Mathematical model

General information

Multiplicity of goals

There never exists of single model: a model is always related so that one wants to make some. The same object, for example a mouse, will not be modélisable in the same way according to whether one rather is interested in

his intellectual performances;
its diseases and their care, even those of a group of related animals but broader (all mammals of which the Man);
the way of drawing it in a convincing way within the framework of a video game.

In the same way, a model is never perfect and completely representative of reality, and thus one more or less directs the parameters to study certain results in particular. For the same model one can parameterize very differently to highlight different things.

Multiplicity of modelings

Even when the goal is fixed, there are always several possible models, which all can be as valid the ones as the others.

In any modeling there is a choice a priori mathematical space being used to locate the whole of the phenomena. Mathematical space is not often identifiable with the reality of physics.

Thus and for example, in Physical, one can find convenient to use an Euclidean three-dimensional space, or a “curved” space, or a space with 4,5,11 or 26 Dimension S, or a Espace of Hilbert, etc And, in general, one can show that these various representations are perfectly equivalent, but more or less adapted to such or such case. All these models are only that: models. They are useful to treat reality, but they should not be taken for reality. If a physicist affirms for example that “the universe is expanding”, it should well be understood that implicitly it indicates “compared to my mathematical framework, all this master key like if… ”. Another physicist can affirm that “the universe is not expanding”: if it uses the same mathematical framework, they are contradicted, but if the second uses another mathematical framework, they can be in fact perfectly of agreement.

The same remark applies to all modelings, and in particular to economic and countable modelings, which will have economic consequences and tax important: prototype of modeling economic being Land register Tax and them bases of taxation real, everyone knows well that they are “false”, i.e. it reflect only very imperfectly the actual value which is supposed to be used as reference.

All this without abolishing reality: the model of bridge can affirm well that all well will occur, it is possible that the bridge collapses nevertheless (in opposite direction, it is doubtful that a bridge is built if the model indicates that it will crumble…).

Typology of model: according to the direction of modeling

Modeling can be exerted

model towards reality: they are the predictive model

These mathematical models are used to anticipate events or situations, like envisaging time with the weather, considering the prices potential of the financial credit with the model of evaluation in Finance, or to prevent the epidemics. One speaks about model predictive , in which known variables, called “explanatory”, will be used to determine unknown variables, known as “to explain”.

of real towards the model: they are the model descriptions

In this case, the models are used to represent historical data. One speaks about model descriptions . The objective is to return account, in an interpretable way, a mass of information. The prototype of these models is the Comptabilité: it describes in a simplified way the real economic events in their affecting an account, i.e. a supposed “label” to characterize them. These accounts are then aggregate to present in a standard way the economic situation of the companies and the countries.

Of course, the two types of models are perfectly dependant: a good prediction supposes at least the prediction of the last and current situation, i.e. a good description. Conversely, a good description would be perfectly vain if it were not used at least of diagnosis, or chart, to identify the action to be taken.

It is interesting to note that the same mathematical model can be applicable to many situations, not having inevitably a quite obvious report/ratio. For example, of the generators of landscapes are able to create the realistic shapes of objects as different as from the mountains, trees, rocks, grass, shell or snowflakes, with only one general model, while at the same time the processes of growth and of constructions of its objects are very diverse. If, instead of creating a new model, one is able to bring closer a problem to an old known model, one obtains a very useful mass of data immediately. Most of work is thus to recognize that a known model applies, or to extend the known properties of a particularly useful class of model (property which one will be able to then use more largely).

Qualities of a model

In preliminary, it is important to understand that mathematical complexity is not a sufficient criterion to judge if a model is relevant or not: there exist classes of models which call upon complex mathematical tools, the such Operations research or the Game theory; other classes, the Accounts Department for example, are of a childish mathematical access (additions, subtractions). But, with comparable result, it is of course the simplest model which is preferable.

A model is relevant

if it covers the field of the real problem well

e.g. a financial model which would not integrate the phenomenon of barter would not be usable to evaluate the companies of ex-Europe of the East.

if it makes it possible to obtain the result discounted : description of the phenomenon with the level of detail or synthesis wished, or right forecasts appearing a posteriori .
in the time desired

One thinks of the joke which promises precise forecasts weather at one week but which ask for one month of calculation.

incidentally, if it is reusable

the investment to describe a model is in general so important that it is seldom justified on a single operation.

How to create a model?

It is not question in an article so court of presenting a methodology applicable to all the situations (if there is one!), but some essential points.

1. The starting point is always a question which one puts on a situation future and/or so complex that one does not find the answer there in an obvious way.

e.g.: is my company viable? This material is it worth the asking price? This drug is it effective? What should be made so that the situation improves?

2. To find the answer, it is necessary to limit the field of the problem by seeking the data which one imagines to have a direct link with the question. Too much to limit made run the risk not to model a phenomenon which has weight in the context, but to open too much involves a dispersion of the means and an accumulation of nonrelevant data which it will be necessary to draw aside by justifying the choices. This stage is most delicate for the quality of the model: it is subjected to the a priori of the modelisator, its lacks of knowledge - sometimes of method - and to the means of which he lays out (time, money, access to the data). During this stage, one chooses the type of general model which one will use, in particular according to the data one thinks of having.

3. It is necessary then to build the model :

to filter the data in order to extract from them the additional “noises”, these irregularities or these events which mask essence;
possibly, to reconstitute the lacks, i.e. the objects which miss to ensure the coherence of the unit (e.g. the operation of a parameter whose one knows the existence but on which one does not have data)

It is there that intervene the mathematical and data-processing tools, which allow a filtering and a construction with a minimum of subjectivity in a minimum of time.

4. The “substrate” remaining constitutes the model, together of rules or equations . These rules should be described most completely possible: their relative importance, data input and exit, the mathematical tools used, the stages by which it is necessary to pass, check-points.

5. The last stage consists with to validate the model : by applying to the filtered data the rules of the model, is the initial situation found? If the variation is too important, it is necessary to rest the question of the limits which one fixed, or of the relevance of the tools used for modeling.

The mathematical tools most current

They are primarily statistical tools and probabilities, of differential calculus (partial derivative equation and ordinary). More precisely,

For the predictive models:

the projection , which consists in predicting the value of a continuous numerical size starting from the values passed, for example by using the methods of regression (linear or not);

For all the models:

the classification , or categorization , which makes it possible to locate an observation (event or individual) in a reduced number of preset classes ;
the chart, which gives a visual image;
the use of the centered variable , where a variable is supposed to represent all the others (e.g. the average);
the correlation , which makes it possible to associate several variables when they have a common behavior;
the clusterisation , which consists in presenting the observations per the most homogeneous possible packages (the clusters );
the reduction of dimensionality , which consists in creating, starting from a whole of observations, a reduced whole of observations (i.e. fewer) which is famous to behave like the initial population.

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