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Added Levenberg-Marquardt algorithm and svd Golub-Kahan #513

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margarita0303 merged 35 commits from dev into dev 2023-06-19 16:11:59 +03:00
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kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/LevenbergMarquardtAlgorithm.kt
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@ -131,14 +131,14 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
var p = inputData.startParameters var p = inputData.startParameters
val t = inputData.independentVariables val t = inputData.independentVariables
val Npar = length(p) // number of parameters val Npar = length(p) // number of parameters
val Npnt = length(inputData.realValues) // number of data points val Npnt = length(inputData.realValues) // number of data points
var pOld = zeros(ShapeND(intArrayOf(Npar, 1))).as2D() // previous set of parameters var pOld = zeros(ShapeND(intArrayOf(Npar, 1))).as2D() // previous set of parameters
var yOld = zeros(ShapeND(intArrayOf(Npnt, 1))).as2D() // previous model, y_old = y_hat(t;p_old) var yOld = zeros(ShapeND(intArrayOf(Npnt, 1))).as2D() // previous model, y_old = y_hat(t;p_old)
var X2 = 1e-3 / eps // a really big initial Chi-sq value var X2 = 1e-3 / eps // a really big initial Chi-sq value
var X2Old = 1e-3 / eps // a really big initial Chi-sq value var X2Old = 1e-3 / eps // a really big initial Chi-sq value
var J = zeros(ShapeND(intArrayOf(Npnt, Npar))).as2D() // Jacobian matrix var J = zeros(ShapeND(intArrayOf(Npnt, Npar))).as2D() // Jacobian matrix
val DoF = Npnt - Npar // statistical degrees of freedom val DoF = Npnt - Npar // statistical degrees of freedom
var weight = fromArray(ShapeND(intArrayOf(1, 1)), doubleArrayOf(inputData.weight)).as2D() var weight = fromArray(ShapeND(intArrayOf(1, 1)), doubleArrayOf(inputData.weight)).as2D()
if (inputData.nargin < 5) { if (inputData.nargin < 5) {
@ -165,16 +165,15 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
} }
var maxIterations = inputData.maxIterations var maxIterations = inputData.maxIterations
var epsilon1 = inputData.epsilons[0] // convergence tolerance for gradient var epsilon1 = inputData.epsilons[0]
var epsilon2 = inputData.epsilons[1] // convergence tolerance for parameters var epsilon2 = inputData.epsilons[1]
var epsilon3 = inputData.epsilons[2] // convergence tolerance for Chi-square var epsilon3 = inputData.epsilons[2]
var epsilon4 = inputData.epsilons[3] // determines acceptance of a L-M step var epsilon4 = inputData.epsilons[3]
var lambda0 = inputData.lambdas[0] // initial value of damping paramter, lambda var lambda0 = inputData.lambdas[0]
var lambdaUpFac = inputData.lambdas[1] // factor for increasing lambda var lambdaUpFac = inputData.lambdas[1]
var lambdaDnFac = inputData.lambdas[2] // factor for decreasing lambda var lambdaDnFac = inputData.lambdas[2]
var updateType = inputData.updateType // 1: Levenberg-Marquardt lambda update var updateType = inputData.updateType
// 2: Quadratic update
// 3: Nielsen's lambda update equations
if (inputData.nargin < 9) { if (inputData.nargin < 9) {
maxIterations = 10 * Npar maxIterations = 10 * Npar
epsilon1 = 1e-3 epsilon1 = 1e-3
@ -194,7 +193,7 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
dp = ones(ShapeND(intArrayOf(Npar, 1))).div(1 / dp[0, 0]).as2D() dp = ones(ShapeND(intArrayOf(Npar, 1))).div(1 / dp[0, 0]).as2D()
} }
var stop = false // termination flag var stop = false // termination flag
if (weight.shape.component1() == 1 || variance(weight) == 0.0) { // identical weights vector if (weight.shape.component1() == 1 || variance(weight) == 0.0) { // identical weights vector
weight = ones(ShapeND(intArrayOf(Npnt, 1))).div(1 / kotlin.math.abs(weight[0, 0])).as2D() weight = ones(ShapeND(intArrayOf(Npnt, 1))).div(1 / kotlin.math.abs(weight[0, 0])).as2D()
@ -218,39 +217,35 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
var lambda = 1.0 var lambda = 1.0
var nu = 1 var nu = 1
when (updateType) {
1 -> lambda = lambda0 // Marquardt: init'l lambda if (updateType == 1) {
else -> { // Quadratic and Nielsen lambda = lambda0 // Marquardt: init'l lambda
lambda = lambda0 * (makeColumnFromDiagonal(JtWJ)).max()!! }
nu = 2 else {
} lambda = lambda0 * (makeColumnFromDiagonal(JtWJ)).max()
nu = 2
} }
X2Old = X2 // previous value of X2 X2Old = X2 // previous value of X2
var h: DoubleTensor var h: DoubleTensor
while (!stop && settings.iteration <= maxIterations) { //--- Start Main Loop while (!stop && settings.iteration <= maxIterations) {
settings.iteration += 1 settings.iteration += 1
// incremental change in parameters // incremental change in parameters
h = when (updateType) { h = if (updateType == 1) { // Marquardt
1 -> { // Marquardt val solve = solve(JtWJ.plus(makeMatrixWithDiagonal(makeColumnFromDiagonal(JtWJ)).div(1 / lambda)).as2D(), JtWdy)
val solve = solve.asDoubleTensor()
solve(JtWJ.plus(makeMatrixWithDiagonal(makeColumnFromDiagonal(JtWJ)).div(1 / lambda)).as2D(), JtWdy) } else { // Quadratic and Nielsen
solve.asDoubleTensor() val solve = solve(JtWJ.plus(lmEye(Npar).div(1 / lambda)).as2D(), JtWdy)
} solve.asDoubleTensor()
else -> { // Quadratic and Nielsen
val solve = solve(JtWJ.plus(lmEye(Npar).div(1 / lambda)).as2D(), JtWdy)
solve.asDoubleTensor()
}
} }
var pTry = (p + h).as2D() // update the [idx] elements var pTry = (p + h).as2D() // update the [idx] elements
pTry = smallestElementComparison(largestElementComparison(minParameters, pTry.as2D()), maxParameters) // apply constraints pTry = smallestElementComparison(largestElementComparison(minParameters, pTry.as2D()), maxParameters) // apply constraints
var deltaY = inputData.realValues.minus(evaluateFunction(inputData.func, t, pTry, inputData.exampleNumber)) // residual error using p_try var deltaY = inputData.realValues.minus(evaluateFunction(inputData.func, t, pTry, inputData.exampleNumber)) // residual error using p_try
for (i in 0 until deltaY.shape.component1()) { // floating point error; break for (i in 0 until deltaY.shape.component1()) { // floating point error; break
for (j in 0 until deltaY.shape.component2()) { for (j in 0 until deltaY.shape.component2()) {
@ -264,21 +259,20 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
settings.funcCalls += 1 settings.funcCalls += 1
val tmp = deltaY.times(weight) val tmp = deltaY.times(weight)
var X2Try = deltaY.as2D().transpose().dot(tmp) // Chi-squared error criteria var X2Try = deltaY.as2D().transpose().dot(tmp) // Chi-squared error criteria
val alpha = 1.0 val alpha = 1.0
if (updateType == 2) { // Quadratic if (updateType == 2) { // Quadratic
// One step of quadratic line update in the h direction for minimum X2 // One step of quadratic line update in the h direction for minimum X2
val alpha = JtWdy.transpose().dot(h) / ((X2Try.minus(X2)).div(2.0).plus(2 * JtWdy.transpose().dot(h))) val alpha = JtWdy.transpose().dot(h) / ((X2Try.minus(X2)).div(2.0).plus(2 * JtWdy.transpose().dot(h)))
h = h.dot(alpha) h = h.dot(alpha)
pTry = p.plus(h).as2D() // update only [idx] elements pTry = p.plus(h).as2D() // update only [idx] elements
pTry = smallestElementComparison(largestElementComparison(minParameters, pTry), maxParameters) // apply constraints pTry = smallestElementComparison(largestElementComparison(minParameters, pTry), maxParameters) // apply constraints
deltaY = inputData.realValues.minus(evaluateFunction(inputData.func, t, pTry, inputData.exampleNumber)) // residual error using p_try deltaY = inputData.realValues.minus(evaluateFunction(inputData.func, t, pTry, inputData.exampleNumber)) // residual error using p_try
settings.funcCalls += 1 settings.funcCalls += 1
X2Try = deltaY.as2D().transpose().dot(deltaY.times(weight)) // Chi-squared error criteria X2Try = deltaY.as2D().transpose().dot(deltaY.times(weight)) // Chi-squared error criteria
} }
val rho = when (updateType) { // Nielsen val rho = when (updateType) { // Nielsen
@ -287,7 +281,6 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
.dot(makeMatrixWithDiagonal(makeColumnFromDiagonal(JtWJ)).div(1 / lambda).dot(h).plus(JtWdy)) .dot(makeMatrixWithDiagonal(makeColumnFromDiagonal(JtWJ)).div(1 / lambda).dot(h).plus(JtWdy))
X2.minus(X2Try).as2D()[0, 0] / abs(tmp.as2D()).as2D()[0, 0] X2.minus(X2Try).as2D()[0, 0] / abs(tmp.as2D()).as2D()[0, 0]
} }
else -> { else -> {
val tmp = h.transposed().dot(h.div(1 / lambda).plus(JtWdy)) val tmp = h.transposed().dot(h.div(1 / lambda).plus(JtWdy))
X2.minus(X2Try).as2D()[0, 0] / abs(tmp.as2D()).as2D()[0, 0] X2.minus(X2Try).as2D()[0, 0] / abs(tmp.as2D()).as2D()[0, 0]
@ -303,7 +296,6 @@ public fun DoubleTensorAlgebra.levenbergMarquardt(inputData: LMInput): LMResultI
lmMatxAns = lmMatx(inputData.func, t, pOld, yOld, dX2.toInt(), J, p, inputData.realValues, weight, dp, settings) lmMatxAns = lmMatx(inputData.func, t, pOld, yOld, dX2.toInt(), J, p, inputData.realValues, weight, dp, settings)
// decrease lambda ==> Gauss-Newton method // decrease lambda ==> Gauss-Newton method
JtWJ = lmMatxAns[0] JtWJ = lmMatxAns[0]
JtWdy = lmMatxAns[1] JtWdy = lmMatxAns[1]
X2 = lmMatxAns[2][0, 0] X2 = lmMatxAns[2][0, 0]
@ -519,7 +511,7 @@ private fun lmMatx(func: (MutableStructure2D<Double>, MutableStructure2D<Double>
yDat: MutableStructure2D<Double>, weight: MutableStructure2D<Double>, dp:MutableStructure2D<Double>, settings:LMSettings) : Array<MutableStructure2D<Double>> yDat: MutableStructure2D<Double>, weight: MutableStructure2D<Double>, dp:MutableStructure2D<Double>, settings:LMSettings) : Array<MutableStructure2D<Double>>
{ {
// default: dp = 0.001 // default: dp = 0.001
val Npar = length(p) // number of parameters val Npar = length(p) // number of parameters
val yHat = evaluateFunction(func, t, p, settings.exampleNumber) // evaluate model using parameters 'p' val yHat = evaluateFunction(func, t, p, settings.exampleNumber) // evaluate model using parameters 'p'
settings.funcCalls += 1 settings.funcCalls += 1
@ -558,8 +550,8 @@ private fun lmFdJ(func: (MutableStructure2D<Double>, MutableStructure2D<Double>,
dp: MutableStructure2D<Double>, settings: LMSettings): MutableStructure2D<Double> { dp: MutableStructure2D<Double>, settings: LMSettings): MutableStructure2D<Double> {
// default: dp = 0.001 * ones(1,n) // default: dp = 0.001 * ones(1,n)
val m = length(y) // number of data points val m = length(y) // number of data points
val n = length(p) // number of parameters val n = length(p) // number of parameters
val ps = p.copyToTensor().as2D() val ps = p.copyToTensor().as2D()
val J = BroadcastDoubleTensorAlgebra.zeros(ShapeND(intArrayOf(m, n))).as2D() // initialize Jacobian to Zero val J = BroadcastDoubleTensorAlgebra.zeros(ShapeND(intArrayOf(m, n))).as2D() // initialize Jacobian to Zero
@ -568,7 +560,7 @@ private fun lmFdJ(func: (MutableStructure2D<Double>, MutableStructure2D<Double>,
for (j in 0 until n) { for (j in 0 until n) {
del[j, 0] = dp[j, 0] * (1 + kotlin.math.abs(p[j, 0])) // parameter perturbation del[j, 0] = dp[j, 0] * (1 + kotlin.math.abs(p[j, 0])) // parameter perturbation
p[j, 0] = ps[j, 0] + del[j, 0] // perturb parameter p(j) p[j, 0] = ps[j, 0] + del[j, 0] // perturb parameter p(j)
val epsilon = 0.0000001 val epsilon = 0.0000001
if (kotlin.math.abs(del[j, 0]) > epsilon) { if (kotlin.math.abs(del[j, 0]) > epsilon) {