Two fundamental characteristics of forecast verification problems—complexity and dimensionality—are described. To develop quantitative definitions of these characteristics, a general framework for the problem of absolute verification (AV) is extended to the problem of comparative verification (CV). Absolute verification focuses on the performance of individual forecasting systems (or forecasters), and it is based on the bivariate distribution of forecasts and observations and its two possible factorizations into conditional and marginal distributions.
Comparative verification compares the performance of two or more forecasting systems, which may produce forecasts under 1) identical conditions or 2) different conditions. The first type of CV is matched comparative verification, and it is based on a 3-yariable distribution with possible factorizations. The second and more complicated type of CV is unmatched comparative verification, and it is based on a 4-variable distribution with 24 possible factorizations.
Complexity can be defined in terms of the number of factorizations, the number of basic factors (conditional and marginal distributions) in each factorization, or the total number of basic factors associated with the respective frameworks. These definitions provide quantitative insight into basic differences in complexity among AV and CV problems. Verification problems involving probabilistic and nonprobabilistic forecasts are of equal complexity.
Dimensionality is defined as the number of probabilities that must be specified to reconstruct the basic distribution of forecasts and observations. It is one less than the total number of distinct combinations of forecasts and observations. Thus, CV problems are of higher dimensionality than AV problems, and problems involving probabilistic forecasts or multivalued nonprobabilistic forecasts exhibit particularly high dimensionality.
Issues related to the implications of these concepts for verification procedures and practices are discussed, including the reduction of complexity and/or dimensionality. Comparative verification problems can be reduced in complexity by making forecasts under identical conditions or by assuming conditional or unconditional independence when warranted. Dimensionality can be reduced by parametric statistical modeling of the distributions of forecasts and/or observations.
Failure to take account of the complexity and dimensionality of verification problems may lead to an incomplete and inefficient body of verification methodology and, thereby, to erroneous conclusions regarding the absolute and relative quality and/or value of forecasting systems.