Contenido

Se podría decir que un modelo de las ciencias físicas es una traducción de la realidad física de un sistema físico en términos matemáticos, es decir, una forma de representar cada uno de los tipos de entidades que intervienen en un cierto proceso físico mediante objetos matemáticos.Las relaciones matemáticas formales entre los objetos del modelo, deben representar de alguna manera las relaciones reales existentes entre las diferentes entidades o aspectos del sistema u objeto real. Así una vez "traducido" o "representado" cierto problema en forma de modelo matemático, se pueden aplicar el cálculo, el álgebra y otras herramientas matemáticas para deducir el comportamiento del sistema bajo estudio. Un modelo físico requerirá por tanto que se pueda seguir el camino inverso al modelado, permitiendo reinterpretar en la realidad las predicciones del modelo.

According to input information

With respect to the function of the origin of the information used to build the models, they can be classified in other ways. We can distinguish between heuristic models and empirical models:

• - Heuristic models (from the Greek euriskein 'find, invent'). They are those that are based on explanations of the natural causes or mechanisms that give rise to the phenomenon studied.

• - Empirical models (from the Greek empeirikos relating to 'experience'). They are those that use direct observations or the results of experiments of the phenomenon studied.

Depending on the type of representation

Furthermore, mathematical models find different names in their various applications. A possible classification may address whether they intend to make qualitative predictions or whether they intend to quantify aspects of the system that is being modeled:

• - Qualitative or conceptual models, these can use figures, graphs or causal descriptions, in general they are content with predicting whether the state of the system will go in a certain direction or whether it will increase or decrease some magnitude, without exactly caring about the specific magnitude of most aspects.

• - Quantitative or numerical models, use numbers to represent aspects of the modeled system, and generally include more or less complex mathematical formulas and algorithms that relate numerical values. The calculation with them allows representing the physical process or the quantitative changes of the modeled system.

According to the randomness

Another classification independent of the previous one, depending on whether or not a specific input or initial situation can correspond to various outputs or results, in this case the models are classified as:.

• - Deterministic. The form of the result is known in a timely manner, since there is no uncertainty. Furthermore, the data used to feed the model is completely known and determined.

• - Stochastic. Probabilistic, in which the expected result is not known, but rather its probability and therefore there is uncertainty.

Classification according to application or objective

Due to their use, they are usually used in the following three areas, however, there are many others such as finance, science, etc.

• - Simulation model or descriptive, of situations that can be measured precisely or randomly, for example with aspects of linear programming when it is precise, and probabilistic or heuristic when it is random. This type of models aims to predict what happens in a given specific situation.

• - Optimization model. To determine the exact point to resolve any administrative, production, or any other situation. When the optimization is integer or non-linear, combined, it refers to mathematical models that are not very predictable, but that can be coupled to some existing alternative that is approximate in its quantification. This type of model requires comparing various conditions, cases or possible values ​​of a parameter and seeing which of them is optimal according to the chosen criterion.

• - Control model. To know precisely how something is doing in an organization, research, area of ​​operation, etc. This model aims to help decide which new measures, variables or parameters must be adjusted to achieve a specific result or state of the modeled system.

Un modelo mixto operacional estadístico es una teoría o situación causal de hechos y expresado con símbolos de formato matemático. Por ejemplo las tablas de contingencia. De hecho los modelos matemáticos se construyen con varios niveles de significación y con diferentes variables.

Kendall y Buckland catalogan hasta 40 tipos diferentes de modelos matemáticos. Ejemplos: Rapoport en modelo matemático e interacción social en 1961 y Bugeda en Sociología matemática en 1970. Por un principio de isomorfismo hay una equivalencia, a conseguir, entre un modelo y una teoría. Además teoría y modelo son sinónimos.

Examples of models by type

If the classification of models is used according to their application or objective (descriptive or simulation models, optimization or optimal selection models, control or treatment models) and depending on whether they are determistic or probabilistic models, some illustrative examples can be given:

Mathematical hydrological simulation model

They are used to study extreme situations, which are difficult to observe in reality, such as the effects of very intense and prolonged rainfall in hydrographic basins, in their natural state, or in which there has been intervention with works such as canals, dams, dams, bridges, etc.

The hydrographic basin is divided into sub-basins considered homogeneous from the point of view: soil type, slope, and plant cover. The number and type of hydrological variables involved in the model are a function of the specific objective for which it is developed.

In many cases, the construction or creation of useful mathematical models follows a series of well-defined phases:

1. • Identification of a complex problem or situation that needs to be simulated, optimized or controlled and therefore would require a predictive mathematical model to make it effective.

2. • Choice of type of model, this requires specifying what type of response is intended to be obtained, what the input data or relevant factors are, and what the model is intended to be used for. This choice must be simple enough to allow an affordable mathematical treatment with the available resources. This phase also requires identifying the greatest number of reliable data, labeling and classifying the unknowns (independent and dependent variables) and establishing physical, chemical, geometric considerations, etc., that adequately represent the phenomenon under study.

3. • Formalization of the model in which the form of the input data will be detailed, what type of mathematical tool will be used, how they adapt to the existing prior information. It could also include the creation of algorithms, assembly of computer files, etc. In this phase, sufficient simplifications may also be introduced to make the mathematical modeling problem computationally tractable.

4. • Comparison of results: the results obtained as predictions need to be compared with the observed facts to see if the model is predicting well. If the results do not fit well, it is common to return to phase 1.

It is important to mention that the vast majority of mathematical models are not exact and have a high degree of idealization and simplification, since a very exact modeling can be more complicated to deal with than a convenient simplification, and therefore be less useful.

It is also important to remember that the mechanism with which a mathematical model is developed has an impact on the development of other knowledge techniques focused on the sociocultural area.

• - System dynamics.

• - Scientific model.

• - Physical model.

• - Climate model.

• - Modeling.

• - Aquifer modeling.

• - MC-14, scientific method in 14 stages.

Literature

• - Definitions.

• - Types of mathematical models.

• - Introduction to modeling (googlepages.com).

Contenido

Se podría decir que un modelo de las ciencias físicas es una traducción de la realidad física de un sistema físico en términos matemáticos, es decir, una forma de representar cada uno de los tipos de entidades que intervienen en un cierto proceso físico mediante objetos matemáticos.Las relaciones matemáticas formales entre los objetos del modelo, deben representar de alguna manera las relaciones reales existentes entre las diferentes entidades o aspectos del sistema u objeto real. Así una vez "traducido" o "representado" cierto problema en forma de modelo matemático, se pueden aplicar el cálculo, el álgebra y otras herramientas matemáticas para deducir el comportamiento del sistema bajo estudio. Un modelo físico requerirá por tanto que se pueda seguir el camino inverso al modelado, permitiendo reinterpretar en la realidad las predicciones del modelo.

According to input information

With respect to the function of the origin of the information used to build the models, they can be classified in other ways. We can distinguish between heuristic models and empirical models:

• - Heuristic models (from the Greek euriskein 'find, invent'). They are those that are based on explanations of the natural causes or mechanisms that give rise to the phenomenon studied.

• - Empirical models (from the Greek empeirikos relating to 'experience'). They are those that use direct observations or the results of experiments of the phenomenon studied.

Depending on the type of representation

Furthermore, mathematical models find different names in their various applications. A possible classification may address whether they intend to make qualitative predictions or whether they intend to quantify aspects of the system that is being modeled:

• - Qualitative or conceptual models, these can use figures, graphs or causal descriptions, in general they are content with predicting whether the state of the system will go in a certain direction or whether it will increase or decrease some magnitude, without exactly caring about the specific magnitude of most aspects.

• - Quantitative or numerical models, use numbers to represent aspects of the modeled system, and generally include more or less complex mathematical formulas and algorithms that relate numerical values. The calculation with them allows representing the physical process or the quantitative changes of the modeled system.

According to the randomness

Another classification independent of the previous one, depending on whether or not a specific input or initial situation can correspond to various outputs or results, in this case the models are classified as:.

• - Deterministic. The form of the result is known in a timely manner, since there is no uncertainty. Furthermore, the data used to feed the model is completely known and determined.

• - Stochastic. Probabilistic, in which the expected result is not known, but rather its probability and therefore there is uncertainty.

Classification according to application or objective

Due to their use, they are usually used in the following three areas, however, there are many others such as finance, science, etc.

• - Simulation model or descriptive, of situations that can be measured precisely or randomly, for example with aspects of linear programming when it is precise, and probabilistic or heuristic when it is random. This type of models aims to predict what happens in a given specific situation.

• - Optimization model. To determine the exact point to resolve any administrative, production, or any other situation. When the optimization is integer or non-linear, combined, it refers to mathematical models that are not very predictable, but that can be coupled to some existing alternative that is approximate in its quantification. This type of model requires comparing various conditions, cases or possible values ​​of a parameter and seeing which of them is optimal according to the chosen criterion.

• - Control model. To know precisely how something is doing in an organization, research, area of ​​operation, etc. This model aims to help decide which new measures, variables or parameters must be adjusted to achieve a specific result or state of the modeled system.

Un modelo mixto operacional estadístico es una teoría o situación causal de hechos y expresado con símbolos de formato matemático. Por ejemplo las tablas de contingencia. De hecho los modelos matemáticos se construyen con varios niveles de significación y con diferentes variables.

Kendall y Buckland catalogan hasta 40 tipos diferentes de modelos matemáticos. Ejemplos: Rapoport en modelo matemático e interacción social en 1961 y Bugeda en Sociología matemática en 1970. Por un principio de isomorfismo hay una equivalencia, a conseguir, entre un modelo y una teoría. Además teoría y modelo son sinónimos.

Examples of models by type

If the classification of models is used according to their application or objective (descriptive or simulation models, optimization or optimal selection models, control or treatment models) and depending on whether they are determistic or probabilistic models, some illustrative examples can be given:

Mathematical hydrological simulation model

They are used to study extreme situations, which are difficult to observe in reality, such as the effects of very intense and prolonged rainfall in hydrographic basins, in their natural state, or in which there has been intervention with works such as canals, dams, dams, bridges, etc.

The hydrographic basin is divided into sub-basins considered homogeneous from the point of view: soil type, slope, and plant cover. The number and type of hydrological variables involved in the model are a function of the specific objective for which it is developed.

In many cases, the construction or creation of useful mathematical models follows a series of well-defined phases:

1. • Identification of a complex problem or situation that needs to be simulated, optimized or controlled and therefore would require a predictive mathematical model to make it effective.

2. • Choice of type of model, this requires specifying what type of response is intended to be obtained, what the input data or relevant factors are, and what the model is intended to be used for. This choice must be simple enough to allow an affordable mathematical treatment with the available resources. This phase also requires identifying the greatest number of reliable data, labeling and classifying the unknowns (independent and dependent variables) and establishing physical, chemical, geometric considerations, etc., that adequately represent the phenomenon under study.

3. • Formalization of the model in which the form of the input data will be detailed, what type of mathematical tool will be used, how they adapt to the existing prior information. It could also include the creation of algorithms, assembly of computer files, etc. In this phase, sufficient simplifications may also be introduced to make the mathematical modeling problem computationally tractable.

4. • Comparison of results: the results obtained as predictions need to be compared with the observed facts to see if the model is predicting well. If the results do not fit well, it is common to return to phase 1.

It is important to mention that the vast majority of mathematical models are not exact and have a high degree of idealization and simplification, since a very exact modeling can be more complicated to deal with than a convenient simplification, and therefore be less useful.

It is also important to remember that the mechanism with which a mathematical model is developed has an impact on the development of other knowledge techniques focused on the sociocultural area.

• - System dynamics.

• - Scientific model.

• - Physical model.

• - Climate model.

• - Modeling.

• - Aquifer modeling.

• - MC-14, scientific method in 14 stages.

Literature

• - Definitions.

• - Types of mathematical models.

• - Introduction to modeling (googlepages.com).