Maurizio d’Amato
Property Valuation and Investment Technical University
-1st Faculty of Engineering - Politecnico di Bari – Italy - Tel. +39 80 59633339
Fax +39 80 5963348 email:
madamato@interfree.it
research center:
www.noaves.com
website:http://mdamato.altervista.org
Rough Set Theory as Automated Valuation
Methodology: The Whole Story
LOOKING FOR AN IMPROVEMENT OF ACTUAL MASS
APPRAISAL METHODOLOGIES…WHY?
Two different points of view
1.
The former is neoclassical one whose mathematical foundations were originally proposed by Arrow and Debreu (1954) ; the mile stone is solving maximization problems. a representative agent with unlimited computational and predictional abilities, while the human mind, has serious structural limitations to their “power”. The application of hedonic price theory (Griliches 1971; Rosen 1974) is based on assumptions of general equilibrium, and the driving logic is based on homo oeconomicus behaviour in a static framework. Therein lies the weakness of this approach. Its underlying assumptions, notably the smooth, continuous and linear relationships between the variables under study, and (in the economic sense) rational behaviour of the buyer and seller, may be not always realistic, as the market operates within a variety of constrains, and the individual market actors suffer from inconsistency and idiosyncracy, as well as information and power imbalances.2.
Simon’s contributions on bounded rationality and problem-solving (Simon,1957; 1979;1981). Human mind performances cannot fit the standards of such “olympic” perfection and consequently the economic agent, when faced with a problem, will find solutions commonly not optimal but “satisficing” according to his subjective and modifiable aspiration level using a “bounded rationality”.IF THIS IT IS TRUE WE MAY NEED METHODOLOGIES WHICH CAN BE HELPFUL WHEN HEDONIC PRICE THEORY MAY BE INADEQUATE…OR THAT MAY BE CLOSER TO HUMAN REASONING.
Maurizio d'Amato
ROUGH SET THEORY: UNIVERSE, OBJECTS, ATTRIBUTE,
INFORMATION FUNCTION,INDISCERNIBILITY RELATION
:
( , )
qf U Q
× →
V and f x q
∈
V
∀ ∈
q
Q and x
∈
U
Maurizio (object Q) belongs to a group of Professors in Real Estate
(selected universe S) among the professors in Real Estate of the
world (finite universe U) with a neck tie which can not be more than
one (Vq) assuming a specific information function
An information system is composed by…
S
=
∑
U Q V
, ,
q
,
f
⌡
Q = {C
∪
D } where C is condition attributes and D is a set of
ROUGH SET THEORY: UNIVERSE, OBJECTS, ATTRIBUTE,
INFORMATION FUNCTION,INDISCERNIBILITY RELATION
{
( , )
:
( ),
}
N
q
I
=
x y
∈ ×
U
U
f
y
q
∈
N
B 1 yes 110 B 0 no 110 B 0 no 110 B 1 yes 110 A 0 no 90 B 0 no 100 B 0 no 100 A 1 yes 90 B 1 yes 90 A 0 no 90 PRICE (D) PARKING (C) COMM AREA (C){
}{
}
{
1
,
4
,
5
,
6
,
8
,
9
2
,
3
,
10
,
7
}
)
(
PARKING
=
IND
{
}{ }{
}
{
1
,
2
,
3
,
6
4
,
5
10
,
9
,
8
,
7
}
)
_
(
COM
AREA
=
IND
{
}
(
PARKING
_
COMM
_
AREA
) { }{ }{ }{ }{ }
=
{
1
,
6
2
,
3
4
,
5
7
,
10
8
,
9
}
IND
Maurizio d'Amato
ROUGH SET THEORY: LOWER AND UPPER APPROXIMATIONS
{
}
( )
:
( )
N X
−
= ∈
x U N x
⊆
X
{
}
( )
: ( )
0
N X
−= ∈
x U N x
∩ ≠
X
{ }
1
,
6
{ }
2
,
3
Here you can find the lower and the upper approximation for a property value
equal to A – class of values . A way to represent reality closer to human
behaviour…
CRISP SET, ROUGH SET OR FUZZY SETS?
The illustration help us in
distinguishing fuzzy sets
from rough sets
D 1 yes 110 B 0 no 110 B 0 no 110 D 1 yes 110 A 0 no 90 B 0 no 100 D 220 215 B 0 no 100 C 215 210 A 1 yes 90 B 210 205 B 1 yes 90 A 205 200 A 0 no 90 PRICE (D) PARKING (C) COMM AREA (C)
7, 10 (110,yes) 8, 9 (110,no) 1, 6 (90,no 4, 5 (100,no) 2, 3 (90,yes)
SQM, PARKING
2, 3, 7, 10 1, 4, 5, 6, 8, 9PARKING
7, 8, 9, 10 4, 5 1, 2, 3 ,6SQM
conditional
attribute
classes of equivalence 7,10 - D 2,4,5,8,9 -B 1,3,6 - APRICE
decisional attribute
classes of equivalenceGETTING USED TO RST AS PROPERTY VALUATION METHOD
(article 2002)…STEP 2- THE DECISIONAL TABLE
D
C
I
I
IF
D
C
→
⊆
The born of a rule in the RST
Maurizio d'Amato
GETTING USED TO RST AS PROPERTY VALUATION METHOD
(article 2002)…STEP 3 - THE RULES
B
PRICE
no
park
sqm
If
=
100
∧
=
⇒
=
We are interested only in deterministic rule, our job require precise information.
The rule must be closer to the object of our universe, therefore we must choose
the ruile with the highest number of attribute because we deal with complex
information
{
}
(
PARKING
_
COMM
_
AREA
) { }
=
4
,
5
⊂
D
B=
{
2
,
4
,
5
,
8
,
9
}
IND
B
PRICE
no
park
sqm
If
=
110
∧
=
⇒
=
{
}
(
PARKING
_
COMM
_
AREA
) { }
=
8
,
9
⊂
D
B=
{
2
,
4
,
5
,
8
,
9
}
GETTING USED TO RST AS PROPERTY VALUATION METHOD
(article 2002)…STEP 4 - THE RULES AND THE OBJECT
B
PRICE
no
park
sqm
If
=
100
∧
=
⇒
=
THE RELATION BETWEEN THE OBJECT AND THE RULE IT IS CRISP IN THE WORK OF 2002. BUT YOU MAY FIND A PROPERTY WHO DOES NOT HAVE THE SAME ATTRIBUTE OF THE RULE…DOES NOT FIT THE COLOURS…
RULE
?
OBJECT
SQM
Maurizio d'Amato
max(0, min( ( ),
( ))
max( ( ),
( )))
( , )
j j j j jc x c y
k
c x c y
R x y
k
+ −
=
GETTING USED TO RST AS PROPERTY VALUATION METHOD
(articles 2003-2007)…THE RULE AND THE OBJECT…from crisp
indiscernibility relation to VTR
VALUED TOLERANCE RELATION
(
)
0
10
0
10
)
60
;
0
max(
10
)
190
10
120
;
0
max(
;
b=
+
−
=
−
=
=
ac
c
R
( , )
max(0;120 10 125)
max(0;5)
5
0, 5
10
10
10
R a b
=
+ −
=
=
=
No similar, according k=10
Similar, according k=10, at 0.5 level
An object may belong or not to a set therefore a
rule must or must not be applied…Too strong for
the real estate market. The Rough Set may
become fuzzy, may have a membership relation
with different values…THE VALUE
TOLERANCE RELATION
GETTING USED TO RST AS PROPERTY VALUATION METHOD
(articles 2003-2007)…THE RULES AND THE OBJECT
If we use a VTR we must develop a ranking system, as we may have different grade
of “approximation” for different attributes We need criteria to rank the relationship
between object and rule
1
( , )
min(
( , ))
n j j jR x
ρ
R x
ρ
==
1 1( , )
stmax(
( , ))
m j criteria j jR x
ρ
R x
ρ
==
2 1 1( , )
ndmax(
( , ))
n m j criteria j j jR x
ρ
R x
ρ
= ==
∑
This means the we are looking for the union of all the set (Tsoukiàs A., Vincke
Ph.(2000): A Characterization of PQI Interval Orders, to appear in Discrete Applied
Mathematics).But the may have a lot of objects and rules! Therefore we must
Maurizio d'Amato
GETTING USED TO RST AS PROPERTY VALUATION METHOD
(articles 2003-2007) The first application – 2004 – k – subjective. The
second application 2007 k measured as stand.deviation…
In the first work (d’Amato,2004) I applied a subjective k threshold. In the forthcoming (d’Amato,2007) and in this work an objective measure is given: the k threshold should be the standard deviation of attribute of the object componing the sample of properties to be valued
