GEOG 4/517:  Geographic Data Analysis
Nonparametric regression

Scatter-diagram smoothing involves drawing a smooth curve on a scatter diagram to summarize a relationship, in a fashion that makes few assumptions initially about the form or strength of the relationship.  It is related to (and is a special case of) nonparametric regression, in which the objective  is to represent the relationship between a response variable and one or more predictor variables, again in way that makes few assumptions about the form of the relationship.  In other words, in contrast to "standard" linear regression analysis, no assumption is made that the relationship is represented by a straight line (although one could certainly think of a straight line as a special case of nonparametric regression). 

Another way of looking at scatter diagram smoothing is as a way of depicting the "local" relationship between a response variable and a predictor variable over parts of their ranges, which may differ from a "global" relationship determined using the whole data set.  (And again, the idea of "local" as opposed to "global" relationships has an obvious geographical analogy.)

Specific and general cases of smoothing and nonparametric regression

• global and local regression comparison

Examples of non-parametric regression

• lowess and loess
• loess surfaces
• other bivariate smoothers

The various smoothers can be summarized as follows:

	Smoother	Form	Influence of individual points
fewest assumptions	loess	no assumptions	unusual points discounted
	smoothing spline	smooth curve	some discounting of unusual points
	robust, robust MM	straight line	unusual points discounted
	least squares (curvilinear)	curve	all points influential
most assumptions	least squares (linear)	straight line	all points influential

Readings:

Kuhnert & Venebles (An Introduction...):  p. 120-128; Cleveland (Visualizing Data.) Ch. 3