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Java example source code file (leastsquares.xml)
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The leastsquares.xml Java example source code
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<?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?>
<document url="fitting.html">
<properties>
<title>The Commons Math User Guide - Least squares
</properties>
<body>
<section name="14 Least squares">
<subsection name="14.1 Overview">
<p>
The least squares package fits a parametric model to a set of observed
values by minimizing a cost function with a specific form.
The fitting basically consists in finding the values
for some parameters p<sub>k such that a cost function
J = sum(w<sub>i(targeti - modeli)2) is
minimized. The various (target<sub>i - modeli(pk))
terms are called residuals. They represent the deviation between a set of
target values target<sub>i and theoretical values computed from
models model<sub>i depending on free parameters pk.
The w<sub>i factors are weights. One classical use case is when the
target values are experimental observations or measurements.
</p>
<p>
Two engines devoted to least-squares problems are available. The first one is
based on the <a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/GaussNewtonOptimizer.html">
Gauss-Newton</a> method. The second one is the = fi(p1, p2, ... pk),
this is code</li>
<li>the target (or observed) components: targeti, this is data
<li>the start values for all pk parameters: sk, this is data
<li>optionally a validator for the pk parameters, this is code
<li>optionally the weight for sample point i: wi, this is data and defaults to 1.0 if not provided
<li>a maximum number of iterations, this is data
<li>a maximum number of evaluations, this is data
<li>a convergence criterion, this is code
</ul>
</p>
<p>
The elements of the list above can be provided as an implementation of the
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresProblem.html">
LeastSquaresProblem</a> interface. However, this is cumbersome to do directly, so some helper
classes are available. The first helper is a mutable builder:
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresBuilder.html">
LeastSquaresBuilder</a>. The second helper is an utility factory:
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresFactory.html">
LeastSquaresFactory</a>.
</p>
<p>
The builder class is better suited when setting the various elements of the least squares
problem is done progressively in different places in the user code. In this case, the user
would create first an empty builder andconfigure it progressively by calling its methods
(<code>start, target, model, ...). Once the configuration
is complete, calling the <code>build method would create the least squares problem.
</p>
<p>
The factory utility is better suited when the various elements of the least squares
problem are all known at one place and the problem can be built in just one sweep, calling
to one of the static <code>LeastSquaresFactory.create method.
</p>
</subsection>
<subsection name="14.3 Model Function">
<p>
The model function is used by the least squares engine to evaluate the model components
model<sub>i given some test parameters pk. It is therefore a multivariate
function (it depends on the various p<sub>k) and it is vector-valued (it has several
components model<sub>i). There must be exactly one component modeli for
each target (or observed) component target<sub>i, otherwise some residuals to be
squared and summed could not be computed. In order for the problem to be well defined, the
number of parameters p<sub>k must be less than the number of components modeli.
Failing to ensure this may lead to the engine throwing an exception as the underlying linear
algebra operations may encounter singular matrices. It is not unusual to have a large number
of components (several thousands) and only a dozen parameters. There are no limitations on these
numbers, though.
</p>
<p>
As the least squares engine needs to create Jacobians matrices for the model function, both
its value and its derivatives <em>with respect to the pk parameters must
be available. There are two ways to provide this:
<ul>
<li>provide one
<a href="../apidocs/org/apache/commons/math3/analysis/MultivariateVectorFunction.html">MultivariateVectorFunction
instance for computing the components values and one
<a href="../apidocs/org/apache/commons/math3/analysis/MultivariateMatrixFunction.html">MultivariateMatrixFunction
instance for computing the components derivatives (i.e. the Jacobian matrix) with
respect to the parameters,</li>
<li>or provide one
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/MultivariateJacobianFunction.html">MultivariateJacobianFunction
instance for computing both the components values and their derivatives simultaneously.</li>
</ul>
The first alternative is best suited for models which are not computationally intensive
as it allows more modularized code with one method for each type of computation. The second
alternative is best suited for models which are computationally intensive and evaluating
both the values and derivatives in one sweep saves a lot of work.
</p>
<p>
The <code>point parameter of the value methods in the
<a href="../apidocs/org/apache/commons/math3/analysis/MultivariateVectorFunction.html">MultivariateVectorFunction,
<a href="../apidocs/org/apache/commons/math3/analysis/MultivariateMatrixFunction.html">MultivariateMatrixFunction,
or <a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/MultivariateJacobianFunction.html">MultivariateJacobianFunction
interfaces will contain the parameters p<sub>k. The values will be the model components
model<sub>i and the derivatives will be the derivatives of the model components
with respect to the parameters dmodel<sub>i/dpk.
</p>
<p>
There are no requirements on how to compute value and derivatives. The
<a href="../apidocs/org/apache/commons/math3/analysis/differentiation/DerivativeStructure.html">
DerivativeStructure</a> class may be useful to compute analytically derivatives in
difficult cases, but this class is not mandated by the API which only expects the derivatives
as a Jacobian matrix containing primitive double entries.
</p>
<p>
One non-obvious feature provided by both the builder and the factory is lazy evaluation. This feature
allows to defer calls to the model functions until they are really needed by the engine. This
can save some calls for engines that evaluate the value and the Jacobians in different loops
(this is the case for Levenberg-Marquardt). However, lazy evaluation is possible <em>only
if the model functions are themselves separated, i.e. it can be used only with the first
alternative above. Setting up the <code>lazyEvaluation flag to true in the builder
or factory and setting up the model function as one
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/MultivariateJacobianFunction.html">MultivariateJacobianFunction
instance at the same time will trigger an illegal state exception telling that the model function
misses required functionality.
</p>
</subsection>
<subsection name="14.4 Parameters Validation">
<p>
In some cases, the model function requires parameters to lie within a specific domain. For example
a parameter may be used in a square root and needs to be positive, or another parameter represents
the sine of an angle and should be within -1 and +1, or several parameters may need to remain in
the unit circle and the sum of their squares must be smaller than 1. The least square solvers available
in Apache Commons Math currently don't allow to set up constraints on the parameters. This is a
known missing feature. There are two ways to circumvent this.
</p>
<p>
Both ways are achieved by setting up a
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/ParameterValidator.html">ParameterValidator
instance. The input of the value and jacobian model functions will always be the output of
the parameter validator if one exists.
</p>
<p>
One way to constrain parameters is to use a continuous mapping between the parameters that the
least squares solver will handle and the real parameters of the mathematical model. Using mapping
functions like <code>logit and sigmoid, one can map a finite range to the
infinite real line. Using mapping functions based on <code>log and exp, one
can map a semi-infinite range to the infinite real line. It is possible to use such a mapping so
that the engine will always see unbounded parameters, whereas on the other side of the mapping the
mathematical model will always see parameters mapped correctly to the expected range. Care must be
taken with derivatives as one must remember that the parameters have been mapped. Care must also
be taken with convergence status. This may be tricky.
</p>
<p>
Another way to constrain parameters is to simply truncate the parameters back to the domain when
one search point escapes from it and not care about derivatives. This works <em>only if the
solution is expected to be inside the domain and not at the boundary, as points out of the domain
will only be temporary test points with a cost function higher than the real solution and will soon
be dropped by the underlying engine. As a rule of thumb, these conditions are met only when the
domain boundaries correspond to unrealistic values that will never be achieved (null distances,
negative masses, ...) but they will not be met when the domain boundaries are more operational
limits (a maximum weight that can be handled by a device, a minimum temperature that can be
sustained by an instrument, ...).
</p>
</subsection>
<subsection name="14.5 Tuning">
<p>
Among the elements to be provided to the least squares problem builder or factory
are some tuning parameters for the solver.
</p>
<p>
The maximum number of iterations refers to the engine algorithm main loop, whereas the
maximum number of iterations refers to the number of calls to the model method. Some
algorithms (like Levenberg-Marquardt) have two embedded loops, with iteration number
being incremented at outer loop level, but a new evaluation being done at each inner
loop. In this case, the number of evaluations will be greater than the number of iterations.
Other algorithms (like Gauss-Newton) have only one level of loops. In this case, the
number of evaluations will equal to the number of iterations. In any case, the maximum
numbers are really only intended as safeguard to prevent infinite loops, so the exact
value of the limit is not important so it is common to select some almost arbitrary number
much larger than the expected number of evaluations and use it for both
<code>maxIterations and maxEvaluations. As an example, if the least
squares solver usually finds a solution in 50 iterations, setting a maximum value to 1000
is probably safe and prevents infinite loops. If the least squares solver needs several
hundreds of evaluations, it would probably be safer to set the maximum value to 10000 or
even 1000000 to avoid failures in slightly more demanding cases. Very fine tuning of
these maximum numbers is often worthless, they are only intended as safeguards.
</p>
<p>
Convergence checking is delegated to a dedicated interface from the <code>optim
package: <a href="../apidocs/org/apache/commons/math3/optim/ConvergenceChecker.html">
ConvergenceChecker</a>, parameterized with either the specific
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresProblem.Evaluation.html">Evaluation
class used for least squares problems or the general
<a href="../apidocs/org/apache/commons/math3/optim/PointVectorValuePair.html">PointVectorValuePair.
Each time convergence is checked, both the previous
and the current evaluations of the least squares problem are provided, so the checker can
compare them and decide whereas convergence has been reached or not. The predefined convergence
checker implementations that can be useful for least squares fitting are:
<ul>
<li>EvaluationRmsChecker,
which uses only the normalized cost (square-root of the sum of squared of the residuals,
divided by the number of measurements),</li>
<li>SimpleVectorValueChecker,
which uses the model components themselves (<em>not the residuals),
<li>SimplePointChecker<PointVectorValuePair>,
which uses the parameters.</li>
</ul>
Of course, users can also provide their own implementation of the
<a href="../apidocs/org/apache/commons/math3/optim/ConvergenceChecker.html">ConvergenceChecker
interface.
</p>
</subsection>
<subsection name="14.6 Optimization Engine">
<p>
Once the least squares problem has been created, using either the builder or the factory,
it is passed to an optimization engine for solving. Two engines devoted to least-squares
problems are available. The first one is
based on the <a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/GaussNewtonOptimizer.html">
Gauss-Newton</a> method. The second one is the methods that all return a new optimizer instance. Only the
final fully configured instance is used. As an example, setting up a Levenberg-Marquardt with
all configuration set to default except the cost relative tolerance and parameter relative tolerance
would be done as follows:
</p>
<source>
LeastSquaresOptimizer optimizer = new LevenbergMarquardtOptimizer().
withCostRelativeTolerance(1.0e-12).
withParameterRelativeTolerance(1.0e-12);
</source>
<p>
As another example, setting up a Gauss-Newton optimizer and forcing the decomposition to SVD (the
default is QR decomposition) would be done as follows:
</p>
<source>
LeastSquaresOptimizer optimizer = new GaussNewtonOptimizer().
withwithDecomposition(GaussNewtonOptimizer.Decomposition.QR);
</source>
</subsection>
<subsection name="14.7 Solving">
<p>
Solving the least squares problem is done by calling the <code>optimize method of the
optimizer and passing the least squares problem as the single parameter:
</p>
<source>
LeastSquaresOptimizer.Optimum optimum = optimizer.optimize(leastSquaresProblem);
</source>
<p>
The <a
href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresOptimizer.Optimum.html">
LeastSquaresOptimizer.Optimum</a> class is a specialized
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresProblem.Evaluation.html">Evaluation
with additional methods te retrieve the number of evaluations and number of iterations performed.
The most important methods are inherited from the
<a href="../apidocs/org/apache/commons/math3/fitting/leastsquares/LeastSquaresProblem.Evaluation.html">Evaluation
class and correspond to the point (i.e. the parameters), cost, Jacobian, RMS, covariance ...
</p>
</subsection>
<subsection name="14.8 Example">
<p>
The following simple example shows how to find the center of a circle of known radius to
to best fit observed 2D points. It is a simplified version of one of the JUnit test cases.
In the complete test case, both the circle center and its radius are fitted, here the
radius is fixed.
</p>
<source>
final double radius = 70.0;
final Vector2D[] observedPoints = new Vector2D[] {
new Vector2D( 30.0, 68.0),
new Vector2D( 50.0, -6.0),
new Vector2D(110.0, -20.0),
new Vector2D( 35.0, 15.0),
new Vector2D( 45.0, 97.0)
};
// the model function components are the distances to current estimated center,
// they should be as close as possible to the specified radius
MultivariateJacobianFunction distancesToCurrentCenter = new MultivariateJacobianFunction() {
public Pair<RealVector, RealMatrix> value(final RealVector point) {
Vector2D center = new Vector2D(point.getEntry(0), point.getEntry(1));
RealVector value = new ArrayRealVector(observedPoints.length);
RealMatrix jacobian = new Array2DRowRealMatrix(observedPoints.length, 2);
for (int i = 0; i < observedPoints.length; ++i) {
Vector2D o = observedPoints[i];
double modelI = Vector2D.distance(o, center);
value.setEntry(i, modelI);
// derivative with respect to p0 = x center
jacobian.setEntry(i, 0, (center.getX() - o.getX()) / modelI);
// derivative with respect to p1 = y center
jacobian.setEntry(i, 1, (center.getX() - o.getX()) / modelI);
}
return new Pair<RealVector, RealMatrix>(value, jacobian);
}
};
// the target is to have all points at the specified radius from the center
double[] prescribedDistances = new double[observedPoints.length];
Arrays.fill(prescribedDistances, radius);
// least squares problem to solve : modeled radius should be close to target radius
LeastSquaresProblem problem = new LeastSquaresBuilder().
start(new double[] { 100.0, 50.0 }).
model(distancesToCurrentCenter).
target(prescribedDistances).
lazyEvaluation(false).
maxEvaluations(1000).
maxIterations(1000).
build();
LeastSquaresOptimizer.Optimum optimum = new LevenbergMarquardtOptimizer().optimize(problem);
Vector2D fittedCenter = new Vector2D(optimum.getPoint().getEntry(0), optimum.getPoint().getEntry(1));
System.out.println("fitted center: " + fittedCenter.getX() + " " + fittedCenter.getY());
System.out.println("RMS: " + optimum.getRMS());
System.out.println("evaluations: " + optimum.getEvaluations());
System.out.println("iterations: " + optimum.getIterations());
</source>
</subsection>
</section>
</body>
</document>
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