Editing Mathematical model (section)

==Classifications==

Mathematical models are of different types:
* Linear vs. nonlinear. If all the operators in a mathematical model exhibit [[linear]]ity, 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, in a [[linear model|statistical linear model]], 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 [[differential operator]]s, but it can still have nonlinear expressions in it.  In a [[Optimization (mathematics)|mathematical programming]] model, if the objective functions and constraints are represented entirely by [[linear equation]]s, then the model is regarded as a linear model.  If one or more of the objective functions or constraints are represented with a [[nonlinearity|nonlinear]] equation, then the model is known as a nonlinear model.<br>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.<br>Nonlinearity, even in fairly simple systems, is often associated with phenomena such as [[Chaos theory|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 equation]]s or [[difference equation]]s.
* 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 [[Newton's method]] or [[Broyden's method]]. In such a case the model is said to be ''implicit''. For example, a [[jet engine]]'s 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 modeling|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 system|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 [[probability]] distributions.
* Deductive, inductive, or floating. A {{visible anchor|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.<ref>{{cite book |author-link=Stanislav Andreski |first=Stanislav |last=Andreski |year=1972 |title=Social Sciences as Sorcery |publisher=[[St. Martin’s Press]] |isbn=0-14-021816-5 }}</ref> Application of [[catastrophe theory]] in science has been characterized as a floating model.<ref>{{cite book |author-link=Clifford Truesdell |first=Clifford |last=Truesdell |year=1984 |title=An Idiot's Fugitive Essays on Science |pages=121–7 |publisher=Springer |isbn=3-540-90703-3 }}</ref>
* 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 concept]]s such as [[Nash equilibrium]]. An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.<ref>Li, C., Xing, Y., He, F., & Cheng, D. (2018). A Strategic Learning Algorithm for State-based Games. ArXiv.</ref>