BoundsOnBinomialProportions.java
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package org.apache.datasketches.common;
/**
* Confidence intervals for binomial proportions.
*
* <p>This class computes an approximation to the Clopper-Pearson confidence interval
* for a binomial proportion. Exact Clopper-Pearson intervals are strictly
* conservative, but these approximations are not.</p>
*
* <p>The main inputs are numbers <i>n</i> and <i>k</i>, which are not the same as other things
* that are called <i>n</i> and <i>k</i> in our sketching library. There is also a third
* parameter, numStdDev, that specifies the desired confidence level.</p>
* <ul>
* <li><i>n</i> is the number of independent randomized trials. It is given and therefore known.</li>
* <li><i>p</i> is the probability of a trial being a success. It is unknown.</li>
* <li><i>k</i> is the number of trials (out of <i>n</i>) that turn out to be successes. It is
* a random variable governed by a binomial distribution. After any given
* batch of <i>n</i> independent trials, the random variable <i>k</i> has a specific
* value which is observed and is therefore known.</li>
* <li><i>pHat</i> = <i>k</i> / <i>n</i> is an unbiased estimate of the unknown success
* probability <i>p</i>.</li>
* </ul>
*
* <p>Alternatively, consider a coin with unknown heads probability <i>p</i>. Where
* <i>n</i> is the number of independent flips of that coin, and <i>k</i> is the number
* of times that the coin comes up heads during a given batch of <i>n</i> flips.
* This class computes a frequentist confidence interval [lowerBoundOnP, upperBoundOnP] for the
* unknown <i>p</i>.</p>
*
* <p>Conceptually, the desired confidence level is specified by a tail probability delta.</p>
*
* <p>Ideally, over a large ensemble of independent batches of trials,
* the fraction of batches in which the true <i>p</i> lies below lowerBoundOnP would be at most
* delta, and the fraction of batches in which the true <i>p</i> lies above upperBoundOnP
* would also be at most delta.
*
* <p>Setting aside the philosophical difficulties attaching to that statement, it isn't quite
* true because we are approximating the Clopper-Pearson interval.</p>
*
* <p>Finally, we point out that in this class's interface, the confidence parameter delta is
* not specified directly, but rather through a "number of standard deviations" numStdDev.
* The library effectively converts that to a delta via delta = normalCDF (-1.0 * numStdDev).</p>
*
* <p>It is perhaps worth emphasizing that the library is NOT merely adding and subtracting
* numStdDev standard deviations to the estimate. It is doing something better, that to some
* extent accounts for the fact that the binomial distribution has a non-gaussian shape.</p>
*
* <p>In particular, it is using an approximation to the inverse of the incomplete beta function
* that appears as formula 26.5.22 on page 945 of the "Handbook of Mathematical Functions"
* by Abramowitz and Stegun.</p>
*
* @author Kevin Lang
*/
public final class BoundsOnBinomialProportions { // confidence intervals for binomial proportions
private BoundsOnBinomialProportions() {}
/**
* Computes lower bound of approximate Clopper-Pearson confidence interval for a binomial
* proportion.
*
* <p>Implementation Notes:<br>
* The approximateLowerBoundOnP is defined with respect to the right tail of the binomial
* distribution.</p>
* <ul>
* <li>We want to solve for the <i>p</i> for which sum<sub><i>j,k,n</i></sub>bino(<i>j;n,p</i>)
* = delta.</li>
* <li>We now restate that in terms of the left tail.</li>
* <li>We want to solve for the p for which sum<sub><i>j,0,(k-1)</i></sub>bino(<i>j;n,p</i>)
* = 1 - delta.</li>
* <li>Define <i>x</i> = 1-<i>p</i>.</li>
* <li>We want to solve for the <i>x</i> for which I<sub><i>x(n-k+1,k)</i></sub> = 1 - delta.</li>
* <li>We specify 1-delta via numStdDevs through the right tail of the standard normal
* distribution.</li>
* <li>Smaller values of numStdDevs correspond to bigger values of 1-delta and hence to smaller
* values of delta. In fact, usefully small values of delta correspond to negative values of
* numStdDevs.</li>
* <li>return <i>p</i> = 1-<i>x</i>.</li>
* </ul>
*
* @param n is the number of trials. Must be non-negative.
* @param k is the number of successes. Must be non-negative, and cannot exceed n.
* @param numStdDevs the number of standard deviations defining the confidence interval
* @return the lower bound of the approximate Clopper-Pearson confidence interval for the
* unknown success probability.
*/
public static double approximateLowerBoundOnP(final long n, final long k, final double numStdDevs) {
checkInputs(n, k);
if (n == 0) { return 0.0; } // the coin was never flipped, so we know nothing
else if (k == 0) { return 0.0; }
else if (k == 1) { return (exactLowerBoundOnPForKequalsOne(n, deltaOfNumStdevs(numStdDevs))); }
else if (k == n) { return (exactLowerBoundOnPForKequalsN(n, deltaOfNumStdevs(numStdDevs))); }
else {
final double x = abramowitzStegunFormula26p5p22((n - k) + 1, k, (-1.0 * numStdDevs));
return (1.0 - x); // which is p
}
}
/**
* Computes upper bound of approximate Clopper-Pearson confidence interval for a binomial
* proportion.
*
* <p>Implementation Notes:<br>
* The approximateUpperBoundOnP is defined with respect to the left tail of the binomial
* distribution.</p>
* <ul>
* <li>We want to solve for the <i>p</i> for which sum<sub><i>j,0,k</i></sub>bino(<i>j;n,p</i>)
* = delta.</li>
* <li>Define <i>x</i> = 1-<i>p</i>.</li>
* <li>We want to solve for the <i>x</i> for which I<sub><i>x(n-k,k+1)</i></sub> = delta.</li>
* <li>We specify delta via numStdDevs through the right tail of the standard normal
* distribution.</li>
* <li>Bigger values of numStdDevs correspond to smaller values of delta.</li>
* <li>return <i>p</i> = 1-<i>x</i>.</li>
* </ul>
* @param n is the number of trials. Must be non-negative.
* @param k is the number of successes. Must be non-negative, and cannot exceed <i>n</i>.
* @param numStdDevs the number of standard deviations defining the confidence interval
* @return the upper bound of the approximate Clopper-Pearson confidence interval for the
* unknown success probability.
*/
public static double approximateUpperBoundOnP(final long n, final long k, final double numStdDevs) {
checkInputs(n, k);
if (n == 0) { return 1.0; } // the coin was never flipped, so we know nothing
else if (k == n) { return 1.0; }
else if (k == (n - 1)) {
return (exactUpperBoundOnPForKequalsNminusOne(n, deltaOfNumStdevs(numStdDevs)));
}
else if (k == 0) {
return (exactUpperBoundOnPForKequalsZero(n, deltaOfNumStdevs(numStdDevs)));
}
else {
final double x = abramowitzStegunFormula26p5p22(n - k, k + 1, numStdDevs);
return (1.0 - x); // which is p
}
}
/**
* Computes an estimate of an unknown binomial proportion.
* @param n is the number of trials. Must be non-negative.
* @param k is the number of successes. Must be non-negative, and cannot exceed n.
* @return the estimate of the unknown binomial proportion.
*/
public static double estimateUnknownP(final long n, final long k) {
checkInputs(n, k);
if (n == 0) { return 0.5; } // the coin was never flipped, so we know nothing
else { return ((double) k / (double) n); }
}
private static void checkInputs(final long n, final long k) {
if (n < 0) { throw new SketchesArgumentException("N must be non-negative"); }
if (k < 0) { throw new SketchesArgumentException("K must be non-negative"); }
if (k > n) { throw new SketchesArgumentException("K cannot exceed N"); }
}
/**
* Computes an approximation to the erf() function.
* @param x is the input to the erf function
* @return returns erf(x), accurate to roughly 7 decimal digits.
*/
public static double erf(final double x) {
if (x < 0.0) { return (-1.0 * (erf_of_nonneg(-1.0 * x))); }
else { return (erf_of_nonneg(x)); }
}
/**
* Computes an approximation to normalCDF(x).
* @param x is the input to the normalCDF function
* @return returns the approximation to normalCDF(x).
*/
public static double normalCDF(final double x) {
return (0.5 * (1.0 + (erf(x / (Math.sqrt(2.0))))));
}
//@formatter:off
// Abramowitz and Stegun formula 7.1.28, p. 88; Claims accuracy of about 7 decimal digits
private static double erf_of_nonneg(final double x) {
// The constants from the book
final double a1 = 0.07052_30784;
final double a3 = 0.00927_05272;
final double a5 = 0.00027_65672;
final double a2 = 0.04228_20123;
final double a4 = 0.00015_20143;
final double a6 = 0.00004_30638;
final double x2 = x * x; // x squared, x cubed, etc.
final double x3 = x2 * x;
final double x4 = x2 * x2;
final double x5 = x2 * x3;
final double x6 = x3 * x3;
final double sum = ( 1.0
+ (a1 * x)
+ (a2 * x2)
+ (a3 * x3)
+ (a4 * x4)
+ (a5 * x5)
+ (a6 * x6) );
final double sum2 = sum * sum; // raise the sum to the 16th power
final double sum4 = sum2 * sum2;
final double sum8 = sum4 * sum4;
final double sum16 = sum8 * sum8;
return (1.0 - (1.0 / sum16));
}
//@formatter:on
private static double deltaOfNumStdevs(final double kappa) {
return (normalCDF(-1.0 * kappa));
}
// Formula 26.5.22 on page 945 of Abramowitz & Stegun, which is an approximation
// of the inverse of the incomplete beta function I_x(a,b) = delta
// viewed as a scalar function of x.
// In other words, we specify delta, and it gives us x (with a and b held constant).
// However, delta is specified in an indirect way through yp which
// is the number of stdDevs that leaves delta probability in the right
// tail of a standard gaussian distribution.
// We point out that the variable names correspond to those in the book,
// and it is worth keeping it that way so that it will always be easy to verify
// that the formula was typed in correctly.
private static double abramowitzStegunFormula26p5p22(final double a, final double b,
final double yp) {
final double b2m1 = (2.0 * b) - 1.0;
final double a2m1 = (2.0 * a) - 1.0;
final double lambda = ((yp * yp) - 3.0) / 6.0;
final double htmp = (1.0 / a2m1) + (1.0 / b2m1);
final double h = 2.0 / htmp;
final double term1 = (yp * (Math.sqrt(h + lambda))) / h;
final double term2 = (1.0 / b2m1) - (1.0 / a2m1);
final double term3 = (lambda + (5.0 / 6.0)) - (2.0 / (3.0 * h));
final double w = term1 - (term2 * term3);
final double xp = a / (a + (b * (Math.exp(2.0 * w))));
return xp;
}
// Formulas for some special cases.
private static double exactUpperBoundOnPForKequalsZero(final double n, final double delta) {
return (1.0 - Math.pow(delta, (1.0 / n)));
}
private static double exactLowerBoundOnPForKequalsN(final double n, final double delta) {
return (Math.pow(delta, (1.0 / n)));
}
private static double exactLowerBoundOnPForKequalsOne(final double n, final double delta) {
return (1.0 - Math.pow((1.0 - delta), (1.0 / n)));
}
private static double exactUpperBoundOnPForKequalsNminusOne(final double n, final double delta) {
return (Math.pow((1.0 - delta), (1.0 / n)));
}
}