Abstract
The ability of fuzzy logic systems to make use of human interpretable linguistic terms makes them very attractive for many applications. These systems make use of membership functions to represent the linguistic variables. The choice of type of the membership function to use and the associated parameters affect the performance of the system. The current types of membership functions like triangular, trapezoidal, Gaussian functions etc., make use of parameters that are possibilistic in nature thus requiring expert knowledge or other sophisticated methods in order to choose the parameters. In this paper, we propose a new type of membership function constructed based on fuzzy estimators. This membership function depends entirely on well-known statistical parameters like the mean, standard deviation and confidence intervals and thus the parameters are easier to choose. Another advantage of this membership function is that it is suitable for modeling systems that exhibit both randomness and fuzziness. Furthermore, in contrast to the parameters of other membership functions, in the case where the parameters are tuned and optimized for a particular application, the final parameters of the proposed membership function can have useful statistical interpretations and provide a better understanding of the system.
Keywords
Introduction
The type of the membership function used and the parameters of the function play an important role in the design, interpretability and overall performance of a fuzzy inference system [7].
There are many types of membership functions to choose from like piecewise linear functions, triangular, trapezoidal or smoother functions such as the symmetric Gaussian function, the generalized Bell curve, the non-symmetric sigmoidal function, but the parameters of these functions are difficult to select. This is because the parameters are possibilistic in nature and depend on the particular application. They therefore require expert knowledge or other sophisticated methods like histogram based methods, transformation of probability distributions to possibilistic distributions, heuristic methods, neural networks, clustering methods, genetic algorithms, etc. to either generate the membership function or select the parameters. Moreover, the membership functions generated and their parameters do not have any underlying additional meaning, apart from the possibilistic interpretation, thus making them difficult to evaluate or interpret in a probabilistic setting.
Probability and possibility theories can be used to model fuzziness and randomness respectively, but recently there has been considerable effort made to integrate statistics with fuzzy set theory [4]. One of the most promising of such attempts is the construction of fuzzy numbers for the parameters of probability density functions which have been estimated from random samples using confidence intervals. These statistical fuzzy numbers are called fuzzy estimators(FE). There are many ways of constructing fuzzy estimators [3, 5] but the most common approach is the one introduced by Buckley [4] and generalized in [6]. Although fuzzy estimators have been around for a while, their potential to be used in a fuzzy inference system has not been investigated till date. The main reason is that they have generally been constructed by superimposed confidence intervals without an explicit membership function. Also, the most popular one introduced by Buckley did not have compact support. Another limitation is the small spread they produce thus making it difficult to have patches (overlap) when used as fuzzy sets, which is a desirable property of fuzzy systems. This has limited their use to statistical applications. Today, explicit membership functions for fuzzy estimators with compact support have been derived, [5, 6] thus overcoming the first two problems. In this paper, we propose a new hybrid type of membership function constructed using fuzzy estimators but modified to possess the desirable property of overlap of other fuzzy membership functions. The proposed membership function is a symmetric trapezoid but can be made into a triangle depending on the parameters chosen. This membership function depends entirely on well-known statistical parameters like the mean, standard deviation and confidence intervals and thus the parameters are easier to choose, fairly robust to the type application and do not require expert knowledge. Another advantage of this membership function is that it is suitable for modeling systems that exhibit both randomness and fuzziness. Furthermore, in contrast to the parameters of other membership functions, in the case where the parameters are tuned and optimized for a particular application, the final parameters of the proposed membership function can have useful statistical interpretations and provide a better understanding of the system. The main contribution of the paper is to demonstrate that membership functions constructed from fuzzy estimators(fuzzy-statistical membership functions) can be used in fuzzy inference systems and thus can be considered an alternative to other known(fuzzy) types of membership functions. The motivation for these kinds of membership functions is the quick selection of parameters for known systems or better statistical interpretation of final(optimized or not) selected parameters for un-known systems. The remainder of the paper is organized as follows. In Section 2 we provide an introduction to fuzzy estimators and their construction. Section 3 describes the proposed membership function, how it is related to fuzzy estimators and ordinary fuzzy sets. We also explain how to choose the parameters and how each parameter affects the shape of the function. In Section 4, we present a simple case study to illustrate how the proposed membership function can be used to represent linguistic variables in a fuzzy inference system. Our conclusions and further research follows in Section 5.
Construction of fuzzy estimators
Let us first introduce some important mathematical notions and fuzzy set that will be used in the construction of the estimators.
A is normal, that is there exist x ∈ R such that A (x) =1. A is a convex fuzzy set, that is for every t ∈ [0, 1] and x1, x2, we have:
A is upper semi-continuous on R, i.e. for ∀x0 ∈ R and for ∀ɛ > 0 there exist a neighborhood V (x0) such that U (x) ≤ U (x0) + ɛ ∀x ∈ V (x0). The support of A is compact.
A
L
(α) is non-decreasing and left continuous. A
R
(α) is non-increasing and left continuous. A
L
(α) ≤ A
R
(α).
The above definitions can be found in [6].
Buckley’s fuzzy estimators
Now, we will describe the existing method [4] of constructing fuzzy estimators. Let X be a random variable with probability density function f (x ; θ) for single parameter θ. Assume that θ is unknown and it must be estimated from a random sample X1, …, X
n
. Denote (1-β)100% confidence intervals for θ by [θ1 (β) , θ2 (β)] for all 0.01 ≤ β < 1. Starting at 0.01 is arbitrary and you could begin at 0.10 or 0.05 or 0.005, etc. Add to this the interval [θ*, θ*] for the 0% confidence interval for θ. Then place these confidence intervals, one on top of the other, to produce a triangular shaped fuzzy number θ whose α-cuts are the following confidence intervals:
According to Buckley, all that is needed is to finish the “bottom” of to make it a complete fuzzy number. We simply drop the graph of straight down to complete its α-cuts, so
Taheri fuzzy estimators
Let X1, X2, …, X
n
be a random sample and let x1, x2, … x
n
be sample values assumed by the sample. If the sample size is large enough, then
All the above fuzzy sets do not have a compact support. Therefore, it is not possible to be used in operations. To make them fuzzy numbers with compact support, Buckley simply drops the graph straight down, so that where amin is the lowest α-cut. The graph of such a fuzzy number is depicted below:
In the next section, we present a more natural way of constructing these numbers in order to achieve compact support while not changing the shape of the curve.
Let X1, X2, …, X n be a random sample and let x1, x2, … x n be sample values assumed by the sample. Let also β ∈ [0, 1). If the sample size is large enough, then
Where
The spreads of the fuzzy estimators presented above are very small because as n increases, the membership function tends to zero as values go farther from the mean. For a small standard deviation, only values extremely close to the mean have non-zero membership values. The small spreads do not allow for overlaps (patches) in the fuzzy sets. We will make a slight modification to the non-asymptotic fuzzy estimators to allow the membership function to accommodate more values close to the mean. Specifically, values between and can be included in the fuzzy set of , where β ∈ [0, 1). Also to make sure that the proposed membership function is robust enough, we will make it trapezoidal shape instead of the regular triangular shape of fuzzy estimators. We introduce a second statistical parameter β1 which will control how wide the top of the trapezoid is. To enhance the statistical interpretation of the top of the trapezoid, the width will be exactly the (1 - β1) 100% confidence interval for the mean, . When β1 = 1, the trapezoid reduces to a triangle. The proposed membership function thus has four statistical parameters, μ, σ, β, β1. The first two parameters can be easily calculated from the sample. The third parameter controls how many standard deviations from the mean are assumed. For instance we know from statistics that if the data is normally distributed then 99.7% of the data are within 3 standard deviations, 95% are within 2 standard deviations while 68% are within one standard deviation. The last parameter is used to statistically control the width of the top of the trapezoid. The intuition behind the trapezoid is that all values within the (1 - β) 100% confidence interval for the mean should take membership function values, 1 equal to that of the mean of the sample. The final membership function is given below
In this case study, we use the proposed customized membership function based on fuzzy estimators to solve two well-known problems and compare the output of the FIS with those of available membership functions in Matlab. The two examples and the parameters used can be found in the Matlab fuzzy logic tool box [10]. The aim of this case study is to show that the proposed membership function is fully compatible with Matlab and can be used either as a standalone custom function or combined with other membership functions. This might be useful when a system contains some inputs or outputs that are exhibit probabilistic characteristics.
A simulation of 100 random integer inputs in the range of [0 10] was used to test the suitability of the proposed membership function for use in a Fuzzy Inference System in the tipper problem. The water tank level control problem was simulated using the Matlab Simulink environment. The result shows that the proposed membership function produces similar results to that of an FIS that uses normal fuzzy membership functions. The added advantage of our membership function is that we now know exactly which statistical properties of the system produced this result instead of just relying on expert opinion only.The goal of the two examples provided below(Fig. 4–9), is to show that the fuzzy inference system constructed using expert knowledge only, can also have a fuzzy-statistical equivalent interpretation. In these examples, the parameters for the normal fuzzy membership functions (i.e. parameters constructed by expert or optimized using any algorithm) are given and are assumed to be correct according to an expert or after optimization of said parameters, using known optimization techniques [8, 9]. So the parameters for the fuzzy-statistical membership functions were chosen to mimic the shape of the normal fuzzy membership functions like the triangular and the Gaussian membership functions (Tables 1 & 2). We chose β1 = 1 for simplicity since both the normal triangular and the Gaussian membership functions can be represented with a triangular fuzzy-statistical membership function. For more advanced applications where the confidence intervals of the mean is of interest for some input or output variable then, β1 can be set to a value say β1 = 0.05 for the 95% confidence interval for the mean. The other parameters μ, and σ, and β are selected based on the center and spread of both the Gaussian and the triangular membership functions such that the spreads are the same for both the normal fuzzy membership functions (triangular, Gaussian) and the corresponding fuzzy statistical membership function we propose. Our model and parameters are essentially the statistical representation of the expert opinion (via the membership functions). In practice, the parameters μ, and σ are estimated from data as the sample mean and sample variance while the parameters, β (how many standard deviations from the mean to include, say 2σ) and β1 (confidence interval for the mean,e.g. 95%) are both statistical decision making parameters (with their corresponding probabilities) rather than expert opinions,so, they are easier to choose. In applications where triangular membership functions are preferred, β1 = 1, so the only parameter left to choose is β.
In both examples, we showed that the output of the system (Fig. 9), using expert opinion (via normal triangular and Gaussian membership functions) is almost the same with the output of the system using the proposed fuzzy-statistical membership function (Figs. 4a,b, 5 and 7a,b), hence both systems can be considered equivalent. The advantage is that, we now know exactly which statistical parameters of inputs and outputs (mean, standard deviation, etc.) were used by the expert in his model. Our model, thus offers a statistical explanation of all decisions made by the expert when building the fuzzy model. In addition to being a standalone membership function, our proposed membership function can be combined with other well-known membership functions like triangular, Gaussian etc. in a single FIS so both the opinion of experts(experience) and statistics can be used to model or control a system. Thus, this type of membership function can be useful for systems that exhibit randomness and fuzziness in some of the inputs or outputs or both. For example, we might want to answer some interesting statistical questions like,’ With what mean, μand standard deviation, σ, of an input, i, does the system perform best (e.g. after optimization of parameters with expert method or optimization algorithm like PSO [9])? Or, what is the performance of the system, when all inputs are within 2σof their corresponding means?(Assuming there were no experts)
We note that expert opinion or any type of parameter optimization technique can complement the proposed membership function, so for example the proposed membership function can be used to estimate the mean, standard deviation of the inputs and outputs of a system if they are previously unknown or cannot be estimated due to lack of data. So, if we know the final parameters produced by the expert or by optimization, give the best results, then we know exactly under what statistical condition, the system performance is optimal.
Another good combined application would be to first model or control the system using ordinary fuzzy membership functions like triangular,Gaussian etc. (whether with optimized parameters or not), then use the errors of the output(assuming the errors are random and normally distributed) as a new input for the model. The membership function of this new input can be constructed directly from the errors of the output of the model using the proposed membership function with fuzzy estimators. This new model could be re-run again to produce better results since it takes the errors made by the expert or the optimization technique into account.
Conclusion
In this paper, we have shown that membership functions formed by fuzzy estimators can be used in fuzzy systems. The results produced are similar to existing membership functions but the proposed membership functions have more advantages as all the parameters are statistically calculated. Also, if the parameters are tuned using an optimization algorithm, the final parameters can be interpreted statistically making the results easier to understand. The proposed membership function is fully compatible with Matlab custom functions and thus can be used for modelling any fuzzy system. In future research, we hope to use the proposed membership function in real life applications like traffic control, process control, financial applications etc. We will also use the proposed membership function in ANFIS and with other optimization algorithms, so that the parameters can be tuned for better performance.