# How to visually show the convergence of a process?

I have some set of measurements that I have represented as vectors $x^t$ for $t \in \{ 1, 2, ...\}$.

I want to test "convergence" of the process (visually) in some sense.

I thought maybe I could run PCA on $x^t$ and then plot the projections to the first principal component, 2nd, etc... and see whether these plots converge to something.

What is the right way to do it? There is variance in between samples, of course, so just plotting the first PCA component will not necessarily converge to something. But what should I look for, and how can I make it visually compelling? (Or maybe there is another, better way of testing the "convergence" of these $x^t$?)

(Please note: the $x^t$ are not coming from an MCMC process, but maybe tools from there could be used?)

• I suggest here to graph the coefficient of variation for the separate series. – Andy W Jun 30 '13 at 16:08
• Can you clarify what you mean by $x^t$? Are you raising some variable to the power of the trial number? Or are you indexing some variable by trial number (i.e., $x_t$)? – gung Jun 30 '13 at 16:12
• sorry, $x^t \in \mathbb{R}^d$ where $t$ is a time index. – process Jun 30 '13 at 16:18
• @AndyW thanks. I looked up the CoV and your answer -- but how do you plot it for a time series? it seems like it can be calculated for the whole sample. or do you mean plotting it after binning the data into X samples per bin? What would be X? Would the bins be a moving window, or disjoint bins? – process Jun 30 '13 at 16:22
• Just to be clear: by "convergence" do you mean that there appears to exist some vector $x_0$ such that some measure of distance between $x^t$ and $x_0$ tends to get smaller as $t$ grows larger? If that is the case, the very definition suggests estimating $x_0$ and plotting this distance against $t$; if it is not the case, then please provide a definition of "convergence." – whuber Jun 30 '13 at 17:22