forked from kscience/kmath
Empirical Mode Decomposition for Double series
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/*
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* Copyright 2018-2024 KMath contributors.
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* Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
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*/
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package space.kscience.kmath.series
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import space.kscience.kmath.structures.*
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import space.kscience.plotly.*
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import space.kscience.plotly.models.Scatter
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import kotlin.math.sin
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fun main(): Unit = with(Double.seriesAlgebra()) {
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val signal = DoubleArray(800) {
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sin(it.toDouble() / 10.0) + 3.5 * sin(it.toDouble() / 60.0)
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}.asBuffer().moveTo(0)
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val emd = empiricalModeDecomposition(
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sConditionThreshold = 1,
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maxSiftIterations = 15,
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nModes = 4
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).decompose(signal)
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println("EMD: ${emd.modes.size} modes extracted, terminated because ${emd.terminatedBecause}")
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fun Plot.series(name: String, buffer: Buffer<Double>, block: Scatter.() -> Unit = {}) {
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this.scatter {
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this.name = name
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this.x.numbers = buffer.offsetIndices
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this.y.doubles = buffer.toDoubleArray()
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block()
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}
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}
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Plotly.plot {
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series("Signal", signal)
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emd.modes.forEachIndexed { index, it ->
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series("Mode ${index+1}", it)
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}
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}.makeFile(resourceLocation = ResourceLocation.REMOTE)
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}
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@ -14,6 +14,7 @@ kotlin.sourceSets {
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dependencies {
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api(projects.kmathCoroutines)
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//implementation(spclibs.atomicfu)
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api(project(":kmath-functions"))
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}
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}
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/*
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* Copyright 2018-2024 KMath contributors.
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* Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
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*/
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package space.kscience.kmath.series
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import space.kscience.kmath.interpolation.SplineInterpolator
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import space.kscience.kmath.interpolation.interpolate
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import space.kscience.kmath.operations.*
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import space.kscience.kmath.operations.Float64BufferOps.Companion.div
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import space.kscience.kmath.operations.Float64BufferOps.Companion.pow
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import space.kscience.kmath.structures.Buffer
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import space.kscience.kmath.structures.last
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import space.kscience.kmath.structures.slice
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import kotlin.math.sign
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/**
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* Empirical mode decomposition of a signal represented as a [Series].
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*
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* @param seriesAlgebra context to perform operations in.
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* @param maxSiftIterations number of iterations in the mode extraction (sifting) process after which
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* the result is returned even if no other conditions are met.
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* @param sConditionThreshold one of the possible criteria for sifting process termination:
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* how many times in a row should a proto-mode satisfy the s-condition (the number of zeros differs from
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* the number of extrema no more than by 1) to be considered an empirical mode.
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* @param siftingDelta one of the possible criteria for sifting process termination: if relative difference of
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* two proto-modes obtained in a sequence is less that this number, the last proto-mode is considered
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* an empirical mode and returned.
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* @param nModes how many modes should be extracted at most. The algorithm may return fewer modes if it was not
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* possible to extract more modes from the signal.
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*/
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public class EmpiricalModeDecomposition<BA, L> (
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private val seriesAlgebra: SeriesAlgebra<Double, *, BA, L>,
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private val sConditionThreshold: Int = 15,
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private val maxSiftIterations: Int = 20,
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private val siftingDelta: Double = 1e-2,
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private val nModes: Int = 6
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) where BA: BufferAlgebra<Double, *>, BA: RingOps<Buffer<Double>>, L : Number {
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/**
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* Take a signal, construct an upper and a lower envelope, find the mean value of two,
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* represented as a series.
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* @param signal Signal to compute on
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*
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* @return null is returned if the signal has not enough extrema to construct envelopes.
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*/
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private fun findMean(signal: Series<Double>): Series<Double>? = with(seriesAlgebra) {
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val interpolator = SplineInterpolator(Float64Field)
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fun generateEnvelope(extrema: List<Int>): Series<Double> {
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val envelopeFunction = interpolator.interpolate(
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Buffer(extrema.size) { signal.labels[extrema[it]].toDouble() },
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Buffer(extrema.size) { signal[extrema[it]] }
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)
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return signal.mapWithLabel { _, label ->
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// For some reason PolynomialInterpolator is exclusive and the right boundary
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// TODO Notify interpolator authors
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envelopeFunction(label.toDouble()) ?: signal.last()
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// need to make the interpolator yield values outside boundaries?
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}
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}
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val maxima = signal.maxima()
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val minima = signal.minima()
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return if (maxima.size < 3 || minima.size < 3) null else {
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val upperEnvelope = generateEnvelope(maxima)
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val lowerEnvelope = generateEnvelope(minima)
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return (upperEnvelope + lowerEnvelope).map { it * 0.5 }
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}
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}
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/**
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* Extract a single empirical mode from a signal. This process is called sifting, hence the name.
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* @param signal Signal to extract a mode from. The first mode is extracted from the initial signal,
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* subsequent modes are extracted from the residuals between the signal and all previous modes.
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*
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* @return [SiftingResult.NotEnoughExtrema] is returned if the signal has too few extrema to extract a mode.
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* Success of an appropriate type (See [SiftingResult.Success] class) is returned otherwise.
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*/
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private fun sift(signal: Series<Double>): SiftingResult = with(seriesAlgebra) {
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val mean = findMean(signal) ?: return SiftingResult.NotEnoughExtrema
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val protoMode = signal.zip(mean) { s, m -> s - m }
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val sNumber = if (protoMode.sCondition()) 1 else 0
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return if (maxSiftIterations == 1) SiftingResult.MaxIterationsReached(protoMode)
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else if (sNumber >= sConditionThreshold) SiftingResult.SNumberReached(protoMode)
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else if (relativeDifference(signal, protoMode) < siftingDelta * signal.size) {
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SiftingResult.DeltaReached(protoMode)
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} else siftInner(protoMode, 2, sNumber)
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}
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/**
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* Compute a single iteration of the sifting process.
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*/
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private fun siftInner(
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prevMode: Series<Double>,
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iterationNumber: Int,
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sNumber: Int
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): SiftingResult = with(seriesAlgebra) {
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val mean = findMean(prevMode) ?: return SiftingResult.SignalFlattened(prevMode)
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val mode = prevMode.zip(mean) { p, m -> p - m }
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val newSNumber = if (mode.sCondition()) sNumber + 1 else sNumber
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return if (iterationNumber >= maxSiftIterations) SiftingResult.MaxIterationsReached(mode)
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else if (sNumber >= sConditionThreshold) SiftingResult.SNumberReached(mode)
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else if (relativeDifference(prevMode, mode) < siftingDelta * mode.size) SiftingResult.DeltaReached(mode)
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else siftInner(mode, iterationNumber + 1, newSNumber)
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}
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/**
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* Extract [nModes] empirical modes from a signal represented by a time series.
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* @param signal Signal to decompose.
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* @return [EMDecompositionResult] with an appropriate reason for algorithm termination
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* (see [EMDTerminationReason] for possible reasons).
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* Modes returned in a list which contains as many modes as it was possible
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* to extract before triggering one of the termination conditions.
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*/
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public fun decompose(signal: Series<Double>): EMDecompositionResult = with(seriesAlgebra) {
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val modes = mutableListOf<Series<Double>>()
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val mode = when(val r = sift(signal)) {
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SiftingResult.NotEnoughExtrema ->
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return EMDecompositionResult(EMDTerminationReason.SIGNAL_TOO_FLAT, modes)
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is SiftingResult.Success -> r.result
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}
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modes.add(mode)
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var residual = signal.zip(mode) { l, r -> l - r }
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repeat(nModes - 1) {
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val nextMode = when(val r = sift(residual)) {
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SiftingResult.NotEnoughExtrema ->
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return EMDecompositionResult(EMDTerminationReason.ALL_POSSIBLE_MODES_EXTRACTED, modes)
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is SiftingResult.Success -> r.result
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}
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modes.add(nextMode)
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residual = residual.zip(nextMode) { l, r -> l - r }
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}
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return EMDecompositionResult(EMDTerminationReason.MAX_MODES_REACHED, modes)
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}
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}
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/**
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* Shortcut to retrieve a decomposition factory from a [SeriesAlgebra] scope.
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* @receiver scope to perform operations in.
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* @param maxSiftIterations number of iterations in the mode extraction (sifting) process after which
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* the result is returned even if no other conditions are met.
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* @param sConditionThreshold one of the possible criteria for sifting process termination:
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* how many times in a row should a proto-mode satisfy the s-condition (the number of zeros differs from
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* the number of extrema no more than by 1) to be considered an empirical mode.
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* @param siftingDelta one of the possible criteria for sifting process termination: if relative difference of
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* two proto-modes obtained in a sequence is less that this number, the last proto-mode is considered
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* an empirical mode and returned.
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* @param nModes how many modes should be extracted at most. The algorithm may return fewer modes if it was not
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* possible to extract more modes from the signal.
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*/
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public fun <L, BA> SeriesAlgebra<Double, *, BA, L>.empiricalModeDecomposition(
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sConditionThreshold: Int = 15,
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maxSiftIterations: Int = 20,
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siftingDelta: Double = 1e-2,
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nModes: Int = 3
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): EmpiricalModeDecomposition<BA, L>
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where BA: BufferAlgebra<Double, *>, BA: RingOps<Buffer<Double>>, L: Number = EmpiricalModeDecomposition(
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seriesAlgebra = this,
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sConditionThreshold = sConditionThreshold,
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maxSiftIterations = maxSiftIterations,
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siftingDelta = siftingDelta,
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nModes = nModes
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)
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/**
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* Brute force count all zeros in the series.
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*/
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private fun Series<Double>.countZeros(): Int {
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require(size >= 2) { "Expected series with at least 2 elements, but got $size elements" }
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return fold(Pair(get(0), 0)) { acc: Pair<Double, Int>, it: Double ->
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if (sign(acc.first) != sign(it)) Pair(it, acc.second + 1) else Pair(it, acc.second)
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}.second
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}
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/**
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* Compute relative difference of two series.
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*/
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private fun <L, BA> SeriesAlgebra<Double, *, BA, L>.relativeDifference(
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current: Series<Double>,
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previous: Series<Double>
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):Double where BA: BufferAlgebra<Double, *>, BA: RingOps<Buffer<Double>>, L: Number {
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return (current - previous).pow(2)
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.div(previous pow 2)
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.fold(0.0) { acc, d -> acc + d } // to avoid unnecessary boxing, but i may be wrong
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}
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private fun <T: Comparable<T>> isExtreme(prev: T, elem: T, next: T): Boolean =
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(elem > prev && elem > next) || (elem < prev && elem < next)
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/**
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* Brute force count all extrema of a series.
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*/
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private fun Series<Double>.countExtrema(): Int {
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require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
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return slice(1..< size - 1).asIterable().foldIndexed(0) { index, acc, it ->
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if (isExtreme(get(index), it, get(index + 2))) acc + 1 else acc
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}
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}
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/**
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* Retrieve indices of knot points used to construct an upper envelope,
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* namely maxima together with the first last point in a series.
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*/
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private fun<T> Series<T>.maxima(): List<Int> where T: Comparable<T> {
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require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
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val maxima = mutableListOf(0)
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slice(1..< size - 1).asIterable().forEachIndexed { index, it ->
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if (it > get(index) && it > get(index + 2)) { // weird offset, is there a way to do it better?
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maxima.add(index + 1)
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}
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}
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maxima.add(size - 1)
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return maxima
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}
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/**
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* Retrieve indices of knot points used to construct a lower envelope,
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* namely minima together with the first last point in a series.
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*/
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private fun<T> Series<T>.minima(): List<Int> where T: Comparable<T> {
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require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
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val minima = mutableListOf(0)
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slice(1..< size - 1).asIterable().forEachIndexed { index, it ->
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if (it < get(index) && it < get(index + 2)) { // weird offset, is there a way to do it better?
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minima.add(index + 1)
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}
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}
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minima.add(size - 1)
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return minima
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}
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/**
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* Check whether the numbers of zeroes and extrema of a series differ by no more than 1.
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* This is a necessary condition of an empirical mode.
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*/
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private fun Series<Double>.sCondition(): Boolean = (countExtrema() - countZeros()) in -1..1
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internal sealed interface SiftingResult {
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/**
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* Represents a condition when a mode has been successfully
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* extracted in a sifting process.
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*/
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open class Success(val result: Series<Double>): SiftingResult
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/**
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* Returned when no termination condition was reached and the proto-mode
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* has become too flat (with not enough extrema to build envelopes)
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* after several sifting iterations.
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*/
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class SignalFlattened(result: Series<Double>) : Success(result)
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/**
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* Returned when sifting process has been terminated due to the
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* S-number condition being reached.
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*/
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class SNumberReached(result: Series<Double>) : Success(result)
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/**
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* Returned when sifting process has been terminated due to the
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* delta condition (Cauchy criterion) being reached.
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*/
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class DeltaReached(result: Series<Double>) : Success(result)
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/**
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* Returned when sifting process has been terminated after
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* executing the maximum number of iterations (specified when creating an instance
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* of [EmpiricalModeDecomposition]).
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*/
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class MaxIterationsReached(result: Series<Double>): Success(result)
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/**
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* Returned when the submitted signal has not enough extrema to build envelopes,
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* i.e. when [SignalFlattened] condition has already been reached before the first sifting iteration.
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*/
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data object NotEnoughExtrema : SiftingResult
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}
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public enum class EMDTerminationReason {
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/**
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* Returned when the signal is too flat, i.e. there are too few extrema
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* to build envelopes necessary to extract modes.
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*/
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SIGNAL_TOO_FLAT,
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/**
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* Returned when there has been extracted as many modes as
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* specified when creating the instance of [EmpiricalModeDecomposition]
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*/
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MAX_MODES_REACHED,
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/**
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* Returned when the algorithm terminated after finding impossible
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* to extract more modes from the signal and the maximum number
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* of modes (specified when creating an instance of [EmpiricalModeDecomposition])
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* has not yet been reached.
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*/
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ALL_POSSIBLE_MODES_EXTRACTED
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}
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public data class EMDecompositionResult(
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val terminatedBecause: EMDTerminationReason,
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val modes: List<Series<Double>>
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)
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