Generate CleanCurve = sum of three exponential curves and NoisyCurve=CleanCurve with 5% proportional Gaussian noise. Fit, the sum of three exponentials with amplitudes and decay rates found by optimization, attempts to fit the NoisyCurve. Parameters and confidence limts are compared with MonteCarlo estimates.

Description

Three exponential curves are summed to form CleanCurve. CleanCurve
is modified to include 5% proportional Gaussian noise producing
NoisyCurve.

Step 01: From the Project GUI, delete FivePercent (data set). 
Step 02: From the RunTime GUI, Run the model.
Step 03: From Plot pages (3) select CreateNoisyandClean.
Step 04: [Plot page GUI->File->Store Project Data Set] store data
         set under name FivePercent.
Step 05: [Plot page Select DataAndFit]. 
Step 06: For Curve 1, use [???] to select FivePercent, plotting the
         NoisyCurve. The solid black line represents the fit to the
         Noisy Curve and the dashed lines represent the three 
         exponentials that are summed to produce Fit.
Step 07: Go to the optimizer GUI. [Run Time GUI->Optimizer GUI].
Step 08: Inspect the Optimizer Gui. It has been preset for you to
         optimize the six parameters controlling the Fit.
Step 09: Under [Data Set][???] Select FivePercent.
Step 10: The OK boxes allow the parameters to be fit. Experiment
         with fitting with 1, 2, or 3 exponentials.
Step 11: To reset the parameters [Optimizer GUI->Config->ParSet->
         Load Project Parameter Set] Select Start.
Step 12: Run the optimizer twice. The first time you run it with
         this data set and initial value of parameters, you will
         get the Error message: "aborted: java.lang.RuntimeException: 
         Matrix is singular." Run the optimizer again.
Step 13: [Optimizer GUI->Report GUI] You will see that the Standard
         Deviation on some of the parameters is extremely large. The
         condition number of the correlation matrix indicates that
         the matrix is nearly singular. 
         The correlation matrix gives the correlation between Af1 and kf1
         as 1 and between Af3 and kf3 as 0.9988. This indicates that
         two of the four parameters are redundant. Remove kf1 and kf3
         from the optimization [uncheck their OK boxes by switching back
         to the Config GUI.] Run the Optimizer again.
Step 14: Recheck the values from the Report GUI. The problem with fitting
         3 exponentials is that there are close to being a set of linearly
         dependent functions.
Step 15: When optimizing, if the point weight (Pwgt) is set to 1/Fit^(1.5),
         the tail of the fit is dramatically improved. If the point weight
         is 1, the optimizer gets stuck in local minima. Where the optimization
         starts from turns out to be important. With perfect data and a
         perfect fit, the correlation between parameters cannot be computed.
 
Fitting 3 exponentials when the data are sparse or noisy is mathematically 
challenging. Test the capability to identify the decay rates when they 
differ by a factor of 3 instead of 5. What happens when the magnitudes of
the components are disparate? When do two exponentials fit the curve as well 
as a 3-exp curve? Consider Akaike information criteria re choosing how many
exponentials to use in fitting. Consider how these functions behave in the
frequncy domain or LaPlace transforms. See Gardner [1963].

In each case, examine the correlation matrix and the parameter confidence 
ranges after each optimization. Look for correlations amongst parameters

Run Monte Carlo Analysis to determine error bounds on the estimation of
amplitudes and time constants. Compare these with the confidence 
ranges estimated from the covariance Matrix: correlation matrix = covariance 
matrix divided by the variance.

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.