Elements of a mathematical model

 

Elements of a mathematical model

Mathematical models can take many forms, including differential or These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include 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.

In the physical sciences, a traditional mathematical model contains most of the following elements:

1. Governing equations
2. Supplementary sub-models
   1. Defining equations
   2. Constitutive equations
3. Assumptions and constraints

Classifications

Mathematical models are usually composed of relationships and variables . Relationships can be described by operators , such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters  of interest, that can be quantified  Several classification criteria can be used for mathematical models according to their structure:

• Linear vs. nonlinear: If all the operators in a mathematical model exhibit , the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example,  it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear  but it can still have nonlinear expressions in it. In a mathematical programming  model, if the objective functions and constraints are represented entirely by linear equations  then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear  equation, then the model is known as a nonlinear model.
  Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently and/or analyzed at a different scale and the results obtained will remain valid for the initial problem when recomposed and rescaled.
  Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization , but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by differential equations or differnce equaions or difference equations.
• Explicit vs. implicit: If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newtons method  or broudens method . In such a case the model is said to be implicit. For example, a jet emgines  physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous: A  discrete model  treats objects as discrete, such as the particles in a molecular model  or the states in a statistical model; while a continuous model  represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic): A deterministic  model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probality distributions.
• Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory  in science has been characterized as a floating model.
• Strategic vs non-strategic Models used in game theory  are different in a sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computing solution concepts  such as Nash equillibrium . An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.

Construction

In  business  and  engineering , mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision varibales, state variables, exogenous  variables, and random variables .

Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters pr constants  The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).

Objectives and constraints  of the system and its users can be represented as function of the output variables or state variables. The objective functions  will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases.

For example, economists  often apply linear algebra  when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of  vector where one symbol represents several variables.

A priori information

Mathematical modeling problems are often classified into black box or white box models, according to how much a priori  information on the system is available. A black-box model is a system of 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, so this concept is useful only as an intuitive guide for deciding which approach to take.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying innction. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means 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. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks  which usually do not make assumptions about incoming data. Alternatively the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.

Subjective information

Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intution, experience, or expert  option,  or based on convenience of mathematical form.  bayesion statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution (which can be subjective), and then update this distribution based on empirical data.

An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.