Determining Functional Relations in Multivariate Oceanographic Systems: Model II Multiple Linear Regression

A method for estimating multivariate functional relationships between sets of measured oceanographic, meteorological, and other field data is presented. Model II regression is well known for describing functional relationships between two variables. However, there is little accessible guidance for the researcher wishing to apply model II methods to a multivariate system consisting of three or more variables. This paper describes a straightforward method to extend model II regression to the case of three or more variables. The multiple model II procedure is applied to an analysis of the optical spectral scattering coefficient measured in the coastal ocean. The spectral scattering coefficient is regressed against both suspended mineral particle concentration and suspended organic particle concentration. The regression coefficients from this analysis provide adjusted estimates of the mineral particle scattering cross section and the organic particle scattering cross section. Greater accuracy and efficiency of the coefficients from this analysis, compared to semiempirical coefficients, is demonstrated. Examples of multivariate data are presented that have been analyzed by partitioning the variables into arbitrary bivariate models. However, in a true multivariate system with correlated predictors, such as a coupled biogeochemical cycle, these bivariate analyses yield incorrect coefficient estimates and may result in large unexplained variance. Employing instead a multivariate model II analysis can alleviate these problems and may be a better choice in these situations.