6 - The shape of a random walk

Summary

The notion and quantification of shape

Shape is an intuitively accessible notion. We organize visual information in terms of shapes, and the shape of an object represents one of the first of its qualities referred to in an informal descriptive rendering of it. While our language presents us with a wide repertoire of verbal images for the approximate portrayal of the shape of a physical entity (“round,” “oblong,” “crescent,” “stellate” …) the precise characterization of a shape, in terms of a number, or set of numbers, has remained elusive. This is with good reason. It is well-known to mathematicians that the class consisting of the set of all curves is a higher order of infinity than the set of all real numbers. This means that there can be no one-to-one correspondence between curves and real numbers. As shapes, intuitively at least, bear a conceptual relationship to curves, it is plausible that the set of all shapes dwarfs in magnitude the set of real numbers, or of finite sets of real numbers.

On the other hand, if one is willing to content oneself with a general paradigm for the measurement of shape, there are ways of quantifying it in terms of numbers that have a certain descriptive and predictive utility. In fact, the numerical specification of shapes has acquired a certain urgency of late, in light of the widespread use of computer imaging and the concomitant focus on the development of codes for the creation and manipulation of pictorial quantities.

In this chapter, we will look at different ways of characterizing and measuring the shape of a random walk.