There is a class of nonlinear regression problems that can be best
expressed by omitting the dependent variable (i.e., the variable
on the left of the equal sign). To understand what this means,
first consider the normal regression case with a dependent
variable. For each observation, the function is evaluated, and the
computed function value is subtracted from the corresponding value of the
dependent variable for that observation. This residual value is
then squared and added to the other squared residual values. The
goal is to minimize the total sum of squared residuals.
In the
case where the dependent variable is omitted, the function is
computed for each observation and the value of the function is
squared (i.e., it is treated as the residual) and added to the
other squared values. The goal is to minimize the sum of the
squared values of the function. Thus, for a perfect fit the
computed value of the function for every observation would be
zero.
To perform this type of analysis omit the dependent variable and
equal sign from the left side of the function specification in
your NLREG program.
As an example of this type of analysis, consider the problem of
fitting a circle to a set of points that form a roughly circular
pattern (i.e., a "circular regression"). Our goal is to determine
the center point of the circle (Xc,Yc) and the radius (R) which
will make the circle best fit the points so that the sum of the
squared distances between the points and the perimeter of the
circle is minimized (the points are as close to the perimeter of
the circle as possible).
For this problem, we have three parameters whose values are to be
determined: Xc, Yc, and R. There will be one data observation for
each point to which the circle is being fitted. For each point
there are two variables, Xp and Yp, the X and Y coordinates of the
point's position.
Since our goal is to minimize the sum of the squared distances
from the points to the perimeter of the circle, we need a function
that will compute this distance for each point. If the center of
the circle is at (Xc,Yc) and the position of a point is (Xp,Yp)
then, from the theorem of Pythagoras, we know the distance from
the center to the point is
sqrt((Xp-Xc)^2 + (Yp-Yc)^2)
But we are interested in the distance from the perimeter to the
point. Since the radius of the circle is R, the distance from the
perimeter to the point (along a straight line from the center to
the point) is
sqrt((Xp-Xc)^2 + (Yp-Yc)^2) - R
That is, the distance from the perimeter to the point is equal to
the distance from the center to the point less the distance from
the center to the perimeter (the radius). The distance will be
positive or negative depending on whether the point is outside or
inside the circle, but this does not matter since the value is
squared as part of the minimization process.
The NLREG statements for this analysis are as follows:
Title "Fit circle to group of points";
Variable Xp; // X coordinate of point
Variable Yp; // Y coordinate of point
Parameter Xc; // X position of circle center
Parameter Yc; // Y position of circle center
Parameter R; // Radius of circle
Function sqrt((Xp-Xc)^2 + (Yp-Yc)^2) - R;
Data;
[ data goes here ]
Note that there is no dependent variable or equal sign to the left
of the function. NLREG will determine the values of the
parameters Yp, Yc, and R such that the sum of the squared values
of the function (i.e., the sum of the squared distances) is
minimized.
