feat: add Adwin, an adaptive window that cuts itself when the stream changes (#7606)

ADWIN keeps a window of recent values and, after every sample, looks for a split into an old and a recent part whose means differ by more than a variance sensitive Hoeffding bound. When it finds one the old part is dropped, so the window grows while the stream is stationary and collapses as soon as it moves, and its length becomes an estimate of how long the current regime has lasted rather than a parameter to tune.

The window is stored as an exponential histogram, buckets of 1, 2, 4 ... elements with at most five of each size, so a window of n elements needs O(log n) buckets and cuts are tried only at bucket boundaries. The bucket merge carries the sum and the variance exactly, which the tests check against a direct computation over 2000 samples.

Signed-off-by: alxkm <19151554+alxkm@users.noreply.github.com>
Co-authored-by: alxkm <19151554+alxkm@users.noreply.github.com>

Author: alxkm

src/main/java/com/thealgorithms/streaming/Adwin.java

@@ -0,0 +1,354 @@
+package com.thealgorithms.streaming;
+
+/**
+ * <b>ADWIN</b>, adaptive windowing, after Bifet and Gavalda: an average over a window whose length is
+ * not a parameter but a result.
+ *
+ * <p>Every windowed estimator forces the same bad choice. A long window is accurate while nothing
+ * changes and hopelessly slow once something does; a short one reacts immediately and is noisy the
+ * rest of the time. ADWIN refuses the choice: it keeps a window of recent values and, after every
+ * sample, looks for a way to split it into an old part and a recent part whose means are too far
+ * apart to be explained by chance. When it finds one, the old part is dropped. The window therefore
+ * grows on its own while the stream is stationary and collapses as soon as the stream moves, and the
+ * length it settles at is an estimate of how long the current regime has been running.
+ *
+ * <p>"Too far apart" is a variance sensitive Hoeffding bound. For a cut into sub-windows of
+ * {@code n0} and {@code n1} elements, with {@code v} the variance of the whole window:
+ *
+ * <pre>
+ * m       = 1 / (n0 - minLength + 1) + 1 / (n1 - minLength + 1)
+ * d       = ln( 2 * ln(width) / delta )
+ * epsilon = sqrt(2 * m * v * d) + 2/3 * d * m
+ * cut when |mean0 - mean1| &gt; epsilon
+ * </pre>
+ *
+ * <p>The {@code delta} parameter is a confidence level: the probability of cutting a window that
+ * never changed is bounded by it, which is the guarantee that makes the window length trustworthy.
+ * Smaller values make the detector more conservative and slower.
+ *
+ * <p>Keeping every sample would cost O(n) memory, so the window is stored as an exponential
+ * histogram: buckets of 1, 2, 4, 8 ... elements, at most {@code MAX_BUCKETS} of each size, each
+ * holding the sum and the variance of the elements it covers. That is O(log n) buckets for a window
+ * of n elements, and cuts are only tried at bucket boundaries, which is what keeps a sample O(log n)
+ * instead of O(n) while costing only a bounded loss of resolution.
+ *
+ * <h2>Usage</h2>
+ *
+ * <pre>{@code
+ * Adwin window = new Adwin(0.002);
+ * for (double sample : stream) {
+ *     if (window.accept(sample)) {
+ *         alert(window.estimate(), window.width());
+ *     }
+ * }
+ * }</pre>
+ *
+ * <p>This class is not thread-safe.
+ *
+ * @see CusumDetector
+ * @see <a href="https://en.wikipedia.org/wiki/Concept_drift">Concept drift</a>, the problem ADWIN was written for; the algorithm is due to A. Bifet and R. Gavalda, Learning from Time-Changing Data with Adaptive Windowing, SDM 2007
+ */
+public final class Adwin {
+
+    /** Confidence level used when none is given. */
+    public static final double DEFAULT_DELTA = 0.002;
+
+    /** How many buckets of the same size the histogram holds before merging the two oldest. */
+    public static final int MAX_BUCKETS = 5;
+
+    private static final int MIN_SUBWINDOW = 5;
+    private static final long MIN_WIDTH_FOR_DETECTION = 2L * MIN_SUBWINDOW;
+
+    private final double delta;
+
+    private Row newest;
+    private Row oldest;
+
+    private long width;
+    private double total;
+    private double variance;
+    private int bucketCount;
+    private long count;
+    private long changeCount;
+
+    /**
+     * Creates a window with the customary confidence level of {@code 0.002}.
+     */
+    public Adwin() {
+        this(DEFAULT_DELTA);
+    }
+
+    /**
+     * Creates a window.
+     *
+     * @param delta the confidence level, in {@code (0, 1)}; smaller values cut less eagerly
+     * @throws IllegalArgumentException if {@code delta} is outside {@code (0, 1)}
+     */
+    public Adwin(double delta) {
+        if (!(delta > 0.0) || !(delta < 1.0)) {
+            throw new IllegalArgumentException("The delta must lie in (0, 1), but was " + delta);
+        }
+        this.delta = delta;
+        start();
+    }
+
+    /**
+     * Feeds one sample into the window.
+     *
+     * @param value the incoming sample
+     * @return {@code true} if the window was cut, that is if the stream changed
+     * @throws IllegalArgumentException if {@code value} is NaN or infinite
+     */
+    public boolean accept(double value) {
+        if (!Double.isFinite(value)) {
+            throw new IllegalArgumentException("Samples must be finite, but was " + value);
+        }
+        count++;
+        insert(value);
+        return detectChange();
+    }
+
+    /**
+     * Runs the window over a whole signal.
+     *
+     * @param signal the samples to inspect
+     * @return a new array of the same length saying for every sample whether it cut the window
+     * @throws IllegalArgumentException if any sample is NaN or infinite
+     * @throws NullPointerException if {@code signal} is {@code null}
+     */
+    public boolean[] scan(double[] signal) {
+        boolean[] cuts = new boolean[signal.length];
+        for (int i = 0; i < signal.length; i++) {
+            cuts[i] = accept(signal[i]);
+        }
+        return cuts;
+    }
+
+    /**
+     * Returns the current estimate of the level of the stream.
+     *
+     * @return the mean of the window, {@code 0} while the window is empty
+     */
+    public double estimate() {
+        return width == 0 ? 0.0 : total / width;
+    }
+
+    /**
+     * Returns the length of the window, which is how many recent samples the estimate rests on.
+     *
+     * @return the window width
+     */
+    public long width() {
+        return width;
+    }
+
+    /**
+     * Returns the variance of the samples inside the window.
+     *
+     * @return the window variance, {@code 0} while the window holds fewer than two samples
+     */
+    public double variance() {
+        return width < 2 ? 0.0 : variance / width;
+    }
+
+    /**
+     * Returns how many buckets the histogram holds, which grows like the logarithm of the width.
+     *
+     * @return the bucket count
+     */
+    public int bucketCount() {
+        return bucketCount;
+    }
+
+    /**
+     * Returns how many samples have been inspected since the last reset.
+     *
+     * @return the sample count
+     */
+    public long count() {
+        return count;
+    }
+
+    /**
+     * Returns how many times the window has been cut since the last reset.
+     *
+     * @return the number of detected changes
+     */
+    public long changeCount() {
+        return changeCount;
+    }
+
+    /**
+     * Returns the configured confidence level.
+     *
+     * @return the delta given at construction time
+     */
+    public double delta() {
+        return delta;
+    }
+
+    /**
+     * Empties the window.
+     */
+    public void reset() {
+        start();
+        count = 0;
+        changeCount = 0;
+    }
+
+    @Override
+    public String toString() {
+        return "Adwin{width=" + width + ", estimate=" + estimate() + ", buckets=" + bucketCount + ", changes=" + changeCount + "}";
+    }
+
+    private void start() {
+        newest = new Row(0);
+        oldest = newest;
+        width = 0;
+        total = 0.0;
+        variance = 0.0;
+        bucketCount = 0;
+    }
+
+    private void insert(double value) {
+        width++;
+        newest.add(value, 0.0);
+        bucketCount++;
+        if (width > 1) {
+            double deviation = value - total / (width - 1);
+            variance += (width - 1) * deviation * deviation / width;
+        }
+        total += value;
+        compress();
+    }
+
+    /**
+     * Merges the two oldest buckets of every row that has run out of room into one bucket of the next
+     * row, which is what keeps the number of buckets logarithmic in the width.
+     */
+    private void compress() {
+        Row row = newest;
+        while (row != null && row.size > MAX_BUCKETS) {
+            if (row.older == null) {
+                row.older = new Row(row.level + 1);
+                row.older.newer = row;
+                oldest = row.older;
+            }
+            long size = 1L << row.level;
+            double firstMean = row.totals[0] / size;
+            double secondMean = row.totals[1] / size;
+            double merged = size * size * (firstMean - secondMean) * (firstMean - secondMean) / (size + size);
+            row.older.add(row.totals[0] + row.totals[1], row.variances[0] + row.variances[1] + merged);
+            row.removeOldest(2);
+            bucketCount--;
+            row = row.older;
+        }
+    }
+
+    /**
+     * Tries every cut the histogram allows, from the oldest boundary inwards, and drops the oldest
+     * bucket whenever a cut is significant. Repeats until no cut is left.
+     *
+     * @return whether anything was dropped
+     */
+    private boolean detectChange() {
+        boolean changed = false;
+        boolean searching = true;
+
+        while (searching && width >= MIN_WIDTH_FOR_DETECTION) {
+            searching = false;
+            long oldWidth = 0;
+            double oldTotal = 0.0;
+
+        outer:
+            for (Row row = oldest; row != null; row = row.newer) {
+                for (int bucket = 0; bucket < row.size; bucket++) {
+                    if (row.newer == null && bucket == row.size - 1) {
+                        break outer;
+                    }
+                    oldWidth += 1L << row.level;
+                    oldTotal += row.totals[bucket];
+                    long recentWidth = width - oldWidth;
+                    double recentTotal = total - oldTotal;
+                    if (recentWidth < MIN_SUBWINDOW) {
+                        break outer;
+                    }
+                    if (oldWidth >= MIN_SUBWINDOW && isSignificant(oldWidth, recentWidth, oldTotal / oldWidth - recentTotal / recentWidth)) {
+                        changed = true;
+                        searching = true;
+                        changeCount++;
+                        dropOldestBucket();
+                        break outer;
+                    }
+                }
+            }
+        }
+        return changed;
+    }
+
+    private boolean isSignificant(long oldWidth, long recentWidth, double difference) {
+        double harmonic = 1.0 / (oldWidth - MIN_SUBWINDOW + 1) + 1.0 / (recentWidth - MIN_SUBWINDOW + 1);
+        double confidence = Math.log(2.0 * Math.log(width) / delta);
+        double windowVariance = variance / width;
+        double epsilon = Math.sqrt(2.0 * harmonic * windowVariance * confidence) + 2.0 / 3.0 * confidence * harmonic;
+        return Math.abs(difference) > epsilon;
+    }
+
+    private void dropOldestBucket() {
+        Row row = oldest;
+        long size = 1L << row.level;
+        double bucketTotal = row.totals[0];
+
+        width -= size;
+        total -= bucketTotal;
+        if (width > 0) {
+            double bucketMean = bucketTotal / size;
+            double difference = bucketMean - total / width;
+            variance -= row.variances[0] + size * width * difference * difference / (size + width);
+        } else {
+            variance = 0.0;
+        }
+        if (variance < 0.0) {
+            variance = 0.0;
+        }
+
+        row.removeOldest(1);
+        bucketCount--;
+        if (row.size == 0 && row.newer != null) {
+            oldest = row.newer;
+            oldest.older = null;
+        }
+    }
+
+    /**
+     * One row of the exponential histogram: up to {@code MAX_BUCKETS + 1} buckets that each cover
+     * {@code 2^level} samples, the oldest at index zero.
+     */
+    private static final class Row {
+
+        private final int level;
+        private final double[] totals = new double[MAX_BUCKETS + 1];
+        private final double[] variances = new double[MAX_BUCKETS + 1];
+        private int size;
+        private Row older;
+        private Row newer;
+
+        Row(int level) {
+            this.level = level;
+        }
+
+        void add(double total, double variance) {
+            totals[size] = total;
+            variances[size] = variance;
+            size++;
+        }
+
+        void removeOldest(int buckets) {
+            for (int i = buckets; i < size; i++) {
+                totals[i - buckets] = totals[i];
+                variances[i - buckets] = variances[i];
+            }
+            size -= buckets;
+        }
+    }
+}