package org.apache.samoa.moa.core;
/*
* #%L
* SAMOA
* %%
* Copyright (C) 2014 - 2015 Apache Software Foundation
* %%
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
* #L%
*/
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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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.
*/
public 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
s = Math.exp(t);
}
return s;
}
}