Mathematical Metamodel

A Mathematical Metamodel is a metamodel that uses mathematical equations to define more constrained mathematical models.

• AKA: Mathematical Model Template, Mathematical Model Family.
• Context:
  • It can typically contain Mathematical Model Variables that represent mathematical variable quantities.
  • It can typically incorporate Mathematical Model Parameters that represent mathematical constant quantities.
  • It can typically express Mathematical Model Relationships through mathematical equations.
  • It can typically generate Mathematical Model Instances by assigning mathematical parameter values.
  • It can often support Mathematical Model Analysis through mathematical analytical techniques.
  • It can often enable Mathematical Model Verification through mathematical proofs.
  • It can often facilitate Mathematical Model Comparison through mathematical goodness-of-fit metrics.
  • It can range from being a Simple Mathematical Metamodel to being a Complex Mathematical Metamodel, depending on its mathematical equation complexity.
  • It can range from being a Parametric Mathematical Metamodel to being a Nonparametric Mathematical Metamodel, depending on its mathematical parameter dependency.
  • It can range from being a Descriptive Mathematical Metamodel to being a Predictive Mathematical Metamodel, depending on its mathematical application purpose.
  • ...
• Examples:
  • Mathematical Metamodel Types, such as:
    • Statistical Mathematical Metamodels, such as:
      • Regression Mathematical Metamodel for mathematical data relationship modeling.
      • Distribution Mathematical Metamodel for mathematical probability modeling.
    • Functional Mathematical Metamodels, such as:
      • Linear Equation Mathematical Metamodel for mathematical linear system modeling.
      • Polynomial Mathematical Metamodel for mathematical curved relationship modeling.
      • Logistic Mathematical Metamodel for mathematical binary outcome modeling.
    • Dynamical Mathematical Metamodels, such as:
      • State-Space Mathematical Metamodel for mathematical system dynamics modeling.
      • Differential Equation Mathematical Metamodel for mathematical continuous process modeling.
    • Software-Based Mathematical Metamodels, such as:
      • Analytica Mathematical Metamodel for mathematical visual modeling.
  • ...
• Counter-Examples:
  • Mathematical Functions, which represent specific mathematical relationships rather than mathematical model families.
  • Instance-Based Mathematical Models, which lack mathematical metamodel analytical structures and rely on mathematical data point comparisons.
  • k-Nearest Neighbor Mathematical Models, which use mathematical instance comparisons rather than mathematical parameter estimation.
• See: Mathematical Modeling, Statistical Model, Mathematical Function Family, Metamodeling Approach.

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References

• http://scholar.google.com/scholar?q=%22a+mathematical+model%22

2014

• http://en.wikipedia.org/wiki/Mathematical_model
  • A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.

    Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models, as far as logic is taken as a part of mathematics. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.

1974

• (Baird, 1974) ⇒ Yonathan Bard. (1974). “Nonlinear Parameter Estimation." Academic Press. ISBN:0120782502
  • QUOTE: We refer to the relations which supposedly describe a certain physical situation, as a model. Typically, a model consists of one or more equations. The quantities appearing in the equations we classify into variables and parameters. The distinction between these is not always clear cut, and it frequently depends on the context in which the variables appear. Usually a model is designed to explain the relationships that exist among quantities which can be measured independently in an experiment; these are the variables of the model. To formulate these relationships, however, one frequently introduces “constants" which stand for inherent properties of nature (or of the materials and equipment used in a given experiment). These are the parameters.

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