Mathematical model
A mathematical model is a simplified, abstract representation of a real-world object, phenomenon, or process that uses mathematical language and concepts to describe its essential characteristics and functional principles.
A mathematical model serves as the primary tool for Mathematical modeling and allows for the analysis of the behavior of studied systems, prediction of their development, and justification of decisions.
Key Characteristics
A typical mathematical model consists of the following elements:
- Object of modeling: The real-world system, process, or phenomenon being studied.
- Variables: Quantities that characterize the state of the object and its changes (e.g., inputs, outputs, internal states). They can be dependent or independent.
- Parameters: Quantities, usually considered constant for a given model, that define the specific properties of the object or system (e.g., mass, friction coefficient, interest rate, geometric dimensions).
- Mathematical relationships (Structure): Equations (algebraic, differential, difference, etc.), inequalities, logical rules, or algorithms that describe the connections between variables and parameters, as well as the object's operational principles.
Mathematical models are subject to requirements that determine their quality and suitability:
- Adequacy: The ability of the model to accurately reflect the properties of interest of the real object within the scope of the problem and the assumptions made. Adequacy is always relative and is verified through validation.
- Accuracy: The degree of quantitative agreement between the modeling results and real-world data.
- Simplicity (Parsimony): The model should be as simple as possible to achieve the modeling goal, avoiding unnecessary complexity.
- Robustness (Stability): Low sensitivity of the modeling results to small changes in input data and parameters.
- Efficiency: The feasibility of studying the model (analytically, numerically, or through simulation) with acceptable computational and time resource costs.
- Correctness (mathematical): For some classes of models, it is important that the mathematical problem has a solution, and preferably a unique one, under the given conditions.
- Interpretability: The ability to provide a clear explanation of the model's structure and its results in the terminology of the subject domain.
Types of Mathematical Models
Mathematical models are classified based on various criteria:
By the nature of variables:
- Deterministic — without random factors;
- Stochastic — accounting for random disturbances.
By the method of description:
- Analytical — systems of equations (differential, algebraic, etc.);
- Numerical — require the use of computational methods to obtain a solution.
- Simulation — models based on algorithms that reproduce the object's behavior.
By spatio-temporal scale:
- Lumped-parameter systems — properties depend only on time;
- Distributed-parameter systems — properties depend on spatial coordinates and time.
By the linearity of mathematical relationships:
- Linear: Described by linear equations.
- Nonlinear: Contain nonlinear relationships between variables.
Model Construction and Validation
A mathematical model does not appear on its own but is the result of the mathematical modeling process. The key stages of this process include:
- Problem formulation: Defining the objectives and the object of modeling.
- Conceptualization: Identifying essential factors, variables, parameters, and relationships.
- Formalization: Expressing the model in mathematical language.
- Parameter identification: Determining the values of parameters (calibration) based on data.
- Model analysis: Solving equations, investigating properties.
- Validation: Checking the model's correspondence with real-world data that was not used during its construction.
It is important to understand that a model that has not passed validation has limited predictive power and practical value.
Limitations of Mathematical Models
When using mathematical models, it is important to consider their inherent limitations:
- Simplification: Any model omits certain details and aspects of the real world.
- Assumptions: A model is correct only to the extent that the assumptions made during its construction are valid.
- Scope of applicability: A model is adequate only for a specific range of conditions, parameters, and tasks.
- Error: Modeling results always contain some degree of error (model error).
Related Concepts
- Mathematical modeling: A mathematical model is a key tool and result of the mathematical modeling process.
- System model: A mathematical model is a formalized representation of a system model using mathematics.
- Formalization: Building a mathematical model is the process of formalizing knowledge and hypotheses about an object.