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        Channel Y = (polynomial expression of Channel X)

After you select the X and Y channels (as described for linear regression), the program computes and displays either a default polynomial of 3 degrees, or (if the Polynomial option has been run since program launch) the last polynomial degree used.  Regression statistics (mean and SD of X and Y, residual Y variance, and r²) as well as the polynomial coefficients are shown.   You can then select other degrees using the pop-up menu.  In this example, a 5-degree polynomial was used to predict the temperature of a small sunlit sphere from shade air temperature; the resulting equation -- shown at the right of the window -- explains about 69% of the variance in sphere temperature.

Note:   In some conditions (particularly if you are computing a high-degree polynomial when the magnitudes of the X and Y channels are large or differ greatly), it is possible to exceed the numeric limits of the program.  Results are unpredictable when theis happens, but usually the value of r² is set to zero or is lower than with a lower degree polynomial, and some or all of the coefficients will have extremely large or small values.

After completing the analysis, several options are available.  The 'Calculate Y' button opens a small window in which you can enter an X value and obtain the Y value derived from the computed polynimlal.   The 'New X and Y' button lets you change the X and Y variables.  The 'Make New Channel' button (only available if the number of channels is less than 40) generates a new channel by applying the current polynomial equation to the data in the X channel -- in other words, it computes and saves a new Y value for every sample in the X channel.

The 'Show All r^2' button helps you select the most appropriate degree for the polynomial equation.  The predictive value of a polynomial always increases as the degree increases.  However, the increase in accuracy usually plateaus, so it is reasonable to use the simplest equation consistent with good predictive power.  When the 'Show All r^2' button is clicked, the program computes the r² value for all degrees between 1 and 9, and then shows a bar graph of the results.  [You can remove this plot by clicking the 'Show All r^2' button again, or click the plot's standard 'close' control at the top left.]

In the example shown at right, predictive power increased until the degree of the polynomial exceeded 4, with little additional change at higher degrees (you can select the color of bar graph plots in the Plot Style option in the VIEW menu.)

Additional considerations:

• As with any regression method, interpolation is 'safer' than extrapolation in terms of predictive accuracy.

• You cannot send results to disk by hitting the 'p' key, as usual.  Instead, use the 'Print' button (this button is available only if output has been selected from the FILE menu).

• This option DOES NOT print to a tabular file (the output format is incompatible).