2.13 Type-I Fuzzy Sets
2.13.1 Fuzzy Inference System
A fuzzy inferencing system (FIS) is a rule-based system that uses fuzzy logic, rather than
boolean logic to reason about data. A FIS can also be called a fuzzy expert system (FES)
or a fuzzy logic controller (FLC) dependent upon the area of its application. The four
main components of a FIS are:
• A fuzzifier, which translates real-valued inputs into fuzzy inputs
• An inference engine, which applies a fuzzy reasoning mechanism to obtain a fuzzy
output using a knowledge base
• A defuzzifier, which translates the fuzzy output into a crisp value
• A rule base, which contains an ensemble of fuzzy rules
The fundamental elements of a FIS can be seen in Figure 2.12. The two most common in-
ferencing methods are Mamdani’s fuzzy inference method and the Takagi-Sugeno-Kang
2.13. Type-I Fuzzy Sets 45
are aggregated to produce an output fuzzy set and the final output is obtained after de-
fuzzification over all fuzzy outputs. Where as in TSK which is also called singleton the
output is the weighted average of each rule’s output [81]. Each input of the FIS is asso-
ciated with one or more membership functions. There are several types of membership
function available, for example, gaussian, triangular, trapezoidal etc. These membership
functions are evaluated with fuzzy rules to find the output. A good classification is de-
pendent upon adequate rules, linguistic variables and membership functions. One of the
main advantages of a medical based FIS is that it is possible to include the knowledge
of specialists, even if statistics data is not available. In the case of medical data, there is
a high level of uncertainty and a FIS may help in providing an optimal solution in such
cases [43, 87]. Rule base Inference Engine Membership Functions Fuzzification Defuzzification IIInput Output
Figure 2.12: Structure of a Fuzzy Inferencing System (taken from [86])
A brief literature review of FIS used with spectral data in medical science follows.
Castanys et al. [17] described a three phase case-based reasoning system (CBR) to
identify unknown materials by means of the automatic recognition of their Raman spectra.
The first phase consists of dimensionality reduction by means of PCA. The second phase
consists of defining similarity measures to objectively quantify the spectral similarity with
2.13. Type-I Fuzzy Sets 46
validation of the results. The total number of rules developed for the fuzzy logic system
was 4. The results indicated that the proposed system worked well and has potential to be
used for other problems like identifying pigment mixtures etc.
Evsukoff et al. [29] presented a frame work for intelligent data analysis of spectral data
in classification and regression problems. In this frame work, the number of interpolation
functions was computed using spectral analysis. Each function was then associated with
a symbol to generate a fuzzy rule. Each symbol was related with a prototype that can be
computed using a clustering algorithm. A rule induction algorithm was used to determine
the minimum number of rules for the frame work. The proposed frame work was tested
on various data sets including iris, cancer, wine etc. The results indicate that the frame
work performed well in classification of these data sets. The authors concluded that the
frame work had the potential to be used in complex problems although there are areas like
finding optimal number of rules that require more in depth research.
Cernuda et al. [18] proposed a specific fuzzy system called the TSK fuzzy system
for calibrating the chemometric models based on NIR spectra. This fuzzy system was
used to model the non-linearity contained in the production process of polytheracrylat
(PEA). The TSK based fuzzy system was used to express non-linearities contained in
the mapping between NIR spectra samples and measured concentrations or target values.
The calibration results obtained by the proposed method were also compared with other
state of the art methods. The results showed that the proposed system outperformed other
methods in terms of properties associated with calibration and was also computationally
comparable.
Mahmoodabadi et al. [66] presented a fully automated system in order to analyse and
classify magnetic resonance spectroscopy (MRS) signals of patients with metabolic brain
diseases. The selected features from MRS constituted the universe of discourse (input).
Every input feature was fuzzified using low (L), normal (N) or high (H) group with a
membership value in range [-1,1]. Only normal (N) had a membership value starting
2.13. Type-I Fuzzy Sets 47
functions were used in the classifier to categorize the metabolic brain diseases. The au-
thors stated that the use of specific membership functions was able to increase accuracy
and interpretability of the system.
Zhengmao Ye [121] used livers, lungs, kidneys and glands Raman spectra and created
and artificial intelligence approach along with fuzzy logic filtering to categorise them. For
the fuzzy logic part of the method, from Raman spectra, consecutive intensity differences
between any point and its adjacent points are normalsised and linear combination of dif-
ference terms was considered as crisp inputs to the fuzzy logic filter. Positive or negative
signs of intensity differences were considered for rule making. A Mamdani type fuzzy
system was created. Linguistic variables were expressed as fuzzy sets of Negative big
(NB), Negative small (NS), Zero (ZE), Positive small (PS), Positive big (PB)]. Centroid
defuzzification was used. The authors concluded that their method was able to perform
well on these data sets and argued that the method had potential to be used for various
cancer cell classifications as well.
Similarly, Pueyo et al. [88] and Kong et al. [56] have also developed fuzzy systems
for spectral data sets.
FIS has also been used for various breast cancer data sets in general other than spectral
data sets. A brief literature review of such examples follows.
Reyes [87] used a FIS with evolutionary algorithm for development of an automated
method, and later on created Fuzzy CoCo (a Fuzzy modelling technique with evolution-
ary algorithms) and applied it to the Wisconsin Breast Cancer Database (WBCD). How-
ever, no experiments were performed on real data sets, therefore, the authors themselves
stressed the need for a more practical approach to understand the actual performance of
the model.
Uriarte and Castillo [33] compared the results of FCM Clustering algorithm and a FIS
based on a combination of FCM and a Genetic Algorithm (GA) on WBCD. This data
base consists of 569 cases, 357 benign and 212 malignant. For each case, there are 10
2.13. Type-I Fuzzy Sets 48
sultant grouping after the FCM clustering algorithm was applied. In each group, average,
minimum and maximum values obtained from the FCM were used as membership func-
tion values. A genetic algorithm (GA) was used to find the rules. For the comparison
of the methods, the final number of grouping was considered as a measure of accuracy.
After training the FIS, four rules were selected. The results of both the methods were
good although the overall FCM clustering algorithm was more accurate at 99.315% as
compared to the FIS which achieved 80.136%.
Jain and Abraham [46] created four fuzzy rule generation methods and compared
their performance on WBCD data set for breast cancer diagnosis. In the first method, a
single fuzzy if then rule was generated for each class using the mean and the standard
deviation of attribute values. For each attribute of the data set, 20 membership functions
were created and a fuzzy partition matrix was used to create the histogram. In the second
method, histogram attribute values were normalised to 1 and used for rule generation
and a single fuzzy if then rule was used as in the first method. In the third method,
rules were created by homogeneously partitioning each attribute creating a simple fuzzy
grid was created. Each attribute had multiple rules instead of a single rule. The last
method was a modified version of the fuzzy grid approach. In this method the shape of
the membership functions is modified by partitioning only areas which are overlapping.
The results showed that modified fuzzy grid approach provided a high classification rate of
99.73% where as modified grade achieved the lowest accuracy of all methods at 62.57%.
Auephanwiriyakul et al. [5] also used FIS to detect abnormalities in mammograms.
One abnormality was microcalcification which is a small deposit of calcium and the other
was mass which is a lump of fat detected in mammograms by expert radiologists, some-
times a small presence of these abnormalities can be ignored. Real mammograms were
used for the experiments. Two FIS were proposed by the authors based on Mamdani’s sys-
tem and their performances were compared. The first system was called the microcalcifi-
cation detection system and it consisted of four features extracted which were parameters