Model fitting by least squares

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## Damped solution

Another way to say is to say is small, or is small. This does not solve the problem of going to zero, so we need the idea that does not get too big. To find , we minimize the quadratic function in .
 (2)

The second term is called a damping factor,'' because it prevents from going to when . Set , which gives:
 (3)

Equation (3) yields our earlier common-sense guess . It also leads us to wider areas of application in which the elements are complex vectors and matrices.

With Fourier transforms, the signal is a complex number at each frequency . Therefore we generalize equation (2) to:

 (4)

To minimize , we could use a real-values approach, where we express in terms of two real values and , and then set and . The approach we take, however, is to use complex values, where we set and . Let us examine :
 (5)

The derivative is the complex conjugate of . Therefore, if either is zero, the other is also zero. Thus, we do not need to specify both and . I usually set equal to zero. Solving equation (5) for gives equation (1).

Equation (1) solves for , giving the solution for what is called the deconvolution problem with a known wavelet .'' Analogously, we can use when the filter is unknown, but the input and output are given. Simply interchange and in the derivation and result.

 Model fitting by least squares

Next: Smoothing the denominator spectrum Up: UNIVARIATE LEAST SQUARES Previous: Dividing by zero smoothly

2014-12-01