Chapter List

The Prediction Problem

An important problem in signal processing is the estimation of the next term in a sequence of numbers from knowledge of the previous values. This is called the prediction problem. The numbers might be the values at closing of a certain stock market index; knowing what has happened up to today, can we predict, with some accuracy, tomorrow s closing value? The numbers might describe the position in space of a missile; knowing where it has been for the past few minutes, can we predict where it will be for the next few? The numbers might be the noon-time temperature in New York City on successive days; can we predict tomorrow s temperature from our knowledge of the temperatures on previous days? It is helpful, in weather prediction and elsewhere, to use not only the previous values of the sequence of interest, but those of related sequences; the recent temperatures in Pittsburgh might be helpful in predicting tomorrow s weather in New York City. In this chapter we begin a discussion of the prediction problem.

Prediction through Interpolation

Suppose that our data are the real numbers x ₁ , , x _m, corresponding to times t = 1 , , m. Our...