Mathematical model

An abstract set of numerical data, with or without equations, which defines connections between data sub-sets and the behavior of a system. The model can even be just a set of equations. The data can be provided as a matrix, with equations defining the relationship between matrix segments (a spreadsheet or even a graph) a spreadsheet or even a graph. Some mathematical models are very complex, especially system dynamics models, which depict the dynamic relationships between variables in a multidimensional model, such as one that relates demographic characteristics with economic and ecological resources. Mathematical models are used particularly in the natural sciences and engineering disciplines but also in the social sciences (such as economics, sociology and political science). When engineers analyze a system to be controlled or optimized, they build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships among the variables. The values of the variables can be practically anything, real or integer numbers, for example. The variables represent some properties of the system. Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information is available about the system. A black-box model is a system for which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models. Usually it is preferable to use as much a priori information as possible to make the model more accurate. Often some parameters have to be estimated before one can use the model. In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. The problem with using a large set of functions to describe a system is that estimating the parameters becomes increasingly difficult when the number of parameters (and different types of functions) increases. Complexity. Adding elements to a model in order to obtain a white box might result in a huge amount of detail inhibiting the usage of that model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. This allows a more robust and simple model. Training. Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it describes. If the modeling is done by a neural network, the optimization of parameters is called training. In more conventional modeling through explicitly given mathematical functions, parameters are determined by curve fitting. Model evaluation. If the model was built well, it will adequately show the relations between system variables for the measurements at hand. In order to know if the model describes the properties of the system well, we split the measured data into two parts: training data and verification data. The training data are used to estimate the model parameters. The verification data are used to evaluate model performance. A model does not need to accurately describe events outside the measured data of the system in order to be quite sufficient.