Model fitting by least squares

Smoothing the denominator spectrum

Equation (2.1) gives us one way to divide by zero. Another way is stated by the equation

$$X(\omega) \eq { \overline{F(\omega)} Y(\omega) \over \langle \overline{F(\omega)} F(\omega) \rangle }$$ (6)

where the strange notation in the denominator means that the spectrum there should be smoothed a little. Such smoothing fills in the holes in the spectrum where zero-division is a danger, filling not with an arbitrary numerical value $\epsilon$ but with an average of nearby spectral values. Additionally, if the denominator spectrum $\overline{F(\omega)} F(\omega)$ is rough, the smoothing creates a shorter autocorrelation function.

Both divisions, equation (2.1) and equation (2.6), irritate us by requiring us to specify a parameter, but for the latter, the parameter has a clear meaning. In the latter case we smooth a spectrum with a smoothing window of width, say $\Delta\omega$ which this corresponds inversely to a time interval over which we smooth. Choosing a numerical value for $\epsilon$ has not such a simple interpretation.

We jump from simple mathematical theorizing towards a genuine practical application when I grab some real data, a function of time and space from another textbook. Let us call this data $f(t,x)$ and its 2-D Fourier transform $F(\omega, k_x)$. The data and its autocorrelation are in Figure 2.1.

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Figure 1. 2-D data (right) and a quadrant of its autocorrelation (left). Notice the longest nonzero time lag on the data is about 5.5 sec which is the latest nonzero signal on the autocorrelation.

The autocorrelation $a(t,x)$ of $f(t,x)$ is the inverse 2-D Fourier Transform of $\overline{F(\omega, k_x)} F(\omega, k_x)$. Autocorrelations $a(x,y)$ satisfy the symmetry relation $a(x,y)=a(-x,-y)$. Figure 2.2 shows only the interesting quadrant of the two independent quadrants. We see the autocorrelation of a 2-D function has some resemblance to the function itself but differs in important ways.

Instead of messing with two different functions $X$ and $Y$ to divide, let us divide $F$ by itself. This sounds like $1=F/F$ but we will watch what happens when we do the division carefully avoiding zero division in the ways we usually do.

Figure 2.2 shows what happens with

$$1 \eq F/F \quad\approx\quad { { \overline{F} F }\over{ \overline{F} F + \epsilon^2 } }$$ (7)

and with

$$1 \eq F/F \quad\approx\quad { { \overline{F} F }\over{ \langle \overline{F} F \rangle } }$$ (8)

From Figure 2.2 we notice that both methods of avoiding zero division give similar results. By playing with the $\epsilon$ and the smoothing width the pictures could be made even more similar. My preference, however, is the smoothing. It is difficult to make physical sense of choosing a numerical value for $\epsilon$. It is much easier to make physical sense of choosing a smoothing window. The smoothing window is in $(\omega,k_x)$ space, but Fourier transformation tells us its effect in $(t,x)$ space.

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Figure 2. Equation (2.7) (left) and equation (2.8) (right). Both ways of dividing by zero give similar results.

Model fitting by least squares