Added Levenberg-Marquardt algorithm and svd Golub-Kahan #513
@ -14,6 +14,8 @@ import space.kscience.kmath.tensors.core.BroadcastDoubleTensorAlgebra.dot
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import space.kscience.kmath.tensors.core.BroadcastDoubleTensorAlgebra.map
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import space.kscience.kmath.tensors.core.BroadcastDoubleTensorAlgebra.transposed
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import space.kscience.kmath.tensors.core.DoubleTensorAlgebra
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import space.kscience.kmath.tensors.core.internal.LMSettings
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import kotlin.reflect.KFunction3
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/**
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* Common linear algebra operations. Operates on [Tensor].
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@ -117,4 +119,10 @@ public interface LinearOpsTensorAlgebra<T, A : Field<T>> : TensorPartialDivision
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* @return the square matrix x which is the solution of the equation.
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*/
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public fun solve(a: MutableStructure2D<Double>, b: MutableStructure2D<Double>): MutableStructure2D<Double>
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public fun lm(
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func: KFunction3<MutableStructure2D<Double>, MutableStructure2D<Double>, LMSettings, MutableStructure2D<Double>>,
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p_input: MutableStructure2D<Double>, t_input: MutableStructure2D<Double>, y_dat_input: MutableStructure2D<Double>,
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weight_input: MutableStructure2D<Double>, dp_input: MutableStructure2D<Double>, p_min_input: MutableStructure2D<Double>, p_max_input: MutableStructure2D<Double>,
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c_input: MutableStructure2D<Double>, opts_input: DoubleArray, nargin: Int, example_number: Int): Double
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}
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@ -9,6 +9,7 @@
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package space.kscience.kmath.tensors.core
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import space.kscience.kmath.PerformancePitfall
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import space.kscience.kmath.linear.transpose
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import space.kscience.kmath.nd.*
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import space.kscience.kmath.operations.DoubleBufferOps
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import space.kscience.kmath.operations.DoubleField
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@ -17,10 +18,8 @@ import space.kscience.kmath.tensors.api.AnalyticTensorAlgebra
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import space.kscience.kmath.tensors.api.LinearOpsTensorAlgebra
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import space.kscience.kmath.tensors.api.Tensor
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import space.kscience.kmath.tensors.core.internal.*
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import kotlin.math.abs
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import kotlin.math.ceil
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import kotlin.math.floor
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import kotlin.math.sqrt
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import kotlin.math.*
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import kotlin.reflect.KFunction3
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/**
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* Implementation of basic operations over double tensors and basic algebra operations on them.
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@ -717,6 +716,368 @@ public open class DoubleTensorAlgebra :
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val aInverse = aSvd.third.dot(s).dot(aSvd.first.transposed())
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return aInverse.dot(b).as2D()
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}
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override fun lm(
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func: KFunction3<MutableStructure2D<Double>, MutableStructure2D<Double>, LMSettings, MutableStructure2D<Double>>,
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p_input: MutableStructure2D<Double>, t_input: MutableStructure2D<Double>, y_dat_input: MutableStructure2D<Double>,
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weight_input: MutableStructure2D<Double>, dp_input: MutableStructure2D<Double>, p_min_input: MutableStructure2D<Double>, p_max_input: MutableStructure2D<Double>,
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c_input: MutableStructure2D<Double>, opts_input: DoubleArray, nargin: Int, example_number: Int): Double {
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var result_chi_sq = 0.0
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val tensor_parameter = 0
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val eps:Double = 2.2204e-16
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var settings = LMSettings(0, 0, example_number)
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settings.func_calls = 0 // running count of function evaluations
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var p = p_input
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val y_dat = y_dat_input
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val t = t_input
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val Npar = length(p) // number of parameters
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val Npnt = length(y_dat) // number of data points
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var p_old = zeros(ShapeND(intArrayOf(Npar, 1))).as2D() // previous set of parameters
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var y_old = zeros(ShapeND(intArrayOf(Npnt, 1))).as2D() // previous model, y_old = y_hat(t;p_old)
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var X2 = 1e-3 / eps // a really big initial Chi-sq value
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var X2_old = 1e-3 / eps // a really big initial Chi-sq value
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var J = zeros(ShapeND(intArrayOf(Npnt, Npar))).as2D() // Jacobian matrix
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val DoF = Npnt - Npar + 1 // statistical degrees of freedom
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var corr_p = 0
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var sigma_p = 0
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var sigma_y = 0
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var R_sq = 0
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var cvg_hist = 0
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if (length(t) != length(y_dat)) {
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println("lm.m error: the length of t must equal the length of y_dat")
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val length_t = length(t)
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val length_y_dat = length(y_dat)
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X2 = 0.0
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corr_p = 0
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sigma_p = 0
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sigma_y = 0
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R_sq = 0
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cvg_hist = 0
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// if (tensor_parameter != 0) { // Зачем эта проверка?
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// return
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// }
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}
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var weight = weight_input
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if (nargin < 5) {
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weight = fromArray(ShapeND(intArrayOf(1, 1)), doubleArrayOf((y_dat.transpose().dot(y_dat)).as1D()[0])).as2D()
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}
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var dp = dp_input
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if (nargin < 6) {
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dp = fromArray(ShapeND(intArrayOf(1, 1)), doubleArrayOf(0.001)).as2D()
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}
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var p_min = p_min_input
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if (nargin < 7) {
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p_min = p
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p_min.abs()
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p_min = p_min.div(-100.0).as2D()
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}
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var p_max = p_max_input
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if (nargin < 8) {
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p_max = p
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p_max.abs()
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p_max = p_max.div(100.0).as2D()
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}
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var c = c_input
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if (nargin < 9) {
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c = fromArray(ShapeND(intArrayOf(1, 1)), doubleArrayOf(1.0)).as2D()
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}
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var opts = opts_input
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if (nargin < 10) {
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opts = doubleArrayOf(3.0, 10.0 * Npar, 1e-3, 1e-3, 1e-1, 1e-1, 1e-2, 11.0, 9.0, 1.0)
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}
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val prnt = opts[0] // >1 intermediate results; >2 plots
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val MaxIter = opts[1].toInt() // maximum number of iterations
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val epsilon_1 = opts[2] // convergence tolerance for gradient
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val epsilon_2 = opts[3] // convergence tolerance for parameters
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val epsilon_3 = opts[4] // convergence tolerance for Chi-square
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val epsilon_4 = opts[5] // determines acceptance of a L-M step
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val lambda_0 = opts[6] // initial value of damping paramter, lambda
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val lambda_UP_fac = opts[7] // factor for increasing lambda
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val lambda_DN_fac = opts[8] // factor for decreasing lambda
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val Update_Type = opts[9].toInt() // 1: Levenberg-Marquardt lambda update
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// 2: Quadratic update
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// 3: Nielsen's lambda update equations
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val plotcmd = "figure(102); plot(t(:,1),y_init,''-k'',t(:,1),y_hat,''-b'',t(:,1),y_dat,''o'',''color'',[0,0.6,0],''MarkerSize'',4); title(sprintf(''\\chi^2_\\nu = %f'',X2/DoF)); drawnow"
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p_min = make_column(p_min)
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p_max = make_column(p_max)
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if (length(make_column(dp)) == 1) {
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dp = ones(ShapeND(intArrayOf(Npar, 1))).div(1 / dp[0, 0]).as2D()
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}
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val idx = get_zero_indices(dp) // indices of the parameters to be fit
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val Nfit = idx?.shape?.component1() // number of parameters to fit
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var stop = false // termination flag
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val y_init = feval(func, t, p, settings) // residual error using p_try
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if (weight.shape.component1() == 1 || variance(weight) == 0.0) { // identical weights vector
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weight = ones(ShapeND(intArrayOf(Npnt, 1))).div(1 / kotlin.math.abs(weight[0, 0])).as2D() // !!! need to check
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println("using uniform weights for error analysis")
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}
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else {
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weight = make_column(weight)
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weight.abs()
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}
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// initialize Jacobian with finite difference calculation
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var lm_matx_ans = lm_matx(func, t, p_old, y_old,1, J, p, y_dat, weight, dp, settings)
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var JtWJ = lm_matx_ans[0]
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var JtWdy = lm_matx_ans[1]
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X2 = lm_matx_ans[2][0, 0]
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var y_hat = lm_matx_ans[3]
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J = lm_matx_ans[4]
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if ( abs(JtWdy).max()!! < epsilon_1 ) {
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println(" *** Your Initial Guess is Extremely Close to Optimal ***\n")
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println(" *** epsilon_1 = %e\n$epsilon_1")
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stop = true
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}
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var lambda = 1.0
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var nu = 1
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when (Update_Type) {
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1 -> lambda = lambda_0 // Marquardt: init'l lambda
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else -> { // Quadratic and Nielsen
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lambda = lambda_0 * (diag(JtWJ)).max()!!
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nu = 2
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}
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}
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X2_old = X2 // previous value of X2
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var cvg_hst = ones(ShapeND(intArrayOf(MaxIter, Npar + 3))) // initialize convergence history
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var h = JtWJ.copyToTensor()
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var dX2 = X2
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while (!stop && settings.iteration <= MaxIter) { //--- Start Main Loop
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settings.iteration += 1
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// incremental change in parameters
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h = when (Update_Type) {
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1 -> { // Marquardt
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val solve = solve(JtWJ.plus(make_matrx_with_diagonal(diag(JtWJ)).div(1 / lambda)).as2D(), JtWdy)
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solve.asDoubleTensor()
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}
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else -> { // Quadratic and Nielsen
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val solve = solve(JtWJ.plus(lm_eye(Npar).div(1 / lambda)).as2D(), JtWdy)
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solve.asDoubleTensor()
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}
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}
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// big = max(abs(h./p)) > 2; % this is a big step
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// --- Are parameters [p+h] much better than [p] ?
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var p_try = (p + h).as2D() // update the [idx] elements
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p_try = smallest_element_comparison(largest_element_comparison(p_min, p_try.as2D()), p_max) // apply constraints
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var delta_y = y_dat.minus(feval(func, t, p_try, settings)) // residual error using p_try
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// TODO
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//if ~all(isfinite(delta_y)) // floating point error; break
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// stop = 1;
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// break
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//end
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settings.func_calls += 1
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val tmp = delta_y.times(weight)
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var X2_try = delta_y.as2D().transpose().dot(tmp) // Chi-squared error criteria
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val alpha = 1.0
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if (Update_Type == 2) { // Quadratic
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// One step of quadratic line update in the h direction for minimum X2
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// TODO
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// val alpha = JtWdy.transpose().dot(h) / ((X2_try.minus(X2)).div(2.0).plus(2 * JtWdy.transpose().dot(h)))
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// alpha = JtWdy'*h / ( (X2_try - X2)/2 + 2*JtWdy'*h ) ;
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// h = alpha * h;
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//
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// p_try = p + h(idx); % update only [idx] elements
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// p_try = min(max(p_min,p_try),p_max); % apply constraints
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//
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// delta_y = y_dat - feval(func,t,p_try,c); % residual error using p_try
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// func_calls = func_calls + 1;
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// тX2_try = delta_y' * ( delta_y .* weight ); % Chi-squared error criteria
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}
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val rho = when (Update_Type) { // Nielsen
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1 -> {
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val tmp = h.transposed().dot(make_matrx_with_diagonal(diag(JtWJ)).div(1 / lambda).dot(h).plus(JtWdy))
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X2.minus(X2_try).as2D()[0, 0] / abs(tmp.as2D()).as2D()[0, 0]
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}
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else -> {
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val tmp = h.transposed().dot(h.div(1 / lambda).plus(JtWdy))
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X2.minus(X2_try).as2D()[0, 0] / abs(tmp.as2D()).as2D()[0, 0]
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}
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}
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println()
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println("rho = " + rho)
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if (rho > epsilon_4) { // it IS significantly better
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val dX2 = X2.minus(X2_old)
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X2_old = X2
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p_old = p.copyToTensor().as2D()
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y_old = y_hat.copyToTensor().as2D()
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p = make_column(p_try) // accept p_try
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lm_matx_ans = lm_matx(func, t, p_old, y_old, dX2.toInt(), J, p, y_dat, weight, dp, settings)
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// decrease lambda ==> Gauss-Newton method
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JtWJ = lm_matx_ans[0]
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JtWdy = lm_matx_ans[1]
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X2 = lm_matx_ans[2][0, 0]
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y_hat = lm_matx_ans[3]
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J = lm_matx_ans[4]
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lambda = when (Update_Type) {
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1 -> { // Levenberg
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max(lambda / lambda_DN_fac, 1e-7);
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}
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2 -> { // Quadratic
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max( lambda / (1 + alpha) , 1e-7 );
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}
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else -> { // Nielsen
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nu = 2
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lambda * max( 1.0 / 3, 1 - (2 * rho - 1).pow(3) )
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}
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}
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// if (prnt > 2) {
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// eval(plotcmd)
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// }
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}
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else { // it IS NOT better
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X2 = X2_old // do not accept p_try
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if (settings.iteration % (2 * Npar) == 0 ) { // rank-1 update of Jacobian
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lm_matx_ans = lm_matx(func, t, p_old, y_old,-1, J, p, y_dat, weight, dp, settings)
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JtWJ = lm_matx_ans[0]
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JtWdy = lm_matx_ans[1]
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dX2 = lm_matx_ans[2][0, 0]
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y_hat = lm_matx_ans[3]
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J = lm_matx_ans[4]
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}
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// increase lambda ==> gradient descent method
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lambda = when (Update_Type) {
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1 -> { // Levenberg
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min(lambda * lambda_UP_fac, 1e7)
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}
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2 -> { // Quadratic
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lambda + abs(((X2_try.as2D()[0, 0] - X2) / 2) / alpha)
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}
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else -> { // Nielsen
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nu *= 2
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lambda * (nu / 2)
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}
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}
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}
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if (prnt > 1) {
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val chi_sq = X2 / DoF
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println("Iteration $settings.iteration, func_calls $settings.func_calls | chi_sq=$chi_sq | lambda=$lambda")
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print("param: ")
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for (pn in 0 until Npar) {
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print(p[pn, 0].toString() + " ")
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}
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print("\ndp/p: ")
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for (pn in 0 until Npar) {
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print((h.as2D()[pn, 0] / p[pn, 0]).toString() + " ")
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}
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result_chi_sq = chi_sq
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}
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// update convergence history ... save _reduced_ Chi-square
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// cvg_hst(iteration,:) = [ func_calls p' X2/DoF lambda ];
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if (abs(JtWdy).max()!! < epsilon_1 && settings.iteration > 2) {
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println(" **** Convergence in r.h.s. (\"JtWdy\") ****")
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println(" **** epsilon_1 = $epsilon_1")
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stop = true
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}
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if ((abs(h.as2D()).div(abs(p) + 1e-12)).max() < epsilon_2 && settings.iteration > 2) {
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println(" **** Convergence in Parameters ****")
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println(" **** epsilon_2 = $epsilon_2")
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stop = true
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}
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if (X2 / DoF < epsilon_3 && settings.iteration > 2) {
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println(" **** Convergence in reduced Chi-square **** ")
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println(" **** epsilon_3 = $epsilon_3")
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stop = true
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}
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if (settings.iteration == MaxIter) {
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println(" !! Maximum Number of Iterations Reached Without Convergence !!")
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stop = true
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}
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} // --- End of Main Loop
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// --- convergence achieved, find covariance and confidence intervals
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// ---- Error Analysis ----
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// if (weight.shape.component1() == 1 || weight.variance() == 0.0) {
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// weight = DoF / (delta_y.transpose().dot(delta_y)) * ones(intArrayOf(Npt, 1))
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// }
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// if (nargout > 1) {
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// val redX2 = X2 / DoF
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// }
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//
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// lm_matx_ans = lm_matx(func, t, p_old, y_old, -1, J, p, y_dat, weight, dp)
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// JtWJ = lm_matx_ans[0]
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// JtWdy = lm_matx_ans[1]
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// X2 = lm_matx_ans[2][0, 0]
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// y_hat = lm_matx_ans[3]
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// J = lm_matx_ans[4]
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//
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// if (nargout > 2) { // standard error of parameters
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// covar_p = inv(JtWJ);
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// siif nagma_p = sqrt(diag(covar_p));
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// }
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//
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// if (nargout > 3) { // standard error of the fit
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// /// sigma_y = sqrt(diag(J * covar_p * J')); // slower version of below
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// sigma_y = zeros(Npnt,1);
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// for i=1:Npnt
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// sigma_y(i) = J(i,:) * covar_p * J(i,:)';
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// end
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// sigma_y = sqrt(sigma_y);
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// }
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//
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// if (nargout > 4) { // parameter correlation matrix
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// corr_p = covar_p ./ [sigma_p*sigma_p'];
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// }
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//
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// if (nargout > 5) { // coefficient of multiple determination
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// R_sq = corr([y_dat y_hat]);
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// R_sq = R_sq(1,2).^2;
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// }
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//
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// if (nargout > 6) { // convergence history
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// cvg_hst = cvg_hst(1:iteration,:);
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// }
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return result_chi_sq
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}
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}
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public val Double.Companion.tensorAlgebra: DoubleTensorAlgebra get() = DoubleTensorAlgebra
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@ -6,11 +6,11 @@
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package space.kscience.kmath.tensors.core
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|
||||
|
||||
import space.kscience.kmath.nd.ShapeND
|
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import space.kscience.kmath.nd.contentEquals
|
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import space.kscience.kmath.nd.get
|
||||
import space.kscience.kmath.nd.*
|
||||
import space.kscience.kmath.operations.invoke
|
||||
import space.kscience.kmath.tensors.core.internal.LMSettings
|
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import space.kscience.kmath.testutils.assertBufferEquals
|
||||
import kotlin.math.roundToInt
|
||||
import kotlin.test.Test
|
||||
import kotlin.test.assertEquals
|
||||
import kotlin.test.assertFalse
|
||||
@ -207,4 +207,83 @@ internal class TestDoubleTensorAlgebra {
|
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assertTrue { ShapeND(5, 5) contentEquals res.shape }
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assertEquals(2.0, res[4, 4])
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}
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|
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@Test
|
||||
fun testLM() = DoubleTensorAlgebra {
|
||||
fun lm_func(t: MutableStructure2D<Double>, p: MutableStructure2D<Double>, settings: LMSettings): MutableStructure2D<Double> {
|
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val m = t.shape.component1()
|
||||
var y_hat = DoubleTensorAlgebra.zeros(ShapeND(intArrayOf(m, 1)))
|
||||
|
||||
if (settings.example_number == 1) {
|
||||
y_hat = DoubleTensorAlgebra.exp((t.times(-1.0 / p[1, 0]))).times(p[0, 0]) + t.times(p[2, 0]).times(
|
||||
DoubleTensorAlgebra.exp((t.times(-1.0 / p[3, 0])))
|
||||
)
|
||||
}
|
||||
else if (settings.example_number == 2) {
|
||||
val mt = t.max()
|
||||
y_hat = (t.times(1.0 / mt)).times(p[0, 0]) +
|
||||
(t.times(1.0 / mt)).pow(2).times(p[1, 0]) +
|
||||
(t.times(1.0 / mt)).pow(3).times(p[2, 0]) +
|
||||
(t.times(1.0 / mt)).pow(4).times(p[3, 0])
|
||||
}
|
||||
else if (settings.example_number == 3) {
|
||||
y_hat = DoubleTensorAlgebra.exp((t.times(-1.0 / p[1, 0])))
|
||||
.times(p[0, 0]) + DoubleTensorAlgebra.sin((t.times(1.0 / p[3, 0]))).times(p[2, 0])
|
||||
}
|
||||
|
||||
return y_hat.as2D()
|
||||
}
|
||||
|
||||
val lm_matx_y_dat = doubleArrayOf(
|
||||
19.6594, 18.6096, 17.6792, 17.2747, 16.3065, 17.1458, 16.0467, 16.7023, 15.7809, 15.9807,
|
||||
14.7620, 15.1128, 16.0973, 15.1934, 15.8636, 15.4763, 15.6860, 15.1895, 15.3495, 16.6054,
|
||||
16.2247, 15.9854, 16.1421, 17.0960, 16.7769, 17.1997, 17.2767, 17.5882, 17.5378, 16.7894,
|
||||
17.7648, 18.2512, 18.1581, 16.7037, 17.8475, 17.9081, 18.3067, 17.9632, 18.2817, 19.1427,
|
||||
18.8130, 18.5658, 18.0056, 18.4607, 18.5918, 18.2544, 18.3731, 18.7511, 19.3181, 17.3066,
|
||||
17.9632, 19.0513, 18.7528, 18.2928, 18.5967, 17.8567, 17.7859, 18.4016, 18.9423, 18.4959,
|
||||
17.8000, 18.4251, 17.7829, 17.4645, 17.5221, 17.3517, 17.4637, 17.7563, 16.8471, 17.4558,
|
||||
17.7447, 17.1487, 17.3183, 16.8312, 17.7551, 17.0942, 15.6093, 16.4163, 15.3755, 16.6725,
|
||||
16.2332, 16.2316, 16.2236, 16.5361, 15.3721, 15.3347, 15.5815, 15.6319, 14.4538, 14.6044,
|
||||
14.7665, 13.3718, 15.0587, 13.8320, 14.7873, 13.6824, 14.2579, 14.2154, 13.5818, 13.8157
|
||||
)
|
||||
|
||||
var example_number = 1
|
||||
val p_init = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(4, 1)), doubleArrayOf(5.0, 2.0, 0.2, 10.0)
|
||||
).as2D()
|
||||
|
||||
var t = ones(ShapeND(intArrayOf(100, 1))).as2D()
|
||||
for (i in 0 until 100) {
|
||||
t[i, 0] = t[i, 0] * (i + 1)
|
||||
}
|
||||
|
||||
val y_dat = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(100, 1)), lm_matx_y_dat
|
||||
).as2D()
|
||||
|
||||
val weight = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(1, 1)), DoubleArray(1) { 4.0 }
|
||||
).as2D()
|
||||
|
||||
val dp = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(1, 1)), DoubleArray(1) { -0.01 }
|
||||
).as2D()
|
||||
|
||||
val p_min = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(4, 1)), doubleArrayOf(-50.0, -20.0, -2.0, -100.0)
|
||||
).as2D()
|
||||
|
||||
val p_max = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(4, 1)), doubleArrayOf(50.0, 20.0, 2.0, 100.0)
|
||||
).as2D()
|
||||
|
||||
val consts = BroadcastDoubleTensorAlgebra.fromArray(
|
||||
ShapeND(intArrayOf(1, 1)), doubleArrayOf(0.0)
|
||||
).as2D()
|
||||
|
||||
val opts = doubleArrayOf(3.0, 100.0, 1e-3, 1e-3, 1e-1, 1e-1, 1e-2, 11.0, 9.0, 1.0)
|
||||
|
||||
val chi_sq = lm(::lm_func, p_init, t, y_dat, weight, dp, p_min, p_max, consts, opts, 10, example_number)
|
||||
assertEquals(0.9131, (chi_sq * 10000).roundToInt() / 10000.0)
|
||||
}
|
||||
}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user