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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  
18  package org.apache.commons.math3.random;
19  
20  import java.io.BufferedReader;
21  import java.io.File;
22  import java.io.FileInputStream;
23  import java.io.IOException;
24  import java.io.InputStream;
25  import java.io.InputStreamReader;
26  import java.net.URL;
27  import java.nio.charset.Charset;
28  import java.util.ArrayList;
29  import java.util.List;
30  
31  import org.apache.commons.math3.distribution.AbstractRealDistribution;
32  import org.apache.commons.math3.distribution.ConstantRealDistribution;
33  import org.apache.commons.math3.distribution.NormalDistribution;
34  import org.apache.commons.math3.distribution.RealDistribution;
35  import org.apache.commons.math3.exception.MathIllegalStateException;
36  import org.apache.commons.math3.exception.MathInternalError;
37  import org.apache.commons.math3.exception.NullArgumentException;
38  import org.apache.commons.math3.exception.OutOfRangeException;
39  import org.apache.commons.math3.exception.ZeroException;
40  import org.apache.commons.math3.exception.NotStrictlyPositiveException;
41  import org.apache.commons.math3.exception.util.LocalizedFormats;
42  import org.apache.commons.math3.stat.descriptive.StatisticalSummary;
43  import org.apache.commons.math3.stat.descriptive.SummaryStatistics;
44  import org.apache.commons.math3.util.FastMath;
45  import org.apache.commons.math3.util.MathUtils;
46  
47  /**
48   * <p>Represents an <a href="http://http://en.wikipedia.org/wiki/Empirical_distribution_function">
49   * empirical probability distribution</a> -- a probability distribution derived
50   * from observed data without making any assumptions about the functional form
51   * of the population distribution that the data come from.</p>
52   *
53   * <p>An <code>EmpiricalDistribution</code> maintains data structures, called
54   * <i>distribution digests</i>, that describe empirical distributions and
55   * support the following operations: <ul>
56   * <li>loading the distribution from a file of observed data values</li>
57   * <li>dividing the input data into "bin ranges" and reporting bin frequency
58   *     counts (data for histogram)</li>
59   * <li>reporting univariate statistics describing the full set of data values
60   *     as well as the observations within each bin</li>
61   * <li>generating random values from the distribution</li>
62   * </ul>
63   * Applications can use <code>EmpiricalDistribution</code> to build grouped
64   * frequency histograms representing the input data or to generate random values
65   * "like" those in the input file -- i.e., the values generated will follow the
66   * distribution of the values in the file.</p>
67   *
68   * <p>The implementation uses what amounts to the
69   * <a href="http://nedwww.ipac.caltech.edu/level5/March02/Silverman/Silver2_6.html">
70   * Variable Kernel Method</a> with Gaussian smoothing:<p>
71   * <strong>Digesting the input file</strong>
72   * <ol><li>Pass the file once to compute min and max.</li>
73   * <li>Divide the range from min-max into <code>binCount</code> "bins."</li>
74   * <li>Pass the data file again, computing bin counts and univariate
75   *     statistics (mean, std dev.) for each of the bins </li>
76   * <li>Divide the interval (0,1) into subintervals associated with the bins,
77   *     with the length of a bin's subinterval proportional to its count.</li></ol>
78   * <strong>Generating random values from the distribution</strong><ol>
79   * <li>Generate a uniformly distributed value in (0,1) </li>
80   * <li>Select the subinterval to which the value belongs.
81   * <li>Generate a random Gaussian value with mean = mean of the associated
82   *     bin and std dev = std dev of associated bin.</li></ol></p>
83   *
84   * <p>EmpiricalDistribution implements the {@link RealDistribution} interface
85   * as follows.  Given x within the range of values in the dataset, let B
86   * be the bin containing x and let K be the within-bin kernel for B.  Let P(B-)
87   * be the sum of the probabilities of the bins below B and let K(B) be the
88   * mass of B under K (i.e., the integral of the kernel density over B).  Then
89   * set P(X < x) = P(B-) + P(B) * K(x) / K(B) where K(x) is the kernel distribution
90   * evaluated at x. This results in a cdf that matches the grouped frequency
91   * distribution at the bin endpoints and interpolates within bins using
92   * within-bin kernels.</p>
93   *
94   *<strong>USAGE NOTES:</strong><ul>
95   *<li>The <code>binCount</code> is set by default to 1000.  A good rule of thumb
96   *    is to set the bin count to approximately the length of the input file divided
97   *    by 10. </li>
98   *<li>The input file <i>must</i> be a plain text file containing one valid numeric
99   *    entry per line.</li>
100  * </ul></p>
101  *
102  */
103 public class EmpiricalDistribution extends AbstractRealDistribution {
104 
105     /** Default bin count */
106     public static final int DEFAULT_BIN_COUNT = 1000;
107 
108     /** Character set for file input */
109     private static final String FILE_CHARSET = "US-ASCII";
110 
111     /** Serializable version identifier */
112     private static final long serialVersionUID = 5729073523949762654L;
113 
114     /** RandomDataGenerator instance to use in repeated calls to getNext() */
115     protected final RandomDataGenerator randomData;
116 
117     /** List of SummaryStatistics objects characterizing the bins */
118     private final List<SummaryStatistics> binStats;
119 
120     /** Sample statistics */
121     private SummaryStatistics sampleStats = null;
122 
123     /** Max loaded value */
124     private double max = Double.NEGATIVE_INFINITY;
125 
126     /** Min loaded value */
127     private double min = Double.POSITIVE_INFINITY;
128 
129     /** Grid size */
130     private double delta = 0d;
131 
132     /** number of bins */
133     private final int binCount;
134 
135     /** is the distribution loaded? */
136     private boolean loaded = false;
137 
138     /** upper bounds of subintervals in (0,1) "belonging" to the bins */
139     private double[] upperBounds = null;
140 
141     /**
142      * Creates a new EmpiricalDistribution with the default bin count.
143      */
144     public EmpiricalDistribution() {
145         this(DEFAULT_BIN_COUNT);
146     }
147 
148     /**
149      * Creates a new EmpiricalDistribution with the specified bin count.
150      *
151      * @param binCount number of bins. Must be strictly positive.
152      * @throws NotStrictlyPositiveException if {@code binCount <= 0}.
153      */
154     public EmpiricalDistribution(int binCount) {
155         this(binCount, new RandomDataGenerator());
156     }
157 
158     /**
159      * Creates a new EmpiricalDistribution with the specified bin count using the
160      * provided {@link RandomGenerator} as the source of random data.
161      *
162      * @param binCount number of bins. Must be strictly positive.
163      * @param generator random data generator (may be null, resulting in default JDK generator)
164      * @throws NotStrictlyPositiveException if {@code binCount <= 0}.
165      * @since 3.0
166      */
167     public EmpiricalDistribution(int binCount, RandomGenerator generator) {
168         this(binCount, new RandomDataGenerator(generator));
169     }
170 
171     /**
172      * Creates a new EmpiricalDistribution with default bin count using the
173      * provided {@link RandomGenerator} as the source of random data.
174      *
175      * @param generator random data generator (may be null, resulting in default JDK generator)
176      * @since 3.0
177      */
178     public EmpiricalDistribution(RandomGenerator generator) {
179         this(DEFAULT_BIN_COUNT, generator);
180     }
181 
182     /**
183      * Creates a new EmpiricalDistribution with the specified bin count using the
184      * provided {@link RandomDataImpl} instance as the source of random data.
185      *
186      * @param binCount number of bins
187      * @param randomData random data generator (may be null, resulting in default JDK generator)
188      * @since 3.0
189      * @deprecated As of 3.1. Please use {@link #EmpiricalDistribution(int,RandomGenerator)} instead.
190      */
191     @Deprecated
192     public EmpiricalDistribution(int binCount, RandomDataImpl randomData) {
193         this(binCount, randomData.getDelegate());
194     }
195 
196     /**
197      * Creates a new EmpiricalDistribution with default bin count using the
198      * provided {@link RandomDataImpl} as the source of random data.
199      *
200      * @param randomData random data generator (may be null, resulting in default JDK generator)
201      * @since 3.0
202      * @deprecated As of 3.1. Please use {@link #EmpiricalDistribution(RandomGenerator)} instead.
203      */
204     @Deprecated
205     public EmpiricalDistribution(RandomDataImpl randomData) {
206         this(DEFAULT_BIN_COUNT, randomData);
207     }
208 
209     /**
210      * Private constructor to allow lazy initialisation of the RNG contained
211      * in the {@link #randomData} instance variable.
212      *
213      * @param binCount number of bins. Must be strictly positive.
214      * @param randomData Random data generator.
215      * @throws NotStrictlyPositiveException if {@code binCount <= 0}.
216      */
217     private EmpiricalDistribution(int binCount,
218                                   RandomDataGenerator randomData) {
219         super(randomData.getRandomGenerator());
220         if (binCount <= 0) {
221             throw new NotStrictlyPositiveException(binCount);
222         }
223         this.binCount = binCount;
224         this.randomData = randomData;
225         binStats = new ArrayList<SummaryStatistics>();
226     }
227 
228     /**
229      * Computes the empirical distribution from the provided
230      * array of numbers.
231      *
232      * @param in the input data array
233      * @exception NullArgumentException if in is null
234      */
235     public void load(double[] in) throws NullArgumentException {
236         DataAdapter da = new ArrayDataAdapter(in);
237         try {
238             da.computeStats();
239             // new adapter for the second pass
240             fillBinStats(new ArrayDataAdapter(in));
241         } catch (IOException ex) {
242             // Can't happen
243             throw new MathInternalError();
244         }
245         loaded = true;
246 
247     }
248 
249     /**
250      * Computes the empirical distribution using data read from a URL.
251      *
252      * <p>The input file <i>must</i> be an ASCII text file containing one
253      * valid numeric entry per line.</p>
254      *
255      * @param url url of the input file
256      *
257      * @throws IOException if an IO error occurs
258      * @throws NullArgumentException if url is null
259      * @throws ZeroException if URL contains no data
260      */
261     public void load(URL url) throws IOException, NullArgumentException, ZeroException {
262         MathUtils.checkNotNull(url);
263         Charset charset = Charset.forName(FILE_CHARSET);
264         BufferedReader in =
265             new BufferedReader(new InputStreamReader(url.openStream(), charset));
266         try {
267             DataAdapter da = new StreamDataAdapter(in);
268             da.computeStats();
269             if (sampleStats.getN() == 0) {
270                 throw new ZeroException(LocalizedFormats.URL_CONTAINS_NO_DATA, url);
271             }
272             // new adapter for the second pass
273             in = new BufferedReader(new InputStreamReader(url.openStream(), charset));
274             fillBinStats(new StreamDataAdapter(in));
275             loaded = true;
276         } finally {
277            try {
278                in.close();
279            } catch (IOException ex) { //NOPMD
280                // ignore
281            }
282         }
283     }
284 
285     /**
286      * Computes the empirical distribution from the input file.
287      *
288      * <p>The input file <i>must</i> be an ASCII text file containing one
289      * valid numeric entry per line.</p>
290      *
291      * @param file the input file
292      * @throws IOException if an IO error occurs
293      * @throws NullArgumentException if file is null
294      */
295     public void load(File file) throws IOException, NullArgumentException {
296         MathUtils.checkNotNull(file);
297         Charset charset = Charset.forName(FILE_CHARSET);
298         InputStream is = new FileInputStream(file);
299         BufferedReader in = new BufferedReader(new InputStreamReader(is, charset));
300         try {
301             DataAdapter da = new StreamDataAdapter(in);
302             da.computeStats();
303             // new adapter for second pass
304             is = new FileInputStream(file);
305             in = new BufferedReader(new InputStreamReader(is, charset));
306             fillBinStats(new StreamDataAdapter(in));
307             loaded = true;
308         } finally {
309             try {
310                 in.close();
311             } catch (IOException ex) { //NOPMD
312                 // ignore
313             }
314         }
315     }
316 
317     /**
318      * Provides methods for computing <code>sampleStats</code> and
319      * <code>beanStats</code> abstracting the source of data.
320      */
321     private abstract class DataAdapter{
322 
323         /**
324          * Compute bin stats.
325          *
326          * @throws IOException  if an error occurs computing bin stats
327          */
328         public abstract void computeBinStats() throws IOException;
329 
330         /**
331          * Compute sample statistics.
332          *
333          * @throws IOException if an error occurs computing sample stats
334          */
335         public abstract void computeStats() throws IOException;
336 
337     }
338 
339     /**
340      * <code>DataAdapter</code> for data provided through some input stream
341      */
342     private class StreamDataAdapter extends DataAdapter{
343 
344         /** Input stream providing access to the data */
345         private BufferedReader inputStream;
346 
347         /**
348          * Create a StreamDataAdapter from a BufferedReader
349          *
350          * @param in BufferedReader input stream
351          */
352         StreamDataAdapter(BufferedReader in){
353             super();
354             inputStream = in;
355         }
356 
357         /** {@inheritDoc} */
358         @Override
359         public void computeBinStats() throws IOException {
360             String str = null;
361             double val = 0.0d;
362             while ((str = inputStream.readLine()) != null) {
363                 val = Double.parseDouble(str);
364                 SummaryStatistics stats = binStats.get(findBin(val));
365                 stats.addValue(val);
366             }
367 
368             inputStream.close();
369             inputStream = null;
370         }
371 
372         /** {@inheritDoc} */
373         @Override
374         public void computeStats() throws IOException {
375             String str = null;
376             double val = 0.0;
377             sampleStats = new SummaryStatistics();
378             while ((str = inputStream.readLine()) != null) {
379                 val = Double.parseDouble(str);
380                 sampleStats.addValue(val);
381             }
382             inputStream.close();
383             inputStream = null;
384         }
385     }
386 
387     /**
388      * <code>DataAdapter</code> for data provided as array of doubles.
389      */
390     private class ArrayDataAdapter extends DataAdapter {
391 
392         /** Array of input  data values */
393         private double[] inputArray;
394 
395         /**
396          * Construct an ArrayDataAdapter from a double[] array
397          *
398          * @param in double[] array holding the data
399          * @throws NullArgumentException if in is null
400          */
401         ArrayDataAdapter(double[] in) throws NullArgumentException {
402             super();
403             MathUtils.checkNotNull(in);
404             inputArray = in;
405         }
406 
407         /** {@inheritDoc} */
408         @Override
409         public void computeStats() throws IOException {
410             sampleStats = new SummaryStatistics();
411             for (int i = 0; i < inputArray.length; i++) {
412                 sampleStats.addValue(inputArray[i]);
413             }
414         }
415 
416         /** {@inheritDoc} */
417         @Override
418         public void computeBinStats() throws IOException {
419             for (int i = 0; i < inputArray.length; i++) {
420                 SummaryStatistics stats =
421                     binStats.get(findBin(inputArray[i]));
422                 stats.addValue(inputArray[i]);
423             }
424         }
425     }
426 
427     /**
428      * Fills binStats array (second pass through data file).
429      *
430      * @param da object providing access to the data
431      * @throws IOException  if an IO error occurs
432      */
433     private void fillBinStats(final DataAdapter da)
434         throws IOException {
435         // Set up grid
436         min = sampleStats.getMin();
437         max = sampleStats.getMax();
438         delta = (max - min)/((double) binCount);
439 
440         // Initialize binStats ArrayList
441         if (!binStats.isEmpty()) {
442             binStats.clear();
443         }
444         for (int i = 0; i < binCount; i++) {
445             SummaryStatistics stats = new SummaryStatistics();
446             binStats.add(i,stats);
447         }
448 
449         // Filling data in binStats Array
450         da.computeBinStats();
451 
452         // Assign upperBounds based on bin counts
453         upperBounds = new double[binCount];
454         upperBounds[0] =
455         ((double) binStats.get(0).getN()) / (double) sampleStats.getN();
456         for (int i = 1; i < binCount-1; i++) {
457             upperBounds[i] = upperBounds[i-1] +
458             ((double) binStats.get(i).getN()) / (double) sampleStats.getN();
459         }
460         upperBounds[binCount-1] = 1.0d;
461     }
462 
463     /**
464      * Returns the index of the bin to which the given value belongs
465      *
466      * @param value  the value whose bin we are trying to find
467      * @return the index of the bin containing the value
468      */
469     private int findBin(double value) {
470         return FastMath.min(
471                 FastMath.max((int) FastMath.ceil((value - min) / delta) - 1, 0),
472                 binCount - 1);
473     }
474 
475     /**
476      * Generates a random value from this distribution.
477      * <strong>Preconditions:</strong><ul>
478      * <li>the distribution must be loaded before invoking this method</li></ul>
479      * @return the random value.
480      * @throws MathIllegalStateException if the distribution has not been loaded
481      */
482     public double getNextValue() throws MathIllegalStateException {
483 
484         if (!loaded) {
485             throw new MathIllegalStateException(LocalizedFormats.DISTRIBUTION_NOT_LOADED);
486         }
487 
488         return sample();
489     }
490 
491     /**
492      * Returns a {@link StatisticalSummary} describing this distribution.
493      * <strong>Preconditions:</strong><ul>
494      * <li>the distribution must be loaded before invoking this method</li></ul>
495      *
496      * @return the sample statistics
497      * @throws IllegalStateException if the distribution has not been loaded
498      */
499     public StatisticalSummary getSampleStats() {
500         return sampleStats;
501     }
502 
503     /**
504      * Returns the number of bins.
505      *
506      * @return the number of bins.
507      */
508     public int getBinCount() {
509         return binCount;
510     }
511 
512     /**
513      * Returns a List of {@link SummaryStatistics} instances containing
514      * statistics describing the values in each of the bins.  The list is
515      * indexed on the bin number.
516      *
517      * @return List of bin statistics.
518      */
519     public List<SummaryStatistics> getBinStats() {
520         return binStats;
521     }
522 
523     /**
524      * <p>Returns a fresh copy of the array of upper bounds for the bins.
525      * Bins are: <br/>
526      * [min,upperBounds[0]],(upperBounds[0],upperBounds[1]],...,
527      *  (upperBounds[binCount-2], upperBounds[binCount-1] = max].</p>
528      *
529      * <p>Note: In versions 1.0-2.0 of commons-math, this method
530      * incorrectly returned the array of probability generator upper
531      * bounds now returned by {@link #getGeneratorUpperBounds()}.</p>
532      *
533      * @return array of bin upper bounds
534      * @since 2.1
535      */
536     public double[] getUpperBounds() {
537         double[] binUpperBounds = new double[binCount];
538         for (int i = 0; i < binCount - 1; i++) {
539             binUpperBounds[i] = min + delta * (i + 1);
540         }
541         binUpperBounds[binCount - 1] = max;
542         return binUpperBounds;
543     }
544 
545     /**
546      * <p>Returns a fresh copy of the array of upper bounds of the subintervals
547      * of [0,1] used in generating data from the empirical distribution.
548      * Subintervals correspond to bins with lengths proportional to bin counts.</p>
549      *
550      * <strong>Preconditions:</strong><ul>
551      * <li>the distribution must be loaded before invoking this method</li></ul>
552      *
553      * <p>In versions 1.0-2.0 of commons-math, this array was (incorrectly) returned
554      * by {@link #getUpperBounds()}.</p>
555      *
556      * @since 2.1
557      * @return array of upper bounds of subintervals used in data generation
558      * @throws NullPointerException unless a {@code load} method has been
559      * called beforehand.
560      */
561     public double[] getGeneratorUpperBounds() {
562         int len = upperBounds.length;
563         double[] out = new double[len];
564         System.arraycopy(upperBounds, 0, out, 0, len);
565         return out;
566     }
567 
568     /**
569      * Property indicating whether or not the distribution has been loaded.
570      *
571      * @return true if the distribution has been loaded
572      */
573     public boolean isLoaded() {
574         return loaded;
575     }
576 
577     /**
578      * Reseeds the random number generator used by {@link #getNextValue()}.
579      *
580      * @param seed random generator seed
581      * @since 3.0
582      */
583     public void reSeed(long seed) {
584         randomData.reSeed(seed);
585     }
586 
587     // Distribution methods ---------------------------
588 
589     /**
590      * {@inheritDoc}
591      * @since 3.1
592      */
593     @Override
594     public double probability(double x) {
595         return 0;
596     }
597 
598     /**
599      * {@inheritDoc}
600      *
601      * <p>Returns the kernel density normalized so that its integral over each bin
602      * equals the bin mass.</p>
603      *
604      * <p>Algorithm description: <ol>
605      * <li>Find the bin B that x belongs to.</li>
606      * <li>Compute K(B) = the mass of B with respect to the within-bin kernel (i.e., the
607      * integral of the kernel density over B).</li>
608      * <li>Return k(x) * P(B) / K(B), where k is the within-bin kernel density
609      * and P(B) is the mass of B.</li></ol></p>
610      * @since 3.1
611      */
612     public double density(double x) {
613         if (x < min || x > max) {
614             return 0d;
615         }
616         final int binIndex = findBin(x);
617         final RealDistribution kernel = getKernel(binStats.get(binIndex));
618         return kernel.density(x) * pB(binIndex) / kB(binIndex);
619     }
620 
621     /**
622      * {@inheritDoc}
623      *
624      * <p>Algorithm description:<ol>
625      * <li>Find the bin B that x belongs to.</li>
626      * <li>Compute P(B) = the mass of B and P(B-) = the combined mass of the bins below B.</li>
627      * <li>Compute K(B) = the probability mass of B with respect to the within-bin kernel
628      * and K(B-) = the kernel distribution evaluated at the lower endpoint of B</li>
629      * <li>Return P(B-) + P(B) * [K(x) - K(B-)] / K(B) where
630      * K(x) is the within-bin kernel distribution function evaluated at x.</li></ol>
631      * If K is a constant distribution, we return P(B-) + P(B) (counting the full
632      * mass of B).</p>
633      *
634      * @since 3.1
635      */
636     public double cumulativeProbability(double x) {
637         if (x < min) {
638             return 0d;
639         } else if (x >= max) {
640             return 1d;
641         }
642         final int binIndex = findBin(x);
643         final double pBminus = pBminus(binIndex);
644         final double pB = pB(binIndex);
645         final RealDistribution kernel = k(x);
646         if (kernel instanceof ConstantRealDistribution) {
647             if (x < kernel.getNumericalMean()) {
648                 return pBminus;
649             } else {
650                 return pBminus + pB;
651             }
652         }
653         final double[] binBounds = getUpperBounds();
654         final double kB = kB(binIndex);
655         final double lower = binIndex == 0 ? min : binBounds[binIndex - 1];
656         final double withinBinCum =
657             (kernel.cumulativeProbability(x) -  kernel.cumulativeProbability(lower)) / kB;
658         return pBminus + pB * withinBinCum;
659     }
660 
661     /**
662      * {@inheritDoc}
663      *
664      * <p>Algorithm description:<ol>
665      * <li>Find the smallest i such that the sum of the masses of the bins
666      *  through i is at least p.</li>
667      * <li>
668      *   Let K be the within-bin kernel distribution for bin i.</br>
669      *   Let K(B) be the mass of B under K. <br/>
670      *   Let K(B-) be K evaluated at the lower endpoint of B (the combined
671      *   mass of the bins below B under K).<br/>
672      *   Let P(B) be the probability of bin i.<br/>
673      *   Let P(B-) be the sum of the bin masses below bin i. <br/>
674      *   Let pCrit = p - P(B-)<br/>
675      * <li>Return the inverse of K evaluated at <br/>
676      *    K(B-) + pCrit * K(B) / P(B) </li>
677      *  </ol></p>
678      *
679      * @since 3.1
680      */
681     @Override
682     public double inverseCumulativeProbability(final double p) throws OutOfRangeException {
683         if (p < 0.0 || p > 1.0) {
684             throw new OutOfRangeException(p, 0, 1);
685         }
686 
687         if (p == 0.0) {
688             return getSupportLowerBound();
689         }
690 
691         if (p == 1.0) {
692             return getSupportUpperBound();
693         }
694 
695         int i = 0;
696         while (cumBinP(i) < p) {
697             i++;
698         }
699 
700         final RealDistribution kernel = getKernel(binStats.get(i));
701         final double kB = kB(i);
702         final double[] binBounds = getUpperBounds();
703         final double lower = i == 0 ? min : binBounds[i - 1];
704         final double kBminus = kernel.cumulativeProbability(lower);
705         final double pB = pB(i);
706         final double pBminus = pBminus(i);
707         final double pCrit = p - pBminus;
708         if (pCrit <= 0) {
709             return lower;
710         }
711         return kernel.inverseCumulativeProbability(kBminus + pCrit * kB / pB);
712     }
713 
714     /**
715      * {@inheritDoc}
716      * @since 3.1
717      */
718     public double getNumericalMean() {
719        return sampleStats.getMean();
720     }
721 
722     /**
723      * {@inheritDoc}
724      * @since 3.1
725      */
726     public double getNumericalVariance() {
727         return sampleStats.getVariance();
728     }
729 
730     /**
731      * {@inheritDoc}
732      * @since 3.1
733      */
734     public double getSupportLowerBound() {
735        return min;
736     }
737 
738     /**
739      * {@inheritDoc}
740      * @since 3.1
741      */
742     public double getSupportUpperBound() {
743         return max;
744     }
745 
746     /**
747      * {@inheritDoc}
748      * @since 3.1
749      */
750     public boolean isSupportLowerBoundInclusive() {
751         return true;
752     }
753 
754     /**
755      * {@inheritDoc}
756      * @since 3.1
757      */
758     public boolean isSupportUpperBoundInclusive() {
759         return true;
760     }
761 
762     /**
763      * {@inheritDoc}
764      * @since 3.1
765      */
766     public boolean isSupportConnected() {
767         return true;
768     }
769 
770     /**
771      * {@inheritDoc}
772      * @since 3.1
773      */
774     @Override
775     public void reseedRandomGenerator(long seed) {
776         randomData.reSeed(seed);
777     }
778 
779     /**
780      * The probability of bin i.
781      *
782      * @param i the index of the bin
783      * @return the probability that selection begins in bin i
784      */
785     private double pB(int i) {
786         return i == 0 ? upperBounds[0] :
787             upperBounds[i] - upperBounds[i - 1];
788     }
789 
790     /**
791      * The combined probability of the bins up to but not including bin i.
792      *
793      * @param i the index of the bin
794      * @return the probability that selection begins in a bin below bin i.
795      */
796     private double pBminus(int i) {
797         return i == 0 ? 0 : upperBounds[i - 1];
798     }
799 
800     /**
801      * Mass of bin i under the within-bin kernel of the bin.
802      *
803      * @param i index of the bin
804      * @return the difference in the within-bin kernel cdf between the
805      * upper and lower endpoints of bin i
806      */
807     @SuppressWarnings("deprecation")
808     private double kB(int i) {
809         final double[] binBounds = getUpperBounds();
810         final RealDistribution kernel = getKernel(binStats.get(i));
811         return i == 0 ? kernel.cumulativeProbability(min, binBounds[0]) :
812             kernel.cumulativeProbability(binBounds[i - 1], binBounds[i]);
813     }
814 
815     /**
816      * The within-bin kernel of the bin that x belongs to.
817      *
818      * @param x the value to locate within a bin
819      * @return the within-bin kernel of the bin containing x
820      */
821     private RealDistribution k(double x) {
822         final int binIndex = findBin(x);
823         return getKernel(binStats.get(binIndex));
824     }
825 
826     /**
827      * The combined probability of the bins up to and including binIndex.
828      *
829      * @param binIndex maximum bin index
830      * @return sum of the probabilities of bins through binIndex
831      */
832     private double cumBinP(int binIndex) {
833         return upperBounds[binIndex];
834     }
835 
836     /**
837      * The within-bin smoothing kernel. Returns a Gaussian distribution
838      * parameterized by {@code bStats}, unless the bin contains only one
839      * observation, in which case a constant distribution is returned.
840      *
841      * @param bStats summary statistics for the bin
842      * @return within-bin kernel parameterized by bStats
843      */
844     protected RealDistribution getKernel(SummaryStatistics bStats) {
845         if (bStats.getN() == 1 || bStats.getVariance() == 0) {
846             return new ConstantRealDistribution(bStats.getMean());
847         } else {
848             return new NormalDistribution(randomData.getRandomGenerator(),
849                 bStats.getMean(), bStats.getStandardDeviation(),
850                 NormalDistribution.DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
851         }
852     }
853 }