001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.math3.stat.inference;
018
019import org.apache.commons.math3.distribution.ChiSquaredDistribution;
020import org.apache.commons.math3.exception.DimensionMismatchException;
021import org.apache.commons.math3.exception.MaxCountExceededException;
022import org.apache.commons.math3.exception.NotPositiveException;
023import org.apache.commons.math3.exception.NotStrictlyPositiveException;
024import org.apache.commons.math3.exception.OutOfRangeException;
025import org.apache.commons.math3.exception.ZeroException;
026import org.apache.commons.math3.exception.util.LocalizedFormats;
027import org.apache.commons.math3.util.FastMath;
028import org.apache.commons.math3.util.MathArrays;
029
030/**
031 * Implements <a href="http://en.wikipedia.org/wiki/G-test">G Test</a>
032 * statistics.
033 *
034 * <p>This is known in statistical genetics as the McDonald-Kreitman test.
035 * The implementation handles both known and unknown distributions.</p>
036 *
037 * <p>Two samples tests can be used when the distribution is unknown <i>a priori</i>
038 * but provided by one sample, or when the hypothesis under test is that the two
039 * samples come from the same underlying distribution.</p>
040 *
041 * @version $Id: GTest.java 1416643 2012-12-03 19:37:14Z tn $
042 * @since 3.1
043 */
044public class GTest {
045
046    /**
047     * Computes the <a href="http://en.wikipedia.org/wiki/G-test">G statistic
048     * for Goodness of Fit</a> comparing {@code observed} and {@code expected}
049     * frequency counts.
050     *
051     * <p>This statistic can be used to perform a G test (Log-Likelihood Ratio
052     * Test) evaluating the null hypothesis that the observed counts follow the
053     * expected distribution.</p>
054     *
055     * <p><strong>Preconditions</strong>: <ul>
056     * <li>Expected counts must all be positive. </li>
057     * <li>Observed counts must all be &ge; 0. </li>
058     * <li>The observed and expected arrays must have the same length and their
059     * common length must be at least 2. </li></ul></p>
060     *
061     * <p>If any of the preconditions are not met, a
062     * {@code MathIllegalArgumentException} is thrown.</p>
063     *
064     * <p><strong>Note:</strong>This implementation rescales the
065     * {@code expected} array if necessary to ensure that the sum of the
066     * expected and observed counts are equal.</p>
067     *
068     * @param observed array of observed frequency counts
069     * @param expected array of expected frequency counts
070     * @return G-Test statistic
071     * @throws NotPositiveException if {@code observed} has negative entries
072     * @throws NotStrictlyPositiveException if {@code expected} has entries that
073     * are not strictly positive
074     * @throws DimensionMismatchException if the array lengths do not match or
075     * are less than 2.
076     */
077    public double g(final double[] expected, final long[] observed)
078            throws NotPositiveException, NotStrictlyPositiveException,
079            DimensionMismatchException {
080
081        if (expected.length < 2) {
082            throw new DimensionMismatchException(expected.length, 2);
083        }
084        if (expected.length != observed.length) {
085            throw new DimensionMismatchException(expected.length, observed.length);
086        }
087        MathArrays.checkPositive(expected);
088        MathArrays.checkNonNegative(observed);
089
090        double sumExpected = 0d;
091        double sumObserved = 0d;
092        for (int i = 0; i < observed.length; i++) {
093            sumExpected += expected[i];
094            sumObserved += observed[i];
095        }
096        double ratio = 1d;
097        boolean rescale = false;
098        if (Math.abs(sumExpected - sumObserved) > 10E-6) {
099            ratio = sumObserved / sumExpected;
100            rescale = true;
101        }
102        double sum = 0d;
103        for (int i = 0; i < observed.length; i++) {
104            final double dev = rescale ?
105                    FastMath.log((double) observed[i] / (ratio * expected[i])) :
106                        FastMath.log((double) observed[i] / expected[i]);
107            sum += ((double) observed[i]) * dev;
108        }
109        return 2d * sum;
110    }
111
112    /**
113     * Returns the <i>observed significance level</i>, or <a href=
114     * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> p-value</a>,
115     * associated with a G-Test for goodness of fit</a> comparing the
116     * {@code observed} frequency counts to those in the {@code expected} array.
117     *
118     * <p>The number returned is the smallest significance level at which one
119     * can reject the null hypothesis that the observed counts conform to the
120     * frequency distribution described by the expected counts.</p>
121     *
122     * <p>The probability returned is the tail probability beyond
123     * {@link #g(double[], long[]) g(expected, observed)}
124     * in the ChiSquare distribution with degrees of freedom one less than the
125     * common length of {@code expected} and {@code observed}.</p>
126     *
127     * <p> <strong>Preconditions</strong>: <ul>
128     * <li>Expected counts must all be positive. </li>
129     * <li>Observed counts must all be &ge; 0. </li>
130     * <li>The observed and expected arrays must have the
131     * same length and their common length must be at least 2.</li>
132     * </ul></p>
133     *
134     * <p>If any of the preconditions are not met, a
135     * {@code MathIllegalArgumentException} is thrown.</p>
136     *
137     * <p><strong>Note:</strong>This implementation rescales the
138     * {@code expected} array if necessary to ensure that the sum of the
139     *  expected and observed counts are equal.</p>
140     *
141     * @param observed array of observed frequency counts
142     * @param expected array of expected frequency counts
143     * @return p-value
144     * @throws NotPositiveException if {@code observed} has negative entries
145     * @throws NotStrictlyPositiveException if {@code expected} has entries that
146     * are not strictly positive
147     * @throws DimensionMismatchException if the array lengths do not match or
148     * are less than 2.
149     * @throws MaxCountExceededException if an error occurs computing the
150     * p-value.
151     */
152    public double gTest(final double[] expected, final long[] observed)
153            throws NotPositiveException, NotStrictlyPositiveException,
154            DimensionMismatchException, MaxCountExceededException {
155
156        final ChiSquaredDistribution distribution =
157                new ChiSquaredDistribution(expected.length - 1.0);
158        return 1.0 - distribution.cumulativeProbability(
159                g(expected, observed));
160    }
161
162    /**
163     * Returns the intrinsic (Hardy-Weinberg proportions) p-Value, as described
164     * in p64-69 of McDonald, J.H. 2009. Handbook of Biological Statistics
165     * (2nd ed.). Sparky House Publishing, Baltimore, Maryland.
166     *
167     * <p> The probability returned is the tail probability beyond
168     * {@link #g(double[], long[]) g(expected, observed)}
169     * in the ChiSquare distribution with degrees of freedom two less than the
170     * common length of {@code expected} and {@code observed}.</p>
171     *
172     * @param observed array of observed frequency counts
173     * @param expected array of expected frequency counts
174     * @return p-value
175     * @throws NotPositiveException if {@code observed} has negative entries
176     * @throws NotStrictlyPositiveException {@code expected} has entries that are
177     * not strictly positive
178     * @throws DimensionMismatchException if the array lengths do not match or
179     * are less than 2.
180     * @throws MaxCountExceededException if an error occurs computing the
181     * p-value.
182     */
183    public double gTestIntrinsic(final double[] expected, final long[] observed)
184            throws NotPositiveException, NotStrictlyPositiveException,
185            DimensionMismatchException, MaxCountExceededException {
186
187        final ChiSquaredDistribution distribution =
188                new ChiSquaredDistribution(expected.length - 2.0);
189        return 1.0 - distribution.cumulativeProbability(
190                g(expected, observed));
191    }
192
193    /**
194     * Performs a G-Test (Log-Likelihood Ratio Test) for goodness of fit
195     * evaluating the null hypothesis that the observed counts conform to the
196     * frequency distribution described by the expected counts, with
197     * significance level {@code alpha}. Returns true iff the null
198     * hypothesis can be rejected with {@code 100 * (1 - alpha)} percent confidence.
199     *
200     * <p><strong>Example:</strong><br> To test the hypothesis that
201     * {@code observed} follows {@code expected} at the 99% level,
202     * use </p><p>
203     * {@code gTest(expected, observed, 0.01)}</p>
204     *
205     * <p>Returns true iff {@link #gTest(double[], long[])
206     *  gTestGoodnessOfFitPValue(expected, observed)} < alpha</p>
207     *
208     * <p><strong>Preconditions</strong>: <ul>
209     * <li>Expected counts must all be positive. </li>
210     * <li>Observed counts must all be &ge; 0. </li>
211     * <li>The observed and expected arrays must have the same length and their
212     * common length must be at least 2.
213     * <li> {@code 0 < alpha < 0.5} </li></ul></p>
214     *
215     * <p>If any of the preconditions are not met, a
216     * {@code MathIllegalArgumentException} is thrown.</p>
217     *
218     * <p><strong>Note:</strong>This implementation rescales the
219     * {@code expected} array if necessary to ensure that the sum of the
220     * expected and observed counts are equal.</p>
221     *
222     * @param observed array of observed frequency counts
223     * @param expected array of expected frequency counts
224     * @param alpha significance level of the test
225     * @return true iff null hypothesis can be rejected with confidence 1 -
226     * alpha
227     * @throws NotPositiveException if {@code observed} has negative entries
228     * @throws NotStrictlyPositiveException if {@code expected} has entries that
229     * are not strictly positive
230     * @throws DimensionMismatchException if the array lengths do not match or
231     * are less than 2.
232     * @throws MaxCountExceededException if an error occurs computing the
233     * p-value.
234     * @throws OutOfRangeException if alpha is not strictly greater than zero
235     * and less than or equal to 0.5
236     */
237    public boolean gTest(final double[] expected, final long[] observed,
238            final double alpha)
239            throws NotPositiveException, NotStrictlyPositiveException,
240            DimensionMismatchException, OutOfRangeException, MaxCountExceededException {
241
242        if ((alpha <= 0) || (alpha > 0.5)) {
243            throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL,
244                    alpha, 0, 0.5);
245        }
246        return gTest(expected, observed) < alpha;
247    }
248
249    /**
250     * Calculates the <a href=
251     * "http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">Shannon
252     * entropy</a> for 2 Dimensional Matrix.  The value returned is the entropy
253     * of the vector formed by concatenating the rows (or columns) of {@code k}
254     * to form a vector. See {@link #entropy(long[])}.
255     *
256     * @param k 2 Dimensional Matrix of long values (for ex. the counts of a
257     * trials)
258     * @return Shannon Entropy of the given Matrix
259     *
260     */
261    private double entropy(final long[][] k) {
262        double h = 0d;
263        double sum_k = 0d;
264        for (int i = 0; i < k.length; i++) {
265            for (int j = 0; j < k[i].length; j++) {
266                sum_k += (double) k[i][j];
267            }
268        }
269        for (int i = 0; i < k.length; i++) {
270            for (int j = 0; j < k[i].length; j++) {
271                if (k[i][j] != 0) {
272                    final double p_ij = (double) k[i][j] / sum_k;
273                    h += p_ij * Math.log(p_ij);
274                }
275            }
276        }
277        return -h;
278    }
279
280    /**
281     * Calculates the <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">
282     * Shannon entropy</a> for a vector.  The values of {@code k} are taken to be
283     * incidence counts of the values of a random variable. What is returned is <br/>
284     * &sum;p<sub>i</sub>log(p<sub>i</sub><br/>
285     * where p<sub>i</sub> = k[i] / (sum of elements in k)
286     *
287     * @param k Vector (for ex. Row Sums of a trials)
288     * @return Shannon Entropy of the given Vector
289     *
290     */
291    private double entropy(final long[] k) {
292        double h = 0d;
293        double sum_k = 0d;
294        for (int i = 0; i < k.length; i++) {
295            sum_k += (double) k[i];
296        }
297        for (int i = 0; i < k.length; i++) {
298            if (k[i] != 0) {
299                final double p_i = (double) k[i] / sum_k;
300                h += p_i * Math.log(p_i);
301            }
302        }
303        return -h;
304    }
305
306    /**
307     * <p>Computes a G (Log-Likelihood Ratio) two sample test statistic for
308     * independence comparing frequency counts in
309     * {@code observed1} and {@code observed2}. The sums of frequency
310     * counts in the two samples are not required to be the same. The formula
311     * used to compute the test statistic is </p>
312     *
313     * <p>{@code 2 * totalSum * [H(rowSums) + H(colSums) - H(k)]}</p>
314     *
315     * <p> where {@code H} is the
316     * <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">
317     * Shannon Entropy</a> of the random variable formed by viewing the elements
318     * of the argument array as incidence counts; <br/>
319     * {@code k} is a matrix with rows {@code [observed1, observed2]}; <br/>
320     * {@code rowSums, colSums} are the row/col sums of {@code k}; <br>
321     * and {@code totalSum} is the overall sum of all entries in {@code k}.</p>
322     *
323     * <p>This statistic can be used to perform a G test evaluating the null
324     * hypothesis that both observed counts are independent </p>
325     *
326     * <p> <strong>Preconditions</strong>: <ul>
327     * <li>Observed counts must be non-negative. </li>
328     * <li>Observed counts for a specific bin must not both be zero. </li>
329     * <li>Observed counts for a specific sample must not all be  0. </li>
330     * <li>The arrays {@code observed1} and {@code observed2} must have
331     * the same length and their common length must be at least 2. </li></ul></p>
332     *
333     * <p>If any of the preconditions are not met, a
334     * {@code MathIllegalArgumentException} is thrown.</p>
335     *
336     * @param observed1 array of observed frequency counts of the first data set
337     * @param observed2 array of observed frequency counts of the second data
338     * set
339     * @return G-Test statistic
340     * @throws DimensionMismatchException the the lengths of the arrays do not
341     * match or their common length is less than 2
342     * @throws NotPositiveException if any entry in {@code observed1} or
343     * {@code observed2} is negative
344     * @throws ZeroException if either all counts of
345     * {@code observed1} or {@code observed2} are zero, or if the count
346     * at the same index is zero for both arrays.
347     */
348    public double gDataSetsComparison(final long[] observed1, final long[] observed2)
349            throws DimensionMismatchException, NotPositiveException, ZeroException {
350
351        // Make sure lengths are same
352        if (observed1.length < 2) {
353            throw new DimensionMismatchException(observed1.length, 2);
354        }
355        if (observed1.length != observed2.length) {
356            throw new DimensionMismatchException(observed1.length, observed2.length);
357        }
358
359        // Ensure non-negative counts
360        MathArrays.checkNonNegative(observed1);
361        MathArrays.checkNonNegative(observed2);
362
363        // Compute and compare count sums
364        long countSum1 = 0;
365        long countSum2 = 0;
366
367        // Compute and compare count sums
368        final long[] collSums = new long[observed1.length];
369        final long[][] k = new long[2][observed1.length];
370
371        for (int i = 0; i < observed1.length; i++) {
372            if (observed1[i] == 0 && observed2[i] == 0) {
373                throw new ZeroException(LocalizedFormats.OBSERVED_COUNTS_BOTTH_ZERO_FOR_ENTRY, i);
374            } else {
375                countSum1 += observed1[i];
376                countSum2 += observed2[i];
377                collSums[i] = observed1[i] + observed2[i];
378                k[0][i] = observed1[i];
379                k[1][i] = observed2[i];
380            }
381        }
382        // Ensure neither sample is uniformly 0
383        if (countSum1 == 0 || countSum2 == 0) {
384            throw new ZeroException();
385        }
386        final long[] rowSums = {countSum1, countSum2};
387        final double sum = (double) countSum1 + (double) countSum2;
388        return 2 * sum * (entropy(rowSums) + entropy(collSums) - entropy(k));
389    }
390
391    /**
392     * Calculates the root log-likelihood ratio for 2 state Datasets. See
393     * {@link #gDataSetsComparison(long[], long[] )}.
394     *
395     * <p>Given two events A and B, let k11 be the number of times both events
396     * occur, k12 the incidence of B without A, k21 the count of A without B,
397     * and k22 the number of times neither A nor B occurs.  What is returned
398     * by this method is </p>
399     *
400     * <p>{@code (sgn) sqrt(gValueDataSetsComparison({k11, k12}, {k21, k22})}</p>
401     *
402     * <p>where {@code sgn} is -1 if {@code k11 / (k11 + k12) < k21 / (k21 + k22))};<br/>
403     * 1 otherwise.</p>
404     *
405     * <p>Signed root LLR has two advantages over the basic LLR: a) it is positive
406     * where k11 is bigger than expected, negative where it is lower b) if there is
407     * no difference it is asymptotically normally distributed. This allows one
408     * to talk about "number of standard deviations" which is a more common frame
409     * of reference than the chi^2 distribution.</p>
410     *
411     * @param k11 number of times the two events occurred together (AB)
412     * @param k12 number of times the second event occurred WITHOUT the
413     * first event (notA,B)
414     * @param k21 number of times the first event occurred WITHOUT the
415     * second event (A, notB)
416     * @param k22 number of times something else occurred (i.e. was neither
417     * of these events (notA, notB)
418     * @return root log-likelihood ratio
419     *
420     */
421    public double rootLogLikelihoodRatio(final long k11, long k12,
422            final long k21, final long k22) {
423        final double llr = gDataSetsComparison(
424                new long[]{k11, k12}, new long[]{k21, k22});
425        double sqrt = FastMath.sqrt(llr);
426        if ((double) k11 / (k11 + k12) < (double) k21 / (k21 + k22)) {
427            sqrt = -sqrt;
428        }
429        return sqrt;
430    }
431
432    /**
433     * <p>Returns the <i>observed significance level</i>, or <a href=
434     * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
435     * p-value</a>, associated with a G-Value (Log-Likelihood Ratio) for two
436     * sample test comparing bin frequency counts in {@code observed1} and
437     * {@code observed2}.</p>
438     *
439     * <p>The number returned is the smallest significance level at which one
440     * can reject the null hypothesis that the observed counts conform to the
441     * same distribution. </p>
442     *
443     * <p>See {@link #gTest(double[], long[])} for details
444     * on how the p-value is computed.  The degrees of of freedom used to
445     * perform the test is one less than the common length of the input observed
446     * count arrays.</p>
447     *
448     * <p><strong>Preconditions</strong>:
449     * <ul> <li>Observed counts must be non-negative. </li>
450     * <li>Observed counts for a specific bin must not both be zero. </li>
451     * <li>Observed counts for a specific sample must not all be 0. </li>
452     * <li>The arrays {@code observed1} and {@code observed2} must
453     * have the same length and their common length must be at least 2. </li>
454     * </ul><p>
455     * <p> If any of the preconditions are not met, a
456     * {@code MathIllegalArgumentException} is thrown.</p>
457     *
458     * @param observed1 array of observed frequency counts of the first data set
459     * @param observed2 array of observed frequency counts of the second data
460     * set
461     * @return p-value
462     * @throws DimensionMismatchException the the length of the arrays does not
463     * match or their common length is less than 2
464     * @throws NotPositiveException if any of the entries in {@code observed1} or
465     * {@code observed2} are negative
466     * @throws ZeroException if either all counts of {@code observed1} or
467     * {@code observed2} are zero, or if the count at some index is
468     * zero for both arrays
469     * @throws MaxCountExceededException if an error occurs computing the
470     * p-value.
471     */
472    public double gTestDataSetsComparison(final long[] observed1,
473            final long[] observed2)
474            throws DimensionMismatchException, NotPositiveException, ZeroException,
475            MaxCountExceededException {
476        final ChiSquaredDistribution distribution = new ChiSquaredDistribution(
477                (double) observed1.length - 1);
478        return 1 - distribution.cumulativeProbability(
479                gDataSetsComparison(observed1, observed2));
480    }
481
482    /**
483     * <p>Performs a G-Test (Log-Likelihood Ratio Test) comparing two binned
484     * data sets. The test evaluates the null hypothesis that the two lists
485     * of observed counts conform to the same frequency distribution, with
486     * significance level {@code alpha}. Returns true iff the null
487     * hypothesis can be rejected  with 100 * (1 - alpha) percent confidence.
488     * </p>
489     * <p>See {@link #gDataSetsComparison(long[], long[])} for details
490     * on the formula used to compute the G (LLR) statistic used in the test and
491     * {@link #gTest(double[], long[])} for information on how
492     * the observed significance level is computed. The degrees of of freedom used
493     * to perform the test is one less than the common length of the input observed
494     * count arrays. </p>
495     *
496     * <strong>Preconditions</strong>: <ul>
497     * <li>Observed counts must be non-negative. </li>
498     * <li>Observed counts for a specific bin must not both be zero. </li>
499     * <li>Observed counts for a specific sample must not all be 0. </li>
500     * <li>The arrays {@code observed1} and {@code observed2} must
501     * have the same length and their common length must be at least 2. </li>
502     * <li>{@code 0 < alpha < 0.5} </li></ul></p>
503     *
504     * <p>If any of the preconditions are not met, a
505     * {@code MathIllegalArgumentException} is thrown.</p>
506     *
507     * @param observed1 array of observed frequency counts of the first data set
508     * @param observed2 array of observed frequency counts of the second data
509     * set
510     * @param alpha significance level of the test
511     * @return true iff null hypothesis can be rejected with confidence 1 -
512     * alpha
513     * @throws DimensionMismatchException the the length of the arrays does not
514     * match
515     * @throws NotPositiveException if any of the entries in {@code observed1} or
516     * {@code observed2} are negative
517     * @throws ZeroException if either all counts of {@code observed1} or
518     * {@code observed2} are zero, or if the count at some index is
519     * zero for both arrays
520     * @throws OutOfRangeException if {@code alpha} is not in the range
521     * (0, 0.5]
522     * @throws MaxCountExceededException if an error occurs performing the test
523     */
524    public boolean gTestDataSetsComparison(
525            final long[] observed1,
526            final long[] observed2,
527            final double alpha)
528            throws DimensionMismatchException, NotPositiveException,
529            ZeroException, OutOfRangeException, MaxCountExceededException {
530
531        if (alpha <= 0 || alpha > 0.5) {
532            throw new OutOfRangeException(
533                    LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5);
534        }
535        return gTestDataSetsComparison(observed1, observed2) < alpha;
536    }
537}