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Empirical Mode Decomposition for Double series

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Igor Dunaev 2024-04-24 16:58:35 +03:00
parent e0125f0b26
commit 8fa42450b2
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40
examples/src/main/kotlin/space/kscience/kmath/series/emdDemo.kt Normal file
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@ -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)
}

1
kmath-stat/build.gradle.kts
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@ -14,6 +14,7 @@ kotlin.sourceSets {
dependencies { dependencies {
api(projects.kmathCoroutines) api(projects.kmathCoroutines)
//implementation(spclibs.atomicfu) //implementation(spclibs.atomicfu)
api(project(":kmath-functions"))
} }
} }

307
kmath-stat/src/commonMain/kotlin/space/kscience/kmath/series/EmpiricalModeDecomposition.kt Normal file
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/*
* 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>>
)