In this paper, we present our fuzzy classifier, built from scratch, and test how well it performs a task of classifying cities to either 'better' or 'worse' category, based on their numerical ratings of various aspects of living there. We check several combinations of norms and defuzzyfying methods, compare the results with three diferent fixed classifications - based on weighted sum, GDP per capita and human expert judgement. We also tried out few diferent classifiers, such as KNN, naive Bayes and soft set based classifier to see which one yields best results for this task.
attributes of the examined city and produces a
numerical value from 1 to 100 alongside with linguistical value
The latest trends in the creation of modern IT systems are ’better’ or ’worse’, all based on comparison with expert
based on the use of artificial intelligence systems [
In this day and the age, due to modern technology and they are described with words, not numbers, therefore
expansion of the Internet, people have access to much some kind of system must exist in order to connect
numore data than ever before[
(1) (2) numerical and linguistical interpretations, for exam- 2.3. Membership functions ple ratings of Housing and Taxation. Consequents refer to potential outcomes from the system, for example Good, Average, Bad.
Degree of membership of each variable to a linguistic
category is contained within a numerical value from range
[
bining all the consequent membership functions and maximas of their corresponding rule activation results. Based on this function’s set of values, given defuzzification method generates single numerical value. It is the outcome of the procedure.
Flow of the fuzzy logic is presented in figure 1.
membership functions for each linguistic value within a category (figure 2). It is sometimes a good practice to do such a thing, especially when operating on a few thin triangular functions, as there is a need to cover wide range of extreme values, which we want to guarantee to yield as a result (figure 3).
operate in very diferent ways. For our research, we were focusing on FOM (First Of Maxima), MOM (Middle Of Maxima), LOM (Last Of Maxima) and centroid middle of area. The first three return as their result the first, mean of or last occurance of highest activated function values, and the centroid one operates in a way described below: ∑︀=1 () ∑︀
=1 ()
2.5. Norms
26–32
Single rule takes linguistic values of observation’s attributes, processes them checking multiple conditions linked with themselves by mathematical logic and if all conditions are fulfilled, certain linguistic value that determines class is returned.
Our rules system is based on division of attributes on two main categories – important and less important ones. We assessed that some of observation’s qualities are absolutely essential for every citizen (Housing, Cost of living, Safety, Healthcare, Travel connectivity) and some are not (Education, Environmental quality, Economy, Taxation, Internet access, Leisure and culture, Commute). For example healthcare is something that a lot of people are going to use in a critical situation that threathen them with death while education concerns only the younger citizens and culture is something that does not decide about survival and is additional bonus, not strategical factor. We created nine rules, where each three are built under the same pattern and the only varying element is linguistic value. First rule states that any three among the five essential dataset’s label - GDP per capita - was obtained from exqualities, if their linguistic value is identical, determine ternal source. The last dataset’s label was defined by returned linguistic value: authors by themselves and was based solely on their knowledge and intuition.
If (1 and 2 and 3 ) or We checked results which were given by various combinations of norms (Zadeh norm, modified Manger norm, (1 and 2 and 4 ) or ... Łukasiewicz norm) with defuzzification methods (First then of maxima, Last of maxima, Middle of maxima, centroid middle of area) applied within fuzzy system and a few (wvqyauh1laue,leriyte.2ie,(syx,31,,iyxs42r,,eyxt5u3,r,nyxe64 d,, yxv75a))luaearaernedfiveseeivssescneenrttianaiolnnql-iuenasgsluietinisetistiai,cl wcoiteshiemornec,alnraesfcsoaiufiellrrsabn(aKdsiNcsNer an,tsNeistaiïovvfietmyB.oadEyeexlscq,euSpoatlfifttosyer:t)aa.ccBccuuyrrraaeccsyyu,,lptwrseeThat rule contains all possible variations of three chosen decided to prioritize high scores of precision due to our attributes among the total five. attempts to achieve as little false positive cases as possiWhen four or five qualities have the same linguistic ble. The reason was that we did not want to propose a value, according to this logic that fact also determines city as a potential proper location to live while it is not. returned value so it does not need to be handled by On all of the plots letter ’Z’ denotes Zadeh norm, letseparate rule because it is intercepted by the first rule. ter ’M’ denotes modified Manger norm, letter ’L’ denotes Second rule states that if two essential qualities have Łukasiewicz norm. the same value, any two among the seven non-essential Results (figures 5 and 6) from the first dataset (weighted attributes with same value guarantee returned value sum) show us a decent outcome from all of the diferequalled to theirs: ent kinds of fuzzy system. They are very repetable and depend on used defuzzification methods, regardless of norms. Comparing it to the rest of classifiers, one may If ((1 and 2 ) or (2 and 3 ) or ...) notice considerably lower performance of KNN, similar and ((1 and 2 ) or (2 and 3 ) or ...) one of Softset and high result of Bayes.
then
the five and seven accordingly.
Third rule states that if only one essential quality have certain value, we need five non-essential attributes with this value to qualify returned value as identical to theirs: If ((1 and 2 and 3 and 4 and 5 ) or ...) and ((1 and 2 and 3 and 4 and 5 ) or ...)
then
To test our fuzzy system properly, we used three datasets, to which labels were assigned diferently every time. Label of first dataset was created by constructing weighted sum of attributes where each quality did not contribute equally. Weight of the most of them was equal to 1, but some - more important to final decision in our opinion - were equal to 1.5 and one was equal to 0.5. Second
Second dataset (GDP per capita, figures 7 and 8) generated lower results of fuzzy system than that based on weighted sum. Again, individual cases are repetitive in similar way. This time KNN achieved better performance while Softset remains close to fuzzy outcomes. Bayes is still unmatched. All non-fuzzy models reached higher precision rates.
Third dataset (subjective classes, figures 9 and 10) produced similar outcomes to second one for fuzzy system decent accuracy with poor precision. KNN and SoftSet have close results to themselves and Bayes as always turned out to be the best model.
First of maxima and Middle of maxima defuzzification methods. Diferent norms were not a considerable factor, defuzzification pretty much defined the result by itself. The rest of the models reached mostly decent outcomes, sometimes even outperforming fuzzy system. Particularly Bayesian classifier turned out to have the highest score in nearly every case (considering accuracy and precision collectively). However, fuzzy system would likely achieve higher results in case of designing more accurate and factful system of rules by real expert or more adequate division into classes.
Although fuzzy system did not bring the best results from the bunch, we would argue that it provided good software, that would be able to capitalize on its strengths, such as flexibility of rulesets, enabling diferent user criteria, and no reliance on other data samples. 26–32