Chapter 11

Bivariate Smoothing

Consider the bivariate data shown in Figure 11.1. If asked, virtually anyone could draw a smooth curve that fits the data well, yet most would find it surprisingly difficult to describe precisely how they had done it. We focus here on a variety of methods for this task, called scatterplot smoothing.

Effective smoothing methods for bivariate data are usually much simpler than for higher-dimensional problems; therefore we initially limit consideration to the case of n bivariate data points (x_{i}, y_{i}), i = 1, . . ., n. Chapter 12 covers smoothing multivariate data.

The goal of smoothing is different for predictor–response data than for general bivariate data. With predictor–response data, the random response variable Y is assumed to be a function (probably stochastic) of the value of a predictor variable X. For example, a model commonly assumed for predictor–response data is Y_{i} = s (x_{i}) + _{i}, where the _{i} are zero-mean stochastic noise and s is a smooth function. In this case, the conditional distribution of Y|x describes how Y depends ...