Uncertainty Management in Engineering: A Model for the Simulation and Evaluation of the Operations Effectiveness in Land Use and Planning

Abstract

The quality of the environment is essential for our health, our economy and our well-being. However, it faces a number of major challenges, not least those related to climate change, unsustainable consumption and production, and various types of pollution.Spatial planning policies (and EU legislation) protect natural habitats, keep water and air clean, ensure adequate waste disposal, improve knowledge of toxic substances and support the transition of businesses towards a sustainable economyThe goal of the work is to develop a standardized methodology for the monitoring and management of spatial information as the basis for spatial planning. The present work makes use of data analysis methods in spatial planning, where the proposed “mathematical” model is of help in supporting decision making. In fact, certain decisions often arise only from the evaluation of certain parameters, which are always small; it is necessary to consider them all, even in a disaggregated way, and give the right weight to each one. The proposed model describes the territorial system as an interaction between the physical system and the social system; it interpret needs and identify problems concerning the physical system and the social system; and formulate purposes and deduce objectives expressed in quantitative magnitude; formulate forecasts on the consequences of decisions to change the uses of the physical system, through electronic processing. The model could be used to evaluate alternative guidelines for change; andto choose from among the possible alternatives the one that is believed to contribute most to the pursuit of the objectives.

“Uncertainty is the natural habitat of human life, although the hope of escaping it is the engine of human activities”

Zygmunt Bauman

1. Introduction

Environmental policies and EU legislation protect natural habitats, keep water and air clean, ensure adequate waste disposal, improve knowledge of toxic substances and support the transition of businesses to a sustainable economy. While there is not a single Community planning policy because the “planning” has more weight, but in essence it is left to the responsibility of national Governments; on the other hand certainly you can have recourse to an important series of strategies, in fact, the European Union has moved so far mainly at the level of standards (Directives and Regulations) and less at the level of plans and programs. There are “strategic” guidelines of political value and certainly not a “European environmental planning”, because the U. E. sets only a reference framework in the various sectors, indicates priorities, suggests the most appropriate means and instruments, but does not interfere with the role of the Member States in terms of planning, except to suggest a general criterion: “integrate planning sustainable in community policy”¹. This point is important for two reasons:

• because planning is considered a necessary and useful political tool as long as it is “sustainable”;

• because national planning cannot ignore the community framework.

One of the strategies, for example, concerns the importance of soil protection and was the subject of attention by the European Commission in 2002. Following the publication in 2001 of the sixth European Environment Action Plan, the European Commission adopted the COM [1] “Towards a Thematic Strategy for Soil Protection”, building on a specific political commitment to this issue. Soil performs many environmental key-functions which are: biomass production, storage, filtering, buffering and transformation and plays a central role in water protection and the exchange of gases with the atmosphere. It is also a habitat and gene pool, an element of the landscape and cultural heritage, and a provider of raw materials. In order to perform its many functions, it is necessary to maintain soil condition². The soil, however, is an integral part of the territory and the territorial dimension is particularly important in the “planning” processes. So land assessment as a spatial planning tool promotes sustainable development of space and settlements; it serves to defend the soil resource, to protect it from pollution, contributing to the conservation and improvement of the quality of the environment. The assessment of soils is of particular importance in areas that are already heavily contaminated and ecologically sensitive in which there continues to be a huge demand for spaces intended for construction and expansion.

The present work makes use of data analysis methods in spatial planning. The problem faced in first appeal sets as objective to define the social and economic orders of the area by the point of view of the situation of the state of fact. In these cases, the task is really difficult as it is necessary to take into account a multiplicity of aspects of different nature (economic, social, historical, urban and environmental) and of various dimensions, moreover it is necessary to summarize them to obtain useful information for make assessments that meet community goals. Far this aim, I follow a methodology based on the analysis of the data and in particular on the method of the principal components, on the cluster analysis and on coverings and partitions with fuzzy sets [2]. In such an approach, I have ascertained that a lot of importance it’s to attribute to the changes of the general system on which one intervenes, kept account that the means that we have to disposition often result inadequate to gather the entity of the aforesaid changes.

The method is generally adaptable; however, to achieve satisfactory results, it should be used directly to related information with the insights that you pursue.

It is considered that a correct methodological approach to the problems of decision resides in multidimensional analysis. In fact in such type of analysis all the technical, economic, social and environmental aspects, are opportunely valued and balanced.

It is recognized by many people that any intervention on the system baits a complex connection of phenomena, necessity emerges then to favor the orientation and the control of the scales through the use of a model that aim at rationalizing the complexity of the situations and to reduce uncertainty that permeates the planning process. Of it the planning and the evaluation of the interventions have a lot of importance. The evaluation is a tool of control (in progress and ex post) of uncertainty on the evolution of the several elements constituent the general context, kept account of all the different demands expressed from consumers, decision men and technicians, and considering the multiplicity of the preset objectives, the potentialities of the available technologies and the constraints [3].

The government policy of territorial planning implemented in our country as well as in other European and non-European nations has made the problem of finding more valid tools for forecasting and decision-making urgent, with the necessary deepening of the study of methodologies linked to urban culture. The goal then is to implement a methodology adapted for the control and management of information concerning the territory in order to provide a scientific setting of the work steps that precede the action of spatial planning. Such an approach starts after the Second World War, especially in the Anglo-Saxon and US world with the study of urban systems, activity patterns and founding principles of their balance, spatial organization and environmental design models. A stop in the search for patterns of systematic and ordered study has had prevailed when the reductive opinion of mathematical studies to be able to exclusively deal with solving quantitative problems.

In this respect, starting from the eighties onwards.

Explains and justifies the denial explicated by certain architects towards mathematical methods, attitude further motivated error exchange for schematismo previously established the need for systematic, order, organic and its symbolism of a scientific theory.

The main role of the scientific disciplines that coordinate and assist the construction of such a setting, it must identify a methodology to build up the logic of abstraction and verification process. In conclusion, the method is always applicable; however, the results must be interpreted in relation to the particular circumstances in which the investigation takes place.

Nowadays the new sciences of complexity, the increasingly heated debate among the philosophers of science, from which relativistic and localistic positions seem to emerge, do not leave much room for the revival of scientific paradigms with models of absolute validity.“Local scientific approaches do not cooperate harmoniously with an image, with a theory of knowledge and the universe, but on the contrary they intersect, overlap, ignoreeachother, contrast, integrate, split”.

Complexity arises from the awareness that “the type of problems that contemporary society faces cannot be standardized, like the problems faced par excellence by strong disciplines. As already mentioned, the use of procedural standards is as of little use as proposing physical standards. In reality, this awareness is not new, but can be traced back to the 1980s, when the weakening of the nation-state form, the crisis of political units of great territorial dimension and of the concept of territory as a univocal entity, make the “gravitational” spatial paradigm obsolete, which presupposes a hierarchical structuring of space.

The “restitution” of complexity is strongly linked to the processing of information: moving in a field of consolidated knowledge practices, one is forced to discard everything that is not compatible or can be tamed with knownmeans, reducing the complexity of the phenomena. It’s necessary to abandon the old certainties to take new paths, such as those traced by the complexity of relationships and continuous interactions between animate and inanimate components of the same world [4]. The theory of complexity, first of all, is not a scientific theory in the strict sense. It would be better to speak (and indeed some authors do) of “complexity challenge” or “complexity thought” or, better still, “complexity epistemology”. It is precisely as an epistemological perspective, in fact, that complexity plays a crucial role in contemporary thought. This is because complexity involves three equally elevant epistemological innovations: a new alliance between philosophy and science, a new way of doing science, a new conception of natural evolution [5]. This chapter makes use of data analysis methods in spatial planning. The exploratory methods of multivariate analysis make it possible to arrive at territorial types (at different scales) that are also significant at an environmental level. The methods of data analysis and the main components in the work aim to describe the territorial system as an interaction between the physical system and the social system; interpreting needs, evaluating alternative directions of change; support the choice between the alternatives of what you think may contribute most to achieving the objectives of preservation [6].

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 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:

1. Definition of the collective or universe “U”. It must be chosen in such a way as to be suitable to the purposes of the research; in this specific case it seemed appropriate to choose, as the statistical units for the phenomenon to be studied, small parts of the territory (municipalities), either because they have bureaucratic and organizational structures, either because they are compared to known statistical summaries. In general, the statistical universe that is going to be investigated and which you want to investigate the interrelationships and structure, is the set $\mathrm{U}=\left\{\mathrm{O}1,\mathrm{O}2,\mathrm{O}3,\dots \mathrm{Om}\right\}$ of “statistical units” or “Objects”.

2. Definition of the phenomenon. A phenomenon is what is observed in the elements of a collective. For example in the collective “group of municipalities” the hierarchical structure, the socio-economic profile, etc., are phenomena And the variables selected for analysis are intended to clarify what should be observed to study the evolution of the phenomenon.

3. Choice of variables. In this regard, it introduces the concept of a statistical nature. If “U” is the collective statistics that you consider, in the case study is the set of municipalities, and V is any set, is said character or statistical variable defined in U and set of values or V mode, each X function defined in U and values in V. If V is contained in the set R of real numbers the statistical variable is said real. In the case study, we have been taken into consideration 10 variables represented by the following territorial real variables:

X1 = number of inhabitants.

X2 = variation of population divided by the total population;

X3 = percentage divided by total active population;

X4 = occupied housing divided by the total housing;

X5 = agricultural area divided by the municipal land area;

X6 = Industry insiders divided by the total number of employees;

X7 = active in agriculture divided by total assets;

X8 = Percentage tertiary sector divided by the total number of employees;

X9 = Active public administration divided by total assets;

X10 = trade workers divided the total number of employees;

In general, given the phenomenon, to describe it is considered the ‘ordered set $\mathrm{X}=\left(\mathrm{X}1,\mathrm{X}2,\dots .\mathrm{Xn}\right)$ 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)

$$“\text{objects}‐\text{characters}”:\mathrm{OC}1=\left[\begin{array}{ccccc}{Z}_1& Z_2& Z_3& \dots & Z_N\\ z_{11}& z_{12}& z_{13}& \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ z_{m1}& z_{m2}& z_{m3}& \dots & z_{\mathit{mn}}\end{array}\right]\begin{array}{c}{O}_1\\ O_2\\ O_3\\ \dots \\ O_m\end{array}$$E1

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 $\mathrm{Y}=\left[\mathrm{Y}1,\dots \mathrm{Yr}\right]$, $\mathrm{Z}=\left[\mathrm{Z}1,\dots .\mathrm{Zn}\right]$ the Y = TZ is written in full:

$$\mathrm{TZ}=\left[\begin{array}{ccccc}{\mathrm{y}}_1& =& {\mathrm{t}}_{11}{z}_1& \dots & t_{1n}{z}_n\\ \dots & =& \dots & \dots & \dots \\ y_r& =& t_{r2}{z}_1& \dots & t_{\mathit{rn}}{z}_n\end{array}\right]$$E2

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:

$$\mathrm{Y}=\mathrm{TZ}$$E3

follows (the transpose inverse):

$$\mathrm{Z}={\mathrm{T}}^{\left(-1\right)}\mathrm{Y}$$E4

From (3) is obtained, in particular the relationship

$${\left(\mathrm{OC}2\right)}^{\mathrm{t}}=\mathrm{T}{\left(\mathrm{OC}1\right)}^{\mathrm{t}}$$E5

and (4) the

$${\left(\mathrm{OC}1\right)}^{\mathrm{t}}={\mathrm{T}}^{\left(-1\right)}{\left(\mathrm{OC}2\right)}^{\mathrm{t}}$$E6

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.

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

$${\mathrm{T}}^{\left(-1\right)}={\mathrm{T}}^{\mathrm{t}}$$E7

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

$$\left({\mathrm{OC}}_2\right)=\left({\mathrm{OC}}_1\right){\mathrm{T}}^{\mathrm{t}},\left({\mathrm{OC}}_1\right)=\left({\mathrm{OC}}_2\right)\mathrm{T}$$E8

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 $\mathrm{y}1,\mathrm{y}2,\dots \mathrm{yn}$-called “factors” such that only some of them $\mathrm{y}1,\mathrm{y}2,\mathrm{y}3,\dots \mathrm{ys}$, (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.

3. An application of factorial analysis methods for the analysis of an urban system

One could refer to one of the detected information on statistical units study outline, there is the method of principal components if T is orthogonal and the matrix of variances and covariances, is a diagonal matrix, = diag (l1 … ln) with them, elements of the diagonal, arranged in descending order. The Yi obtained by the relationship Y = TZ in this case are called “main components”. They are factors not related and equipped with various mathematical properties.

The principal component analysis is basically a theory for the study of a phenomenon represented by many random variables $\mathrm{X}1,\mathrm{X}2,\dots .\mathrm{Xn}$, from the point of view of its variability. It is proposed to represent the same phenomenon with new centered variables $\mathrm{Y}1,\mathrm{Y}2,\dots .\mathrm{Yn}$ said main components, not related to each other, with decreasing variability, such that the sum of the variability of Yi is equal to that of Xj and so that already with a few variables explain a large proportion of the variability of the phenomenon.

These variables allow to obtain, inter alia,:

1. simple geometric representations of the phenomenon under examination.

2. simplification of the procedure of multivariate regression of new variables considered.

3. decomposition of the variability in a manner easily understandable phenomenon.

4. formation of homogeneous groups with more simple procedures which includes immediacy with the practical significance.

5. ability to solve problems of multi-objective programming choices.

In our research we studied the “socio-economic balance of the municipalities of the province of Pescara” phenomenon. It was represented by 10 random variables {X1 …. X10} and 46 objects {O1 … O46} with O1, O2 etc. we indicate each of the 46 Municipalities of the Province of Pescara.

For each variable Xj j = 1, ..10, was calculated the average mj and the vector X = (X1 …. X10) has been replaced by S = scraps (S1 … S10); with Sj = Xj -mj; for each j = {1 … 10} was taken as the multiplier value $a\mathrm{j}=\frac{1}{{\sigma }_j}$, with σj standard deviation of Xj.

It is thus obtained a vector Z = (Z1 … Z10), and then there was obtained OC1 matrix. If Mx is object-matrix characters relating to the variables Xj and M it is the matrix that has as its j-th column vector with all components equal to mj, you get

$${\mathrm{OC}}_1=\left({\mathrm{M}}_{\mathrm{x}}‐\mathrm{M}\right){\mathrm{D}}^{-1}$$E9

with $\mathrm{D}=diag\left(\mathrm{s}1\dots \mathrm{sn}\right)$, diagonal matrix of the standard deviations. The matrix of variances and covariances of Z is then

$$\mathrm{R}=\frac{1}{n}{\left({\mathrm{OC}}_1\right)}^{\mathrm{t}}\left({\mathrm{OC}}_1\right),$$E10

generic element

$${\mathrm{r}}_{\mathrm{hk}}=cov\left({\mathrm{Z}}_{\mathrm{h}},{\mathrm{Z}}_{\mathrm{k}}\right)=\frac{1}{n}\sum _{r=1}^n{\mathrm{Z}}_{\mathrm{rh}}{\mathrm{Z}}_{\mathrm{rk}}$$E11

The R is evidently also correlation matrix for both the vector X and the vector Z.

In our work, we were prepared using the formulas (9)(10) calculated with the program Mathematica 2.2.

It is a verification by calculating instead of the R of the variances and covariances S matrix of Xj is also performed given by the formula

$$\mathrm{S}={\left({\mathrm{M}}_{\mathrm{x}}‐\mathrm{M}\right)}^{\mathrm{t}}\left({\mathrm{M}}_{\mathrm{x}}‐\mathrm{M}\right)$$E12

and obtaining

$$\mathrm{R}={\mathrm{D}}^{-1}\mathrm{\Sigma }{\mathrm{D}}^{-1}$$E13

The eigenvalues and eigenvectors of R were then calculated.

Said $\mathrm{L}=diag\left(\mathrm{l}1\dots \mathrm{l}10\right)$ the matrix of the eigenvalues, in ascending order and called A the matrix that has the corresponding eigenvectors for columns is

$$\mathrm{RA}=\mathrm{A\Lambda }$$E14

that is

$${\mathrm{A}}^{\mathrm{t}}\mathrm{RA}=\mathrm{\Lambda }$$E15

Then the matrix T = At is the orthogonal transformation matrix that is passed from Zj to new variables Y equivalent orthogonally variables and uncorrelated.

In fact from Y = TZ it is obtained.

$\mathrm{OC}2=\mathrm{OC}1{\mathrm{T}}^{\mathrm{t}}=\left(\mathrm{OC}1\right)\mathrm{A}$ and therefore the variance and covariance matrix of Y is the

$$\mathrm{\Delta }=\frac{1}{n}{\left({\mathrm{OC}}_2\right)}^{\mathrm{t}}\left({\mathrm{OC}}_2\right)=\frac{1}{n}{\mathrm{A}}^{\mathrm{t}}{\left({\mathrm{OC}}_1\right)}^{\mathrm{t}}\left({\mathrm{OC}}_1\right)\mathrm{A}={\mathrm{A}}^{\mathrm{t}}\mathrm{RA}=\mathrm{\Lambda },$$E16

which is a diagonal matrix

The OC2 matrix provides the values assumed in correspondence with the objects Oi from the main components Yj. It was found that the total variability was practically absorbed by the first three main components, so that the phenomenon observed in the urban system can be sufficiently described by the OC2 submatrix formed by the first three columns.

The profiles show the coordinates of the municipalities and variables with respect to the first three factorial axes. On the basis of these results, a hierarchical classification of the area reported in the following images was also obtained (Figures 1 and 2).

Figure 1.

An hierarchical classification of the area.

Figure 2.

An hierarchical classification of the area.

As evident, the analysis outputs are the principal components that cut the cloud of points (municipalities) in the direction of greatest inertia. The coordinates of the objects (always the municipalities) and the coordinates of the characters (the variables considered)on the factorial axes (always on the same axis system) are important for determining the results (Figures 3–5).

Figure 3.

The first three main components explain respectively 27%, 21.4% and 13.5% for a total of 61.9% of the total system inertia. Each of them gives rise to a particular composition of the municipalities on the factorial axes: first factorial axis.

Figure 4.

Second factorial axis.

Figure 5.

Third factorial axis.

The analysis (not all reported for the sake of brevity) made it possible to obtain “artificial” variables (from 10 to 3) which, unlike the originals, are not correlated but equally provide information on the area. The procedure is valid when the variables to be analyzed are many 40, 50 etc. The first three main components explain respectively 27%, 21.4% and 13.5% for a total of 61.9% of the total system inertia. Each of them gives rise to a particular composition of the municipalities on the factorial axes as can be clearly seen by analyzing the thermometer graphs (Figures 3–5).

The first component explains more the commercial aspect of the areas and determines a different position of the municipalities depending on whether they are negatively or positively correlated with it. The value assumed by it, allows to classify the municipalities in the ranking and gradually group those municipalities with similar values up to constitute.

Homogeneous classes, or at least similar in relation to this aspect, with common characteristics. The same interpretation applies to the variables explained on the second and third factorial axes. But a better representation is obtained with the observation of the factorial plans (Figures 1 and 2).The origin of the axes represents the batricenter of the system of masses that gravitate around them. Urban areas are characterized as places in the territory where various and multiple functions (as well as activities) are concentrated and interacted. The different weight and the different location that define the presence and organization of these functions in the different areas represent the synthesis, the landing point, of all the transformation processes of human activities in the long term.

The first factor explains the eighty 83.4% of the total inertia. The greatest contribution to the formation of territorial morphologies given by the position of the municipalities on this axis is given by groups of motion or variable correlated with each other and linked in complex form to the social structure of the resident populations. It is a group of variables that articulates the tertiary sector in a satisfactory way, useful for recognizing urban and non-urban social forms. The second factor, with a similar distribution of municipalities, explains the aspect linked to industry to the extent of 4.54% of total inertia; the most significant variables are those that sufficiently articulate industry and the production of consumer goods. The representation of the variables on the third factorial axis highlights that the indicative variable of the mechanical industry employees alone explains about 3% of the variability of the system based on this specialization, a hierarchical but sectoral classification of municipalities is created. It is in fact a specific variable that cannot take into account the urban reinforcement of the province but that at least highlights a characterizing aspect: the high concentration of industrial activities in mechanical processing.

In the graphs 4 and 5 the position of Pescara (the largest city of the considered area) is that of the vertex of a tetrahedron, or of a hypertetrahedron, thinking of a space relative to the variables that is not three-dimensional but ten-dimensional (are the variables).

The other municipalities that occupy positions increasingly close to the center of gravity are those that represent characteristics that are increasingly closer to the morphology and less to the characteristics of rural morphologies. The purpose of this analysis is to define a map of the Province that is able to grasp, at a fine territorial scale, the incidence and location of the functions that take place in this territorial area.

The interaction of the various functions examined (residential, industrial, commercial, exception, cc) defines the typology of the different territorial morphologies. From this descriptive approach it is possible to go back, by interpretative way, to synthetic results (the map in fact) which expose the prevailing ways of the interaction between functions (land use) and territorial extension and their mutual influence. In areas with a greater concentration of industrial functions, a greater concentration of pollution of various types (soil, air, water, etc.) may be seen and it will be necessary to intervene with targeted policies to reduce negative emissions. In Figure 6 the concentric circles representing different categories of territorial (spatial) morphologies, and on them you can insert a spatial system in which the centers or the analyzed unit is positioned with its coordinates (derived from the position on the factorial axes); evaluating such a graphical representation, in relation to a first phase (the state of fact analysis) and in a second phase (project) you could observe the impact of a possible intervention on the part of analyzed territory (diachronic representation).

Figure 6.

The concentric circles representing different categories of territorial (spatial) morphologies: urban, productive, agricultural.

4. Conclusions

Finally, the usefulness of such an application lies in the consideration that complex variables can hardly be simplified and assumed uniquely in a model; instead of them we prefer an indicator (factorial axis) composed of different fractions of several variables and therefore more consistent with reality and more representative of the single variables. The profiles that can be outlined starting from the proposed model show the positions (coordinates) of the territories considered (small agglomerated, municipalities or parts of larger and more widespread urbanizations) and the variables, with respect to the factorial axes about these results, a soft hierarchical classification of the area is obtained (that is, non-rigorous nuanced and sharp outlines). You can therefore obtain spatial morphologies classes that are formed with the passage of territorial units considered from one to another cluster. Such shifts are caused perhaps by the simulation of interventions made on the territory or from variation (oscillation) of some input in the initial models.

The urban context, therefore, as a complex system, cannot be simply broken down into its constitutive dimensions (spatial, temporal, environmental, social, and economic), being the sum of the same, something more than the simple superimposition of the single dimensions, in a Gestalt vision which cannot be ignored. It becomes, therefore, the subject of analysis through the definition of the object “relationship between the size”, taking into account that the social characterizes the very existence of the city (of the territory), since there are no cities without human presence.