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STATISTICS

New Indices to Quantify Patterns of Residuals Produced by Model Estimates

^a Research Institute for Industrial Crops, Via di Corticella, 133, 40128 Bologna, Italy
^b DiProVe, Università di Milano, Via Celoria 2, 20133 Milano, Italy

^* Corresponding author (m.donatelli{at}isci.it).

Received for publication June 18, 2002. The evaluation of patterns in the residuals of model estimates vs. other variables can be useful in both model evaluation and parameter calibration. New indices that allow quantifying such patterns (pattern indices) are presented. Groups of residuals are created by dividing the range of the variable under evaluation into two, three, four, or five subranges. Two types of indices are proposed. The first type (PI-type) is based on the absolute value of the maximum difference between pairwise comparisons among average residuals of each group of residuals. A variant of this index is computed by using variance ratios (PI-F type). The subranges of the variable that determines the grouping of residuals may be of equal length (PI) or variable length (PI). In the second case, they are generated by an algorithm that optimizes subranges to maximize patterns. The power of the diverse pattern indices at identifying patterns was investigated, and their effectiveness was compared against the runs test. Critical values for pattern indices were generated by Monte Carlo simulations. Monte Carlo probability tables, the results of power analysis, and the results of using pattern indices at two case studies (i.e., daily radiation and soil water content estimates) were presented. The analysis based on pattern indices provided insight in model structure and parameter calibration. Pattern indices also allowed evaluating model performance and discriminating among alternative models. Higher power in identifying patterns was given by range-based pattern indices than by those based on variance ratios.

Abbreviations: BC, Bristow–Campbell (model) • DB, Donatelli–Bellocchi (model) • doy, day of year • MSE, mean of square error • PI, range-based fixed pattern index • PI-F, F-based fixed pattern index • PI, range-based variable pattern index • PI–F, F-based variable pattern index • RMSE, root mean square error • Tmin, minimum air temperature