Using Multiple Independent Variables
A fit can be computed for data having between 1 and
10 independent variables, or basis dimensions. For example, fitting
a curve to (x,y) pairs estimates the function y(x) and involves 1
independent variable, x, and an observation, y, which is the
observed value at coordinate x. Fitting a surface z(x,y) involves 2
independent variables, x and y. The independent variables are
specified as basis dimension 1 for y(x) and as basis dimensions 1
and 2 for z(x,y). The CLsqFit class accommodates up to 10
basis dimensions, or independent variables. It also fits up to 100
coefficients which may be mixed among the variables. The
interaction between coefficients and basis dimensions is controlled
by the basis
function. Some examples will illustrate this concept.
Coefficients are represented by a[n]
where n is in the range 1 through
100.
- A line is fit by the equation y = a[1] +
a[2] x. This has 1 independent variable, x, and 2
coefficients, a[1] and a[2]. You could force the slope a[2] to,
say, 1.0 and fit only the intercept a[1] using the ForceCoef method. This
still involves setting 2 coefficients but one of them is
subsequently forced.
|
- Fitting a general polynomial curve y = f(x) involves 1 basis
dimension for x but n coefficients in a polynomial of degree n. The
equation being fit is y = a[1] + a[2] x + ... +
a[n] x^(n-1).
|
- Fitting a plane z = a[1] + a[2] x + a[3]
y involves 2 dimensions, x and y, but 3 coefficients, a[1]
through a[3].
|
- A warped plane involves a cross term that couples x and y
through the equation z = a[1] + a[2] x + a[3] y
+ a[4] xy. This involves 2 dimensions and 4
coefficients.
|
- A general 2-dimensional polynomial involving all cross terms
has m x n coefficients in 2 dimensions. An example is illustrated
by the warped plane which has m=2 and n=2, making 4 coefficients in
total. For m=3, n=2, the equation is z = a[1] +
a[2] x + a[3] x^2 + a[4] y + a[5] xy + a[6] x^2y. This
latter equation has dimension 2 and 6 coefficients. A 5x7
polynomial of 2 dimensions would have 35 coefficients. You can
remove coefficients from the fit by forcing them to specific values
(see Forcing Coefficient Values). For example, to remove
all cross terms from this fit, specify a 3x2 (6 term) fit and then
use use ForceCoef( 5, 0 ) and ForceCoef( 6, 0 ) to force a[5]=0 and a[6]=0.
Multiple coefficients are specified for the n-dimensional
polynomial with n > 1 are specified using a table, like {3,2} in
the present example.
|
As can be seen from these examples, higher order
multi-dimensional fits may involve complicated equations. Thus the
CLsqFit class includes 2 useful built-in fitting functions:
an n--dimensional polynomial fit of up to 100 terms including all
cross terms, and a hyperplane function for fitting up to 10
independent variables of first power only. Other basis functions
can be specified by writing them as a script function; see Basis Functions
and the SetBasisFunc method.
Related Topics
CLsqFit class, Basis Functions, Forcing
Coefficient Values
