Curve-Fit Preferences

This option is used to specify the fitting controls that are used for all of TableCurve 2D's Curve-Fit options except the Curve-Fit Peak Functions, the Curve-Fit Transition Functions, and the Curve-Fit Kinetic Equations which have their own separate non-linear controls.

Linear fitting controls include the default fitting procedure, the term significance threshold, the maximum order of equation permitted, forcing a zero intercept, using base 10 rather than natural logarithms, and disabling the degree of freedom normally assigned to the error. Non-linear controls include the maximum allowable number of iterations, the convergence precision, the minimization criteria, and intercept options. You can save a custom fit configuration to disk for recall at any future time.

Linear Fitting Procedure

Unless you have specific matrix methods expertise and wish to select a specific procedure for fitting all equations, the real decision here is whether to use the Fast Std option or the Fast SVD procedure. Both use a mix of methods to minimize processing time. The difference is whether higher order polynomials and rationals are fitted by a conventional method (Gaussian Elimination) or by Singular Value Decomposition (SVD).

Higher order polynomials and rationals tend to produce ill-conditioned design matrices. When solved by conventional algorithms, such as Gaussian Elimination, the coefficients will often span many orders of magnitude and they will tend to be very large in magnitude and alternating in sign. Changing a seemingly insignificant digit in only one of the coefficients could drive the function far from the data.

With SVD, the solution will be more stable. The coefficients are driven toward zero rather than infinity by the ill-conditioning, and the result is that the coefficients will span fewer orders of magnitude and they will tend to be smaller in magnitude. A small change in a single coefficient is less likely to drive the function away from the data and limited extrapolations may even be possible. The disadvantages of SVD are that its results will likely produce a poorer goodness-of-fit statistic and the method requires an order of magnitude greater fitting time. In some cases, the zeroing that occurs within SVD can cause the fit to fail entirely.

The Fast SVD method assumes you wish this additional SVD stability in higher order polynomials and rationals. The Fast Std method assumes you wish the conventional polynomial or rational solution.

If you are using the Fast SVD method and not getting the results you expect for higher order polynomials and rationals, you may want to select the Fast Std, or the specific Gauss-Jordan or LU-Decomposition methods.

Note that selecting a specific method will slightly increase the fitting time and will typically reveal differences only with the higher order polynomial and rational functions. Note also that TableCurve 2D exclusively uses extended double precision (80 bit) floating point math. It is possible that no method selected will match the polynomial coefficients produced by another program which uses double precision (64 bit) or single precision (32 bit) math.

Highest Term Count to Be Fitted

To change the maximum coefficient count permitted when fitting linear equations, enter any value between 2 and 21. The default is 21, which permits all of TableCurve's equations. Note that the maximum term count applies to all linear fits, including the built-in equation sets.

It is recommended that you reset this count back to 21 when you have completed the limited term fitting. This prevents the unfortunate case in which a low term count preference is subsequently forgotten and a succession of All Equations fits are made only to later discover that the higher term equations were not included.

The maximum term count may be overridden to insure a degree of freedom of one, (or zero, if the No DOF For Error option is checked).

The Arrange by Parameter Count option in the Review's Equation List sorts the candidate equations by term count, and is probably a better way to manage the inspection of simpler equations.

Term Significant Digit Threshold

The default Term Significance is 9. This means that a given term in a linear equation must make at least a 1e-9 (0.0000001%) contribution to the average Y value when evaluations are made at Xmin, Xmean, and Xmax. A term significance of 9 will only rarely discard an equation. You may enter any value for term significance from 1 to 15. At a low value such as 2 (1e-2,1%), a considerable number of equations may be excluded. Although there is no direct correlation, you may wish to set the term significance at a value slightly above the number of significant figures in your data.

Force Zero Intercept on Standard Equations

When this box is checked, all of TableCurve's standard equations, Eqns 1-105, 1001-1630, 2001-3330, and 4001-5140, will be forced through the origin. This option reduces the design matrix of the full equation so as to produce a 0.0 value for the intercept term. You should use this option only when it is absolutely necessary for a given equation to pass through the origin. Note that TableCurve's standard equations with log and inverse terms are less likely to produce anything of value when this option is selected. With this option active, the a coefficient of the affected equations will appear as 0.0 and will have no standard error.

Log Equations Ln=Base 10 Log Rather than Natural Log

When this box is checked, all of TableCurve's linear equations containing the natural log function become common or base 10 log functions. Note that while this option may benefit the simple equations relative to subsequently plotting them with TableCurve's base 10 common log graphs, the various log polynomials and rationals are likely to produce poorer fits since a wider dynamic range is being raised to the higher powers.

No DOF For Error (For Fitting Error-Free Data)

This option only affects the equations fitted when there are fewer than 22 points in the data set. When this box is checked, TableCurve does not reserve a degree of freedom for the error of fit. This means it will be possible to fit a polynomial with five coefficients to five data points. Normally, functions with five or more coefficients would not be fitted when only five data points are present. This option should be used only when fitting generated or handbook data known to have negligible error.

Non-Linear Minimization

In addition to standard least-squares minimization, TableCurve 2D's non-linear 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 be a superior solution to fitting the LN(Y) rather than Y or seeking to weight the data so that these low-valued Y points can factor into a least-squares solution.

The other instance in which robust fitting is recommended is if it is known that there are significant outliers within the data. Robust estimation will minimize the impact of outliers.

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 TableCurve's Levenburg-Marquardt non-linear 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.

Maximum Iterations

The default number of iterations for a non-linear equation is 200. You may enter any value between 2 and 9999. There is seldom much to gain beyond 200 iterations unless you are working with a very complex model or one whose parametrization evidences a very slow convergence.

Converge to Significant Digits in r²

The default non-linear convergence precision is 6. This means that the r² coefficient of determination must be unchanging in the sixth decimal place for five consecutive iterations to signal convergence. You may enter any value between 3 and 15. 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.

Intercept Options

Most of TableCurve's non-linear equations have both an intercept and zero-intercept form. There are three intercept options where you can fit only those non-linear equations with intercepts, only those with zero-intercepts, or both versions.

Saving Fit Preferences

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

Reset

The Reset button restores TableCurve's default curve-fit preferences.