# Rough Set Theory for Real Estate Appraisals: An Application to Directional District of Naples

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rough Set Theory: Original Idea and Analytical Aspects

- $S=\langle U,Q,V,\rho \rangle $ is an information system, where U is a finite set of objects, Q is a finite set of attributes, $V={U}_{q\in Q}{V}_{q}$ being V
_{q}a domain of q attribute, and $\rho :U\times Q\to V$ is a function such that $\rho \left(x,q\right)\in {V}_{q}$ for every $q\in Q$ and $x\in U$.$P\subseteq Q$ and $x,y\in U$: x and y are considered as indiscernible objects by the set of attributes P in S if $\rho $ (x,q) = $\rho $ (y,q), for every $q\in P$.The elementary groups of P relationships are defined as equivalence classes in S.Essentially, two objects with similar attributes fall in the same equivalence class and are considered indiscernible. - Two types of attributes exist: “conditional” attributes represent the observations; “decisional” attributes represent the “judgments” detected or assigned for the overall set of conditional attributes, with reference to the specific object.If conditional attributes are equal but decisional attributes are different, the set of objects comes to be in a “rough” region (see Figure 2).The objects that may be distinguished are inserted in different equivalence classes, identified by approximations. Every equivalence class is identified by upper and lower approximation regions.For $P\subseteq Q$ and $Y\subseteq U$, $\underset{\_}{P}Y$ is defined as the lower approximation of Y and $\overline{P}Y$ as the upper approximation of Y:$$\underset{\_}{P}Y=\left\{X\in {P}^{*}andX\subseteq Y\right\}$$$$\overline{P}Y=\left\{X\in {P}^{*}andX\cap \text{}Y\ne 0\right\}$$The lower approximation $\underset{\_}{P}Y$ represents the set of elements U “certainly” included in Y, applying the set of conditional attributes P; the upper approximation $\overline{P}Y$ represents the set of elements U that “if possible” is included in Y, applying the same set of attributes P.The relationship between number of elements of the lower approximation and number of elements of the upper approximation is defined as “accuracy” of approximation:$${\mu}_{p}\left(Y\right)=\frac{card\left(\underset{\_}{P}Y\right)}{card\left(\overline{P}Y\right)}$$$${\gamma}_{P}\left(X\right)=\frac{{{\displaystyle \sum}}_{i=1}^{n}card\left(\underset{\_}{P}{Y}_{i}\right)}{card\left(U\right)}$$The quality classification expresses synthetically the relationship between the numbers of correctly classified objects respect to the total number of objects.
- The choice of the equivalence class is performed by “if then” rules, rules that are measured in terms of “precision” and “coverage” in relation to analyzed objects. The “precision” defines the rule (“if”) and identifies the objects, while “coverage” detects the fraction of objects that responds positively to the rule (then).
- The best rule is, precisely, defined as that which provides the best coverage (better generalization capacity).

## 3. Rough Set Theory Applied to Real Estate Appraisals

## 4. Case Study: Application of RST to Directional District of Naples

^{2}), presence/absence of parking space (represented by a dummy variable: 1 = if present, 0 = if absent), and maintenance status (represented by a dummy variable: 1 = if office unit was recently renovated featuring luxury finishes, 0 = if office unit has a normal state of maintenance). The market price (in euros) is the decisional variable (decisional attribute).

- IF Area = (51, 53) → Price Class = 1;
- IF Area = 50 ˄ Park = 0 → Price Class = 1;
- IF Area = 60 ˄ Maintenance = 0 ˄ Park = 0 → Price Class = 1;
- IF Area = 58 → Price Class = 2;
- IF Area = (50, 55) ˄ Park = 1 → Price Class = 2;
- IF Area = (80, 54, 52, 62, 67, 90) → Price Class = 3;
- IF Area = (120, 140, 108, 110) → Price Class = 5;
- Can also to be defined the “approximate” rules, which are a specificity of RST:
- IF Area = 55 ˄ Park = 0 → Price Class = 1 OR 2;
- IF Area = 60 ˄ Maintenance = 1 → Price Class = 1 OR 3;
- IF Area = 65 → Price Class = 1 OR 3;
- IF Area = 60 ˄ Park = 1 → Price Class = 2 OR 4.

^{2}, with parking space and without luxury maintenance status, we do not have the opportunity to apply the rules identified by RST. For this reason, RST may be integrated with a functional extension also called “Value Tolerance Relation” (VTR).

_{j}is the VTR that may assume continuous values included in the interval [0, 1], x and y are objects, c

_{j}(x) and c

_{j}(y) indicate the measures of attribute j in the objects x and y, max and min represent the intersection and the union of fuzzy sets, and k is the value threshold to distinguish two objects.

_{j}equals 1, the two objects considered are highly similar, while if R

_{j}equals 0 the same two object are completely different.

_{j}and can be applied for all objects. The relationship between all attributes of an object and the conditional part of the rules is determined by intersection of all comparison sets with the rule. There will be several values of R

_{j}(one for each attribute of object and conditional part of rule), and the minimum R

_{j}will be selected between n comparisons among the attribute of the object and conditional part of the rule:

^{2}with a parking space and normal maintenance status, taking into account the rules that have been determined using RST for real estate data and k-threshold as standard deviation (see Table 2).

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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No. | Area (m^{2}) | Maintenance | Parking Space | Price (€) | Price Class |
---|---|---|---|---|---|

1 | 51 | 1 | 0 | 60,828 | 1 |

2 | 53 | 0 | 0 | 71,315 | 1 |

3 | 55 | 0 | 0 | 79,705 | 1 |

4 | 60 | 1 | 0 | 79,705 | 1 |

5 | 60 | 0 | 0 | 79,705 | 1 |

6 | 60 | 0 | 0 | 79,705 | 1 |

7 | 50 | 1 | 0 | 82,222 | 1 |

8 | 65 | 0 | 0 | 82,222 | 1 |

9 | 55 | 0 | 0 | 92,290 | 2 |

10 | 58 | 0 | 0 | 92,290 | 2 |

11 | 60 | 0 | 1 | 96,485 | 2 |

12 | 50 | 1 | 1 | 100,680 | 2 |

13 | 55 | 0 | 1 | 100,680 | 2 |

14 | 50 | 0 | 1 | 109,070 | 2 |

15 | 54 | 0 | 1 | 113,265 | 3 |

16 | 52 | 0 | 1 | 113,265 | 3 |

17 | 80 | 0 | 1 | 117,460 | 3 |

18 | 65 | 0 | 0 | 117,460 | 3 |

19 | 80 | 0 | 0 | 125,850 | 3 |

20 | 62 | 0 | 0 | 125,850 | 3 |

21 | 60 | 1 | 0 | 142,630 | 3 |

22 | 67 | 0 | 1 | 151,020 | 3 |

23 | 90 | 1 | 1 | 159,410 | 3 |

24 | 60 | 0 | 1 | 167,800 | 4 |

25 | 108 | 0 | 1 | 226,530 | 5 |

26 | 110 | 0 | 1 | 234,920 | 5 |

27 | 120 | 0 | 0 | 236,598 | 5 |

28 | 120 | 1 | 0 | 251,700 | 5 |

29 | 140 | 0 | 1 | 251,700 | 5 |

30 | 140 | 1 | 1 | 268,480 | 5 |

Description | Area (m^{2}) | Maintenance (m^{2}) | Parking Space (m^{2}) | Price (€) |
---|---|---|---|---|

Std. Deviation | 27.63 | 0.45 | 0.51 | 62,400.24 |

Median | 60.00 | 0.00 | 0.00 | 113,265.00 |

Average | 73.00 | 0.27 | 0.47 | 133,694.65 |

Min | 50.00 | 0.00 | 0.00 | 60,827.50 |

Max | 140.00 | 1.00 | 1.00 | 268,480.00 |

Price Class | Min | Max | Difference |
---|---|---|---|

1 | 60,000 | 85,000 | 25,000 |

2 | 85,000 | 110,000 | 25,000 |

3 | 110,000 | 165,000 | 50,000 |

4 | 165,000 | 210,000 | 50,000 |

5 | Over 210,000 | - |

List of equivalence classes in which conditional attributes have equal modalities | {1}; {2}; {5, 6}; {7}; {10}; {12}; {13}; {14}; {15}; {16}; {17}; {19}; {20}; {22}; {23}; {25}; {26}; {27}; {28}; {29}; {30}; {3}; {4}; {8}; {9}; {11}; {18}; {21}; {24}. |

LOWER APPROXIMATION CLASS 1: all objects that are indistinguishable as well as having a price belonging to Class 1 | {1}; {2}; {5, 6}; {7}. |

UPPER APPROXIMATION CLASS 1: all objects that may belong to Class 1 | {1}; {2}; {5, 6}; {7}; {3}; {9}; {4}; {21}; {8}; {18}. |

Number of objects in the class: 8 | |

Number of objects in the lower approximation: 5 | |

Number of objects in the upper approximation: 11 | |

Accuracy: 0.4545 |

LOWER APPROXIMATION CLASS 2: all objects that are indistinguishable as well as having a price belonging to Class 2 | {10}; {12}; {13}; {14}. |

UPPER APPROXIMATION CLASS 2: all objects that may belong to Class 2 | {10}; {12}; {13}; {14}; {3}; {9}; {11}; {24}. |

Number of objects in the class: 6 | |

Number of objects in the lower approximation: 4 | |

Number of objects in the upper approximation: 8 | |

Accuracy: 0.5000 |

LOWER APPROXIMATION CLASS 3: all objects that are indistinguishable as well as having a price belonging to Class 3 | {15}; {16}; {17}; {19}; {20}; {22}; {23}. |

UPPER APPROXIMATION CLASS 3: all objects that may belong to Class 3 | {15}; {16}; {17}; {19}; {20}; {22}; {23}; {4}; {21}; {8}; {18}. |

Number of objects in the class: 9 | |

Number of objects in the lower approximation: 7 | |

Number of objects in the upper approximation: 11 | |

Accuracy: 0.6364 |

LOWER APPROXIMATION CLASS 4: all objects that are indistinguishable as well as having a price belonging to Class 4 | |

UPPER APPROXIMATION CLASS 4: all objects that may belong to Class 4 | {11}; {24}. |

Number of objects in the class: 1 | |

Number of objects in the lower approximation: 0 | |

Number of objects in the upper approximation: 2 | |

Accuracy: 0.0000 |

LOWER APPROXIMATION CLASS 5: all objects that are indistinguishable as well as having a price belonging to Class 5 | {25}; {26}; {27}; {28}; {29}; {30}. |

UPPER APPROXIMATION CLASS 5: all objects that may belong to Class 5 | {25}; {26}; {27}; {28}; {29}; {30}. |

Number of objects in the class: 6 | |

Number of objects in the lower approximation: 6 | |

Number of objects in the upper approximation: 6 | |

Accuracy: 1.0000 |

Object | Area = 130 m^{2} | Maintenance = 0 | Parking Space = 1 | R_{j} (Min R_{j} Rule) | Rule Selected |
---|---|---|---|---|---|

k-threshold | 27.63 | 0.45 | 0.51 | ||

R_{j} Rule 1 | 0.0000 | 1.0000 | 0.0000 | 0 | Rule 7 |

R_{j} Rule 2 | 0.0000 | 1.0000 | 0.0000 | 0 | |

R_{j} Rule 3 | 0.0000 | 1.0000 | 0.0000 | 0 | |

R_{j} Rule 4 | 0.0000 | 1.0000 | 0.0000 | 0 | |

R_{j} Rule 5 | 0.0000 | 1.0000 | 1.0000 | 0 | |

R_{j} Rule 6 | 0.0000 | 1.0000 | 0.0000 | 0 | |

R_{j} Rule 7 | 0.2838 | 1.0000 | 0.0000 | 0.2838 |

**Table 11.**Comparison between multiple regression analysis (MRA) and Rough Set Theory (RST) in terms of forecasting performances.

Percentages of Errors (MRA) | ||||

0%–10% | 10%–20% | 20%–30% | more than 30% | Mean Absolute Percentage Error |

46.67% | 23.33% | 20.00% | 10.00% | 14.44% |

Percentages of Errors (RST) | ||||

0%–10% | 10%–20% | 20%–30% | more than 30% | Mean Absolute Percentage Error |

50.00% | 36.66% | 6.67% | 6.67% | 12.29% |

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**MDPI and ACS Style**

Del Giudice, V.; De Paola, P.; Cantisani, G.B.
Rough Set Theory for Real Estate Appraisals: An Application to Directional District of Naples. *Buildings* **2017**, *7*, 12.
https://doi.org/10.3390/buildings7010012

**AMA Style**

Del Giudice V, De Paola P, Cantisani GB.
Rough Set Theory for Real Estate Appraisals: An Application to Directional District of Naples. *Buildings*. 2017; 7(1):12.
https://doi.org/10.3390/buildings7010012

**Chicago/Turabian Style**

Del Giudice, Vincenzo, Pierfrancesco De Paola, and Giovanni Battista Cantisani.
2017. "Rough Set Theory for Real Estate Appraisals: An Application to Directional District of Naples" *Buildings* 7, no. 1: 12.
https://doi.org/10.3390/buildings7010012