com.opengamma.strata.math.impl.statistics.leastsquare.NonLinearLeastSquare Maven / Gradle / Ivy
/*
* Copyright (C) 2009 - present by OpenGamma Inc. and the OpenGamma group of companies
*
* Please see distribution for license.
*/
package com.opengamma.strata.math.impl.statistics.leastsquare;
import static com.opengamma.strata.math.MathUtils.pow2;
import java.util.function.Function;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import com.opengamma.strata.collect.ArgChecker;
import com.opengamma.strata.collect.array.DoubleArray;
import com.opengamma.strata.collect.array.DoubleMatrix;
import com.opengamma.strata.math.MathException;
import com.opengamma.strata.math.impl.differentiation.VectorFieldFirstOrderDifferentiator;
import com.opengamma.strata.math.impl.differentiation.VectorFieldSecondOrderDifferentiator;
import com.opengamma.strata.math.impl.function.ParameterizedFunction;
import com.opengamma.strata.math.impl.linearalgebra.DecompositionFactory;
import com.opengamma.strata.math.impl.linearalgebra.SVDecompositionCommons;
import com.opengamma.strata.math.impl.linearalgebra.SVDecompositionResult;
import com.opengamma.strata.math.impl.matrix.MatrixAlgebra;
import com.opengamma.strata.math.impl.matrix.MatrixAlgebraFactory;
import com.opengamma.strata.math.linearalgebra.Decomposition;
import com.opengamma.strata.math.linearalgebra.DecompositionResult;
/**
* Non linear least square calculator.
*/
// CSOFF: JavadocMethod
public class NonLinearLeastSquare {
private static final Logger LOGGER = LoggerFactory.getLogger(NonLinearLeastSquare.class);
private static final int MAX_ATTEMPTS = 10000;
private static final Function UNCONSTRAINED = new Function() {
@Override
public Boolean apply(DoubleArray x) {
return true;
}
};
private final double _eps;
private final Decomposition _decomposition;
private final MatrixAlgebra _algebra;
public NonLinearLeastSquare() {
this(DecompositionFactory.SV_COMMONS, MatrixAlgebraFactory.OG_ALGEBRA, 1e-8);
}
public NonLinearLeastSquare(Decomposition decomposition, MatrixAlgebra algebra, double eps) {
_decomposition = decomposition;
_algebra = algebra;
_eps = eps;
}
//-------------------------------------------------------------------------
/**
* Use this when the model is in the ParameterizedFunction form and analytic parameter sensitivity is not available.
* @param x Set of measurement points
* @param y Set of measurement values
* @param func The model in ParameterizedFunction form (i.e. takes measurement points and a set of parameters and
* returns a model value)
* @param startPos Initial value of the parameters
* @return A LeastSquareResults object
*/
public LeastSquareResults solve(
DoubleArray x,
DoubleArray y,
ParameterizedFunction func,
DoubleArray startPos) {
ArgChecker.notNull(x, "x");
ArgChecker.notNull(y, "y");
int n = x.size();
ArgChecker.isTrue(y.size() == n, "y wrong length");
return solve(x, y, DoubleArray.filled(n, 1), func, startPos);
}
/**
* Use this when the model is in the ParameterizedFunction form and analytic parameter sensitivity is not available
* but a measurement error is.
* @param x Set of measurement points
* @param y Set of measurement values
* @param sigma y Set of measurement errors
* @param func The model in ParameterizedFunction form (i.e. takes measurement points and a set of parameters and
* returns a model value)
* @param startPos Initial value of the parameters
* @return A LeastSquareResults object
*/
public LeastSquareResults solve(
DoubleArray x,
DoubleArray y,
double sigma,
ParameterizedFunction func,
DoubleArray startPos) {
ArgChecker.notNull(x, "x");
ArgChecker.notNull(y, "y");
ArgChecker.notNull(sigma, "sigma");
int n = x.size();
ArgChecker.isTrue(y.size() == n, "y wrong length");
return solve(x, y, DoubleArray.filled(n, sigma), func, startPos);
}
/**
* Use this when the model is in the ParameterizedFunction form and analytic parameter sensitivity is not available
* but an array of measurements errors is.
* @param x Set of measurement points
* @param y Set of measurement values
* @param sigma Set of measurement errors
* @param func The model in ParameterizedFunction form (i.e. takes measurement points and a set of parameters and
* returns a model value)
* @param startPos Initial value of the parameters
* @return A LeastSquareResults object
*/
public LeastSquareResults solve(
DoubleArray x,
DoubleArray y,
DoubleArray sigma,
ParameterizedFunction func,
DoubleArray startPos) {
ArgChecker.notNull(x, "x");
ArgChecker.notNull(y, "y");
ArgChecker.notNull(sigma, "sigma");
int n = x.size();
ArgChecker.isTrue(y.size() == n, "y wrong length");
ArgChecker.isTrue(sigma.size() == n, "sigma wrong length");
Function func1D = new Function() {
@Override
public DoubleArray apply(DoubleArray theta) {
return DoubleArray.of(x.size(), i -> func.evaluate(x.get(i), theta));
}
};
return solve(y, sigma, func1D, startPos, null);
}
/**
* Use this when the model is in the ParameterizedFunction form and analytic parameter sensitivity.
* @param x Set of measurement points
* @param y Set of measurement values
* @param func The model in ParameterizedFunction form (i.e. takes a measurement points and a set of parameters and
* returns a model value)
* @param grad The model parameter sensitivities in ParameterizedFunction form (i.e. takes a measurement points and a
* set of parameters and returns a model parameter sensitivities)
* @param startPos Initial value of the parameters
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray x,
DoubleArray y,
ParameterizedFunction func,
ParameterizedFunction grad,
DoubleArray startPos) {
ArgChecker.notNull(x, "x");
ArgChecker.notNull(y, "y");
ArgChecker.notNull(x, "sigma");
int n = x.size();
ArgChecker.isTrue(y.size() == n, "y wrong length");
// emcleod 31-1-2011 arbitrary value 1 for now
return solve(x, y, DoubleArray.filled(n, 1), func, grad, startPos);
}
/**
* Use this when the model is in the ParameterizedFunction form and analytic parameter sensitivity and a single
* measurement error are available.
* @param x Set of measurement points
* @param y Set of measurement values
* @param sigma Measurement errors
* @param func The model in ParameterizedFunction form (i.e. takes a measurement points and a set of parameters and
* returns a model value)
* @param grad The model parameter sensitivities in ParameterizedFunction form (i.e. takes a measurement points and a
* set of parameters and returns a model parameter sensitivities)
* @param startPos Initial value of the parameters
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray x,
DoubleArray y,
double sigma,
ParameterizedFunction func,
ParameterizedFunction grad,
DoubleArray startPos) {
ArgChecker.notNull(x, "x");
ArgChecker.notNull(y, "y");
int n = x.size();
ArgChecker.isTrue(y.size() == n, "y wrong length");
return solve(x, y, DoubleArray.filled(n, sigma), func, grad, startPos);
}
/**
* Use this when the model is in the ParameterizedFunction form and analytic parameter sensitivity and measurement
* errors are available.
* @param x Set of measurement points
* @param y Set of measurement values
* @param sigma Set of measurement errors
* @param func The model in ParameterizedFunction form (i.e. takes a measurement points and a set of parameters and
* returns a model value)
* @param grad The model parameter sensitivities in ParameterizedFunction form (i.e. takes a measurement points and a
* set of parameters and returns a model parameter sensitivities)
* @param startPos Initial value of the parameters
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray x,
DoubleArray y,
DoubleArray sigma,
ParameterizedFunction func,
ParameterizedFunction grad,
DoubleArray startPos) {
ArgChecker.notNull(x, "x");
ArgChecker.notNull(y, "y");
ArgChecker.notNull(x, "sigma");
int n = x.size();
ArgChecker.isTrue(y.size() == n, "y wrong length");
ArgChecker.isTrue(sigma.size() == n, "sigma wrong length");
Function func1D = new Function() {
@Override
public DoubleArray apply(DoubleArray theta) {
return DoubleArray.of(x.size(), i -> func.evaluate(x.get(i), theta));
}
};
Function jac = new Function() {
@Override
public DoubleMatrix apply(DoubleArray theta) {
int m = x.size();
double[][] res = new double[m][];
for (int i = 0; i < m; i++) {
DoubleArray temp = grad.evaluate(x.get(i), theta);
res[i] = temp.toArray();
}
return DoubleMatrix.copyOf(res);
}
};
return solve(y, sigma, func1D, jac, startPos, null);
}
/**
* Use this when the model is given as a function of its parameters only (i.e. a function that takes a set of
* parameters and return a set of model values,
* so the measurement points are already known to the function), and analytic parameter sensitivity is not available
* @param observedValues Set of measurement values
* @param func The model as a function of its parameters only
* @param startPos Initial value of the parameters
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray observedValues,
Function func,
DoubleArray startPos) {
int n = observedValues.size();
VectorFieldFirstOrderDifferentiator jac = new VectorFieldFirstOrderDifferentiator();
return solve(observedValues, DoubleArray.filled(n, 1.0), func, jac.differentiate(func), startPos, null);
}
/**
* Use this when the model is given as a function of its parameters only (i.e. a function that takes a set of
* parameters and return a set of model values,
* so the measurement points are already known to the function), and analytic parameter sensitivity is not available
* @param observedValues Set of measurement values
* @param sigma Set of measurement errors
* @param func The model as a function of its parameters only
* @param startPos Initial value of the parameters
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray observedValues,
DoubleArray sigma,
Function func,
DoubleArray startPos) {
VectorFieldFirstOrderDifferentiator jac = new VectorFieldFirstOrderDifferentiator();
return solve(observedValues, sigma, func, jac.differentiate(func), startPos, null);
}
/**
* Use this when the model is given as a function of its parameters only (i.e. a function that takes a set of
* parameters and return a set of model values,
* so the measurement points are already known to the function), and analytic parameter sensitivity is not available
* @param observedValues Set of measurement values
* @param sigma Set of measurement errors
* @param func The model as a function of its parameters only
* @param startPos Initial value of the parameters
* @param maxJumps A vector containing the maximum absolute allowed step in a particular direction in each iteration.
* Can be null, in which case no constant
* on the step size is applied.
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray observedValues,
DoubleArray sigma,
Function func,
DoubleArray startPos,
DoubleArray maxJumps) {
VectorFieldFirstOrderDifferentiator jac = new VectorFieldFirstOrderDifferentiator();
return solve(observedValues, sigma, func, jac.differentiate(func), startPos, maxJumps);
}
/**
* Use this when the model is given as a function of its parameters only (i.e. a function that takes a set of
* parameters and return a set of model values,
* so the measurement points are already known to the function), and analytic parameter sensitivity is available
* @param observedValues Set of measurement values
* @param sigma Set of measurement errors
* @param func The model as a function of its parameters only
* @param jac The model sensitivity to its parameters (i.e. the Jacobian matrix) as a function of its parameters only
* @param startPos Initial value of the parameters
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray observedValues,
DoubleArray sigma,
Function func,
Function jac, DoubleArray startPos) {
return solve(observedValues, sigma, func, jac, startPos, UNCONSTRAINED, null);
}
/**
* Use this when the model is given as a function of its parameters only (i.e. a function that takes a set of
* parameters and return a set of model values,
* so the measurement points are already known to the function), and analytic parameter sensitivity is available
* @param observedValues Set of measurement values
* @param sigma Set of measurement errors
* @param func The model as a function of its parameters only
* @param jac The model sensitivity to its parameters (i.e. the Jacobian matrix) as a function of its parameters only
* @param startPos Initial value of the parameters
* @param maxJumps A vector containing the maximum absolute allowed step in a particular direction in each iteration.
* Can be null, in which case on constant
* on the step size is applied.
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray observedValues,
DoubleArray sigma,
Function func,
Function jac,
DoubleArray startPos,
DoubleArray maxJumps) {
return solve(observedValues, sigma, func, jac, startPos, UNCONSTRAINED, maxJumps);
}
/**
* Use this when the model is given as a function of its parameters only (i.e. a function that takes a set of
* parameters and return a set of model values,
* so the measurement points are already known to the function), and analytic parameter sensitivity is available
* @param observedValues Set of measurement values
* @param sigma Set of measurement errors
* @param func The model as a function of its parameters only
* @param jac The model sensitivity to its parameters (i.e. the Jacobian matrix) as a function of its parameters only
* @param startPos Initial value of the parameters
* @param constraints A function that returns true if the trial point is within the constraints of the model
* @param maxJumps A vector containing the maximum absolute allowed step in a particular direction in each iteration.
* Can be null, in which case on constant
* on the step size is applied.
* @return value of the fitted parameters
*/
public LeastSquareResults solve(
DoubleArray observedValues,
DoubleArray sigma,
Function func,
Function jac,
DoubleArray startPos,
Function constraints,
DoubleArray maxJumps) {
ArgChecker.notNull(observedValues, "observedValues");
ArgChecker.notNull(sigma, " sigma");
ArgChecker.notNull(func, " func");
ArgChecker.notNull(jac, " jac");
ArgChecker.notNull(startPos, "startPos");
int nObs = observedValues.size();
int nParms = startPos.size();
ArgChecker.isTrue(nObs == sigma.size(), "observedValues and sigma must be same length");
ArgChecker.isTrue(nObs >= nParms,
"must have data points greater or equal to number of parameters. #date points = {}, #parameters = {}", nObs, nParms);
ArgChecker.isTrue(constraints.apply(startPos),
"The inital value of the parameters (startPos) is {} - this is not an allowed value", startPos);
DoubleMatrix alpha;
DecompositionResult decmp;
DoubleArray theta = startPos;
double lambda = 0.0; // TODO debug if the model is linear, it will be solved in 1 step
double newChiSqr, oldChiSqr;
DoubleArray error = getError(func, observedValues, sigma, theta);
DoubleArray newError;
DoubleMatrix jacobian = getJacobian(jac, sigma, theta);
oldChiSqr = getChiSqr(error);
// If we start at the solution we are done
if (oldChiSqr == 0.0) {
return finish(oldChiSqr, jacobian, theta, sigma);
}
DoubleArray beta = getChiSqrGrad(error, jacobian);
for (int count = 0; count < MAX_ATTEMPTS; count++) {
alpha = getModifiedCurvatureMatrix(jacobian, lambda);
DoubleArray deltaTheta;
try {
decmp = _decomposition.apply(alpha);
deltaTheta = decmp.solve(beta);
} catch (Exception e) {
throw new MathException(e);
}
DoubleArray trialTheta = (DoubleArray) _algebra.add(theta, deltaTheta);
// acceptable step is found
if (!constraints.apply(trialTheta) || !allowJump(deltaTheta, maxJumps)) {
lambda = increaseLambda(lambda);
continue;
}
newError = getError(func, observedValues, sigma, trialTheta);
newChiSqr = getChiSqr(newError);
// Check for convergence when no improvement in chiSqr occurs
if (Math.abs(newChiSqr - oldChiSqr) / (1 + oldChiSqr) < _eps) {
DoubleMatrix alpha0 = lambda == 0.0 ? alpha : getModifiedCurvatureMatrix(jacobian, 0.0);
// if the model is an exact fit to the data, then no more improvement is possible
if (newChiSqr < _eps) {
if (lambda > 0.0) {
decmp = _decomposition.apply(alpha0);
}
return finish(alpha0, decmp, newChiSqr, jacobian, trialTheta, sigma);
}
SVDecompositionCommons svd = (SVDecompositionCommons) DecompositionFactory.SV_COMMONS;
// add the second derivative information to the Hessian matrix to check we are not at a local maximum or saddle
// point
VectorFieldSecondOrderDifferentiator diff = new VectorFieldSecondOrderDifferentiator();
Function secDivFunc = diff.differentiate(func, constraints);
DoubleMatrix[] secDiv = secDivFunc.apply(trialTheta);
double[][] temp = new double[nParms][nParms];
for (int i = 0; i < nObs; i++) {
for (int j = 0; j < nParms; j++) {
for (int k = 0; k < nParms; k++) {
temp[j][k] -= newError.get(i) * secDiv[i].get(j, k) / sigma.get(i);
}
}
}
DoubleMatrix newAlpha = (DoubleMatrix) _algebra.add(alpha0, DoubleMatrix.copyOf(temp));
SVDecompositionResult svdRes = svd.apply(newAlpha);
double[] w = svdRes.getSingularValues();
DoubleMatrix u = svdRes.getU();
DoubleMatrix v = svdRes.getV();
double[] p = new double[nParms];
boolean saddle = false;
double sum = 0.0;
for (int i = 0; i < nParms; i++) {
double a = 0.0;
for (int j = 0; j < nParms; j++) {
a += u.get(j, i) * v.get(j, i);
}
int sign = a > 0.0 ? 1 : -1;
if (w[i] * sign < 0.0) {
sum += w[i];
w[i] = -w[i];
saddle = true;
}
}
// if a local maximum or saddle point is found (as indicated by negative eigenvalues), move in a direction that
// is a weighted
// sum of the eigenvectors corresponding to the negative eigenvalues
if (saddle) {
lambda = increaseLambda(lambda);
for (int i = 0; i < nParms; i++) {
if (w[i] < 0.0) {
double scale = 0.5 * Math.sqrt(-oldChiSqr * w[i]) / sum;
for (int j = 0; j < nParms; j++) {
p[j] += scale * u.get(j, i);
}
}
}
DoubleArray direction = DoubleArray.copyOf(p);
deltaTheta = direction;
trialTheta = (DoubleArray) _algebra.add(theta, deltaTheta);
int i = 0;
double scale = 1.0;
while (!constraints.apply(trialTheta)) {
scale *= -0.5;
deltaTheta = (DoubleArray) _algebra.scale(direction, scale);
trialTheta = (DoubleArray) _algebra.add(theta, deltaTheta);
i++;
if (i > 10) {
throw new MathException("Could not satify constraint");
}
}
newError = getError(func, observedValues, sigma, trialTheta);
newChiSqr = getChiSqr(newError);
int counter = 0;
while (newChiSqr > oldChiSqr) {
// if even a tiny move along the negative eigenvalue cannot improve chiSqr, then exit
if (counter > 10 || Math.abs(newChiSqr - oldChiSqr) / (1 + oldChiSqr) < _eps) {
LOGGER.warn("Saddle point detected, but no improvement to chi^2 possible by moving away. " +
"It is recommended that a different starting point is used.");
return finish(newAlpha, decmp, oldChiSqr, jacobian, theta, sigma);
}
scale /= 2.0;
deltaTheta = (DoubleArray) _algebra.scale(direction, scale);
trialTheta = (DoubleArray) _algebra.add(theta, deltaTheta);
newError = getError(func, observedValues, sigma, trialTheta);
newChiSqr = getChiSqr(newError);
counter++;
}
} else {
// this should be the normal finish - i.e. no improvement in chiSqr and at a true minimum (although there is
// no guarantee it is not a local minimum)
return finish(newAlpha, decmp, newChiSqr, jacobian, trialTheta, sigma);
}
}
if (newChiSqr < oldChiSqr) {
lambda = decreaseLambda(lambda);
theta = trialTheta;
error = newError;
jacobian = getJacobian(jac, sigma, trialTheta);
beta = getChiSqrGrad(error, jacobian);
oldChiSqr = newChiSqr;
} else {
lambda = increaseLambda(lambda);
}
}
throw new MathException("Could not converge in " + MAX_ATTEMPTS + " attempts");
}
private double decreaseLambda(double lambda) {
return lambda / 10;
}
private double increaseLambda(double lambda) {
if (lambda == 0.0) { // this will happen the first time a full quadratic step fails
return 0.1;
}
return lambda * 10;
}
private boolean allowJump(DoubleArray deltaTheta, DoubleArray maxJumps) {
if (maxJumps == null) {
return true;
}
int n = deltaTheta.size();
for (int i = 0; i < n; i++) {
if (Math.abs(deltaTheta.get(i)) > maxJumps.get(i)) {
return false;
}
}
return true;
}
/**
*
* the inverse-Jacobian where the i-j entry is the sensitivity of the ith (fitted) parameter (a_i) to the jth data
* point (y_j).
* @param sigma Set of measurement errors
* @param func The model as a function of its parameters only
* @param jac The model sensitivity to its parameters (i.e. the Jacobian matrix) as a function of its parameters only
* @param originalSolution The value of the parameters at a converged solution
* @return inverse-Jacobian
*/
public DoubleMatrix calInverseJacobian(
DoubleArray sigma,
Function func,
Function jac,
DoubleArray originalSolution) {
DoubleMatrix jacobian = getJacobian(jac, sigma, originalSolution);
DoubleMatrix a = getModifiedCurvatureMatrix(jacobian, 0.0);
DoubleMatrix bT = getBTranspose(jacobian, sigma);
DecompositionResult decRes = _decomposition.apply(a);
return decRes.solve(bT);
}
private LeastSquareResults finish(
double newChiSqr,
DoubleMatrix jacobian,
DoubleArray newTheta,
DoubleArray sigma) {
DoubleMatrix alpha = getModifiedCurvatureMatrix(jacobian, 0.0);
DecompositionResult decmp = _decomposition.apply(alpha);
return finish(alpha, decmp, newChiSqr, jacobian, newTheta, sigma);
}
private LeastSquareResults finish(
DoubleMatrix alpha,
DecompositionResult decmp,
double newChiSqr,
DoubleMatrix jacobian,
DoubleArray newTheta,
DoubleArray sigma) {
DoubleMatrix covariance = decmp.solve(DoubleMatrix.identity(alpha.rowCount()));
DoubleMatrix bT = getBTranspose(jacobian, sigma);
DoubleMatrix inverseJacobian = decmp.solve(bT);
return new LeastSquareResults(newChiSqr, newTheta, covariance, inverseJacobian);
}
private DoubleArray getError(final Function func, final DoubleArray observedValues, final DoubleArray sigma, final DoubleArray theta) {
int n = observedValues.size();
DoubleArray modelValues = func.apply(theta);
ArgChecker.isTrue(n == modelValues.size(), "Number of data points different between model (" + modelValues.size() + ") and observed (" + n + ")");
return DoubleArray.of(n, i -> (observedValues.get(i) - modelValues.get(i)) / sigma.get(i));
}
private DoubleMatrix getBTranspose(DoubleMatrix jacobian, DoubleArray sigma) {
int n = jacobian.rowCount();
int m = jacobian.columnCount();
double[][] res = new double[m][n];
for (int i = 0; i < n; i++) {
double sigmaInv = 1.0 / sigma.get(i);
for (int k = 0; k < m; k++) {
res[k][i] = jacobian.get(i, k) * sigmaInv;
}
}
return DoubleMatrix.copyOf(res);
}
private DoubleMatrix getJacobian(final Function jac, final DoubleArray sigma, final DoubleArray theta) {
DoubleMatrix res = jac.apply(theta);
double[][] data = res.toArray();
int n = res.rowCount();
int m = res.columnCount();
ArgChecker.isTrue(theta.size() == m, "Jacobian is wrong size");
ArgChecker.isTrue(sigma.size() == n, "Jacobian is wrong size");
for (int i = 0; i < n; i++) {
double sigmaInv = 1.0 / sigma.get(i);
for (int j = 0; j < m; j++) {
data[i][j] *= sigmaInv;
}
}
return DoubleMatrix.ofUnsafe(data);
}
private double getChiSqr(DoubleArray error) {
return _algebra.getInnerProduct(error, error);
}
private DoubleArray getChiSqrGrad(DoubleArray error, DoubleMatrix jacobian) {
return (DoubleArray) _algebra.multiply(error, jacobian);
}
@SuppressWarnings("unused")
private DoubleArray getDiagonalCurvatureMatrix(DoubleMatrix jacobian) {
int n = jacobian.rowCount();
int m = jacobian.columnCount();
double[] alpha = new double[m];
for (int i = 0; i < m; i++) {
double sum = 0.0;
for (int k = 0; k < n; k++) {
sum += pow2(jacobian.get(k, i));
}
alpha[i] = sum;
}
return DoubleArray.copyOf(alpha);
}
private DoubleMatrix getModifiedCurvatureMatrix(DoubleMatrix jacobian, double lambda) {
int m = jacobian.columnCount();
double onePLambda = 1.0 + lambda;
DoubleMatrix alpha = _algebra.matrixTransposeMultiplyMatrix(jacobian);
// scale the diagonal
double[][] data = alpha.toArray();
for (int i = 0; i < m; i++) {
data[i][i] *= onePLambda;
}
return DoubleMatrix.ofUnsafe(data);
}
}
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