feat: add KalmanFilter, a scalar Kalman filter with sensor fusion

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

Author: alxkm

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

@@ -0,0 +1,287 @@
+package com.thealgorithms.streaming;
+
+/**
+ * A scalar (one dimensional) <b>Kalman filter</b>: the optimal way to track a quantity that drifts
+ * slowly while every measurement of it is noisy.
+ *
+ * <p>The filter carries two numbers: the current estimate {@code x} and how much it distrusts that
+ * estimate, the error variance {@code p}. Each step has two halves.
+ *
+ * <pre>
+ * predict:  x &lt;- x + u            p &lt;- p + q
+ * update:   k &lt;- p / (p + r)      x &lt;- x + k * (z - x)      p &lt;- (1 - k) * p
+ * </pre>
+ *
+ * <p>where {@code q} is the process noise (how much the tracked quantity is expected to wander
+ * between two steps), {@code r} the measurement noise, {@code z} the measurement and {@code k} the
+ * Kalman gain. The gain is the whole story: it is the share of the measurement that gets believed.
+ * When the filter is unsure ({@code p} large) or the sensor is good ({@code r} small), {@code k}
+ * approaches 1 and the filter follows the sensor; in the opposite case it clings to its own
+ * prediction. Nothing tunes this by hand, the variances do it.
+ *
+ * <h2>Sensor fusion</h2>
+ *
+ * <p>Fusing several sensors is not a separate algorithm: it is what happens when the same estimate
+ * is corrected once per sensor, each with its own noise level. A cheap sensor with a large {@code r}
+ * nudges the estimate a little, a precise one pulls it a lot, and the result is exactly the
+ * inverse-variance weighted combination that {@link #fuse(double[], double[])} computes in closed
+ * form:
+ *
+ * <pre>{@code
+ * KalmanFilter filter = new KalmanFilter(startingHeight, 1.0, 0.01, 1.0);
+ * for (int t = 0; t < steps; t++) {
+ *     filter.predict();
+ *     filter.update(barometer[t], barometerVariance);
+ *     filter.update(gps[t], gpsVariance);          // second sensor, same estimate
+ *     double height = filter.estimate();
+ * }
+ * }</pre>
+ *
+ * <p>Both steps run in O(1) time and memory. This class is not thread-safe.
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Kalman_filter">Kalman filter</a>
+ */
+public final class KalmanFilter {
+
+    private final double processNoise;
+    private final double measurementNoise;
+
+    private double estimate;
+    private double errorCovariance;
+    private double lastGain;
+
+    private final double initialEstimate;
+    private final double initialErrorCovariance;
+
+    /**
+     * Creates a filter.
+     *
+     * @param initialEstimate the starting guess for the tracked quantity
+     * @param initialErrorCovariance how uncertain that guess is; a large value makes the filter trust
+     *     the first measurements almost completely
+     * @param processNoise variance added on every {@link #predict()}, i.e. how fast the quantity is
+     *     expected to change on its own
+     * @param measurementNoise default variance of a measurement, used by {@link #update(double)}
+     * @throws IllegalArgumentException if any argument is not finite, or if a variance is negative,
+     *     or if {@code measurementNoise} is zero
+     */
+    public KalmanFilter(double initialEstimate, double initialErrorCovariance, double processNoise, double measurementNoise) {
+        requireFinite(initialEstimate, "initialEstimate");
+        requireNonNegativeVariance(initialErrorCovariance, "initialErrorCovariance");
+        requireNonNegativeVariance(processNoise, "processNoise");
+        requirePositiveVariance(measurementNoise, "measurementNoise");
+
+        this.initialEstimate = initialEstimate;
+        this.initialErrorCovariance = initialErrorCovariance;
+        this.processNoise = processNoise;
+        this.measurementNoise = measurementNoise;
+        reset();
+    }
+
+    /**
+     * Advances the model by one step without any control input, growing the uncertainty by the
+     * process noise.
+     *
+     * @return the predicted estimate, unchanged in value for this constant model
+     */
+    public double predict() {
+        return predict(0.0);
+    }
+
+    /**
+     * Advances the model by one step, shifting the estimate by a known control input.
+     *
+     * @param controlInput the change the estimate is expected to undergo, e.g. velocity times the
+     *     time step when tracking a position
+     * @return the predicted estimate
+     * @throws IllegalArgumentException if {@code controlInput} is not finite
+     */
+    public double predict(double controlInput) {
+        requireFinite(controlInput, "controlInput");
+        estimate += controlInput;
+        errorCovariance += processNoise;
+        return estimate;
+    }
+
+    /**
+     * Corrects the estimate with a measurement taken by the default sensor.
+     *
+     * @param measurement the observed value
+     * @return the corrected estimate
+     * @throws IllegalArgumentException if {@code measurement} is not finite
+     */
+    public double update(double measurement) {
+        return update(measurement, measurementNoise);
+    }
+
+    /**
+     * Corrects the estimate with a measurement whose noise differs from the default one. Calling this
+     * several times per step, once per sensor, is the whole of sensor fusion.
+     *
+     * @param measurement the observed value
+     * @param noise variance of this particular measurement, strictly positive
+     * @return the corrected estimate
+     * @throws IllegalArgumentException if {@code measurement} is not finite or {@code noise} is not strictly positive
+     */
+    public double update(double measurement, double noise) {
+        requireFinite(measurement, "measurement");
+        requirePositiveVariance(noise, "noise");
+
+        double innovationVariance = errorCovariance + noise;
+        lastGain = errorCovariance / innovationVariance;
+        estimate += lastGain * (measurement - estimate);
+        // Algebraically this is (1 - gain) * p, but computing 1 - gain cancels away most of the
+        // significant digits whenever the gain is close to one, as it is on the first measurements.
+        errorCovariance = errorCovariance * noise / innovationVariance;
+        return estimate;
+    }
+
+    /**
+     * Runs one full cycle: predict, then correct with the given measurement.
+     *
+     * @param measurement the observed value
+     * @return the filtered estimate
+     * @throws IllegalArgumentException if {@code measurement} is not finite
+     */
+    public double filter(double measurement) {
+        predict();
+        return update(measurement);
+    }
+
+    /**
+     * Filters a whole signal offline, one cycle per sample.
+     *
+     * @param measurements the noisy signal
+     * @return a new array holding the filtered signal, of the same length
+     * @throws IllegalArgumentException if any measurement is not finite
+     * @throws NullPointerException if {@code measurements} is {@code null}
+     */
+    public double[] filter(double[] measurements) {
+        double[] filtered = new double[measurements.length];
+        for (int i = 0; i < measurements.length; i++) {
+            filtered[i] = filter(measurements[i]);
+        }
+        return filtered;
+    }
+
+    /**
+     * Combines simultaneous readings of the same quantity taken by independent sensors, weighting
+     * each by the inverse of its variance. This is the closed form of what repeated
+     * {@link #update(double, double)} calls achieve within one step.
+     *
+     * @param measurements one reading per sensor
+     * @param variances the noise variance of each sensor, strictly positive, same length as {@code measurements}
+     * @return the fused reading together with its variance, which is never larger than the variance of
+     *     the best single sensor
+     * @throws IllegalArgumentException if the arrays are empty, differ in length, hold a non-finite
+     *     measurement or a non-positive variance
+     * @throws NullPointerException if either array is {@code null}
+     */
+    public static Estimate fuse(double[] measurements, double[] variances) {
+        if (measurements.length != variances.length) {
+            throw new IllegalArgumentException("There must be exactly one variance per measurement, but got " + measurements.length + " and " + variances.length);
+        }
+        if (measurements.length == 0) {
+            throw new IllegalArgumentException("At least one measurement is required");
+        }
+
+        double weightSum = 0.0;
+        double weightedSum = 0.0;
+        for (int i = 0; i < measurements.length; i++) {
+            requireFinite(measurements[i], "measurement");
+            requirePositiveVariance(variances[i], "variance");
+            double weight = 1.0 / variances[i];
+            weightSum += weight;
+            weightedSum += weight * measurements[i];
+        }
+        return new Estimate(weightedSum / weightSum, 1.0 / weightSum);
+    }
+
+    /**
+     * Returns the current estimate of the tracked quantity.
+     *
+     * @return the state estimate
+     */
+    public double estimate() {
+        return estimate;
+    }
+
+    /**
+     * Returns the variance of the current estimate; it shrinks with every update and grows with every
+     * prediction.
+     *
+     * @return the error covariance
+     */
+    public double errorCovariance() {
+        return errorCovariance;
+    }
+
+    /**
+     * Returns the Kalman gain used by the most recent update, a number in {@code [0, 1)} telling how
+     * much of that measurement was believed.
+     *
+     * @return the last gain, {@code 0} if no update has happened yet
+     */
+    public double lastGain() {
+        return lastGain;
+    }
+
+    /**
+     * Returns the process noise variance.
+     *
+     * @return the value given at construction time
+     */
+    public double processNoise() {
+        return processNoise;
+    }
+
+    /**
+     * Returns the default measurement noise variance.
+     *
+     * @return the value given at construction time
+     */
+    public double measurementNoise() {
+        return measurementNoise;
+    }
+
+    /**
+     * Restores the state the filter had right after construction.
+     */
+    public void reset() {
+        estimate = initialEstimate;
+        errorCovariance = initialErrorCovariance;
+        lastGain = 0.0;
+    }
+
+    @Override
+    public String toString() {
+        return "KalmanFilter{estimate=" + estimate + ", errorCovariance=" + errorCovariance + ", lastGain=" + lastGain + '}';
+    }
+
+    private static void requireFinite(double value, String name) {
+        if (!Double.isFinite(value)) {
+            throw new IllegalArgumentException("The " + name + " must be finite, but was " + value);
+        }
+    }
+
+    private static void requireNonNegativeVariance(double value, String name) {
+        if (!(value >= 0.0) || !Double.isFinite(value)) {
+            throw new IllegalArgumentException("The " + name + " must be finite and non-negative, but was " + value);
+        }
+    }
+
+    private static void requirePositiveVariance(double value, String name) {
+        if (!(value > 0.0) || !Double.isFinite(value)) {
+            throw new IllegalArgumentException("The " + name + " must be finite and strictly positive, but was " + value);
+        }
+    }
+
+    /**
+     * A value paired with the variance that describes how much it can be trusted.
+     *
+     * @param value the estimated quantity
+     * @param variance the variance of that estimate
+     */
+    public record Estimate(double value, double variance) {
+    }
+}