## Class LevenbergMarquardt

• All Implemented Interfaces:
Serializable, Cloneable, Optimizer

public abstract class LevenbergMarquardt
extends Object
implements Serializable, Cloneable, Optimizer
This class implements a parallel Levenberg-Marquardt non-linear least-squares fit algorithm.

The solver minimizes $$|| f ||_{L_{2}}$$ for a function $$f:\mathbb{R}^n \rightarrow \mathbb{R}^m$$. The solver requires the calculation of a Jacobi-matrix $$J = \frac{\mathrm{d}f}{\mathrm{d}x}$$. The iteration steps are then defined by $\Delta x = H_{\lambda}^{-1} J^T f$ where $$H_{\lambda}$$ is a regularized approximation of the Hessian matrix. The solver supports two different regularizations. For RegularizationMethod.LEVENBERG the solver uses $$H_{\lambda} = J^T J + \lambda I$$. For RegularizationMethod.LEVENBERG_MARQUARDT the solver uses $$H_{\lambda} = J^T J + \lambda \text{diag}(J^T J)$$.

The design avoids the need to define the objective function as a separate class. The objective function is defined by overriding a class method, see the sample code below.

The Levenberg-Marquardt solver is implemented in using multi-threading. The calculation of the derivatives (in case a specific implementation of setDerivatives(double[] parameters, double[][] derivatives) is not provided) may be performed in parallel by setting the parameter numberOfThreads.

To use the solver inherit from it and implement the objective function as setValues(double[] parameters, double[] values) where values has to be set to the value of the objective functions for the given parameters.
You may also provide an a derivative for your objective function by additionally overriding the function setDerivatives(double[] parameters, double[][] derivatives), otherwise the solver will calculate the derivative via finite differences.

To reject a point, it is allowed to set an element of values to Double.NaN in the implementation of setValues(double[] parameters, double[] values). Put differently: The solver handles NaN values in values as an error larger than the current one (regardless of the current error) and rejects the point.
Note, however, that is is an error if the initial parameter guess results in an NaN value. That is, the solver should be initialized with an initial parameter in an admissible region.

The following simple example finds a solution for the equation
 0.0 * x1 + 1.0 * x2 = 5.0 2.0 * x1 + 1.0 * x2 = 10.0
 
LevenbergMarquardt optimizer = new LevenbergMarquardt() {
// Override your objective function here
public void setValues(double[] parameters, double[] values) {
values[0] = parameters[0] * 0.0 + parameters[1];
values[1] = parameters[0] * 2.0 + parameters[1];
}
};

// Set solver parameters
optimizer.setInitialParameters(new double[] { 0, 0 });
optimizer.setWeights(new double[] { 1, 1 });
optimizer.setMaxIteration(100);
optimizer.setTargetValues(new double[] { 5, 10 });

optimizer.run();

double[] bestParameters = optimizer.getBestFitParameters();


See the example in the main method below.

The class can be initialized to use a multi-threaded valuation. If initialized this way the implementation of setValues must be thread-safe. The solver will evaluate the gradient of the value vector in parallel, i.e., use as many threads as the number of parameters.

Note: Iteration steps will be logged (java.util.logging) with LogLevel.FINE
Version:
1.6
Author:
Christian Fries
Serialized Form
• ### Nested Class Summary

Nested Classes
Modifier and Type Class Description
static class  LevenbergMarquardt.RegularizationMethod
The regularization method used to invert the approximation of the Hessian matrix.
• ### Nested classes/interfaces inherited from interface net.finmath.optimizer.Optimizer

Optimizer.ObjectiveFunction
• ### Constructor Summary

Constructors
Constructor Description
LevenbergMarquardt()
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(double[] initialParameters, double[] targetValues, int maxIteration, int numberOfThreads)
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(double[] initialParameters, double[] targetValues, int maxIteration, ExecutorService executorService)
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(int numberOfThreads)
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(List<Number> initialParameters, List<Number> targetValues, int maxIteration, int numberOfThreads)
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(List<Number> initialParameters, List<Number> targetValues, int maxIteration, ExecutorService executorService)
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(LevenbergMarquardt.RegularizationMethod regularizationMethod, double[] initialParameters, double[] targetValues, int maxIteration, int numberOfThreads)
Create a Levenberg-Marquardt solver.
LevenbergMarquardt​(LevenbergMarquardt.RegularizationMethod regularizationMethod, double[] initialParameters, double[] targetValues, int maxIteration, ExecutorService executorService)
Create a Levenberg-Marquardt solver.
• ### Method Summary

All Methods
Modifier and Type Method Description
LevenbergMarquardt clone()
Create a clone of this LevenbergMarquardt optimizer.
double[] getBestFitParameters()
LevenbergMarquardt getCloneWithModifiedTargetValues​(double[] newTargetVaues, double[] newWeights, boolean isUseBestParametersAsInitialParameters)
Create a clone of this LevenbergMarquardt optimizer with a new vector for the target values and weights.
LevenbergMarquardt getCloneWithModifiedTargetValues​(List<Number> newTargetVaues, List<Number> newWeights, boolean isUseBestParametersAsInitialParameters)
Create a clone of this LevenbergMarquardt optimizer with a new vector for the target values and weights.
int getIterations()
Get the number of iterations.
double getLambda()
Get the parameter λ used in the Tikhonov-like regularization of the Hessian matrix, that is the $$\lambda$$ in $$H + \lambda \diag H$$.
double getLambdaDivisor()
Get the divisor applied to lambda (for the next iteration) if the inversion of regularized Hessian succeeds, that is, if $$H + \lambda \diag H$$ is invertable.
double getLambdaMultiplicator()
Get the multiplicator applied to lambda if the inversion of regularized Hessian fails, that is, if $$H + \lambda \diag H$$ is not invertable.
double getMeanSquaredError​(double[] value)
double getRootMeanSquaredError()
static void main​(String[] args)
void run()
Runs the optimization.
void setDerivatives​(double[] parameters, double[][] derivatives)
The derivative of the objective function.
LevenbergMarquardt setErrorTolerance​(double errorTolerance)
Set the error tolerance.
LevenbergMarquardt setInitialParameters​(double[] initialParameters)
Set the initial parameters for the solver.
LevenbergMarquardt setLambda​(double lambda)
Set the parameter λ used in the Tikhonov-like regularization of the Hessian matrix, that is the $$\lambda$$ in $$H + \lambda \diag H$$.
void setLambdaDivisor​(double lambdaDivisor)
Set the divisor applied to lambda (for the next iteration) if the inversion of regularized Hessian succeeds, that is, if $$H + \lambda \diag H$$ is invertable.
void setLambdaMultiplicator​(double lambdaMultiplicator)
Set the multiplicator applied to lambda if the inversion of regularized Hessian fails, that is, if $$H + \lambda \diag H$$ is not invertable.
LevenbergMarquardt setMaxIteration​(int maxIteration)
Set the maximum number of iterations to be performed until the solver gives up.
LevenbergMarquardt setParameterSteps​(double[] parameterSteps)
Set the parameter step for the solver.
LevenbergMarquardt setTargetValues​(double[] targetValues)
Set the target values for the solver.
abstract void setValues​(double[] parameters, double[] values)
The objective function.
LevenbergMarquardt setWeights​(double[] weights)
Set the weight for the objective function.
• ### Methods inherited from class java.lang.Object

equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### LevenbergMarquardt

public LevenbergMarquardt​(LevenbergMarquardt.RegularizationMethod regularizationMethod,
double[] initialParameters,
double[] targetValues,
int maxIteration,
ExecutorService executorService)
Create a Levenberg-Marquardt solver.
Parameters:
regularizationMethod - The regularization method to use. See LevenbergMarquardt.RegularizationMethod.
initialParameters - Initial value for the parameters where the solver starts its search.
targetValues - Target values to achieve.
maxIteration - Maximum number of iterations.
executorService - Executor to be used for concurrent valuation of the derivatives. This is only performed if setDerivative is not overwritten. Warning: The implementation of setValues has to be thread safe!
• #### LevenbergMarquardt

public LevenbergMarquardt​(double[] initialParameters,
double[] targetValues,
int maxIteration,
ExecutorService executorService)
Create a Levenberg-Marquardt solver.
Parameters:
initialParameters - Initial value for the parameters where the solver starts its search.
targetValues - Target values to achieve.
maxIteration - Maximum number of iterations.
executorService - Executor to be used for concurrent valuation of the derivatives. This is only performed if setDerivative is not overwritten. Warning: The implementation of setValues has to be thread safe!
• #### LevenbergMarquardt

public LevenbergMarquardt​(LevenbergMarquardt.RegularizationMethod regularizationMethod,
double[] initialParameters,
double[] targetValues,
int maxIteration,
int numberOfThreads)
Create a Levenberg-Marquardt solver.
Parameters:
regularizationMethod - The regularization method to use. See LevenbergMarquardt.RegularizationMethod.
initialParameters - Initial value for the parameters where the solver starts its search.
targetValues - Target values to achieve.
maxIteration - Maximum number of iterations.
numberOfThreads - Maximum number of threads. Warning: If this number is larger than one, the implementation of setValues has to be thread safe!
• #### LevenbergMarquardt

public LevenbergMarquardt​(double[] initialParameters,
double[] targetValues,
int maxIteration,
int numberOfThreads)
Create a Levenberg-Marquardt solver.
Parameters:
initialParameters - Initial value for the parameters where the solver starts its search.
targetValues - Target values to achieve.
maxIteration - Maximum number of iterations.
numberOfThreads - Maximum number of threads. Warning: If this number is larger than one, the implementation of setValues has to be thread safe!
• #### LevenbergMarquardt

public LevenbergMarquardt​(List<Number> initialParameters,
List<Number> targetValues,
int maxIteration,
ExecutorService executorService)
Create a Levenberg-Marquardt solver.
Parameters:
initialParameters - List of initial values for the parameters where the solver starts its search.
targetValues - List of target values to achieve.
maxIteration - Maximum number of iterations.
executorService - Executor to be used for concurrent valuation of the derivatives. This is only performed if setDerivative is not overwritten. Warning: The implementation of setValues has to be thread safe!
• #### LevenbergMarquardt

public LevenbergMarquardt​(List<Number> initialParameters,
List<Number> targetValues,
int maxIteration,
int numberOfThreads)
Create a Levenberg-Marquardt solver.
Parameters:
initialParameters - Initial value for the parameters where the solver starts its search.
targetValues - Target values to achieve.
maxIteration - Maximum number of iterations.
numberOfThreads - Maximum number of threads. Warning: If this number is larger than one, the implementation of setValues has to be thread safe!
• #### LevenbergMarquardt

public LevenbergMarquardt()
Create a Levenberg-Marquardt solver.
• #### LevenbergMarquardt

public LevenbergMarquardt​(int numberOfThreads)
Create a Levenberg-Marquardt solver.
Parameters:
numberOfThreads - Maximum number of threads. Warning: If this number is larger than one, the implementation of setValues has to be thread safe!
• ### Method Detail

• #### main

public static void main​(String[] args)
throws SolverException,
CloneNotSupportedException
Throws:
SolverException
CloneNotSupportedException
• #### setInitialParameters

public LevenbergMarquardt setInitialParameters​(double[] initialParameters)
Set the initial parameters for the solver.
Parameters:
initialParameters - The initial parameters.
Returns:
A self reference.
• #### setParameterSteps

public LevenbergMarquardt setParameterSteps​(double[] parameterSteps)
Set the parameter step for the solver. The parameter step is used to evaluate the derivatives via finite differences, if analytic derivatives are not provided.
Parameters:
parameterSteps - The parameter step.
Returns:
A self reference.
• #### setTargetValues

public LevenbergMarquardt setTargetValues​(double[] targetValues)
Set the target values for the solver. The solver will solver the equation weights * objectiveFunction = targetValues.
Parameters:
targetValues - The target values.
Returns:
A self reference.
• #### setMaxIteration

public LevenbergMarquardt setMaxIteration​(int maxIteration)
Set the maximum number of iterations to be performed until the solver gives up.
Parameters:
maxIteration - The maximum number of iterations.
Returns:
A self reference.
• #### setWeights

public LevenbergMarquardt setWeights​(double[] weights)
Set the weight for the objective function.
Parameters:
weights - The weights for the objective function.
Returns:
A self reference.
• #### setErrorTolerance

public LevenbergMarquardt setErrorTolerance​(double errorTolerance)
Set the error tolerance. The solver considers the solution "found" if the error is not improving by this given error tolerance.
Parameters:
errorTolerance - The error tolerance.
Returns:
A self reference.
• #### getLambda

public double getLambda()
Get the parameter λ used in the Tikhonov-like regularization of the Hessian matrix, that is the $$\lambda$$ in $$H + \lambda \diag H$$.
Returns:
the parameter $$\lambda$$.
• #### setLambda

public LevenbergMarquardt setLambda​(double lambda)
Set the parameter λ used in the Tikhonov-like regularization of the Hessian matrix, that is the $$\lambda$$ in $$H + \lambda \diag H$$.
Parameters:
lambda - the lambda to set
Returns:
Self reference to this optimizer.
• #### getLambdaMultiplicator

public double getLambdaMultiplicator()
Get the multiplicator applied to lambda if the inversion of regularized Hessian fails, that is, if $$H + \lambda \diag H$$ is not invertable.
Returns:
the lambdaMultiplicator
• #### setLambdaMultiplicator

public void setLambdaMultiplicator​(double lambdaMultiplicator)
Set the multiplicator applied to lambda if the inversion of regularized Hessian fails, that is, if $$H + \lambda \diag H$$ is not invertable. This will make lambda larger, hence let the stepping move slower.
Parameters:
lambdaMultiplicator - the lambdaMultiplicator to set. Should be > 1.

public double getLambdaDivisor()
Get the divisor applied to lambda (for the next iteration) if the inversion of regularized Hessian succeeds, that is, if $$H + \lambda \diag H$$ is invertable.
Returns:

public void setLambdaDivisor​(double lambdaDivisor)
Set the divisor applied to lambda (for the next iteration) if the inversion of regularized Hessian succeeds, that is, if $$H + \lambda \diag H$$ is invertable. This will make lambda smaller, hence let the stepping move faster.
Parameters:
lambdaDivisor - the lambdaDivisor to set. Should be > 1.
• #### getBestFitParameters

public double[] getBestFitParameters()
Description copied from interface: Optimizer
Specified by:
getBestFitParameters in interface Optimizer
Returns:
The best fit parameter.
• #### getRootMeanSquaredError

public double getRootMeanSquaredError()
Specified by:
getRootMeanSquaredError in interface Optimizer
Returns:
the the root mean square error of achieved with the the best fit parameter
• #### getIterations

public int getIterations()
Description copied from interface: Optimizer
Get the number of iterations.
Specified by:
getIterations in interface Optimizer
Returns:
The number of iterations required
• #### setValues

public abstract void setValues​(double[] parameters,
double[] values)
throws SolverException
The objective function. Override this method to implement your custom function.
Parameters:
parameters - Input value. The parameter vector.
values - Output value. The vector of values f(i,parameters), i=1,...,n
Throws:
SolverException - Thrown if the valuation fails, specific cause may be available via the cause() method.
• #### setDerivatives

public void setDerivatives​(double[] parameters,
double[][] derivatives)
throws SolverException
The derivative of the objective function. You may override this method if you like to implement your own derivative.
Parameters:
parameters - Input value. The parameter vector.
derivatives - Output value, where derivatives[i][j] is d(value(j)) / d(parameters(i)
Throws:
SolverException - Thrown if the valuation fails, specific cause may be available via the cause() method.
• #### run

public void run()
throws SolverException
Description copied from interface: Optimizer
Runs the optimization.
Specified by:
run in interface Optimizer
Throws:
SolverException - Thrown if the valuation fails, specific cause may be available via the cause() method.
• #### getMeanSquaredError

public double getMeanSquaredError​(double[] value)
• #### clone

public LevenbergMarquardt clone()
throws CloneNotSupportedException
Create a clone of this LevenbergMarquardt optimizer. The clone will use the same objective function than this implementation, i.e., the implementation of setValues(double[], double[]) and that of setDerivatives(double[], double[][]) is reused.
Overrides:
clone in class Object
Throws:
CloneNotSupportedException
• #### getCloneWithModifiedTargetValues

public LevenbergMarquardt getCloneWithModifiedTargetValues​(double[] newTargetVaues,
double[] newWeights,
boolean isUseBestParametersAsInitialParameters)
throws CloneNotSupportedException
Create a clone of this LevenbergMarquardt optimizer with a new vector for the target values and weights. The clone will use the same objective function than this implementation, i.e., the implementation of setValues(double[], double[]) and that of setDerivatives(double[], double[][]) is reused. The initial values of the cloned optimizer will either be the original initial values of this object or the best parameters obtained by this optimizer, the latter is used only if this optimized signals a done().
Parameters:
newTargetVaues - New array of target values.
newWeights - New array of weights.
isUseBestParametersAsInitialParameters - If true and this optimizer is done(), then the clone will use this.getBestFitParameters() as initial parameters.
Returns:
A new LevenbergMarquardt optimizer, cloning this one except modified target values and weights.
Throws:
CloneNotSupportedException - Thrown if this optimizer cannot be cloned.
• #### getCloneWithModifiedTargetValues

public LevenbergMarquardt getCloneWithModifiedTargetValues​(List<Number> newTargetVaues,
List<Number> newWeights,
boolean isUseBestParametersAsInitialParameters)
throws CloneNotSupportedException
Create a clone of this LevenbergMarquardt optimizer with a new vector for the target values and weights. The clone will use the same objective function than this implementation, i.e., the implementation of setValues(double[], double[]) and that of setDerivatives(double[], double[][]) is reused. The initial values of the cloned optimizer will either be the original initial values of this object or the best parameters obtained by this optimizer, the latter is used only if this optimized signals a done().
Parameters:
newTargetVaues - New list of target values.
newWeights - New list of weights.
isUseBestParametersAsInitialParameters - If true and this optimizer is done(), then the clone will use this.getBestFitParameters() as initial parameters.
Returns:
A new LevenbergMarquardt optimizer, cloning this one except modified target values and weights.
Throws:
CloneNotSupportedException - Thrown if this optimizer cannot be cloned.
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