## Statistical Models

Statistical models are those that do not require detailed information about the plant involved but rely mainly on statistical techniques, such as correlation or regression, relating to the appropriate plant and environmental variables (Norman, 1979). Most statistical models are crop-yield weather models, which are applied to estimate yield over large areas with variable success. The regression coefficients themselves are not necessarily related to the important processes and therefore are highly variable with crop type, region, etc. Many studies are required to produce the regression equations necessary for the widespread application of this kind of model. A great advantage of these simple crop-weather models is that they use readily available weather data. Although results are not very accurate, the statistical model is able to recognize the years that bumper crops and crop failures can be expected, several weeks prior to harvest.

Regression models are attractive because of their simple and straightforward relationship between yield and one or more environmental factors, but these are not accurate enough to be used for other areas and other crops. Despite this limitation, they are used extensively for the prediction of yield of a single crop over a large region, with a variety of soils, agronomic practices, and insect-disease problems. A combination of such factors is still beyond the dynamic simulation models. It is a technique well worth retaining in the arsenal of tools available to the agricultural climatologist (Whisler et al., 1986).

### Dynamic Simulation Models

A dynamic model is the one whose output varies with time and in which processes are characterized. To characterize processes, the state variables must be known. State variables are those necessary to define the state of the system at a point in time. Dynamic simulation crop models predict changes in crop status with time as a function of biogenetic parameters (Hume and Callander, 1990).

Simulation means that the model acts like a real crop, gradually germinating and growing leaves, stems, and roots during the season. In other words, simulation is the process of using a model dynamically by following a system over a time period.

Dynamic models can be classified as preliminary models, comprehensive models, and summary models (Penning de Vries et al., 1989). Preliminary models have structure and data that reflect current scientific knowledge. These are simple because insight is at the exploratory level. A comprehensive model is a model of a system in which essential elements are thoroughly understood and much of this knowledge is incorporated. Summary models are abstracts of comprehensive models and are found at production levels. Comprehensive dynamic models predict yield much closer to reality than do the regression models. However, the more accurate the dynamic model is, the more information is required for initialization and about driving variables. In many cases these may not be available, hence the regression models may still be our best option.

Developing a comprehensive, dynamic crop simulation model requires a multidisciplinary team. Plant physiologists, agronomists, and soil scientists are needed to help define both the overall framework of the problem and the specificities of the environment and plant growth relationship. Entomologists and plant pathologists are required to define the insect and pathogen subsystems that are important parts in the crop ecosystems. An agrometeo-rologist selects and contributes data about weather and microclimate fluxes in and around the plant canopies. A computer programmer selects the computer language and develops the overall framework of the model (Ritche et al., 1986).

After designing the first version of the model and analyzing the results of its output, faults are invariably found, which require changes in the structure of the model. These changes require additional study and verification, and the loop continues. Furthermore, in the process of development, the initial representation of the modeling process is improved upon and the range of experimental data used is widened. In the final verification process, it is important to consider an independent data set that was not used in the development of the model (Guardrian, 1977).

### Verification and Validation

Verification and validation are two commonly used terms in modeling. Verification is the test of truthfulness or correctness. By comparing historical data recorded for real world systems to what the computer gives as the output of the model, verification certifies that the functional relationships modeled are correct. If a model does not behave according to expectations, then some correction of the functional relationship may be necessary or coefficients may need to be adjusted. The latter is called calibration.

Validation is concerned with the comparison of model predictions with results from independent experiments. Models can be considered valid and useful even when there are some differences between experimental data and simulation output. A model is considered valid if the simulated values lie within the projected confidence band.

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