Design Enhancement by Fuzzy Logic in Architecture
Bzer Ciftciogh
Delft University of Technology, Faculty of Architecture
Building Technology, 2628 CR Delft, The Netherlands
Fuzzy logic systems find application especially in
engineering systems due to their suitability for applications
dealing with the concept of Takagi-Sugeno fuzzy model.
However, the fuzzy concept is particularly valid also in the areas
where information is qualitative. Exact science applications deal
with the information by modeling and thereafter identifying the
relationships in the model by suitable computation. In contrast,
the fuzzy logic employment in soft sciences is not as
straightforward as it is in exact sciences and special care should
be taken in the former case. Analogous to exact sciences, the
majority of soft information sources belongs to soft sciences
where the quantities dealt with are usually not measurable in the
engineering sense. Therefore, for soft sciences fuzzy logic is an
important means for dealing with associated imprecise
information processing. In spite of this, the employment of fuzzy
logic in soft sciences is not common. In this work, aspects of
fuzzy logic implementation in the areas of soft sciences is pointed
out. This is exemplified by a design application i n building
technology using a soft design data set from a real life
environment.
Abstract-
I. INTRODUCTION
Since Zadeh's [1,2] introduction of fuzzy sets, many systems
have been designed which use the concept of fuzzy sets.
Depending on the role of fuzzy logic in these systems, they
are collectively referred to as fuzzy logic systems or
mentioned in the context of fuzzy logic. The utilization of
fuzzy logic in engineering systems is greatly simplified by the
Takagi-Sugeno concept in contrast with the Mamdani concept
for fuzzy computation. The main difference lies in the
consequent of Fuzzy rules. Mamdani fuzzy systems use fuzzy
sets as rule consequent whereas TS fuzzy systems employ
linear functions of input variables as rule consequent. Fuzzy
set representation of vague information naturally is very
appealing due to ubiquitous vague information in many
technological areas. Such vague information is especially of
concern in the sot? sciences where direct measurement of the
quantities of interest mostly is not possible. In such situations,
assessment on the subject in fuzzy logic terms plays the
essential role. In this respect io make a vague assessment or
decision in the form of a fuzzy set is more appropriate than to
make an erroneous crisp assessment or decision. The
essential concern of this research is the building technological
data for architectural design where the data come from
various disciplines and therefore they can be qualitative as
much as they might be quantitative. Although both, fuzzy
logic and soft sciences are well established, the direct
interaction of these technologies is not commonly exercised.
0-7803-7810-5/03/$17.00
02003 IEEE
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The reason for this might partly be due to rich mathematical
background behind fuzzy logic making direct access difficult
for users, which are not in the same field. Conversely,
someone from engineering sciences does not easily realize the
problems subject to interest in soft sciences. Because of this
inconvenience, the pace of fuzzy logic employment in soft
sciences is rather slow. With the advances in interdisciplinary
research, it is almost certain that, fuzzy logic is going to play
an important role in information processing tasks of soft
sciences by the coordination of various efforts from different
fields. Today, these types of activities are rapidly increasing
thanks to advances in the information and communication
technology (ICT).
The organization of the paper is as follows. In the next section
TS fuzzy model is briefly explained for the sake of
completeness. In the third section the nature of the data set
from soft sciences is pointed out and data preparation for
fuzzy logic is described. To circumvent certain difficulties in
the pure realm of fuzzy logic, the necessity of implementation
by a neural network equivalent of fuzzy logic is presented.
The same section describes a case study for design
enhancement by fuzzy logic in Architecture. This is followed
by conclusions.
11.
FUZZY MODELING
2.1 Takagi-Sugeno Fuzzy Modeling
Takagi-Sugeno [3] type h z y modeling consists of a set of
fuzzy rules with a local input-output relation in a linear form.
R, :If xi is Ai, and ...x, is A,,
(1)
Then j j = a r x + b , . i=1,2 ,......, K
where Ri is the ith rule, x=[~~q~,.....,x.]~~X
is the vector of
input variables; Ail, An, ...,A, are fuzzy sets andy, is the rule
output; K is the number of rules. The output of the model is
calculated through the weighted average of the rule
consequents of the form
"
y=-
i;P,(4i
i;P,
,=1
(2)
(x)
*=1
In (2), pi(x) is the degree of activation of the ith rule
The IEEE International Conferenceon Fuzzy Systems
where
~~~
~
II
(Y
?-"v,-j,
> R 4 0 . 1 1 is the membershio function of the
I
~
1
fuzzy set A B at the input (antecedent) of Ri.
To form the fuzzy system model from the data set with N
data samples, given by
T
X = [ x , , x2, .... .....&] , y=Iy,,y2, ......,YNI'
(4)
where each data sample has a dimension of n (N>>n). First
the structure is determined and afterwards the parameters of
the structure are identified. The number of rules characterizes
the stnicture of a fuzzy system. The number of rules is
determined by clustering methods. Fuzzy clustering in the
Cartesian product-space XxY is applied to partition the
training data. The partitions correspond to the characteristic
regions where the system's behavior is approximated by local
linear models in the multidimensional space.
r
Given the training data and the number of clusters K, a
suitable clustering algorithm [4] is applied. One of such
clustering algorithms is known as Gustafson-Kessel (GK) [ 5 ] .
As result of the clustering process a fuzzy partition matrix U
is obtained. The fuzzy sets in the antecedent of the rules are
identified by means of the partition matrix U which has
dimensions [ N a ; N is the size of the data set. The ik-th
element of ,&E [0,1] is the membership degree of the i-th data
item in cluster k; that is, the ith row of U contains the point
wise description of a multidimensional fuzzy set. Onedimensional fuzzy sets A , are obtained from the
multidimensional fuzzy sets by projections onto the space of
the input variables xP This is expressed by the point-wise
projection operator of the form
p . 4 , , ( x j k ) = Proij(fiik)
(5)
The point-wise defined fuzzy sets A , are then approximated
by appropriate parametric functions. The consequent
parameters for each rule are obtained by means of linear least
square estimation. For this, consider the matrices
T
X=[xl, ..., x ~ , ] XJX,I](extended matrix [Nx(n+l)] ) ; Ai
(diagonal matrix dimension of [ N x N ] ) and
XE=[(AIX);
(A2X.); ...... (AKXO)] ( [ N M ( n + l ) ] )
where the diagonal matrix Ai consists of
membership degree as its k-th diagonal element
.Pi(+)= xPC
(xk
)
(6)
normalized
(7)
Pj ( x k )
j.1
The parameter vector r9 dimension of [K(n+l)] is given by
s'; ......... ; %TIT
(8)
where 9,T=[a? bi] ( I S i S K ) .
80
Now, if we denote the input and output data sets as XE and Y
respectively then, the fuzzy system can be represented as a
regression model of the matrix form
y=XE 9 + e
(9)
It is interesting to note that, the formulation given by (S), is
the general regression formalism in a linear system modeling
where 9 is model parameter vector; XEregressor matrix; y is
regressant or system output vector in a general multivariable
model and the solution is well established.
2.2. Fuzzy Modeling of Soft Data
A data set from soft sciences is expectedly a mixture of
linguistic and numerical data where linguistic character is
anticipated to be dominant. Before the structure of a fuzzy
model is defined, data should be preliminarily processed as
suitable inputs for the model. For this purpose, each variable
in the input vector
X=[Xl,X2,...,x.1 T
(10)
(X=[x1,x2,
..., X N l 7
and the output vector
Y2, ..., YmlT
T
Y'LYI,
(Y'lYl,Y2,
...> Y N l 1
(1 1)
is normalized to unity so that all input and output variables
are within an n dimensional and an m dimensional unit cubes,
respectively. If we assume that the input vector x represents a
n-dimensional vector of real-valued fuzzy membership
grades: x E [O, ly, then, the output y of the model represents a
m-dimensional vector of corresponding real-valued
membership grades, y ~ [ O , l ] ~This
.
structure actually
performs a non-linear mapping from a n-dimensional hypercube to a m-dimensional hyper-cube: l"=[O,l":x E [0,1]" +
I": y ~ [ O , l ] .~ For each variable fuzzy sets are defined
corresponding to the sub-intervals of the variables. Generally,
about five intervals are optimal since for linguistic variables
more number of intervals is difficult to differentiate the
adjacent fuzzy sets accurately from the fuzzy interpretation
viewpoint. Conversely, less number of intervals causes gross
approximation by the model. The soft data can be categorized
in sub-intervals in different ways. One example is by means
of expert judgment. In this work this is done by means of
users' perceptions related to a metro station in public use.
One of the essential differences between the data set from
an engineering system and a data set from soft sciences is the
number of parameters used in the model. In the engineering
systems for a multi-input and multi-output fuzzy system
(MIMO) generally the number of input variables as well as
number of output variables are both relatively low compared
to those of soft sciences like architectural data. Typical values
are IO (at most) or less for engineering systems and 50 or
more for architectural model. Based on the data matrix
The IEEE International Conference on Fuzzy Systems
.z=[X ;Y]T,
(12)
clustering algorithms performed for a set of soft data are not
generally conclusive for concatenated matrix Z with the
.elements
T
Z=[XITJ2T,... J N T ; Y I Y 2,...YN1 ,
(13)
if the column dimension of the Z is about ten, i.e. n+mzlO,
while N >> n. This is due to ill-conditioned matrices as to
eigenvectors. These are some practical limitations although
there is no limit from the theoretical viewpoint. Such practical
limitations are rather lenient for fuzzy modeling of
engineering systems. However the same limitations are quite
serious in architecture where Z of dimension [Nx(n+m)]as
[200x50]is quite common. For this size of data set to devise a
fuzzy model is a formidable work due to probable failure in
clustering at the initial stage. If the clustering is conclusive,
the accumulated' projection errors make the model highly
inaccurate. These .will be illustrated in the following section.
These limitations for a TS model are equally valid for a fuzzy
[logicmodel with singleton consequents.
2.3. Fuzzy Modeling.of Soft Data by Neural Networks
The above stated practical limitations imposed on fuzzy
modeling of soft data can be circumvented by means of a
fuzzy model implemented by a neural structure, namely radial
hasis function (RBF) network. In this case, an RBF network is
equivalent to a Fuzzy system [ 6 ] where the RBF centers are on
a grid [7] fulfilling the conditions
I . The number ofreceptive
. .filed units is eaual to the number
offizzy $then rules
2. The output of each fuzqv $then rule is composed of a
constant.
3. The membership functions within each rule are chosen as
Gaussianfunctions with the same variance
3. The T-norm operator used to compute each rule's firing
strength is multiplication
5. Both the RBFN and fuzzy inference system under
consideration use the same method (here weighted sum).
The normalization and regular sub-division of each variable
from a soft data set fulfils exactly these grid conditions so that
such a RBF network becomes exactly equivalent of a fuzzy
system with singleton consequents. This equivalence is
further extended [8]by removing the condition 2 above.
In the RBF type neural network structure, to avoid the
convergence problems, the training is performed by means of
orthogonal least squares (OLS) algorithm [9,10].
111.
DESIGN ENHANCEMENT BY FUZZY
MODELWG: A CASE STUDY
3. I . Fuzzy Modeling of Architectural Data
81
Fuzzy modeling is an important component o f sofl computing.
One of the important employment of fuzzy modeling in exact
sciences, is for design. In contrast with this, employment of
soft computing methods in soft sciences is not common. This
is presumably due to rich mathematical involvement of soft
computing making it not easily accessible by soft sciences.
However, fuzzy logic is very appealing especially for design
in architecture. Therefore, the design enhancement by fuzzy
modeling in architecture is aimed in this work. This is
accomplished by using soft data from a metro station (Blaak)
in Rotterdam, in the Netherlands. The design enhancement
details from the architectural viewpoint are described in
another research work [ll]. Here, some details of the
research from fuzzy modeling viewpoint will he presented.
The data set contains n=45 design variables and the number
of data is n=196. The design information made available to
RBF network at input is basically linguistic and qualitative
categorized in five sub-intervals, as mentioned earlier. The
input design variables are given in Table I:
TABLE I
MPUT DESIGN VARIABLES OF FUZZY MODEL11 IL
Overview
Entrance
Train platform
Metro platform
Exchange area
Escape
Possibilities
Distances
Presence of People
Public
control
Few
people
daytime
Lighting
Entrance
Train platform
Metro platform
Exchange area
Dark areas
Safety Surrounding
Safetv in surrounding
I
. .
Few people night
Attractiveness
Color
Material
Spatial proportions
Furniture
Maintenance
Spaciousness entrance
Spaciousness train platform
Svaciousness metro vlatform
Platform length
Platform width
Platform height
Pleasantness entrance
Pleasantness train platform
Pleasantness metro platform
Wayfinding
~
To the station
In station
Placement of signs
Number of signs
Davlight
Pleasantness
Orientation
Physiological
Noise
Temperature winter
Temperature summer
Draft entrance
Draft platforms
Draft exchange areas
Ventilation entrance
Ventilation platforms
The output variables selected are safery and comfort. The
input and output range of the variables between zero and one
The IEEE International Conference on Fuuy Systems
is divided into five equal intervals and gaussian membership
functions are placed at the locations 0.1,0.3,0.5,0.7 and 0.9.
This placement fulfils the grid condition of the neural
equivalence of fuzzy model [7]. The soft data contains only
0.1, 0.3, 0.5, 0.7 and 0.9 as fuzzy assessment of the variables
represented by the associated fuzzy sets, which are
multivariable gaussians in the RBF structure. The selected
gaussians as nodes in the RBF structure, provide the fuzzy
rules. For 43 input variables and two output variables, the
RBF model outcomes for safety and comfort variables is
shown in figure 1.
RBF ovfpufotCOMFORTvarlsbls
I
I ,
08
06
04
0.2
I
I
100
50
200
150
REF oufpul of SAFETY variable
Figure 2. Data set comfort and safety and their estimation by
fuzzy modeling implemented by RBF network where number of
input variables is 43 and number of data is SO
In order to compare the performances of both fuzzy model
approaches, namely, direct fuzzy model and RBF fuzzy
model, we take the number of input variables as low as 3
namely, color, material and spatial proportions and we
consider one variable at the output namely comfort. Also we
consider the number of data as N=60 for easy comparison. In
order to avoid the ill-conditioning due to probable hidden
correlation among the variables, some random noise is added
on the input variables. Such added noise is equivalent to
regularization [I21 in modeling a n d improves the model
performance when model is tested by unknown input data.
For direct fuzzy modeling the data set of three variables is
combined with the comfort data set. The combined set is
subjected to clustering by the Gustafson-Kessel algorithm.
After the clusters have been determined, they are projected on
the input variables. The fuzzy membership functions
established for 4 clusters are shown in figure 3. Note that,
each parameter has four membership functions and each dot
is due to projection which are altogether 60.
Membershap Functions tor COLOR
I
0.2 I
0
M
100
ssquenu,
01 input
200
150
. *:
0.5
panama
.. ..
Figure I . Data set comfort and safety and their estimation by
f u z v modeling implemented by RBF network where number of
input variables is 43 and number of data samples is 196
The RBF modeling outcomes shown in figure 1 are for 100
RBF nodes, that is 100 rules. The same figure in expanded
form with 50 patterns is shown in figure 2. The effectiveness
of this model is demonstrated by the test data and sensitivity
analyses are carried out for the importance gradation of the
input variables with respect to safety and comfort variables at
the output. The results are well-documented elsewhere [ l l ] .
REF outout of COMFORTvanable
0.8
spatial pmpartions
Figure 3. The point-wise projected fuzzy membership
functions of the color, material and spatial proportions
variables after clustering
-
0.6 0.4
-
The cluster means matrix C of dimension [3x4] from the GK
algorithm for these three variables is given by
I
10
20
30
40
50
REF aulputatSAFEN vanable
c=
0.8
0.6
0.4
0.2
'
30
sequsncs Of input puems
20
40
0.6435
0.7166
0.6725
0.6702
0.6676
0.6880
0.6410
0.5119
0.4332
1
The T S fuzzy model with these membership functions is
shown in figure 4 for both local and global models. Local and
global models together with RBF network counterpart are
discussed in another work 1131. As result of the limitation of
I
10
[
0.7635
0.6563
0.7486
$3
82
The IEEE international Conference on Fuzzy Systems
data categorization to five sub-intervals between 0 and 1 there
are no fine details in the data and therefore both models are
almost the same. In the figure, thick broken line represents the
estimation where the data point are presented by dots. The
same data set above is subjected to RBF modeling.
Theoretically, four distinct membership functions for each
variable gives 64 rules. However, as seen from figure 3, some
rules overlap. With this consideration, in the RBF structure 30
rules are considered. The fuzzy model estimation of comfort
variable by RBF network is shown in figure 5 .
COMFORT and 11s local model
I,
I
0.8
0.6 -
0.40.2'
0
20
10
30
40 .
1
50
60
50
1
60
COMFORT and its global model
1,
0.8
0.60.4
0
0
10
.
20
2
30
40
1
COMFORT
Figure 4: Data set of comfort estimated byfuzw modding for
local (a) and global (b) models. In both cases number of input
variables is 3 and number of data is 60
COMFORT variable and its approximation by 60 REF network
1,
0.2'
0
10
20
30
40
50
In figure 5, the dashed line represents the approximation
where data points are shown by dots. The comparison of
figure 4 and figure 5 reveals that, the fuzzy modeling by both
methods have almost the same performance for the given
model formation parameters.
3.2. Fuzzy Modeling for Architectural Design Enhancement
The fuzzy model established with soft data can be used in
various ways in soft sciences. Design enhancement in
architecture is one example. Input variables of the model
constitute a space and in this space one can search an optimal
location producing the most desirable performance of the
output. Since the input variables are not continuous the search
in such a space cannot be carried out in analytic form. For this
purpose, genetic search algorithms are quite suitable. These
algorithms are collectively referred to as evolutionary
algorithms and an important class of such algorithms is
genetic algorithms (GAS) [14,15]. GAS can successfully
handle multidimensional, multiple-criteria problems in
architecture [I61 as well as in a variety of engineering
disciplines. The search is made for optimality, the process is
referred to as combinatorial optimization. In the present
concem of design enhancement the search is made for the
fuzzy model input parameter values that provide a desirable
model output defined by a criterion. In the terminology of
GAS, the criterion is termed a jtness /unction. If we
characterize the fitness function as quality of design, we can
look for a maximum of this design quality, in this study. The
model outputs comfort and safety define a two-dimensional
space. The qualiry of design is defined as a product of these
variables and search is made for the maximization of this
product. For 43 variables at the model input and 196 data
samples, the results are shown in figures 6 and 7.
I
60
100
COMFORT variable and its approximation by 30 REF network
10'
102
COMFORT and SAFETY durTno oenelic search
1
0.9
0.8
0.1
0.2'
0
10
20
30
40
50
100
J
IO'
102
lo3
iteralion number
60
data sequence
Figure 5. Data set comfort and safety and their estimation by
fuzw modeling implemented by RBF nehvork where number of
input variables is 3 and number of data samples is 60
83
Figure 6. Variation of the fitness function and the fitness
function variables comfort and sufety during GA search
In figure 6, in the upper plot the genetic search stam at the
point where design quality is slightly below 0.5 and it
The IEEE International Conferenceon Fuzzy Systems
searches for a maximum. The number of iterations for search
is 150 and the corresponding design quality figure is close to
0.8. If we consider that for the model formation comfort and
safety variables have maximum about 0.9, then the quality of
design figure is reliably close to the maximum. In the same
figure the lower plot shows the comfort and safe& parameters
variations during the genetic search. Here, the fitness function
is designed to be simple for the purpose of demonstration.
However the method GA is suitable for complex fitness
function definitions and multi-objective optimization. These
are important merits for a desipn task.
between the performances of TS and RBF network type fuzzy
model implementations is not significant, as one should
expect, that is, the structural equivalence extends also to their
performances. On the contrary however, for high number of
model variables RBF network approach become imperative as
the satisfactory transparency, accuracy and compactness of
fuzzy logic modeling becomes difficult to maintain.
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,"I
".r,.blacl
",
Figure 7. Optimal pattem of the input variables o f the fuzzy
model for the enhanced design quality
IV. CONCLUSIONS
Design enhancement by soft computing in soft sciences and
particularly in architecture is addressed. For this purpose,
fuzzy modeling is an important aid since the data from soft
sciences may be dominantly soft, i.e., linguistic and
qualitative. The traditional methods cannot process such
information satisfactorily for a maximum gain. A fuzzy
model can serve to gain more insight into the given
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architecture where presently such applications are not
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