There is a fairly well developed theory of trajectories
in 
, which encompasses both the case of
continuous and discrete time, and which is aimed
not at classifying the trajectories but at
predicting, interpolating and smoothing them.
This entails some kind of modelling process for
the production of the function. It is reasonable
to care about this sort of thing even if we are
interested in classifying trajectories and not
predicting, interpolating or smoothing them,
for two
excellent reasons. The first is that we may reasonably
expect to be plagued with noise in whatever
dynamic pattern we are faced with, and the theory
of filters gives some advice on what to do about
this. The second is that any recognition or classification
will surely involve us in some kind of
modelling of the process which generated the trajectories,
and so it is worth looking to see what kinds
of models other workers concerned with dynamic
patterns have thought worth investigating. We
may be able to pinch their ideas.
In the case n = 1 we are talking about predicting, interpolating or smoothing the graph of a function of a single real variable, as they used to call it (and still do in some places). The function is usually referred to as a signal or a time series, depending on whether your source of information is an engineer or a statistician. The business of finding a function which looks to be a good fit to the given graph is called filtering if you are an engineer, regression if you are a statistician, and function-fitting or function approximation if you are just one of the boys.
Conceptually, you are confronted with a sequence of dots, or possibly a wiggly line, on a piece of paper which invites you to take a thick, black brush and to draw a smooth curve through the dots or the wiggly line, not necessarily through each dot, maybe passing in between dots or wiggles. Then you can extend your smooth curve with a flourish (extrapolation or prediction), fill in between points (interpolation) or decide you would prefer your curve to the actual data at some particular x value (smoothing). To do this by eye is to invite confrontation with anybody else who did the same thing, since the two smooth curves will surely differ, at least a bit. This leaves us with the problem of automating the process of getting the smooth curve, of getting the right smooth curve, or at least one which can be agreed upon as acceptable and also computed automatically. After chapter three, it probably won't come as much of a shock to discover that there is no one `right' solution, and doing it by hand and eye and some dash may be performed with a clear conscience. This method is no less rational and defensible than any other; the rationales for the other methods are more complicated but ultimately hinge on you buying some untestable assumption about the data. The advantage of automating the process is not that you can feel any more confidence in its rightness, but that it can be then done to a lot more data than your arm could cope with.
There are two traditions in the area, one is the statistical tradition (time series) going back to Yule and Kolmogorov which talks of AutoRegressive Moving Average (ARMA) models for time series, and the engineering tradition which talks of Finite Impulse Response and Infinite Impulse Response (FIR and IIR) filters for signals. These are the same thing. I have heard tell of a statistician who asked an engineer if he wanted ARMA modelling done in his course on stochastic time series for engineers, and was told `no, we don't need that, just see if you can fit in a bit on FIR and IIR filters, if you have time.' Not knowing what these were, the statistician simply shortened the course.
The engineering terminology confuses the objectives with the methods and details of the source of the problem, that is to say it is cluttered with massive irrelevancies and makes it hard to see the wood for the trees. The statistical terminology by contrast has been made so abstract by some (the French school) that you need a strong background in analysis before you can understand what the subject area is, by which time you may have lost interest.
The basic ideas then, have been obscured by the language in that traditional manner so widely believed to be the main contribution of Mathematics to Science and Engineering. It will sadden the reader to learn that a whole machinery of analysis of speech, Linear Predictive Coding, depends upon it. The reader is advised that if she lacks the training in either of the areas mentioned, she will find her patience sorely tried by the literature. I shall try in this section to explicate the main ideas while avoiding details. As you were warned in the introduction to this chapter, the area is a large one, and the most useful contribution I can make in this book is to sketch out the central ideas in my inimitable way, so as to allow you to decide if you want or need to go further, and to suggest where to look if you do.