Non-Linear Optimization Preferences

These preferences control adjustable elements within AutoSignal's non-linear fitting engine.

n Components

The number of components will initially be set to the number of spectral peaks identified in the procedure. You may choose to fit a smaller number of components. In this instance, those components with the least spectral magnitudes are discarded prior to fitting.

Component Model

The model can be one of the following:

Sine : Y=Ampl*sin(2*Pi*Freq*X+Phase)

Sine, Exp Damped : Y=Ampl*exp(-k*X)*sin(2*Pi*Freq*X+Phase)

Sine, Exp Formation : Y=Ampl*(1-exp(-k*X))*sin(2*Pi*Freq*X+Phase)

Note that both the decay and formation sinusoid use a first order rate constant, k. This is reported as the Damping parameter in the Non-Linear Optimization's Numeric Summary.

Share Across Components

This option allows the Phase to be shared across all components (a single phase value is fitted to all components). If a four parameter model is used, the Rate can also be shared across all components.

Built-In Fn Constraints

These are particularly important when fitting large numbers of components. It is essential that the fitting algorithm exert constraints to assure spectral peaks remain close to where they are originally detected. Non-linear fitting, in and of itself, cannot assure such.

Local minima in multidimensional non-linear fitting refers to a limitation in iterative minimization algorithms. Unlike linear least-squares, where a global minimum always occurs (and within a single step), achieving a global minimum (the optimum fit) in iterative non-linear fitting is not assured. Indeed oscillatory functions are the worst of culprits for functions producing local minima.

When a sinusoid's frequency is shifted sufficiently up or down, a local minima condition develops. It is therefore especially important that frequency values be constrained and not be permitted unlimited freedom of movement.

When the spectral identication algorithms are used as frequency estimators for a linear sinusoidal fit, the estimates that result are usually of sufficient accuracy to avoid local minima in the non-linear optimization. These accurate starrting estimates make it possible to impose constraints on the principal parameter values. The constraints are specified as percents on the parameters.

AutoSignal's algorithm is designed to deal with minimization of constraint violations as well as a goodness of fit minimization. In particularly difficult fits, you may see upwards of several dozen constraints violated on the initial iteration. In time the algorithm should work itself to a position in multidimensional parameter space where no constraints are violated. In conditions where constraints are readily violated, the fit algorithm may spend more iterations working itself free of constraint violations than in resolving the optimum fit parameters.

You may disable constraints by unchecking them. In normal practice, it is recommended that you keep the constraints active.To preserve sign, a constraint must be less than 100%.

Maximum Iterations

The default number of iterations is 500. You may enter any value between 10 and 10000. There is usually very little to gain beyond 500 iterations.

Converge to Significant Digits in Chi-Square

The default non-linear convergence precision is 6. This means that the chi-square (goodness of fit) must be unchanging in the sixth decimal place for five consecutive iterations to signal convergence. You may enter any value between 3 and 14. Note that often only a few additional iterations are needed to converge to a high precision. A convergence precision of 12 may thus require very little additional fitting time than the default of 6.

Robust Minimization

In addition to standard least-squares minimization, AutoSignal's Levenburg-Marquardt non-linear fitting engine is capable of three different Robust Estimations. These minimizations are sometimes referred to as maximum likelihood or m-estimate fitting.

If your data span a large number of orders of magnitude in the Y variable and the low-valued Y points are not factoring into the fit, a robust fit will remedy this problem. This may well be a superior solution to seeking to weight the data so that these low-valued Y points can factor into a least-squares solution.

The other instance where robust fitting is recommended is when it is known that there are significant outliers within the data. Robust estimation will minimize the impact of outliers. This may be important when relatively few points define an individual waveform.

Least-squares corresponds to a Gaussian maximum likelihood distribution of errors. All of the robust minimizations correspond with maximum likelihood probability distributions significantly less compact than the Gaussian. These wider tails mean that the errors associated with outliers are expected. When outliers are suspected or likely, the Lorentzian minimization is highly recommended. Although the same non-linear fitting engine is used for all of these minimizations, you will generally find that a higher number of iterations will be required for a robust fit as compared to least squares.

Saving and Reading Fit Preferences

Use the Save item to save the current preferences to disk. The default file extension is PRF. These are binary files that can only be produced within the program. The current preferences are always saved automatically across sessions. You will want to save preferences to disk if you plan to regularly use more than one fit configuration in your work. Preference files can be recalled at anytime using the Read item.

Reset

The Reset button restores AutoSignal default non-linear fit preferences.