**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 X1, X2, … *:*Xn, from the point of view of its variability. It is proposed to represent the same phenomenon with new centered variables Y1, Y2, … *:*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,:


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 X10) has been replaced by S = scraps (S1 ...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*<sup>j</sup> <sup>¼</sup> <sup>1</sup> *σ 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

$$\mathbf{OC\_1} = (\mathbf{M\_x} \mathbf{-M}) \mathbf{D^{-1}} \tag{9}$$

with D ¼ diag s1 ð Þ … sn , diagonal matrix of the standard deviations. The matrix of variances and covariances of Z is then

$$\mathbf{R} = \frac{1}{n} (\mathbf{OC\_1})^\mathrm{t} (\mathbf{OC\_1}),\tag{10}$$

generic element

$$\mathbf{r}\_{\rm hk} = \text{cov}(\mathbf{Z}\_{\rm h}, \mathbf{Z}\_{\rm k}) = \frac{1}{n} \sum\_{r=1}^{n} \mathbf{Z}\_{\rm rh} \mathbf{Z}\_{\rm rk} \tag{11}$$

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

$$\mathbf{S} = (\mathbf{M}\_{\mathbf{x}} \mathbf{-M})^{t} (\mathbf{M}\_{\mathbf{x}} \mathbf{-M}) \tag{12}$$

and obtaining

$$\mathbf{R} = \mathbf{D}^{-1} \boldsymbol{\Sigma} \mathbf{D}^{-1} \tag{13}$$

The eigenvalues and eigenvectors of R were then calculated.

Said L ¼ diag l1 ð Þ … l10 the matrix of the eigenvalues, in ascending order and called A the matrix that has the corresponding eigenvectors for columns is

$$\mathbf{RA} = \mathbf{A}\boldsymbol{\Lambda} \tag{14}$$

that is

$$\mathbf{A}^{\mathbf{t}} \mathbf{R} \mathbf{A} = \Lambda \tag{15}$$

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.

OC2 <sup>¼</sup> OC1T<sup>t</sup> <sup>¼</sup> ð Þ OC1 A and therefore the variance and covariance matrix of Y is the

$$
\Delta = \frac{1}{n} (\mathbf{OC}\_2)^\dagger (\mathbf{OC}\_2) = \frac{1}{n} \mathbf{A}^\dagger (\mathbf{OC}\_1)^\dagger (\mathbf{OC}\_1) \mathbf{A} = \mathbf{A}^\dagger \mathbf{RA} = \Lambda,\tag{16}
$$

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

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**).

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**).

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

**Figure 1.** *An hierarchical classification of the area.*

**Figure 2.** *An hierarchical classification of the area.*

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

#### **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.*

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

#### **Figure 5.** *Third factorial axis.*

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

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

**Figure 6.**

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

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).
