Review and Equation Selection Q&A¶

What is the Graph Start and List Start all about?¶

The Review utilizes two separate ‘desktops’, the specific organization of windows used to review and select the most appropriate curve-fit equation.

The Graph Start option assumes you are mainly interested in making the selection by stepping through the curve-fit graphs. The default equation list is in a small-popup window that coexists on screen with the graph.

The List Start option assumes you wish to focus considerable attention upon the inspection of the equations within the list. The default equation list is large, and automatically closes after a selection is made. Clicking on the List button restores this large list.

Each option takes shape as you customize it to suit the amount of screen space available and the items you wish to see displayed when stepping through the equations.

How does one effectively inspect equations?¶

Although equation inspection and selection is still somewhat of an art, some guidelines can be given:

Curve-Fit Graph¶

Graphically inspect the quality of the fit across the whole data range (and possibly beyond in the case of extrapolations). The numerical goodness of fit criteria are sometimes poor indicators of fit for points at the lower portion of the Y-scale. The zoom processes and logarithmic axes often aid in this inspection.

Detecting Outliers¶

The coloring of points based upon residuals will readily illustrate points outside 2 and 3 standard errors. These may represent outliers which should be excluded from fitting or treated by a robust non-linear procedure.

Math Errors¶

Be very cautious of selecting an equation where a part of the data region shown is undefined or where major changes in the curve occur. When a math error or singularity is detected, a vertical line will be drawn at this X-value. A vertical error line is not always drawn when a singularity is present. A curve rapidly going off-scale and then returning, without this error bar, may still indicate a value or region of values where the equation is undefined. In the instance of an extremely sharp singularity, the graph may not show even a hint of the singularity.

Rational Singularities¶

Singularities are far and away of greatest concern with rational functions. For these functions, refer to the numeric summary to see if any poles (roots of the denominator) exist within the data range or beyond (in the case of extrapolations). When the rational’s denominator is a polynomial in x, all real roots are found and reported. When the rational denominator is not a polynomial in x, such as is the case for ln(x), half order, even order, Chebyshev, and Fourier series rationals, a partitioning is used to isolate pertinent real roots.

Confidence and Prediction Intervals¶

Graphically inspect the prediction intervals for the fit. This is a superb way to detect outliers in the data. A good equation should have tight and stable intervals, especially in the area of concern. Be cautious of intervals which generate math errors (vertical lines of a different color). The computation of the intervals is more sensitive to breakdown due to machine floating point precision limitations than the curve-fit equations themselves.

Replicate Error Bars¶

If you are fitting replicate data, use error bars to detect observations that should be excluded.

Residuals Inspection¶

Graphically inspect the residuals across the whole data range. An ideal, especially in parametric models, is that the residuals not only be very small but also random in sign (no systematic trend). Note the runs count in the Y-title, the number of sign changes in the residuals.

SNP Plot¶

If you plan on using the confidence limits on the parameters, or the confidence or prediction intervals about the fitted curve, inspect the Residuals SNP plot to insure that the residuals are normally distributed.

Fitting Speed¶

Choose a fast equation if you are going to use it in a program where the speed of the code is important. One of TableCurve 2D’s unique features is the ability to choose an equation based on the floating point execution speed of its own generated code for that equation. The FP values are shown in the equation list.

Evaluation¶

Preview the performance of the equation using the evaluation procedure at pertinent X-values before arriving at a final selection.

I see three tenth order polynomials in the equation list, all with the same goodness of fit. Which one should I use?¶

You are seeing a standard polynomial fitted by the Intel FPU, a high-precision polynomial fitted by the 38-digit math emulator, and a Chebyshev polynomial that has been converted to a standard polynomial. Even when all three show an equal goodness of fit within the equation list, there may be significant differences in the Precision Summary in terms of errors arising due to coefficient precision truncation. This issue is normally only of concern when fitting full precision generated data.

In general, the high-precision fit will have the most accurate coefficients.

Since no inverse exists for the Chebyshev converted polynomials, confidence and prediction intervals will not be available.

Why is the format for the Lotus 123 export different from that if the Excel export?¶

Graphs are included in the Lotus 123 worksheet format. Graphs are not generated in the Excel format. Since each of the products import both formats, use the Lotus 123 format when you want graphs exported, and the Excel format when you do not.

Why is the formula for certain equations missing from Excel and Lotus export?¶

Some of the functions require a full evaluation routine and cannot be coded into a single cell formula. If you use Excel, there is a very attractive alternative, however. For later Excel versions supporting VBA (Visual Basic for Applications), you may wish to add a Basic module containing the curve-fit equation to the spreadsheet. This approach makes the curve-fit function global to Excel just as if it were a built-in Excel function. The on-line help and printed documentation offer step-by-step instructions.

How do I display the Curve-Fit and Residuals in the same graph?¶

The Toggle Residuals on Y2 Axis button offers a fast way to toggle the Y2 Axis Residuals on and off.

You can also use the Set Y2 Axis Content option in the control panel of the Review’s Curve-Fit Graph. You can also display derivatives or cumulative area in the Y2 axis of the graph.

Since the Y and Y2 axis share common X values, the Residuals distribution and SNP plots will not be available. For these graphs, you must use the separate Residuals option.

What is an SNP plot anyway?¶

It is the recommended approach in TableCurve 2D for assuring that errors are normally distributed. A delta stabilized normal probability (SNP) plot uses an arctangent transformation on both X and Y to produce a normal probability plot that uses a linear scale for both the X and Y axes. On such a plot, perfectly normal errors plot as a horizontal y=0 line. Critical limits also appear as horizontal lines.

TableCurve 2D plots 90, 95, 99, and 99.9% critical limit lines on the SNP plot. A 99% critical limit means that in only 1 out of 100 data sets should even a single point violate this limit. You may find the 99% critical limit the most useful. If even a single data point in the SNP violates this 99% limit, it is reasonable to assume that the errors fail this normality test. You should inspect the SNP before attempting to use the parameter confidence statistics or confidence or prediction intervals in any way.

Can I set up a reference from a separate fit defining a standard, and have this plotted for all future fits?¶

To save the standard, simply use the Save Current Equation as Reference option in the Curve-Fit graph’s Reference menu.

In the subsequent fits, use the Equation References option and select the Add One or More Imported Fits to References option. The reference will consist only of the fitted curve.

I know which TableCurve 2D equation number I want. How do I find it in the equation list?¶

The Search for Specific Equation button is used to directly select the TableCurve 2D equation desired.

How do I limit equations to only those which are continuously increasing or continuously decreasing?¶

Filter the equation list by Continuous Trend First Derivative. For the kind of profiles unique to transition data, filter by Constant Sign in First Derivative. Derivative filtering is configured in the Set X-Range for Derivative Filtering option.

This is a convenience for the instance where the Equation List window is not visible.

In the code generation option, is the function code itself portable to different compilers?¶

The C, Fortran 77, and Fortran 90 function code should compile with any compiler.

How do I change the numeric precision used in graph titles and labels?¶

Use the Preferences option in the File menu of the Curve-Fit Graph to change the title precision.

The label precision is set directly in the TableCurve 2D graph, specifically via the 2D View option.

How do I control which of the titles are displayed in Review graphs?¶

This is also set in the TableCurve 2D graph’s 2D View option.

Can I copy the curve-fit graph to the clipboard not as an image but as the raw numeric data which comprises the graph?¶

Use the All Numeric Info as Spreadsheet Block to Clipboard option in the graph’s Copy option. This is available for every program graph.

On certain fits the first equation does not appear in the list upon entering the Review. What happened to it?¶

The Rationals Fully Defined Xmin-Xmax is initially turned on within the Equation List. This setting filters out equations which were detected as having a singularity in the data range. Such equations are generally undesired. To see the excluded equations, simply uncheck this Rationals Fully Defined Xmin-Xmax item. This can be done in either the List menu of the Curve-Fit Graph or in the Filter menu in the Equation List.

Can a Singularity Constrained Non-Linear SVD Rational equation (Eqns 7901-7939) have a pole in the middle of the data range? Also, for equations with no poles in the data range, shouldn't these equations produce the same fit as the linear form?¶

For these Singularity Constrained Non-Linear SVD Rational models, the fit begins with the coefficients for the equivalent linear fit. If a pole is detected in the X range of the data, the coefficients are adjusted in an effort to produce a model without such poles. Then, a true non-linear rational fit is made, with a constraint set to eliminate, if possible, the formation of any pole within the X data range. In a non-linear fit, if the zeroing of singular values produces nonsense, that iteration’s adjustments are ignored. As such, it is generally possible to keep the coefficients within a useful range for the SVD algorithm. This results in the most stable set of coefficients possible.

In general, because of the constraints, the SVD, and the non-linear nature of the rational model, the results from these fits will differ from that of the linearly fitted rationals of the same form, even when no data range singularities are present.

As in all non-linear fitting, constraint-based fitting is hardly foolproof, and this is all the more true when the coefficients must be resolved through SVD. As such, these models are not always successful. In some cases, it is simply not possible to avoid the formation of a pole in the X range for even a modest goodness of fit. When these equations are successful, however, the fit will generally be superior to the linear procedures.

If I see points during the Review that I wish to have excluded from a refit, how do I proceed?¶

The Mark Excluded Points for Refit option is used to automatically select outliers or points where math errors occur. The outliers can also be tagged manually by left clicking on the point desired. Marked points are redrawn with a surrounding circle. To unmark a point that has been marked, click once again with the left mouse button. The surrounding circle will be removed.

When a refit is initiated, marked active points are made inactive while marked inactive points are made active. You can go back and forth between including and excluding a set of marked points by successive refits.