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2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
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5    * The ASF licenses this file to You under the Apache License, Version 2.0
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9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
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14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  package org.apache.commons.math3.stat.inference;
18  
19  import org.apache.commons.math3.distribution.ChiSquaredDistribution;
20  import org.apache.commons.math3.exception.DimensionMismatchException;
21  import org.apache.commons.math3.exception.MaxCountExceededException;
22  import org.apache.commons.math3.exception.NotPositiveException;
23  import org.apache.commons.math3.exception.NotStrictlyPositiveException;
24  import org.apache.commons.math3.exception.OutOfRangeException;
25  import org.apache.commons.math3.exception.ZeroException;
26  import org.apache.commons.math3.exception.util.LocalizedFormats;
27  import org.apache.commons.math3.util.FastMath;
28  import org.apache.commons.math3.util.MathArrays;
29  
30  /**
31   * Implements <a href="http://en.wikipedia.org/wiki/G-test">G Test</a>
32   * statistics.
33   *
34   * <p>This is known in statistical genetics as the McDonald-Kreitman test.
35   * The implementation handles both known and unknown distributions.</p>
36   *
37   * <p>Two samples tests can be used when the distribution is unknown <i>a priori</i>
38   * but provided by one sample, or when the hypothesis under test is that the two
39   * samples come from the same underlying distribution.</p>
40   *
41   * @version $Id: GTest.java 1547633 2013-12-03 23:03:06Z tn $
42   * @since 3.1
43   */
44  public class GTest {
45  
46      /**
47       * Computes the <a href="http://en.wikipedia.org/wiki/G-test">G statistic
48       * for Goodness of Fit</a> comparing {@code observed} and {@code expected}
49       * frequency counts.
50       *
51       * <p>This statistic can be used to perform a G test (Log-Likelihood Ratio
52       * Test) evaluating the null hypothesis that the observed counts follow the
53       * expected distribution.</p>
54       *
55       * <p><strong>Preconditions</strong>: <ul>
56       * <li>Expected counts must all be positive. </li>
57       * <li>Observed counts must all be &ge; 0. </li>
58       * <li>The observed and expected arrays must have the same length and their
59       * common length must be at least 2. </li></ul></p>
60       *
61       * <p>If any of the preconditions are not met, a
62       * {@code MathIllegalArgumentException} is thrown.</p>
63       *
64       * <p><strong>Note:</strong>This implementation rescales the
65       * {@code expected} array if necessary to ensure that the sum of the
66       * expected and observed counts are equal.</p>
67       *
68       * @param observed array of observed frequency counts
69       * @param expected array of expected frequency counts
70       * @return G-Test statistic
71       * @throws NotPositiveException if {@code observed} has negative entries
72       * @throws NotStrictlyPositiveException if {@code expected} has entries that
73       * are not strictly positive
74       * @throws DimensionMismatchException if the array lengths do not match or
75       * are less than 2.
76       */
77      public double g(final double[] expected, final long[] observed)
78              throws NotPositiveException, NotStrictlyPositiveException,
79              DimensionMismatchException {
80  
81          if (expected.length < 2) {
82              throw new DimensionMismatchException(expected.length, 2);
83          }
84          if (expected.length != observed.length) {
85              throw new DimensionMismatchException(expected.length, observed.length);
86          }
87          MathArrays.checkPositive(expected);
88          MathArrays.checkNonNegative(observed);
89  
90          double sumExpected = 0d;
91          double sumObserved = 0d;
92          for (int i = 0; i < observed.length; i++) {
93              sumExpected += expected[i];
94              sumObserved += observed[i];
95          }
96          double ratio = 1d;
97          boolean rescale = false;
98          if (FastMath.abs(sumExpected - sumObserved) > 10E-6) {
99              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 * FastMath.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 * FastMath.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 }