**2. Models for the mathematical representation of the real situations: the territory and the complexity**

I have cited of complexity as a structuring character of our reality and in particular the territory. It is therefore obvious that the "management of complexity" is the fundamental problem for those involved in reading, representation or knowledge of the city and the territory. Hence the need to question those methods consolidated by practice but certainly not by results, with which we are used to working because they are equipped with tested tools capable of selecting and modeling a reality that at the end of the process "must" be verifiable. A real situation usually presents itself in a confusing, complex, vague way. It is not immediately clear how we can formulate a mathematical model to represent the phenomena observed. However after a first process of abstraction, using logical principles and common sense, you can try to get a '"acceptable" mathematical representation of the situation to be studied. From the point of view of the study of the territory, the difficulty lies precisely in transferring a perception that belongs to our mind into a model and therefore into artificial symbols. Multiple internal and external factors exert their influence, one on the other, in a continuous process of dependencies and reciprocal relations thus configuring what can be defined as the "system" [7]. The system is in a symbolic language, the area on which it operates. "At this point it is worthwhile to include the definition already indicated that the models are abstract representations of reality, helping us to perceive meaningful relationships in the real world, to manipulate them and then predict other. In this sense, the model while qualifying as a design tool, as part of a rational methodology should not be confused with the project; the project subtends a model but in support to this penetrates into the elements and relationships in a much more analytical (..) consequently, the model belongs to the moment meta projectual of the design

*Uncertainty Management in Engineering: A Model for the Simulation and Evaluation… DOI: http://dx.doi.org/10.5772/intechopen.96519*

process." [8]. "The elements, relationships and interrelationships to be searched, analyzed and interpreted involve an enormous amount of work [9], since it seems superfluous to assert it, the territory, the city represents the" impact "of the community structures, the projection plane, and the organization of social, economic, administrative, cultural, residential activities etc." [2].

Meanwhile, following this procedure, the following organization chart can be adopted:


X1 = number of inhabitants.


In general, given the phenomenon, to describe it is considered the 'ordered set X ¼ ð Þ X1, X2, … *:*Xn of real variables that you think will adequately represent the phenomenon, and it is called performance of the set of phenomena values assumed by these variables in the various objects of the collective. For each object Oi set U and for each variable Xj is determined so zij the value assumed by the object Oi in the variable Xj. In order to better understand the phenomenon variables Xj are replaced by variables centered Zj = aij (Xj-mj), where I is the average of Xj and aij different from 0 is an appropriate multiplier.

The statistical survey result is thus represented by means of a Eq. (1)

$$\text{\textquotedblleft objects-characters\textquotedblright} : \text{OC1} = \begin{bmatrix} \text{Z}\_1 & \text{Z}\_2 & \text{Z}\_3 & \dots & \text{Z}\_N \\\\ \text{z}\_{11} & \text{z}\_{12} & \text{z}\_{13} & \dots & \dots \\\\ \dots & \dots & \dots & \dots & \dots \\\\ \dots & \dots & \dots & \dots & \dots \\\\ \text{z}\_{m1} & \text{z}\_{m2} & \text{z}\_{m3} & \dots & \dots & \dots \end{bmatrix} \begin{array}{c} \text{O}\_1 \\ \text{O}\_2 \\ \text{O}\_3 \\ \dots \\ \dots \\ \text{O}\_m \end{array} \tag{1}$$

In the case study, it was thus obtained a base matrix of 10 to 46 objects indicators (Municipalities). The MM1 = (U tern, X, OC1) is defined mathematical model of the phenomenon.

Once created the MM1 model, starting from the foregoing considerations, we finally have the means to manage information globally taking into account all significant relationships between the variables.

The MM1 model contains easily visible information such as the values of the objects in the collective U Xj variables and other information hidden or latent. In order to highlight all the information, latent or apparent, contained in the model we introduce the concept of "equivalent models".

Two models MM1 = (U1, Z, OC1) and MM2 = (U2, Y, OC2) are equivalent if that of OC2 derives from the knowledge of the OC1 object-mode matrix and vice versa. MM1 MM2 implies if the knowledge of OC1 to OC2 is deduced.

In this chapter, we consider only pairs of models (MM1, MM2) wherein U1 = also U2 and, once assigned the MM1 model, suppose that MM2 is such that each yj component of the vector Y is a linear function of those of the vector Z, namely that there is a matrix T, the general term tij, such that Y = TZ. In this case we say that MM1 and MM2 linearly implies that T is the transformation matrix from MM1 to MM2.

If Y ¼ ½ � Y1, … Yr , Z ¼ ½ � Z1, … *:*Zn the Y = TZ is written in full:

$$\text{TZ} = \begin{bmatrix} \mathbf{y}\_1 & = & \mathbf{t}\_{11}\mathbf{z}\_1 & \dots & \mathbf{t}\_{1n}\mathbf{z}\_n\\ \dots & = & \dots & \dots & \dots\\ \mathbf{y}\_r & = & \mathbf{t}\_{r2}\mathbf{z}\_1 & \dots & \mathbf{t}\_{rn}\mathbf{z}\_n \end{bmatrix} \tag{2}$$

If the information matrix T is square and invertible then also implies MM1 MM2 linearly and is said to MM1 and MM2 are linearly equivalent in this case by:

$$\mathbf{Y} = \mathbf{T} \mathbf{Z} \tag{3}$$

follows (the transpose inverse):

$$\mathbf{Z} = \mathbf{T}^{(-1)} \mathbf{Y} \tag{4}$$

From (3) is obtained, in particular the relationship

$$\left(\text{OC2}\right)^{\text{t}} = \text{T(OC1)}^{\text{t}} \tag{5}$$

and (4) the

$$(\mathbf{OC1})^{t} = \mathbf{T}^{(-1)} \left( \mathbf{OC2} \right)^{t} \tag{6}$$

By suitably selecting T and then passing from the model MM1 to MM2 linearly equivalent model occurs in general that some properties contained in MM1 but latent, become evident in MM2.

*Uncertainty Management in Engineering: A Model for the Simulation and Evaluation… DOI: http://dx.doi.org/10.5772/intechopen.96519*

If the matrix T is orthogonal then it is invertible and is

$$\mathbf{T}^{(-1)} = \mathbf{T}^{\mathbf{t}} \tag{7}$$

so that (5) and (6) reduce respectively to

$$(\mathbf{OC}\_2) = (\mathbf{OC}\_1)\mathbf{T}^t, (\mathbf{OC}\_1) = (\mathbf{OC}\_2)\mathbf{T} \tag{8}$$

In this case the models MM1 and MM2 call them orthogonally equivalent.

The transition from one model to an equivalent orthogonally can be useful because they are preserved to the particular mathematical properties that allow to better interpret the urban phenomenon.

Consequently, I will adopt some statistical techniques essentially consisting in the passage from the initial MM1 model to specific equivalent and in particular orthogonally equivalent models. These techniques are called factorial analysis.

The techniques mentioned above that contribute to the research of the relationships between variables, have in particular the advantage of highlighting that for some new variables y1, y2, … yn-called "factors" such that only some of them y1, y2, y3, … ys, (with s < < n), are relevant for the explanation of the phenomenon. This, in fact, allows a reduction in the number of variables and thus a simplification of the model.
