Techniques for Developing Large Scale Fuzzy Logic Systems Jon C. Ervin Sema E. Alptekin Apogee Research Group California Polytechnic State University Los Osos, CA 93402, USA San Luis Obispo, CA 93407, USA [email protected] [email protected] property of the formulation can quickly Abstract overwhelm available computing resources in even relatively modest sized applications. One In this paper, we describe various promising extension, Intuitionistic Fuzzy Logic techniques used to make Intuitionistic (IFL), can further exacerbate memory problems Fuzzy Logic Systems amenable to by effectively doubling the number of input operating on applications with large membership functions. Atanassov [1] introduced numbers of inputs. A rule reduction the notion of pairing non-membership functions technique known as Combs method is in IFL with the usual membership functions of combined with an automated tuning Fuzzy Logic. process based on Particle Swarm Optimization. A second stage of tuning A number of researchers have developed on rule weights results in improved techniques to mitigate the problem “exponential performance and further reduction in rule explosion” or “curse of dimensionality” the size of the rule-base. The entire through various means [8, 10, and 14], however, process has been developed to operate these methods generally tend to be complex and within the Matlab software limited to special cases. One promising environment. The technique is tested exception is a technique developed by William against the Wisconsin Breast Cancer Database. The use of these tools shows Combs that changes the problem from one of an great promise in significantly exponential dependence to a linear dependence expanding the range and complexity of on the number of inputs [4]. The use of Combs problems that can be addressed using method also simplifies rule-base generation and Atanassov’s Intuitionistic Fuzzy Logic. makes it easier to automate tuning of Fuzzy systems. Keywords: Atanassov’s Intuitionistic Fuzzy Another issue that must be addressed when Logic, Particle Swarm Optimization, Combs implementing a Fuzzy Logic algorithm for a Method. specific application is constructing membership 1 Introduction functions for each input that provide adequate system performance. A common approach taken The decision making paradigm known as Fuzzy is to interview experts in the particular field and Logic has been implemented in a wide range of use their feedback to build the fuzzy applications since it was first introduced by Dr. membership functions. This process can be Lotfi Zadeh in his seminal paper published in cumbersome and/or completely impractical for 1965 [17]. A number of researchers have since large problems, leading to the need to made major contributions in expanding and implement some sort of automated membership adapting Fuzzy Logic beyond its initial function construction and optimization process. conception. One major Achilles heel for Fuzzy Logic applications has been that in most current A number of methods for automating this algorithms, memory requirements expand process have been published in the literature. A exponentially with linear growth in inputs. This common approach has been to somehow L. Magdalena, M. Ojeda-Aciego, J.L. Verdegay (eds): Proceedings of IPMU'08, pp. 806{811 Torremolinos (M´alaga), June 22{27, 2008 construct membership functions and then use a found in the literature discussing the validity of population based optimization method such as the claims made for the URC and the interested the Genetic Algorithm to tune them. We chose reader is referred to the references [3, 5, 12, and to follow a variation of this approach by 15] for detail on this topic. Our own experience constructing the membership functions and to date has been that the URC performs as well using a method known as Particle Swarm or better than the IRC formulation. Optimization (PSO) for tuning. The PSO algorithm, first developed by Eberhart and We used Combs method to apply Atanassov’s Kennedy [6], was inspired by the coordinated Intuitionistic Fuzzy Logic to the well known group behaviors of animals such as flocks of Wisconsin Breast Cancer Database (WBCD) birds in flight. One important advantage of this [16]. An initial set of four IFL membership and technique is the simple formulation and non-membership functions were developed for implementation of the underlying equations. each of the nine input variables of the WBCD. The 683 unique samples in this database The main focus of our research has been to associate various diagnostic tests as real-valued combine several techniques to simplify the inputs with a binary output of benign/malignant tuning process and make it practical to address for a suspect tissue mass. The WBCD was problems of much greater size and complexity chosen as a test case because it provides a [7]. Recently, we have built on previous relatively large number of inputs, allowing us to research to further improve the optimization demonstrate the rule-base reduction advantage process. We believe our efforts have brought us of the URC method. The entire rule-base closer to realizing an efficient, automated consisted of 36 rules (9 inputs x 4 process to generate Fuzzy Logic systems membership/non-membership functions) versus capable of handling large and complex the 262,144 (49) maximum number of rules that problems. could be used in the IRC method. The following section provides a brief Figure 1 shows the distribution of the values of background of Combs method and the database one of nine WBCD input variables with respect used to verify our approach. The next section to benign/malignancy of the tissue mass. On this describes the PSO optimization process. chart a diagnosis of benign is marked as an “x” Following that we provide some of our results and a diagnosis of malignant is marked with an and end with a section on our conclusions. “o”. The distribution shows a strong correspondence between low values and a 2 Combs URC Method diagnosis of benign, however there are a number of exceptions to this generalization. This William Combs refers to his method as the distribution is characteristic to a greater or lesser union rule configuration (URC) versus what he degree in all nine of the input variables. calls the intersection rule configuration (IRC) of the traditional fuzzy rule-base construct [4]. The main difference between the URC and IRC is 10 o o o o o o oo oo oo ooo ooooo xoo oo oooooo o oooo o oooo ooo o xx o ooo ooooo oooooooooo ooo o oo oooo o that every rule in the rule-base of the URC is oo o ooooooooo oooooo oo o o ooooo o o ooooo o 9 oo o o oo oo o o o o required to have only one antecedent for every o o o o 8 o x o consequent. Initially, this may sound counter- ooo o o o o o x o o oo o o 7 o o o intuitive as a means of reducing the number of o x o oo o o rules, however by imposing this restriction it 6 o o o means that each membership function of all the Value o o 5 o x o o o o o o x x o x input variables is used only once in the o o x x o o o 4 o x oo o x o x x o o o o x antecedent of a rule in the rule base. Each of o o x o o Relative x x o o o o o x x o 3 ox o x o o x o these rules are joined by a logical OR in the o x x o x x o o x xo 2 x o x ox x x x o xo rule-base, hence the designation union rule x xxxx xx x o o xo x o x x x xxo configuration. ooo x x x 1 xxxxxxx xx xx xxx xx xx xx xxxx xxxox xxxxxxxxox xxxx xxxxxxxxxxx xxxoxxxxxxxxoxxxx xxxxo xxxxxxox xx xxx xxx xxxxxx xxxxxo xxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxx xoxxxxxxxxxx xxxxooxoxx x xxxxxxxxxxxx xxx xxxxx xxxxxx xxxxx xx xx xxxxxxxxxxxxx xxxxxxxxx 0 xoxxx x xx xxxxxoxoxx xxx xx xxxxxxxxxx xxxxxxxxx xxxxxxxxxx xxxoxxxxxoxxxxxxxx Combs and his various co-authors show that the 0 100 200 300 400 500 600 700 entire problem space can be accessed by Data Point implementing the URC. A spirited debate can be Figure 1: Distribution of Input “Bare Nuclei” Proceedings of IPMU'08 807 3 Fuzzy Classification Model function described a predominately low or high input value. This action left two points of each In the spirit of simplifying the development and membership function to float freely over the tuning processes, the entire optimization input range and it was these two values that algorithm was created, debugged and executed were subjected to the optimization process. within the Matlab software suite [11]. The intuitionistic fuzzy classifier was created in the The total number of optimization variables was: Matlab Fuzzy Logic Toolbox. A set of (9 input variables) x (2 points per function) x (4 customized Matlab m-files provide overall functions per input variable) = 72 variables. control of the optimization process. The m-files make calls to the PSO and IFL modules, with Figure 2 shows an example of the the program terminating after a user-defined (non)membership function for one input variable number of iterations have been completed. with the 8 points subjected to optimization circled. Here we abbreviate “membership and The Fuzzy Logic Toolbox provides a number of non membership” with “(non)membership”. general default values for the user to choose from in building the fuzzy classification model. For instance, the software allows the user to Not Low choose between two methods, minimum or product, to fill in for the “AND” function in the Not High rule antecedent.