Consider a simple time series:

`> tp <- seq_len(10) > tp [1] 1 2 3 4 5 6 7 8 9 10`

we can compute an adjacency matrix for this time series representing the temporal links between samples. In computing this matrix we add an imaginary site at time 0 and the link between this observation and the first actual observation at time 1 is known as link 0. Between time 1 and time 2, the link is link 1 and so on. Because time is a directional process, sites are connected to (affected by) links that are “upstream” of the site. Hence every site is connected to link 0, but link 9 is only connected to site 10; it occurs temporally after each site except site 10. The adjacency matrix thus defined is created as follows:

`> adjmat <- matrix(0, ncol = length(tp), nrow = length(tp)) > adjmat[lower.tri(adjmat, diag = TRUE)] <- 1 > rownames(adjmat) <- paste("Site", seq_along(tp)) > colnames(adjmat) <- paste("Link", seq_along(tp)-1) > adjmat Link 0 Link 1 Link 2 Link 3 Link 4 Link 5 Link 6 Link 7 Site 1 1 0 0 0 0 0 0 0 Site 2 1 1 0 0 0 0 0 0 Site 3 1 1 1 0 0 0 0 0 Site 4 1 1 1 1 0 0 0 0 Site 5 1 1 1 1 1 0 0 0 Site 6 1 1 1 1 1 1 0 0 Site 7 1 1 1 1 1 1 1 0 Site 8 1 1 1 1 1 1 1 1 Site 9 1 1 1 1 1 1 1 1 Site 10 1 1 1 1 1 1 1 1 Link 8 Link 9 Site 1 0 0 Site 2 0 0 Site 3 0 0 Site 4 0 0 Site 5 0 0 Site 6 0 0 Site 7 0 0 Site 8 0 0 Site 9 1 0 Site 10 1 1`

The SVD provides a decomposition of this matrix into Eigenfunctions of variation as different temporal scales. The figure below shows the extracted functions (from

`SVD$u`

)`> SVD <- svd(adjmat, nu = length(tp), nv = 0)`

The eigenfunctions are periodic components at various temporal scales. Trying

`tp <- seq_len(25)`

(or longer) shows this better than the shorter example I showed above.

Does this sort of analysis have a proper name in statistics?It sounds similar to Singular Spectrum Analysis but that is a decomposition of an embedded time series (a matrix whose columns are lagged versions of the time series).

Background:

I came up with this idea by modifying an idea from spatial ecology called Asymmetric Eigenvector Maps (AEM) which considers a spatial process with known direction and forming an adjacency matrix between a spatial array of samples that contains 1s where a sample can be connected to a link and a 0 where it can’t, under the constraint that links can only be connected “downstream” – hence the asymmetric nature of the analysis. What I described above is a one-dimensional version of the AEM method. A reprint of the AEM method can be found here if you are interested.The figure was produced with:

`layout(matrix(1:12, ncol = 3, nrow = 4)) op <- par(mar = c(3,4,1,1)) apply(SVD$u, 2, function(x, t) plot(t, x, type = "l", xlab = "", ylab = ""), t = tp) par(op) layout(1)`

**Answer**

This looks like a variation on “Principal Component Analysis”.

http://mathworld.wolfram.com/PrincipalComponentAnalysis.html

In structural analysis the eigenvalues of a system are used to look at linear deformations, places where superposition is still valid. The method is called “Modal Analysis”.

http://macl.caeds.eng.uml.edu/macl-pa/modes/modal2.html

**Attribution***Source : Link , Question Author : Gavin Simpson , Answer Author : EngrStudent*