Mathematical Metamodel

A Mathematical Metamodel is a metamodel that uses mathematical equations to define more constrained mathematical models through parametric templates built upon mathematical structures.

AKA: Mathematical Model Template, Mathematical Model Family, Parametric Mathematical Framework.
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 typically utilize Mathematical Structures as mathematical semantic foundations.
- It can typically instantiate Mathematical Model Families through mathematical parameter spaces.
- It can typically preserve Mathematical Structure Properties across mathematical model instances.
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- 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 often inherit Mathematical Constraints from underlying mathematical structures.
- It can often define Mathematical Model Morphisms between mathematical model instances.
- It can often establish Mathematical Model Hierarchies through mathematical specialization relationships.
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- 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.
- It can range from being a Linear Mathematical Metamodel to being a Nonlinear Mathematical Metamodel, depending on its mathematical equation linearity.
- It can range from being a Deterministic Mathematical Metamodel to being a Stochastic Mathematical Metamodel, depending on its mathematical uncertainty handling.
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- It can integrate with Mathematical Structure Theory for mathematical foundational support.
- It can connect to Mathematical Optimization Frameworks for mathematical parameter estimation.
- It can support Mathematical Model Selection through mathematical information criteria.
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Examples:
- Mathematical Metamodel Types, such as:
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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.
- Mathematical Structures, which provide mathematical semantic foundations rather than mathematical parametric templates.
- Mathematical Algorithms, which specify mathematical computational procedures rather than mathematical model frameworks.
See: Mathematical Modeling, Statistical Model, Mathematical Function Family, Metamodeling Approach, Mathematical Structure, Model Theory, Parametric Modeling.

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.

Mathematical Metamodel

References

2014

1974

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