User`s guide

Parametric Fitting

3-9

where X

is the transpose of the design matrix X. Solving for b,

In MATLAB, you can use the backslash operator to solve a system of

simultaneous linear equations for unknown coefficients. Because inverting

X can lead to unacceptable rounding errors, MATLAB uses QR

decomposition with pivoting, which is a very stable algorithm numerically.

Refer to “Arithmetic Operators” in the MATLAB documentation for more

information about the backslash operator and QR decomposition.

You can plug b back into the model formula to get the predicted response

values, .

A hat (circumflex) over a letter denotes an estimate of a parameter or a

prediction from a model. The projection matrix H is called the hat matrix,

because it puts the hat on y.

The residuals are given by

Refer to [1] or [2] for a complete description of the matrix representation of

least squares regression.

Weighted Linear Least Squares

As described in “Basic Assumptions About the Error” on page 3-5, it is usually

assumed that the response data is of equal quality and, therefore, has constant

variance. If this assumption is violated, your fit might be unduly influenced by

data of poor quality. To improve the fit, you can use weighted least squares

regression where an additional scale factor (the weight) is included in the

fitting process. Weighted least squares regression minimizes the error

estimate

X()

1–

Xb Hy==

HXX

X()

1–

ryy

– 1 H–()y==

–()

i 1=

∑