examples/src/main/kotlin/space/kscience/kmath/series/emdDemo.kt

@@ -0,0 +1,40 @@
+/*
+ * Copyright 2018-2024 KMath contributors.
+ * Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
+ */
+
+package space.kscience.kmath.series
+
+import space.kscience.kmath.structures.*
+import space.kscience.plotly.*
+import space.kscience.plotly.models.Scatter
+import kotlin.math.sin
+
+fun main(): Unit = with(Double.seriesAlgebra()) {
+    val signal = DoubleArray(800) {
+        sin(it.toDouble() / 10.0) + 3.5 * sin(it.toDouble() / 60.0)
+    }.asBuffer().moveTo(0)
+
+    val emd = empiricalModeDecomposition(
+        sConditionThreshold = 1,
+        maxSiftIterations = 15,
+        nModes = 4
+    ).decompose(signal)
+    println("EMD: ${emd.modes.size} modes extracted, terminated because ${emd.terminatedBecause}")
+
+    fun Plot.series(name: String, buffer: Buffer<Double>, block: Scatter.() -> Unit = {}) {
+        this.scatter {
+            this.name = name
+            this.x.numbers = buffer.offsetIndices
+            this.y.doubles = buffer.toDoubleArray()
+            block()
+        }
+    }
+
+    Plotly.plot {
+        series("Signal", signal)
+        emd.modes.forEachIndexed { index, it ->
+            series("Mode ${index+1}", it)
+        }
+    }.makeFile(resourceLocation = ResourceLocation.REMOTE)
+}

kmath-stat/build.gradle.kts

@@ -14,6 +14,7 @@ kotlin.sourceSets {
         dependencies {
             api(projects.kmathCoroutines)
             //implementation(spclibs.atomicfu)
+            api(project(":kmath-functions"))
         }
     }
 

kmath-stat/src/commonMain/kotlin/space/kscience/kmath/series/EmpiricalModeDecomposition.kt

@@ -0,0 +1,307 @@
+/*
+ * Copyright 2018-2024 KMath contributors.
+ * Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
+ */
+
+package space.kscience.kmath.series
+
+import space.kscience.kmath.interpolation.SplineInterpolator
+import space.kscience.kmath.interpolation.interpolate
+import space.kscience.kmath.operations.*
+import space.kscience.kmath.operations.Float64BufferOps.Companion.div
+import space.kscience.kmath.operations.Float64BufferOps.Companion.pow
+import space.kscience.kmath.structures.Buffer
+import space.kscience.kmath.structures.last
+import space.kscience.kmath.structures.slice
+import kotlin.math.sign
+
+/**
+ * Empirical mode decomposition of a signal represented as a [Series].
+ *
+ * @param seriesAlgebra context to perform operations in.
+ * @param maxSiftIterations number of iterations in the mode extraction (sifting) process after which
+ * the result is returned even if no other conditions are met.
+ * @param sConditionThreshold one of the possible criteria for sifting process termination:
+ * how many times in a row should a proto-mode satisfy the s-condition (the number of zeros differs from
+ * the number of extrema no more than by 1) to be considered an empirical mode.
+ * @param siftingDelta one of the possible criteria for sifting process termination: if relative difference of
+ * two proto-modes obtained in a sequence is less that this number, the last proto-mode is considered
+ * an empirical mode and returned.
+ * @param nModes how many modes should be extracted at most. The algorithm may return fewer modes if it was not
+ * possible to extract more modes from the signal.
+ */
+public class EmpiricalModeDecomposition<BA, L> (
+    private val seriesAlgebra: SeriesAlgebra<Double, *, BA, L>,
+    private val sConditionThreshold: Int = 15,
+    private val maxSiftIterations: Int = 20,
+    private val siftingDelta: Double = 1e-2,
+    private val nModes: Int = 6
+) where BA: BufferAlgebra<Double, *>,  BA: RingOps<Buffer<Double>>,  L : Number {
+
+    /**
+     * Take a signal, construct an upper and a lower envelope, find the mean value of two,
+     * represented as a series.
+     * @param signal Signal to compute on
+     *
+     * @return null is returned if the signal has not enough extrema to construct envelopes.
+     */
+    private fun findMean(signal: Series<Double>): Series<Double>? = with(seriesAlgebra) {
+        val interpolator = SplineInterpolator(Float64Field)
+        fun generateEnvelope(extrema: List<Int>): Series<Double> {
+            val envelopeFunction = interpolator.interpolate(
+                Buffer(extrema.size) { signal.labels[extrema[it]].toDouble() },
+                Buffer(extrema.size) { signal[extrema[it]] }
+            )
+            return signal.mapWithLabel { _, label ->
+                // For some reason PolynomialInterpolator is exclusive and the right boundary
+                // TODO Notify interpolator authors
+                envelopeFunction(label.toDouble()) ?: signal.last()
+                // need to make the interpolator yield values outside boundaries?
+            }
+        }
+        val maxima = signal.maxima()
+        val minima = signal.minima()
+        return if (maxima.size < 3 || minima.size < 3) null else {
+            val upperEnvelope = generateEnvelope(maxima)
+            val lowerEnvelope = generateEnvelope(minima)
+            return (upperEnvelope + lowerEnvelope).map { it * 0.5 }
+        }
+    }
+
+    /**
+     * Extract a single empirical mode from a signal. This process is called sifting, hence the name.
+     * @param signal Signal to extract a mode from. The first mode is extracted from the initial signal,
+     * subsequent modes are extracted from the residuals between the signal and all previous modes.
+     *
+     * @return [SiftingResult.NotEnoughExtrema] is returned if the signal has too few extrema to extract a mode.
+     * Success of an appropriate type (See [SiftingResult.Success] class) is returned otherwise.
+     */
+    private fun sift(signal: Series<Double>): SiftingResult = with(seriesAlgebra) {
+        val mean = findMean(signal) ?: return SiftingResult.NotEnoughExtrema
+        val protoMode = signal.zip(mean) { s, m -> s - m }
+        val sNumber = if (protoMode.sCondition()) 1 else 0
+        return if (maxSiftIterations == 1) SiftingResult.MaxIterationsReached(protoMode)
+        else if (sNumber >= sConditionThreshold) SiftingResult.SNumberReached(protoMode)
+        else if (relativeDifference(signal, protoMode) < siftingDelta * signal.size) {
+            SiftingResult.DeltaReached(protoMode)
+        } else siftInner(protoMode, 2, sNumber)
+    }
+
+    /**
+     * Compute a single iteration of the sifting process.
+     */
+    private fun siftInner(
+        prevMode: Series<Double>,
+        iterationNumber: Int,
+        sNumber: Int
+    ): SiftingResult = with(seriesAlgebra) {
+        val mean = findMean(prevMode) ?: return SiftingResult.SignalFlattened(prevMode)
+        val mode = prevMode.zip(mean) { p, m -> p - m }
+        val newSNumber = if (mode.sCondition()) sNumber + 1 else sNumber
+        return if (iterationNumber >= maxSiftIterations) SiftingResult.MaxIterationsReached(mode)
+        else if (sNumber >= sConditionThreshold) SiftingResult.SNumberReached(mode)
+        else if (relativeDifference(prevMode, mode) < siftingDelta * mode.size) SiftingResult.DeltaReached(mode)
+        else siftInner(mode, iterationNumber + 1, newSNumber)
+    }
+
+    /**
+     * Extract [nModes] empirical modes from a signal represented by a time series.
+     * @param signal Signal to decompose.
+     * @return [EMDecompositionResult] with an appropriate reason for algorithm termination
+     * (see [EMDTerminationReason] for possible reasons).
+     * Modes returned in a list which contains as many modes as it was possible
+     * to extract before triggering one of the termination conditions.
+     */
+    public fun decompose(signal: Series<Double>): EMDecompositionResult = with(seriesAlgebra) {
+        val modes = mutableListOf<Series<Double>>()
+        val mode = when(val r = sift(signal)) {
+            SiftingResult.NotEnoughExtrema ->
+                return EMDecompositionResult(EMDTerminationReason.SIGNAL_TOO_FLAT, modes)
+            is SiftingResult.Success -> r.result
+        }
+        modes.add(mode)
+
+        var residual = signal.zip(mode) { l, r -> l - r }
+        repeat(nModes - 1) {
+            val nextMode = when(val r = sift(residual)) {
+                SiftingResult.NotEnoughExtrema ->
+                    return EMDecompositionResult(EMDTerminationReason.ALL_POSSIBLE_MODES_EXTRACTED, modes)
+                is SiftingResult.Success -> r.result
+            }
+            modes.add(nextMode)
+            residual = residual.zip(nextMode) { l, r -> l - r }
+        }
+        return EMDecompositionResult(EMDTerminationReason.MAX_MODES_REACHED, modes)
+    }
+}
+
+/**
+ * Shortcut to retrieve a decomposition factory from a [SeriesAlgebra] scope.
+ * @receiver scope to perform operations in.
+ * @param maxSiftIterations number of iterations in the mode extraction (sifting) process after which
+ * the result is returned even if no other conditions are met.
+ * @param sConditionThreshold one of the possible criteria for sifting process termination:
+ * how many times in a row should a proto-mode satisfy the s-condition (the number of zeros differs from
+ * the number of extrema no more than by 1) to be considered an empirical mode.
+ * @param siftingDelta one of the possible criteria for sifting process termination: if relative difference of
+ * two proto-modes obtained in a sequence is less that this number, the last proto-mode is considered
+ * an empirical mode and returned.
+ * @param nModes how many modes should be extracted at most. The algorithm may return fewer modes if it was not
+ * possible to extract more modes from the signal.
+ */
+public fun <L, BA> SeriesAlgebra<Double, *, BA, L>.empiricalModeDecomposition(
+    sConditionThreshold: Int = 15,
+    maxSiftIterations: Int = 20,
+    siftingDelta: Double = 1e-2,
+    nModes: Int = 3
+): EmpiricalModeDecomposition<BA, L>
+where BA: BufferAlgebra<Double, *>, BA: RingOps<Buffer<Double>>, L: Number = EmpiricalModeDecomposition(
+    seriesAlgebra = this,
+    sConditionThreshold = sConditionThreshold,
+    maxSiftIterations = maxSiftIterations,
+    siftingDelta = siftingDelta,
+    nModes = nModes
+)
+
+/**
+ * Brute force count all zeros in the series.
+ */
+private fun Series<Double>.countZeros(): Int {
+    require(size >= 2) { "Expected series with at least 2 elements, but got $size elements" }
+    return fold(Pair(get(0), 0)) { acc: Pair<Double, Int>, it: Double ->
+        if (sign(acc.first) != sign(it)) Pair(it, acc.second + 1) else Pair(it, acc.second)
+    }.second
+}
+
+/**
+ * Compute relative difference of two series.
+ */
+private fun <L, BA> SeriesAlgebra<Double, *, BA, L>.relativeDifference(
+    current: Series<Double>,
+    previous: Series<Double>
+):Double where BA: BufferAlgebra<Double, *>, BA: RingOps<Buffer<Double>>, L: Number {
+    return (current - previous).pow(2)
+        .div(previous pow 2)
+        .fold(0.0) { acc, d -> acc + d } // to avoid unnecessary boxing, but i may be wrong
+}
+
+private fun <T: Comparable<T>> isExtreme(prev: T, elem: T, next: T): Boolean =
+    (elem > prev && elem > next) || (elem < prev && elem < next)
+
+/**
+ * Brute force count all extrema of a series.
+ */
+private fun Series<Double>.countExtrema(): Int {
+    require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
+    return slice(1..< size - 1).asIterable().foldIndexed(0) { index, acc, it ->
+        if (isExtreme(get(index), it, get(index + 2))) acc + 1 else acc
+    }
+}
+
+
+/**
+ * Retrieve indices of knot points used to construct an upper envelope,
+ * namely maxima together with the first last point in a series.
+ */
+private fun<T> Series<T>.maxima(): List<Int> where T: Comparable<T> {
+    require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
+    val maxima = mutableListOf(0)
+    slice(1..< size - 1).asIterable().forEachIndexed { index, it ->
+        if (it > get(index) && it > get(index + 2)) { // weird offset, is there a way to do it better?
+            maxima.add(index + 1)
+        }
+    }
+    maxima.add(size - 1)
+    return maxima
+}
+
+/**
+ * Retrieve indices of knot points used to construct a lower envelope,
+ * namely minima together with the first last point in a series.
+ */
+private fun<T> Series<T>.minima(): List<Int> where T: Comparable<T> {
+    require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
+    val minima = mutableListOf(0)
+    slice(1..< size - 1).asIterable().forEachIndexed { index, it ->
+        if (it < get(index) && it < get(index + 2)) { // weird offset, is there a way to do it better?
+            minima.add(index + 1)
+        }
+    }
+    minima.add(size - 1)
+    return minima
+}
+
+/**
+ * Check whether the numbers of zeroes and extrema of a series differ by no more than 1.
+ * This is a necessary condition of an empirical mode.
+ */
+private fun Series<Double>.sCondition(): Boolean = (countExtrema() - countZeros()) in -1..1
+
+internal sealed interface SiftingResult {
+
+    /**
+     * Represents a condition when a mode has been successfully
+     * extracted in a sifting process.
+     */
+    open class Success(val result: Series<Double>): SiftingResult
+
+    /**
+     * Returned when no termination condition was reached and the proto-mode
+     * has become too flat (with not enough extrema to build envelopes)
+     * after several sifting iterations.
+     */
+    class SignalFlattened(result: Series<Double>) : Success(result)
+
+    /**
+     * Returned when sifting process has been terminated due to the
+     * S-number condition being reached.
+     */
+    class SNumberReached(result: Series<Double>) : Success(result)
+
+    /**
+     * Returned when sifting process has been terminated due to the
+     * delta condition (Cauchy criterion) being reached.
+     */
+    class DeltaReached(result: Series<Double>) : Success(result)
+
+    /**
+     * Returned when sifting process has been terminated after
+     * executing the maximum number of iterations (specified when creating an instance
+     * of [EmpiricalModeDecomposition]).
+     */
+    class MaxIterationsReached(result: Series<Double>): Success(result)
+
+    /**
+     * Returned when the submitted signal has not enough extrema to build envelopes,
+     * i.e. when [SignalFlattened] condition has already been reached before the first sifting iteration.
+     */
+    data object NotEnoughExtrema : SiftingResult
+
+}
+
+public enum class EMDTerminationReason {
+    /**
+     * Returned when the signal is too flat, i.e. there are too few extrema
+     * to build envelopes necessary to extract modes.
+     */
+    SIGNAL_TOO_FLAT,
+
+    /**
+     * Returned when there has been extracted as many modes as
+     * specified when creating the instance of [EmpiricalModeDecomposition]
+     */
+    MAX_MODES_REACHED,
+
+    /**
+     * Returned when the algorithm terminated after finding impossible
+     * to extract more modes from the signal and the maximum number
+     * of modes (specified when creating an instance of [EmpiricalModeDecomposition])
+     * has not yet been reached.
+     */
+    ALL_POSSIBLE_MODES_EXTRACTED
+}
+
+public data class EMDecompositionResult(
+    val terminatedBecause: EMDTerminationReason,
+    val modes: List<Series<Double>>
+)