Multidimensional deconvolution examples

 

Deconvolution is a well-known process with deep and complex theoretical underpinnings. Here we'll begin with a superficial view of the theory and then jump to some examples in two-dimensional space.

One definition of science that distinguishes it from other views of the human condition is that science is those studies that offer demonstrable predictive power. That direction brings us too far afield. Never-the-less, science can be formalized as the construction of predicted data that can be compared with observed data. This requires a theory. Oft times, we really don't have a theory that we have much confidence in. Fortunately, there is an ``all-purpose'' theory of statistical signal processing that often allows us some predictability. Basically, it goes like this.

We have some data. From this data we can estimate a spectrum. By assuming the spectrum remains constant in time or space (stationary assumption), we have a certain predictive power. How much power this amounts to in practice depends on lots of things we won't examine in this brief introduction beyond saying that it depends a lot on both the observed data and the validity of the stationary assumption.

It often happens that the item of interest is not the prediction, but the error in the prediction. For example, we might be given the history of the stock market upon which to estimate the spectrum. Then we might be asked to predict (for each moment in time), the market price tomorrow based on today's price and all previous prices. The display that is often of interest is the error in prediction, did we overshoot or undershoot? Theoretically, it turns out that the prediction error is spectally white. If it had any other spectrum, that would imply suboptimum prediction. This leads to the simplest definition of prediction error, the definition we will adopt here. (The examples here come from one of my other books (GEE). These examples are based not on spectra, but on actual prediction error, but we will not examine the distinction, a distinction that is small compared to the essential ideas discussed here.)