477358631E2,
9.34528527171957607540E2,
1.02755188689515710272E3,
5.57535335369399327526E2
};
double Q[] = {
//1.0
1.32281951154744992508E1,
8.67072140885989742329E1,
3.54937778887819891062E2,
9.75708501743205489753E2,
1.82390916687909736289E3,
2.24633760818710981792E3,
1.65666309194161350182E3,
5.57535340817727675546E2
};
double R[] = {
5.64189583547755073984E-1,
1.27536670759978104416E0,
5.01905042251180477414E0,
6.16021097993053585195E0,
7.40974269950448939160E0,
2.97886665372100240670E0
};
double S[] = {
//1.00000000000000000000E0,
2.26052863220117276590E0,
9.39603524938001434673E0,
1.20489539808096656605E1,
1.70814450747565897222E1,
9.60896809063285878198E0,
3.36907645100081516050E0
};
if( a < 0.0 ) x = -a;
else x = a;
if( x < 1.0 ) return 1.0 - errorFunction(a);
z = -a * a;
if( z < -MAXLOG ) {
if( a < 0 ) return( 2.0 );
else return( 0.0 );
}
z = Math.exp(z);
if( x < 8.0 ) {
p = polevl( x, P, 8 );
q = p1evl( x, Q, 8 );
} else {
p = polevl( x, R, 5 );
q = p1evl( x, S, 6 );
}
y = (z * p)/q;
if( a < 0 ) y = 2.0 - y;
if( y == 0.0 ) {
if( a < 0 ) return 2.0;
else return( 0.0 );
}
return y;
}
/**
* Evaluates the given polynomial of degree <tt>N</tt> at <tt>x</tt>.
* Evaluates polynomial when coefficient of N is 1.0.
* Otherwise same as <tt>polevl()</tt>.
* <pre>
* 2 N
* y = C + C x + C x +...+ C x
* 0 1 2 N
*
* Coefficients are stored in reverse order:
*
* coef[0] = C , ..., coef[N] = C .
* N 0
* </pre>
* The function <tt>p1evl()</tt> assumes that <tt>coef[N] = 1.0</tt> and is
* omitted from the array. Its calling arguments are
* otherwise the same as <tt>polevl()</tt>.
* <p>
* In the interest of speed, there are no checks for out of bounds arithmetic.
*
* @param x argument to the polynomial.
* @param coef the coefficients of the polynomial.
* @param N the degree of the polynomial.
*/
static double p1evl( double x, double coef[], int N ) {
double ans;
ans = x + coef[0];
for(int i=1; i<N; i++) ans = ans*x+coef[i];
return ans;
}
/**
* Evaluates the given polynomial of degree <tt>N</tt> at <tt>x</tt>.
* <pre>
* 2 N
* y = C + C x + C x +...+ C x
* 0 1 2 N
*
* Coefficients are stored in reverse order:
*
* coef[0] = C , ..., coef[N] = C .
* N 0
* </pre>
* In the interest of speed, there are no checks for out of bounds arithmetic.
*
* @param x argument to the polynomial.
* @param coef the coefficients of the polynomial.
* @param N the degree of the polynomial.
*/
static double polevl( double x, double coef[], int N ) {
double ans;
ans = coef[0];
for(int i=1; i<=N; i++) ans = ans*x+coef[i];
return ans;
}
/**
* Returns the Incomplete Gamma function.
* @param a the parameter of the gamma distribution.
* @param x the integration end point.
*/
static double incompleteGamma(double a, double x)
{
double ans, ax, c, r;
if( x <= 0 || a <= 0 ) return 0.0;
if( x > 1.0 && x > a ) return 1.0 - incompleteGammaComplement(a,x);
/* Compute x**a * exp(-x) / gamma(a) */
ax = a * Math.log(x) - x - lnGamma(a);
if( ax < -MAXLOG ) return( 0.0 );
ax = Math.exp(ax);
/* power series */
r = a;
c = 1.0;
ans = 1.0;
do {
r += 1.0;
c *= x/r;
ans += c;
}
while( c/ans > MACHEP );
return( ans * ax/a );
}
/**
* Returns the Complemented Incomplete Gamma function.
* @param a the parameter of the gamma distribution.
* @param x the integration start point.
*/
static double incompleteGammaComplement( double a, double x ) {
double ans, ax, c, yc, r, t, y, z;
double pk, pkm1, pkm2, qk, qkm1, qkm2;
if( x <= 0 || a <= 0 ) return 1.0;
if( x < 1.0 || x < a ) return 1.0 - incompleteGamma(a,x);
ax = a * Math.log(x) - x - lnGamma(a);
if( ax < -MAXLOG ) return 0.0;
ax = Math.exp(ax);
/* continued fraction */
y = 1.0 - a;
z = x + y + 1.0;
c = 0.0;
pkm2 = 1.0;
qkm2 = x;
pkm1 = x + 1.0;
qkm1 = z * x;
ans = pkm1/qkm1;
do {
c += 1.0;
y += 1.0;
z += 2.0;
yc = y * c;
pk = pkm1 * z - pkm2 * yc;
qk = qkm1 * z - qkm2 * yc;
if( qk != 0 ) {
r = pk/qk;
t = Math.abs( (ans - r)/r );
ans = r;
} else
t = 1.0;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if( Math.abs(pk) > big ) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
} while( t > MACHEP );
return ans * ax;
}
/**
* Returns the Gamma function of the argument.
*/
static double gamma(double x) {
double P[] = {
1.60119522476751861407E-4,
1.19135147006586384913E-3,
1.04213797561761569935E-2,
4.76367800457137231464E-2,
2.07448227648435975150E-1,
4.94214826801497100753E-1,
9.99999999999999996796E-1
};
double Q[] = {
-2.31581873324120129819E-5,
5.39605580493303397842E-4,
-4.45641913851797240494E-3,
1.18139785222060435552E-2,
3.58236398605498653373E-2,
-2.34591795718243348568E-1,
7.14304917030273074085E-2,
1.00000000000000000320E0
};
double p, z;
int i;
double q = Math.abs(x);
if( q > 33.0 ) {
if( x < 0.0 ) {
p = Math.floor(q);
if( p == q ) throw new ArithmeticException("gamma: overflow");
i = (int)p;
z = q - p;
if( z > 0.5 ) {
p += 1.0;
z = q - p;
}
z = q * Math.sin( Math.PI * z );
if( z == 0.0 ) throw new ArithmeticException("gamma: overflow");
z = Math.abs(z);
z = Math.PI/(z * stirlingFormula(q) );
return -z;
} else {
return stirlingFormula(x);
}
}
z = 1.0;
while( x >= 3.0 ) {
x -= 1.0;
z *= x;
}
while( x < 0.0 ) {
if( x == 0.0 ) {
throw new ArithmeticException("gamma: singular");
} else
if( x > -1.E-9 ) {
return( z/((1.0 + 0.5772156649015329 * x) * x) );
}
z /= x;
x += 1.0;
}
while( x < 2.0 ) {
if( x == 0.0 ) {
throw new ArithmeticException("gamma: singular");
} else
if( x < 1.e-9 ) {
return( z/((1.0 + 0.5772156649015329 * x) * x) );
}
z /= x;
x += 1.0;
}
if( (x == 2.0) || (x == 3.0) ) return z;
x -= 2.0;
p = polevl( x, P, 6 );
q = polevl( x, Q, 7 );
return z * p / q;
}
/**
* Returns the Gamma function computed by Stirling's formula.
* The polynomial STIR is valid for 33 <= x <= 172.
*/
static double stirlingFormula(double x) {
double STIR[] = {
7.87311395793093628397E-4,
-2.29549961613378126380E-4,
-2.68132617805781232825E-3,
3.47222221605458667310E-3,
8.33333333333482257126E-2,
};
double MAXSTIR = 143.01608;
double w = 1.0/x;
double y = Math.exp(x);
w = 1.0 + w * polevl( w, STIR, 4 );
if( x > MAXSTIR ) {
/* Avoid overflow in Math.pow() */
double v = Math.pow( x, 0.5 * x - 0.25 );
y = v * (v / y);
} else {
y = Math.pow( x, x - 0.5 ) / y;
}
y = SQTPI * y * w;
return y;
}
/**
* Returns the Incomplete Beta Function evaluated from zero to <tt>xx</tt>.
*
* @param aa the alpha parameter of the beta distribution.
* @param bb the beta parameter of the beta distribution.
* @param xx the integration end point.
*/
public static double incompleteBeta( double aa, double bb, double xx ) {
double a, b, t, x, xc, w, y;
boolean flag;
if( aa <= 0.0 || bb <= 0.0 ) throw new
ArithmeticException("ibeta: Domain error!");
if( (xx <= 0.0) || ( xx >= 1.0) ) {
if( xx == 0.0 ) return 0.0;
if( xx == 1.0 ) return 1.0;
throw new ArithmeticException("ibeta: Domain error!");
}
flag = false;
if( (bb * xx) <= 1.0 && xx <= 0.95) {
t = powerSeries(aa, bb, xx);
return t;
}
w = 1.0 - xx;
/* Reverse a and b if x is greater than the mean. */
if( xx > (aa/(aa+bb)) ) {
flag = true;
a = bb;
b = aa;
xc = xx;
x = w;
} else {
a = aa;
b = bb;
xc = w;
x = xx;
}
if( flag && (b * x) <= 1.0 && x <= 0.95) {
t = powerSeries(a, b, x);
if( t <= MACHEP ) t = 1.0 - MACHEP;
else t = 1.0 - t;
return t;
}
/* Choose expansion for better convergence. */
y = x * (a+b-2.0) - (a-1.0);
if( y < 0.0 )
w = incompleteBetaFraction1( a, b, x );
else
w = incompleteBetaFraction2( a, b, x ) / xc;
/* Multiply w by the factor
a b _ _ _
x (1-x) | (a+b) / ( a | (a) | (b) ) . */
y = a * Math.log(x);
t = b * Math.log(xc);
if( (a+b) < MAXGAM && Math.abs(y) < MAXLOG && Math.abs(t) < MAXLOG ) {
t = Math.pow(xc,b);
t *= Math.pow(x,a);
t /= a;
t *= w;
t *= gamma(a+b) / (gamma(a) * gamma(b));
if( flag ) {
if( t <= MACHEP ) t = 1.0 - MACHEP;
else t = 1.0 - t;
}
return t;
}
/* Resort to logarithms. */
y += t + lnGamma(a+b) - lnGamma(a) - lnGamma(b);
y += Math.log(w/a);
if( y < MINLOG )
t = 0.0;
else
t = Math.exp(y);
if( flag ) {
if( t <= MACHEP ) t = 1.0 - MACHEP;
else t = 1.0 - t;
}
return t;
}
/**
* Continued fraction expansion #1 for incomplete beta integral.
*/
static double incompleteBetaFraction1( double a, double b, double x ) {
double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
double k1, k2, k3, k4, k5, k6, k7, k8;
double r, t, ans, thresh;
int n;
k1 = a;
k2 = a + b;
k3 = a;
k4 = a + 1.0;
k5 = 1.0;
k6 = b - 1.0;
k7 = k4;
k8 = a + 2.0;
pkm2 = 0.0;
qkm2 = 1.0;
pkm1 = 1.0;
qkm1 = 1.0;
ans = 1.0;
r = 1.0;
n = 0;
thresh = 3.0 * MACHEP;
do {
xk = -( x * k1 * k2 )/( k3 * k4 );
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
xk = ( x * k5 * k6 )/( k7 * k8 );
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if( qk != 0 ) r = pk/qk;
if( r != 0 ) {
t = Math.abs( (ans - r)/r );
ans = r;
} else
t = 1.0;
if( t < thresh ) return ans;
k1 += 1.0;
k2 += 1.0;
k3 += 2.0;
k4 += 2.0;
k5 += 1.0;
k6 -= 1.0;
k7 += 2.0;
k8 += 2.0;
if( (Math.abs(qk) + Math.abs(pk)) > big ) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
if( (Math.abs(qk) < biginv) || (Math.abs(pk) < biginv) ) {
pkm2 *= big;
pkm1 *= big;
qkm2 *= big;
qkm1 *= big;
}
} while( ++n < 300 );
return ans;
}
/**
* Continued fraction expansion #2 for incomplete beta integral.
*/
static double incompleteBetaFraction2( double a, double b, double x ) {
double xk, pk, pkm1, pkm2, qk, qkm1, qkm2;
double k1, k2, k3, k4, k5, k6, k7, k8;
double r, t, ans, z, thresh;
int n;
k1 = a;
k2 = b - 1.0;
k3 = a;
k4 = a + 1.0;
k5 = 1.0;
k6 = a + b;
k7 = a + 1.0;;
k8 = a + 2.0;
pkm2 = 0.0;
qkm2 = 1.0;
pkm1 = 1.0;
qkm1 = 1.0;
z = x / (1.0-x);
ans = 1.0;
r = 1.0;
n = 0;
thresh = 3.0 * MACHEP;
do {
xk = -( z * k1 * k2 )/( k3 * k4 );
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
xk = ( z * k5 * k6 )/( k7 * k8 );
pk = pkm1 + pkm2 * xk;
qk = qkm1 + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if( qk != 0 ) r = pk/qk;
if( r != 0 ) {
t = Math.abs( (ans - r)/r );
ans = r;
} else
t = 1.0;
if( t < thresh ) return ans;
k1 += 1.0;
k2 -= 1.0;
k3 += 2.0;
k4 += 2.0;
k5 += 1.0;
k6 += 1.0;
k7 += 2.0;
k8 += 2.0;
if( (Math.abs(qk) + Math.abs(pk)) > big ) {
pkm2 *= biginv;
pkm1 *= biginv;
qkm2 *= biginv;
qkm1 *= biginv;
}
if( (Math.abs(qk) < biginv) || (Math.abs(pk) < biginv) ) {
pkm2 *= big;
pkm1 *= big;
qkm2 *= big;
qkm1 *= big;
}
} while( ++n < 300 );
return ans;
}
/**
* Power series for incomplete beta integral.
* Use when b*x is small and x not too close to 1.
*/
static double powerSeries( double a, double b, double x ) {
double s, t, u, v, n, t1, z, ai;
ai = 1.0 / a;
u = (1.0 - b) * x;
v = u / (a + 1.0);
t1 = v;
t = u;
n = 2.0;
s = 0.0;
z = MACHEP * ai;
while( Math.abs(v) > z ) {
u = (n - b) * x / n;
t *= u;
v = t / (a + n);
s += v;
n += 1.0;
}
s += t1;
s += ai;
u = a * Math.log(x);
if( (a+b) < MAXGAM && Math.abs(u) < MAXLOG ) {
t = gamma(a+b)/(gamma(a)*gamma(b));
s = s * t * Math.pow(x,a);
} else {
t = lnGamma(a+b) - lnGamma(a) - lnGamma(b) + u + Math.log(s);
if( t < MINLOG ) s = 0.0;
else