Multiple linear regression may be used to describe the relation of one geologic variable to several other (independent) variables, and may also be used to fit a trend surface to geographically distributed variables. The least-squares estimates of the regression coefficients commonly differ from the true coefficients if the independent variables are correlated. The estimates can be too large in absolute value, and may even have the wrong sign. Also, the least-squares solution may be unstable; replicate samples are likely to give widely differing values of the regression coefficients.

Ridge regression analysis, described by Hoerl and Kennard, is a technique for removing the effect of correlations from the regression analysis. The regression coefficients obtained are biased but have smaller sums of squared deviations between the coefficients and their estimates.

Correlations between geologic variables are common, and multiple regression coefficients based on these data may be suspect. For example, in trend surface analysis correlations between the geographic coordinates may range from zero (gridded data) to different from zero (clustered data) in a linear trend. In addition, when higher order terms are used in the trend the various powers of the geographic coordinates are highly correlated.

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