As I observed a couple of posts ago, the Stahle SWM network can be arranged so that its correlation matrix is closely approximated by a Toeplitz matrix i.e. there is a “natural” linear order. I also noted that results from matrix algebra proving that, under relevant conditions, all the coefficients in the PC1 were of the same sign; there was one sign change for the coefficients in the PC2, two in the PC3 and so on. These resulted in contour maps for the eigenvector coefficients with very distinct geometries – here’s an example of the PC2 in the SWM network showing the strong N-S gradient in this PC.
As soon as one plots the site locations and contours the eigenvector coefficients, the raggedness of the original site geometry is demonstrated. But how much does this raggedness matter? This leads us away from matrix algebra into more complicated mathematics, as the eigenvectors which contour out to the pretty gradients in the earlier diagrams now become eigenfunctions with an integral operator replacing the simpler matrix multiplication.
As an exercise, I thought that it would be interesting to see what happened in an idealized situation in which the network was a line segment with spatial autocorrelation as a negative exponential function of distance between sites.
Because all the circumstances are pretty simple, one presumes that the resulting functions are simple and I presume that there’s a simple derivation somewhere for eigenfunctions in these simple cases.
Since I don’t know these derivations, I went back to more or less brute force methods and did some calculations using N=51; N=101,… , assuming that the sites were uniformly spaced along the line segment, creating a correlation matrix for rho=0.8, carried out SVD on the constructed correlation matrix, plotted the eigenvectors and experimented with fitting elementary functions to the resulting eigenvectors. As shown below, I got excellent fits (with some edge effect) for the following eigenfunctions:
where k = 1,.. and t is over [0,1].
So it looks like the eigenfunctions are pretty simple. One can also see how the number of sign changes increases by 1 as k increases by 1.
Even the plots in the network illustrated yesterday show elements of these idealizations. For example, here’s the PC1 from the combined network. I think that I can persuade myself that there are elements of the sin (kt) k=1 shape from the idealized PC1 coefficient curve illustrated in the top right panel.
Next here’s the PC2 from the combined network. Again I think that I can persuade myself that there are elements of the sin (kt) k=2 shape in this example.
Obviously the functions are elementary and I’m sure that there are any number of elegant derivations of the formulas. But I think that the results are at least a little bit pretty in the context of something as humdrum as the Stahle SWM network. As one goes from a 1-D to a 2-D situation, the geometry is somewhat more complicated, but we’re still going to see well-constrained relatively elementary functions making up the eigenfunctions for squarish and rectangular shaped regions.
Also, here’s a plot of the normalized eigenvalues for N=101. Again, I’m sure that there’s a known distribution for these eigenvalues somewhere in the mathematical literature and would welcome any references.