The LAS 7.0 module which fits two sets of tie points is polyfit. Polyfit uses a tie point location file containing tie point pairs which specify a mapping from the output (reference) coordinates to the input (search) coordinates to derive a pair of bivariate polynomials. The polynomials are calculated using least squares regression analysis in a forward stepping procedure (stepwise linear regression). The user controls the regression fitting by specification of the polynomial degree and statistical significance levels (alpha) parameters and can remove or reinstate tie point pairs from the fitting process.
The degree of the polynomial is restricted to a maximum of 4. The actual degree may be less than that specified, depending on the number and distribution of the tie point pairs. (If a set of tie points describes a lower order transformation than the degree entered, the stepwise regression should eliminate polynomial terms which are not significant.) The following table gives the minimum number of tie point pairs required for each degree:
Number of
Degree Points
1 3
2 6
3 10
4 15
The stepwise linear regression procedure described in the following paragraphs is used by polyfit to fit tie point pairs, by the digitizer interface to compute the digitizer to map fit, and by tiepts to compute rough transformations and tie point residuals.
Stepwise Linear Regression
Stepwise linear regression examines variables incorporated in the model at every stage of the regression. A variable which may have been the best choice to enter the model at an early stage may later be nonsignificant because of the relationships between it and other variables now in the regression. Once a variable is proven to be nonsignificant, it is removed from the model. This process continues until no more variables can be accepted and no more can be rejected.
The process of determining whether or not a variable is significant is based on the F-statistic and a user-entered statistical significance level for variables entering the model and for those exiting the model (commonly referred to as alpha). For example, if the user enters an alpha of (0.05,0.05), a point with a statistical significance in the upper 95 percent of the F-distribution is entered into the model; points in the model may stay in the model if their statistical significance level is in the upper 95 percent of the F-distribution.
Alpha values near 1.0 allow variables to enter and remain in the model that do not significantly help define the relations between the tie point pairs. Alpha values near 0.0 may prevent any variables from entering the model and/or will cause variables to be quickly removed after other variables are entered.
Alpha values of (0.05,0.05) are recommended. Alpha values of (0.999,0.999) closely approximate a common least squares fit.
This procedure is modeled after Draper & Smith's Applied Regression Analysis, Section 6.8 entitled "Computational Method for Stepwise Regression." This stepwise procedure was originated by Efroymson.
Residual Errors
A residual error is the difference between the actual value and the value calculated by the polynomial resulting from the modeling process. The magnitude of the residual is the square root of the sum of the square of the error in the X direction and the square of the error in the Y direction.