Functional Data Analysis

An Overview of Functional Data Analysis

Functional data analysis explores samples of data where each observation arises from a curve or function. We require the data to provide enough information to estimate the curve and its properties that we wish to use. For example, Figure 30.1 displays one of 50 replications of the writing of ‘statistical science’ in Chinese described in Ramsay (2000). The curve is multivariate, being a trajectory in three-dimensional space: two coordinates X and Y on the writing surface plus the vertical coordinate Z. The location of the tip of the pen was recorded 400 times per second over the roughly six seconds required to write the script, with an error level of about 0.5 millimeters. This may sound like a lot of data, but there are 50 pen strokes, some of which have consistently sharp structures, and we have even more highly localized events, such as effects of friction as the pen makes or loses contact with the writing surface. We will find the structure in pen acceleration especially revealing, and estimating second and even third derivatives will be a challenge.

Functional data analysis assumes that the curve being estimated is smooth. Smoothness is closely connected with the concept of energy, and in fact most real world systems that we study in the social and life sciences as well as in chemistry and physics have a limited energy budget for producing change. Later in the chapter, we will see that an astonishing amount of energy is reflected in the second derivatives of the script coordinates, and that this energy is dispatched by the writer's motor cortex with a remarkable level of precision and stability. Even stock market prices, often considered to be intrinsically nonsmooth, reflect the supply of money available for securities transactions, and on a sufficiently fine time scale have limited capacities for change. In fact, without smoothness we would be lost, since a function is a potentially infinite dimensional object, being defined by its value x(t) at each value t on a continuum. Nothing infinite can be estimated accurately with a finite amount of data unless some principle like the conservation of energy guarantees that most of its variation will be low dimensional.

In practice, smoothness means that one or more of the curves's derivatives can be estimated, and it is the many ways in which we make use of derivative information that sets functional data analysis apart from neighboring methodologies in data analysis space. Good derivative estimates take us to [Page 717]an entirely new range of functional models, called dynamical systems, that model change directly by the use of differential equations. Although we admit that it may be some time before models like these appear in undergraduate texts in fields that do not require calculus of undergraduates, we claim that psychology and other social sciences abound in opportunities to model change.

Figure 30.1 One of 50 samples of the X and Y coordinates of pen position during the writing of ‘statistical science’ in Chinese. The script is composed of 50 strokes and takes about six seconds to produce. Gaps in the script correspond to the pen being lifted off the writing plane.

...