Plotting and checking the bivariate distributions of multiple Gaussian data

Abstract

The geostatistical modeling of continuous variables relies heavily on the multivariate Gaussian distribution. It is remarkably tractable. The multivariate Gaussian distribution is adopted for K multiple variables (often K is between 2 and 10) and for N multiple locations (often N is in the tens of millions). Our focus is on the relationship between the K variables. Each variable is transformed to be univariate Gaussian, but the multivariate nature of the data is not necessarily Gaussian after univariate transformation. If multiple data variables are deemed non-Gaussian, then additional steps need to be taken such as linearization by alternating conditional expectation (ACE) or multivariate transformation by the stepwise conditional transformation (SCT). Although all L-variate distributions (1< L≤ K) should be checked, the bivariate distributions are practically important; there are relatively few data in practice to investigate higher order distributions. A quantitative measure of departure from the bivariate Gaussian distribution is established based on quadrants and the distribution of differences from the theoretically expected distribution. Although approximate, the measure of departure is useful for comparing different distributions and guiding the geostatistician to look closer at some data variables. A scatnscores program is shown that will plot all K( K–1)/2 bivariate cross plots associated with K variables. The correlation coefficients, number of data, degree of departure from the bivariate Gaussian distribution, and bivariate Gaussian probability contours associated with specified cumulative probabilities are shown. The data ID numbers can also be shown to help identify outlier or problematic data.