FIT POLYNOMIAL... 4 In some cases,
relationships between different variables are best expressed with a polynomial
regression. LabAnalyst lets
you fit pairs of channels with polynomials of up to 9 degrees. The
model is:
Channel Y = (polynomial expression of Channel X).
After you select the X and Y channels (as described above for linear regression), the program
computes and displays a default polynomial of 3 degrees. Regression statistics (mean and SD of X and Y, residual Y variance, and r^{2}) 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 52% of the variance in sphere temperature.
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
r^{2} 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 Colors and
lines option in the VIEW menu.)
Additional considerations:
- As with any regression method, interpolation is 'safer' than extrapolation in terms of predictive accuracy.
- In some conditions (particularly if you are computing a high-degree
polynomial when the magnitudes of the X and Y channels differ greatly and r^{2} is low),
it is possible to exceed the numeric limits of the program. Usually,
the value of r^{2} is set to zero when this happens, and some or all of the coefficients will have extremely large or small values.
- You cannot send results to disk or printer 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).