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Date: Thu, 20 Dec 2007 17:02:59 +0900
From: koshino kazuhiro <koshino@ri.ncvc.go.jp>
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To: Jakarta Commons Users List <user@commons.apache.org>
Subject: Re: [math] Standard errors in estimated parameters using
 LevenbergMarquardtEstimator
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Hello Luc,

Thanks for your interest in this issue and introducing me to your JIRA 
system.
I created a ticket in JIRA for this issue.

Kind Regards,

Koshino

> Hi Koshino,
> 
> Sorry for this very late reply.
> 
> Could you please open a ticket in our JIRA issue tracking system with 
> your request or enhancement ? This will prevent your suggestion to be 
> forgotten and will automatically keep you informed as it is processed. 
> The tracking system is here: https://issues.apache.org/jira/browse/MATH
> You will have to sign up (its free) to be able to create a new issue.
> 
> Thanks
> Luc
> 
> koshino kazuhiro wrote:
>> Hello,
>>
>> Following code is one idea to obtain errors on estimated parameters.
>> (I ignore consistency with GaussNewtonEstimator via interface Estimator.)
>>
>> This code is based on the example in "[math] Example using 
>> EstimationProblem, WeightedMeasurement from apache.commons.math?".
>>
>> Kind Regards,
>> Koshino
>>
>> import org.apache.commons.math.estimation.EstimatedParameter;
>> import org.apache.commons.math.estimation.EstimationException;
>> import org.apache.commons.math.estimation.LevenbergMarquardtEstimator;
>> import org.apache.commons.math.estimation.SimpleEstimationProblem;
>> import org.apache.commons.math.estimation.WeightedMeasurement;
>>
>> // To calculate covariance matrix
>> import org.apache.commons.math.linear.RealMatrix;
>> import org.apache.commons.math.linear.MatrixUtils;
>>
>> public class QuadraticFitter extends SimpleEstimationProblem {
>>
>>     public static void main(String[] args) {
>>
>>         try {
>>             // create the uninitialized fitting problem
>>             QuadraticFitter fitter = new QuadraticFitter();
>>
>> //            // register the sample points (perfect quadratic case)
>> //            fitter.addPoint(1.0, 5.0, 0.2);
>> //            fitter.addPoint(2.0, 10.0, 0.4);
>> //            fitter.addPoint(3.0, 17.0, 1.0);
>> //            fitter.addPoint(4.0, 26.0, 0.8);
>> //            fitter.addPoint(5.0, 37.0, 0.3);
>>
>>             // register the sample points (practical case)
>>             fitter.addPoint(1.0, 1.0, 0.2);
>>             fitter.addPoint(2.0, 3.0, 0.4);
>>             fitter.addPoint(3.0, 2.0, 1.0);
>>             fitter.addPoint(4.0, 6.0, 0.8);
>>             fitter.addPoint(5.0, 4.0, 0.3);
>>             double centerX = 3.0;
>>
>>             // solve the problem, using a Levenberg-Marquardt algorithm
>>             // with default settings
>>             LevenbergMarquardtEstimator estimator =
>>               new LevenbergMarquardtEstimator();
>>             estimator.estimate(fitter);
>>             System.out.println("RMS after fitting = "
>>                              + estimator.getRMS(fitter));
>>
>>             // ******************************************
>>             // If Jacobian and chi-square are available,
>>             // we obtain errors on estimated parameters.
>>             // ******************************************
>>             RealMatrix jacobian = estimator.getJacobian();
>>             RealMatrix transJ = jacobian.transpose();
>>             RealMatrix covar = transJ.multiply(jacobian).inverse();
>>             double chisq = estimator.getChiSquare();
>>             int nm = fitter.getMeasurements().length;
>>             int np = fitter.getUnboundParameters().length;
>>             int dof = nm - np;
>>             for (int i = 0; i < np; ++i) {
>>               System.out.println(Math.sqrt(covar.getEntry(i, i)) * 
>> chisq / dof);
>>             }}
>>
>>             // display the results
>>             System.out.println("a = " + fitter.getA()
>>                     + ", b = " + fitter.getB()
>>                     + ", c = " + fitter.getC());
>>             System.out.println("y (" + centerX + ") = "
>>                                + fitter.theoreticalValue(centerX));
>>
>>         } catch (EstimationException ee) {
>>             System.err.println(ee.getMessage());
>>         }
>>
>>     }
>>   // skip constructor and methods
>> }
>>
>>> Hello,
>>>
>>>> Oooops. Sorry, my answer does not match your question. You asked 
>>>> about parameters and I answered about observations (measurements).
>>>
>>> The term "standard errors" in my sentences was ambiguity.
>>> I should use covariance matrix rather than standard errors.
>>>
>>> Your suggestion for RMS is very important because my measurements was 
>>> degraded by statistical noises.
>>>
>>>> So the proper answer is no, there is no way to get any information 
>>>> about errors on parameters yet. One method to have an estimate of 
>>>> the quality of the result is to check the eigenvalues related to the 
>>>> parameters. Perhaps we could look at this after having included a 
>>>> Singular Value Decomposition algorithm ? Is this what you want or do 
>>>> you have some other idea ?
>>>
>>> In LevenbergMarquardtEstimator.java (revision 560660),
>>> a jacobian matrix is used to estimate parameters.
>>> Using the jacobian matrix, could we obtain a covariance matrix, i.e. 
>>> errors on parameters?
>>> covariance matrix = (Jt * J)^(-1)
>>> where J, Jt and ^(-1) denotes a jacobian, a transpose matrix of J and 
>>> an inverse operator, respectively.
>>>
>>> Kind Regards,
>>>
>>> Koshino
>>>
>>>> Luc
>>>>
>>>>> the method (because it is your problems which defines both the 
>>>>> model, the
>>>>> parameters and the observations).
>>>>> If you call it before calling estimate, it will use the initial 
>>>>> values of the
>>>>> parameters, if you call it after having called estimate, it will 
>>>>> use the
>>>>> adjusted values.
>>>>>
>>>>> Here is what the javadoc says about this method:
>>>>>
>>>>>    * Get the Root Mean Square value, i.e. the root of the arithmetic
>>>>>    * mean of the square of all weighted residuals. This is related 
>>>>> to the
>>>>>    * criterion that is minimized by the estimator as follows: if
>>>>>    * <em>c</em> is the criterion, and <em>n</em> is the number of
>>>>>    * measurements, then the RMS is <em>sqrt (c/n)</em>.
>>>>>> I think that those values are very important to validate estimated
>>>>>> parameters.
>>>>>
>>>>> It may sometimes be misleading. If your problem model is wrong and too
>>>>> "flexible", and if your observation are bad (measurements errors), 
>>>>> then you may
>>>>> adjust too many parameters and have the bad model really follow the 
>>>>> bad
>>>>> measurements and give you artificially low residuals. Then you may 
>>>>> think
>>>>> everything is perfect which is false. This is about the same kind 
>>>>> of problems
>>>>> knowns as "Gibbs oscillations" for polynomial fitting when you use 
>>>>> a too high
>>>>> degree.
>>>>>
>>>>> Luc
>>>>>
>>>>>> Is use of classes in java.lang.reflect.* only way to get standard 
>>>>>> errors?
>>>>>>
>>>>>> Kind Regards,
>>>>>>
>>>>>> Koshino
>>>>>>
>>>>>> ---------------------------------------------------------------------
>>
>>
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> 
> 
> 
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> 

-- 
Kazuhiro Koshino

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