3.2. Implementation Procedures
Spatial analysis is defined by the urban plan of the Campus project. The result is identification of 62 private cadastral parcels in the research area.
Step 1—Defining criteria for cadastral parcels comparison (input data), linguistic variables, and terms (initial phase)
The definition of cadastral parcels set and their initial analysis identified the criteria (input variables) used in their comparison. The final set includes the seven input variables (
Figure 2). Given that cadastral parcels share some of the same spatial properties, the set of criteria is reduced for those related to location and infrastructural characteristics (microlocation, aspect, infrastructure, ecology, and vegetation).
Figure 5 shows the general structure of the fuzzy logic model; the left shows the defined seven criteria as input variables and the right shows the output variable (the bonitet assessment).
Step 2—Cadastral parcels evaluation by all criteria
By determining the ideal shape of the cadastral parcel, the deviation of each cadastral parcel from its ideal shape according to the criteria selected for their comparison was determined. The difference is defined by a crisp number, but before determining the fuzzy input values, and in order to compare cadastral parcels according to the criteria of different character, it is necessary to carry out their standardization, i.e., reduction to the relation from 0 to 1.
Step 3—Normalization of input data—linear and value functions (input data—“crisp” input)
The input variables (criteria for comparing cadastral parcels) are defined below, and linear and nonlinear (value) functions are determined for the purpose of the standardization procedure. Standardization is carried out for the purpose of definition of input variables values in the range from 0 to 1 which allows their comparison.
Construction utility
The criterion is determined by the official data of the cadastre and land register, and its standardization is defined based on data for the size and construction of the building parcel for the area of the City of Split. Section 2.2.1.1. Paragraph 15 of the Official Gazette of the City of Split [
32] defines the minimum sizes of building parcels for construction in the urban area. Based on the minimum values prescribed by the Official Gazette and expert assessment, the classes of cadastral parcels evaluation are defined:
1 st class: cadastral parcels whose area does not exceed 200 m2. The upper limit is defined in accordance with the provisions in the Official Gazette on the minimum size of the cadastral parcels for buildings that are built in a row,
2 nd class: cadastral parcels with an area in the range of 200 to 400 m2. The boundaries are defined in accordance with the provisions on the minimum size of the cadastral parcel: the lower limit for the buildings in the row (minimum parcel area is 200 m2) and the upper limit for free-standing buildings (minimum parcel area is 400 m2).
3 rd class: cadastral parcels with an area in the range of 400 to 1000 m2,
4 th class: cadastral parcels with an area in the range of 1000 to 2000 m2,
5 th class: cadastral parcels with an area of more than 2000 m2.
Standardized values of construction utility are defined by a linear transformation of the area of cadastral parcels and are given in the range from 0 (cadastral parcels that are not constructionally usable) to 1 (cadastral parcels whose area exceeds 2000 m
2) and are shown by the graph in
Figure 6.
The graph shows five linear functions depending on the defined classes of cadastral parcels area. The linear transformation is defined for each class separately and is expressed by the formula ( is the area of the cadastral parcel j, is the linear function for the i class):
Standardized area values are hereinafter referred to as .
Orientation
The criterion was determined based on slope orientation data for each cadastral parcel towards the sides of the world (showing slope exposure towards N, NE, E, SE, S, SW, W, NW, and flat areas) obtained by the Aster digital relief model analysis. The value of each cell in the output grid defines the direction in which the slope faces each side of the world. It is measured in degrees clockwise from 0° (north direction) to 360° (again north direction). Flat surfaces without slope are given a value of −1. In
Table 1, the sides of the world with defined value ranges [
33] are shown.
Values of orientation are standardized using a fourth-order polynomial value function defined by the expression (
is the orientation of cadastral parcel
j, and
is the value function defined for the orientation):
Standardized orientation values are given in the range from 0 to 1, where grade 1 is defined for the south direction, i.e., 180° within the class south, and grades 0 for the north direction, i.e., 0° to 22.5° and 337.5° to 360° within class north. Grades of 0.5 were assigned to the east (90°) and west (270°) directions.
In
Figure 7, a graphical representation of the specified function is given.
Standardized orientation values are hereinafter referred to as .
Road access
Access to the registered road is defined by a binary evaluation of the criteria:
- -
cadastral parcels that have access to a registered road are assigned a grade of 1.
- -
cadastral parcels that do not have access to a registered road are assigned a grade of 0.
As this is a criterion for evaluating cadastral parcels with two crisp grades (0 and 1), only two MFs are defined, which is a case that is very rarely applied when defining the functions of input variables. It is known that for less than three MFs it is not possible to differentiate the set. However, the experts decided to include this criterion because the absence of the same would not result in a complete analysis and objective approach when comparing cadastral parcels.
Distance from main roads
The distance from the main roads is determined as the shortest distance of the cadastral parcels to one of the four main streets: Matica Hrvatska, Vukovarska Street, Bruno Bušića Street, and Velebitska Street, and it is expressed in meters. Given the linear character of the criteria values, for the purposes of their standardization the linear transformation was defined. The distance from the main roads is defined from 0 to 1 as a criterion of minimum, i.e., grade 1 is given to cadastral parcels whose distance from the road is 0 m, while grade 0 is given to t cadastral parcels that are 1000 meters and more away from roads. The linear function of the distance from the main roads is defined by the expression (
is the distance from the main road of the cadastral parcel j, and
is the linear function defined for the distance):
In
Figure 8, a graphical representation of the linear function for the specified criterion is shown.
Standardized values of distance from main roads are hereinafter referred to as .
Slope
The criterion was determined based on terrain slope data for each cadastral parcel obtained by analysis of the Aster digital relief model and it is expressed in degrees. Given the linear character of the criteria values, for the purposes of their standardization the linear transformation was defined. The slope is defined from 0 to 1 as a criterion of minimum, i.e., grade 1 is given to cadastral parcels that have a slope from 0° to 2°, while grade 0 is given to those cadastral parcels that have a slope of 33° and more. The linear slope function is defined by the expression (
is the slope of the cadastral parcel j, and
is the expression for the linear function):
In
Figure 9, a graphical representation of the linear slope function is shown.
Standardized slope values are hereinafter referred to as .
Shape
The shape of the cadastral parcel was determined based on three independent, one-parameter indices: the index of compactness, the index of rugosity, and the index of boundary points. The choice of criteria is the same as for agricultural land (proposed in the doctoral dissertation of Iva Odak [
34]) with the difference in the choice of the cadastral parcel optimal shape. For an urban area, a solid with all equal sides (a square) is defined as the optimal cadastral parcel shape. Although the shape of the cadastral parcel is determined by the urban plan and the parameters defined in the plan should be taken in a more detailed assessment, to identify general guidelines to assess the cadastral parcels bonitet, the square was chosen as the most favorable form for building on it.
Compactness is determined by the expression [
35]
where
is area of the cadastral parcel
j, the
is the circumference of the cadastral parcel is
j, and the compactness is expressed in the range of values from 0 to 1 (compactness of the circle).
As the optimal shape of the cadastral parcel in urban areas is equal to a square with a side ratio of 1:1, it is necessary to standardize the function in such a way that its maximum value is defined for the compactness of the square that is .
The compactness values are standardized using a sixth-order polynomial value function defined by the expression (
is the compactness of the cadastral parcel
j and
is the value function defined for compactness):
Standardized values, as well as values before standardization, are defined by an absolute grade ranging from 0 (cadastral parcels of the most unfavorable shape) to 1 (cadastral parcels of square shape). Standardized values of compactness are graphically shown in
Figure 10 and are hereinafter referred to as the index of compactness.
Rugosity is defined as the ratio of the circumference of a cadastral parcel convex hull, which represents the smallest circumference of a solid and the circumference of that same cadastral parcel and is defined by the expression [
36]
where
is the circumference of the convex hull of the cadastral parcel
j,
the circumference of the cadastral parcel
j, and
the index of rugosity of the cadastral parcel
j.
The rugosity values are defined in the range 0 to 1, where grade 1 represents those cadastral parcels whose circumference is equal to the circumference of the convex hull (cadastral parcels with optimal circumference).
The number of boundary points is determined based on defining the optimal number of points (sides) of the cadastral parcels and defining their standardized function. The optimal number of points is defined concerning the optimal shape of the cadastral parcel that is equal to a square. Accordingly, the optimal number of boundary points is equal to 4.
The number of boundary points are standardized using a fifth-order polynomial value function defined by the expression (
is the number of boundary points of the cadastral parcel j, and
is the value function defined for the number of boundary points) [
37]:
Standardized values of the number of boundary points are graphically shown in
Figure 11.
The index of shape is ultimately defined as the arithmetic mean of the three listed indices:
where
is the index of compactness of the cadastral parcel
j,
is the index of rugosity of the cadastral parcel
j, and
is the number of boundary points of the cadastral parcel
j. The values of the index of shape are defined in the range from 0 to 1, where grade 1 represents cadastral parcels with optimal shape (square shape), while 0 is assigned to cadastral parcels of extremely irregular shapes.
In
Table 2 the values of the index of compactness, index of rugosity, index of the number of boundary points, and the index of shape are shown. For the sake of transparency, the indices are shown for 10 of 62 cadastral parcels.
Compliance between cadastre and land registry
The compliance of the cadastre and the land register is defined by a binary evaluation of the criteria:
- -
Cadastral parcels for which the data in the cadastre and land register are harmonized have been assigned a grade of 1.
- -
Cadastral parcels for which the data in the cadastre and land register are not harmonized were assigned a grade of 0.
As is the case with the road access criteria, cadastral parcels are evaluated with two crisp grades, 0 and 1, and accordingly, two MFs are defined with clear, crisp boundaries of belonging to a particular set (cadastral parcels belongs or does not belong to a particular set).
The evaluation of cadastral parcels was performed according to the defined criteria and is shown in
Table 3. For the sake of transparency, the evaluation is shown for 10 of 62 cadastral parcels.
Step 4—Defining MFs
A triangular MF has been selected to define the membership of the input variables. The shape, number, and range of MFs of input and output variables are shown below.
Construction utility
The input variable construction utility is defined with six MFs depending on the classes of construction utility evaluation of cadastral parcels shown in step number 3. In
Table 4, fuzzy sets of construction utility and their associated linguistic values are presented.
In
Figure 12, the shape, number, and range of MFs for the construction utility criterion are presented.
Orientation
The input variable orientation is defined by five uniformly distributed MFs with the degree of overlap depending on the value function of the cadastral parcel orientation shown in step 1. In
Table 5, fuzzy sets of cadastral parcels orientation and their associated linguistic values are presented.
In
Figure 13, the shape, number, and range of MFs for the orientation criterion are presented.
Road access
The input variable road access is defined with two functions representing two crisp sets. In
Figure 14 the shape, number, and range of MFs for the input variable road access are shown.
Distance from main roads
The input variable distance from the main roads is defined with six uniformly distributed MFs with the degree of overlap
. In
Table 6, fuzzy sets and their associated linguistic values are shown.
In
Figure 15, the shape, number, and range of MFs for the distance from the main roads criterion are presented.
Slope
The input variable slope is defined with seven MFs depending on the value function of the cadastral parcel slope shown in step 3. In
Table 7, fuzzy sets of cadastral parcels slope and their associated linguistic values are shown.
In
Figure 16, the shape, number, and range of MFs for the slope criterion are presented.
Shape
The input variable shape, as well as the distance from the main roads, is defined with six uniformly distributed MFs with the degree of overlap
. In
Table 8, fuzzy sets and their associated linguistic values are shown.
In
Figure 17, the shape, number, and range of MFs for the shape criterion are presented.
Compliance between cadastre and land registry
The input variable compliance between cadastre and land registry is defined with two functions representing two crisp sets. In
Figure 18, the shape, number, and range of MFs for the input variable compliance between the cadastre and land registry are shown.
Bonitet
The output variable bonitet of the cadastral parcels is defined with eleven uniformly distributed MFs with a degree of overlap
. In
Table 9, fuzzy sets of bonitet and their associated linguistic values are presented. The reason for choosing a larger number of output fuzzy sets is the need for greater accuracy of the output data when comparing cadastral parcels with small differences in the attribute values of the input variables.
In
Figure 19, the shape, number, and range of MFs for the output variable bonitet are shown.
Step 5—Defining the rule base in the knowledge base
As already mentioned, the development of a rule base is a complex and time-consuming process. The total number of rules that need to be defined within this research is
Given the large number of rules that need to be defined, as well as the impossibility of maintaining consistency when defining the relationships of their causal and consequential relations, the methodology used to mathematically define the logical relationship of input and output variables was applied in creating the rule base. The advantage of the proposed methodology is primarily found in maintaining consistency in defining the rules, and then in facilitating and accelerating expert assessment. In contrast to the traditional approach to defining rules based on individual expert assessment of the input–output relationship in logical equations, by application of this methodology expert knowledge was primarily used to define weights, which on the one hand took into account the relative ratio of input variables, and on the other hand defined their association with the output variable. Using the AHP method, the weights (W) of the input variables were defined by expert evaluation. Further, the consistency ratio was determined, CR < 0.1, and for each expert, criteria weights satisfied this condition. In
Table 10, their mean values determined by the arithmetic mean are shown. Averaging by the arithmetic mean was primarily chosen because in such a defined result all input values are equally represented. After defining the weight mean values, their percentage normalization (W ‘) was performed. The distribution of the weighting coefficient value according to the number of membership functions of the output variable (seven MFs) determines the relationship between the individual causal part of the logical equation and its consequences, i.e., its bonitet value. After the elimination procedure, which includes the elimination of rules that do not correspond to the defined data set and the elimination of redundant rules, the rules were reduced to 861.
Table 10 shows a part of the rules that are associated with cadastral parcel 6505/1, i.e., the MFs of its input and output variables.
Step 6—Conversion of input data crisp values into fuzzy values using MFs (fuzzification)
In the process of fuzzification, fuzzy values are assigned to the input crisp values. Fuzzy sets are described by triangular MFs, i.e., for each given crisp value are two fuzzy values. Each input value has got the sum of the membership degrees equal to one. Input values which membership is defined only by one fuzzy set are called singleton set (they membership to that set is one hundred percent, i.e., ). Singleton sets of input variables are
- -
Construction utility:
- -
Orientation:
- -
Distance from the main roads:
- -
Slope:
- -
Shape:
The membership of cadastral parcels for the input variables “Road access” and “Compliance of cadastre and land registry” is defined with two singleton sets (cadastral parcels belong to either one or the other set).
Steps 7–9—Mamdani Fuzzy Inference System
The Mamdani model of local fuzzy inference was used to implement the fuzzy logic model. In the first step of the implementation, the interference operator is selected, i.e., the logical operator T-norm, by which the connection in the logical equations is defined by the intersection of the input variables MFs. The “AND” logical minimum operator is selected because it best defines the character of the input variables. In the second step of the reasoning process, the Mamdani implicator, i.e., the minimum operator, was applied. Accordingly, the resultant value of the causal part of the equation always corresponds to the minimum value of the input variables membership degrees (Step 7).
Clipping method was chosen to define the output fuzzy set. The conclusion of each equation is cut to a height determined by the minimum membership degree of the input variables. By applying the aggregation operator, all output fuzzy sets were combined into a single fuzzy set. For the decision operator, which includes the implication and aggregation operators, the Mamdani decision operator is selected, i.e., the max-min operator. The resultant value is defined as the union of previously defined segments of all fuzzy sets of one output variable (Step 8).
The conversion of the resulting output fuzzy values into their crisp numerical values is called the process of defuzzification, i.e., process of sharpening (Step 9). Defuzzification is performed by the center of gravity method (centroid method).
Step 10—Output data set (“crisp” estimated bonitet value of cadastral parcels)
By determining the ideal shape of the cadastral parcel (parcel with a bonitet value of 1), the deviation of each cadastral parcel from their ideal shape was determined according to the criteria selected for their comparison. The difference is defined by crisp number, i.e., bonitet value in the range from 0 (lowest bonitet value) to 1 (highest bonitet value). The bonitet values for all private cadastral parcels in the project area of the Campus, University of Split are shown in
Table 11.
Values of the private cadastral parcels bonitet values in the project area of the Campus, University of Split range from BV (6528/8) = 0.354 as the minimum value (smallest bonitet value) to BV (6566/2) = 0.900 as the maximum value (highest bonitet value). Most cadastral parcels, more precisely 52 of 62, have a bonitet value greater than 0.5 (84% of all parcels). The obtained bonitet values will be used for land reallocation in Urban Renewal process.
In
Figure 20, the distribution of private cadastral parcels in the project area of the Campus, University of Split with their bonitet values is shown. Light red tones show those cadastral parcels with relatively small bonitet values, while dark red tones indicate the best rated cadastral parcels according to defined criteria (with the highest bonitet values).