Abstract
Identifying patterns in bivariate data on a scatterplot remains a basic statistical problem, with special flavor when both variables are on the same footing. Ideas of double, diagonal, and polar smoothing inspired by Cleveland and McGill's 1984 paper in the Journal of the American Statistical Association are revisited with various examples from environmental datasets. Double smoothing means smoothing both y given x and x given y. Diagonal smoothing means smoothing based on the sum and difference of y and x that treats the two variables symmetrically, possibly under standardization. Polar smoothing is based on the transformation from Cartesian to polar coordinates followed by smoothing and then reverse transformation; here the smoothing is implemented by regression on a series of sine and cosine terms. These methods thus offer exploratory tools for determining the broad structure of bivariate data.