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     */
017    package org.apache.commons.math3.stat.inference;
018    
019    import org.apache.commons.math3.distribution.ChiSquaredDistribution;
020    import org.apache.commons.math3.exception.DimensionMismatchException;
021    import org.apache.commons.math3.exception.MaxCountExceededException;
022    import org.apache.commons.math3.exception.NotPositiveException;
023    import org.apache.commons.math3.exception.NotStrictlyPositiveException;
024    import org.apache.commons.math3.exception.OutOfRangeException;
025    import org.apache.commons.math3.exception.ZeroException;
026    import org.apache.commons.math3.exception.util.LocalizedFormats;
027    import org.apache.commons.math3.util.FastMath;
028    import 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     */
044    public 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    }