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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.operations.algebra
import space.kscience.kmath.operations.bufferAlgebra
import space.kscience.kmath.structures.*
import space.kscience.kmath.operations.invoke
import space.kscience.plotly.*
import space.kscience.plotly.models.Scatter
import kotlin.math.sin
private val customAlgebra = (Double.algebra.bufferAlgebra) { SeriesAlgebra(this) { it.toDouble() } }
fun main(): Unit = (customAlgebra) {
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,
siftingDelta = 1e-2,
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.labels
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)
}

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

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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.structures.Buffer
import space.kscience.kmath.structures.last
/**
* 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<T: Comparable<T>, A: Field<T>, BA, L: T> (
private val seriesAlgebra: SeriesAlgebra<T, A, BA, L>,
private val sConditionThreshold: Int = 15,
private val maxSiftIterations: Int = 20,
private val siftingDelta: T,
private val nModes: Int = 6
) where BA: BufferAlgebra<T, A>, BA: FieldOps<Buffer<T>> {
/**
* Take a signal, construct an upper and a lower envelopes, find the mean value of two,
* represented as a series.
* @param signal Signal to compute on
*
* @return mean [Series] or `null`. `null` is returned in case
* the signal does not have enough extrema to construct envelopes.
*/
private fun findMean(signal: Series<T>): Series<T>? = (seriesAlgebra) {
val interpolator = SplineInterpolator(elementAlgebra)
val makeBuffer = elementAlgebra.bufferFactory
fun generateEnvelope(extrema: List<Int>, paddedExtremeValues: Buffer<T>): Series<T> {
val envelopeFunction = interpolator.interpolate(
makeBuffer(extrema.size) { signal.labels[extrema[it]] },
paddedExtremeValues
)
return signal.mapWithLabel { _, label ->
// For some reason PolynomialInterpolator is exclusive and the right boundary
// TODO Notify interpolator authors
envelopeFunction(label) ?: paddedExtremeValues.last()
// need to make the interpolator yield values outside boundaries?
}
}
// Extrema padding (experimental) TODO padding needs a dedicated function
val maxima = listOf(0) + signal.peaks() + (signal.size - 1)
val maxValues = makeBuffer(maxima.size) { signal[maxima[it]] }
if (maxValues[0] < maxValues[1]) {
maxValues[0] = maxValues[1]
}
if (maxValues.last() < maxValues[maxValues.size - 2]) {
maxValues[maxValues.size - 1] = maxValues[maxValues.size - 2]
}
val minima = listOf(0) + signal.troughs() + (signal.size - 1)
val minValues = makeBuffer(minima.size) { signal[minima[it]] }
if (minValues[0] > minValues[1]) {
minValues[0] = minValues[1]
}
if (minValues.last() > minValues[minValues.size - 2]) {
minValues[minValues.size - 1] = minValues[minValues.size - 2]
}
return if (maxima.size < 3 || minima.size < 3) null else { // maybe make an early return?
val upperEnvelope = generateEnvelope(maxima, maxValues)
val lowerEnvelope = generateEnvelope(minima, minValues)
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<T>): SiftingResult = siftInner(signal, 1, 0)
/**
* Compute a single iteration of the sifting process.
*/
private tailrec fun siftInner(
prevMode: Series<T>,
iterationNumber: Int,
sNumber: Int
): SiftingResult = (seriesAlgebra) {
val mean = findMean(prevMode) ?:
return if (iterationNumber == 1) SiftingResult.NotEnoughExtrema
else SiftingResult.SignalFlattened(prevMode)
val mode = prevMode.zip(mean) { p, m -> p - m }
val newSNumber = if (sCondition(mode)) sNumber + 1 else sNumber
return when {
iterationNumber >= maxSiftIterations -> SiftingResult.MaxIterationsReached(mode)
sNumber >= sConditionThreshold -> SiftingResult.SNumberReached(mode)
relativeDifference(mode, prevMode) < (elementAlgebra) { 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<T>): EMDecompositionResult<T> = (seriesAlgebra) {
val modes = mutableListOf<Series<T>>()
var residual = signal
repeat(nModes) {
val nextMode = when(val r = sift(residual)) {
SiftingResult.NotEnoughExtrema ->
return EMDecompositionResult(
if (it == 0) EMDTerminationReason.SIGNAL_TOO_FLAT
else EMDTerminationReason.ALL_POSSIBLE_MODES_EXTRACTED,
modes,
residual
)
is SiftingResult.Success<*> -> r.result
}
modes.add(nextMode as Series<T>) // TODO remove unchecked cast
residual = residual.zip(nextMode) { l, r -> l - r }
}
return EMDecompositionResult(EMDTerminationReason.MAX_MODES_REACHED, modes, residual)
}
}
/**
* 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 <T: Comparable<T>, L: T, A: Field<T>, BA> SeriesAlgebra<T, A, BA, L>.empiricalModeDecomposition(
sConditionThreshold: Int = 15,
maxSiftIterations: Int = 20,
siftingDelta: T,
nModes: Int = 3
): EmpiricalModeDecomposition<T, A, BA, L>
where BA: BufferAlgebra<T, A>, BA: FieldOps<Buffer<T>> = EmpiricalModeDecomposition(
seriesAlgebra = this,
sConditionThreshold = sConditionThreshold,
maxSiftIterations = maxSiftIterations,
siftingDelta = siftingDelta,
nModes = nModes
)
/**
* Brute force count all zeros in the series.
*/
private fun <T: Comparable<T>, A: Ring<T>, BA> SeriesAlgebra<T, A, BA, *>.countZeros(
signal: Series<T>
): Int where BA: BufferAlgebra<T, A>, BA: FieldOps<Buffer<T>> {
require(signal.size >= 2) { "Expected series with at least 2 elements, but got ${signal.size} elements" }
data class SignCounter(val prevSign: Int, val zeroCount: Int)
fun strictSign(arg: T): Int = if (arg > elementAlgebra.zero) 1 else -1
return signal.fold(SignCounter(strictSign(signal[0]), 0)) { acc, it ->
val currentSign = strictSign(it)
if (acc.prevSign != currentSign) SignCounter(currentSign, acc.zeroCount + 1)
else SignCounter(currentSign, acc.zeroCount)
}.zeroCount
}
/**
* Compute relative difference of two series.
*/
private fun <T, A: Ring<T>, BA> SeriesAlgebra<T, A, BA, *>.relativeDifference(
current: Series<T>,
previous: Series<T>
): T where BA: BufferAlgebra<T, A>, BA: FieldOps<Buffer<T>> = (bufferAlgebra) {
((current - previous) * (current - previous))
.div(previous * previous)
.fold(elementAlgebra.zero) { acc, it -> acc + it}
}
/**
* Brute force count all extrema of a series.
*/
private fun <T: Comparable<T>> Series<T>.countExtrema(): Int {
require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
return peaks().size + troughs().size
}
/**
* 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 <T: Comparable<T>, A: Ring<T>, BA> SeriesAlgebra<T, A, BA, *>.sCondition(
signal: Series<T>
): Boolean where BA: BufferAlgebra<T, A>, BA: FieldOps<Buffer<T>> =
(signal.countExtrema() - countZeros(signal)) in -1..1
internal sealed interface SiftingResult {
/**
* Represents a condition when a mode has been successfully
* extracted in a sifting process.
*/
open class Success<T>(val result: Series<T>): 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<T>(result: Series<T>) : Success<T>(result)
/**
* Returned when sifting process has been terminated due to the
* S-number condition being reached.
*/
class SNumberReached<T>(result: Series<T>) : Success<T>(result)
/**
* Returned when sifting process has been terminated due to the
* delta condition (Cauchy criterion) being reached.
*/
class DeltaReached<T>(result: Series<T>) : Success<T>(result)
/**
* Returned when sifting process has been terminated after
* executing the maximum number of iterations (specified when creating an instance
* of [EmpiricalModeDecomposition]).
*/
class MaxIterationsReached<T>(result: Series<T>): Success<T>(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<T>(
val terminatedBecause: EMDTerminationReason,
val modes: List<Series<T>>,
val residual: Series<T>
)

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@ -169,7 +169,7 @@ public open class SeriesAlgebra<T, out A : Ring<T>, out BA : BufferAlgebra<T, A>
public inline fun Buffer<T>.mapWithLabel(crossinline transform: A.(arg: T, label: L) -> T): Series<T> {
val labels = labels
val buf = elementAlgebra.bufferFactory(size) {
elementAlgebra.transform(getByOffset(it), labels[it])
elementAlgebra.transform(get(it), labels[it])
}
return buf.moveTo(offsetIndices.first)
}

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@ -0,0 +1,89 @@
/*
* 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
public enum class PlateauEdgePolicy {
/**
* Midpoints of plateau are returned, edges not belonging to a plateau are ignored.
*
* A midpoint is the index closest to the average of indices of the left and right edges
* of the plateau:
*
* `val midpoint = ((leftEdge + rightEdge) / 2).toInt`
*/
AVERAGE,
/**
* Both left and right edges are returned.
*/
KEEP_ALL_EDGES,
/**
* Only right edges are returned.
*/
KEEP_RIGHT_EDGES,
/**
* Only left edges are returned.
*/
KEEP_LEFT_EDGES,
/**
* Ignore plateau, only peaks (troughs) with values strictly greater (less)
* than values of the adjacent points are returned.
*/
IGNORE
}
public fun <T: Comparable<T>> Series<T>.peaks(
plateauEdgePolicy: PlateauEdgePolicy = PlateauEdgePolicy.AVERAGE
): List<Int> = findPeaks(plateauEdgePolicy, { other -> this > other }, { other -> this >= other })
public fun <T: Comparable<T>> Series<T>.troughs(
plateauEdgePolicy: PlateauEdgePolicy = PlateauEdgePolicy.AVERAGE
): List<Int> = findPeaks(plateauEdgePolicy, { other -> this < other }, { other -> this <= other })
private fun <T: Comparable<T>> Series<T>.findPeaks(
plateauPolicy: PlateauEdgePolicy = PlateauEdgePolicy.AVERAGE,
cmpStrong: T.(T) -> Boolean,
cmpWeak: T.(T) -> Boolean
): List<Int> {
require(size >= 3) { "Expected series with at least 3 elements, but got $size elements" }
if (plateauPolicy == PlateauEdgePolicy.AVERAGE) return peaksWithPlateau(cmpStrong)
fun peakCriterion(left: T, middle: T, right: T): Boolean = when(plateauPolicy) {
PlateauEdgePolicy.KEEP_LEFT_EDGES -> middle.cmpStrong(left) && middle.cmpWeak(right)
PlateauEdgePolicy.KEEP_RIGHT_EDGES -> middle.cmpWeak(left) && middle.cmpStrong(right)
PlateauEdgePolicy.KEEP_ALL_EDGES ->
(middle.cmpStrong(left) && middle.cmpWeak(right)) || (middle.cmpWeak(left) && middle.cmpStrong(right))
else -> middle.cmpStrong(right) && middle.cmpStrong(left)
}
val indices = mutableListOf<Int>()
for (index in 1 .. size - 2) {
val left = this[index - 1]
val middle = this[index]
val right = this[index + 1]
if (peakCriterion(left, middle, right)) indices.add(index)
}
return indices
}
private fun <T: Comparable<T>> Series<T>.peaksWithPlateau(cmpStrong: T.(T) -> Boolean): List<Int> {
val peaks = mutableListOf<Int>()
tailrec fun peaksPlateauInner(index: Int) {
val nextUnequal = (index + 1 ..< size).firstOrNull { this[it] != this[index] } ?: (size - 1)
val newIndex = if (this[index].cmpStrong(this[index - 1]) && this[index].cmpStrong(this[nextUnequal])) {
peaks.add((index + nextUnequal) / 2)
nextUnequal
} else index + 1
if (newIndex < size - 1) peaksPlateauInner(newIndex)
}
peaksPlateauInner(1)
return peaks
}