Membership functions

Figure 9-25. Membership function for the fuzzy set of numbers close to 3.

On the other hand, if we want to characteri2e objects which are described by the rather fuzzy statement "numbers dose to three", we then need a membership function which describes the doseness to three. An adequate membership function could be the one plotted in Figure 9-25 m x) has its maximum value of m x) = 1 for value x = 3. The greater the distance from x to 3 gets, the smaller is the value of m x). until it reaches its minimum m x) = 0 if the distance from x to 3 is greater than say 2, thus for x > 5 or x < 1. 

An important property of a fuzzy set is its cardinality. While for crisp sets the cardinality is simply the number of elements in a set, the cardinality of a fuzzy set A, CardA, gives the sum of the values of the membership function of A, as in Eq. (9). 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

The central concept of fuzzy set theory is that the membership function /i, like probability theory, can have a value of between 0 and 1. In Figure 10.3, the membership function /i has a linear relationship with the x-axis, called the universe of discourse U. This produces a triangular shaped fuzzy set. 

Square nodes in the ANFIS structure denote parameter sets of the membership functions of the TSK fuzzy system. Circular nodes are static/non-modifiable and perform operations such as product or max/min calculations. A hybrid learning rule is used to accelerate parameter adaption. This uses sequential least squares in the forward pass to identify consequent parameters, and back-propagation in the backward pass to establish the premise parameters. 

Layer 1 contains adaptive nodes that require suitable premise membership functions (triangular, trapezoidal, bell, etc). Hence... 

Fig. 10.38 Adaption of membership function features using genetic algorithms.

Current Membership Function (click on MF to select) Name... 

The input variables state now no longer jumps abruptly from one state to the next, but loses value in one membership function while gaining value in the next. At any one time, the truth value of the indoor or outdoor temperature will almost always be m some degree part of two membership functions ... 

As proposed in [131, 132], the general AFDF scheme can be given in terms of an atomic orbital membership function mk(i) defined as... 

Zadeh [1975] extended the classical set theory to the so-called fuzzy set theory, introducing membership functions that can take on any value between 0 and 1. As illustrated by the intersection of the (hard) reference data set (A) and the fuzzed test data set (C), the intersection (E) shows an agreement of about 80%. Details on application of fuzzy set theory in analytical chemistry can be found in Blaffert [1984], Otto and Bandemer [ 1986a,b] and Otto et al. [1992],... 

A crisp membership function for the determination of volatile liquids. 

Membership functions for the sets "very volatile," "volatile," and "slightly volatile."... 

To deal with this difficulty, we construct a membership function plot, from which memberships can be determined directly (Figure 8.6). The membership function defines an unambiguous relationship between boiling point and membership value, so the latter can then be determined consistently, given the boiling point. 

The x-axis in a plot of a membership function represents the universe of discourse. This is the complete range of values that the independent variable can take the y-axis is the membership value of the fuzzy set. 

The membership functions in Figure 8.6 provide the basis for a consistent determination of membership values for liquids in three sets, "very volatile,"... 

Triangular (Figure 8.7), piecewise linear (Figure 8.8), or trapezoidal (Figure 8.9) functions are commonly used as membership functions because they are easily prepared and computationally fast. 

A membership function that resembles a normal distribution. 

Suppose that we have defined a membership function for the "Low pH" set. Most acid solutions would be, to some degree, members of this fuzzy set. We may want to be able to qualify the description by adding modifiers, such as "very," "slightly," or "extremely" whose use allows us to retain close ties to natural language. The qualifiers that modify the shape of a fuzzy set are known as hedges. We can see the effect of the hedges "very" and "very very" in Figure 8.11. 

The rules that the fuzzy system uses are expressed in terms such as a "high" or a "medium" pH, while the experimental input data are numerical quantities. The first stage in applying these rules is to transform the input data into a degree of membership for each variable in each class through the use of membership functions. 

This presumed membership of 0.5 in the "low" set must now be combined with the output from Rl, which was that the rate is "high" to a degree of 0.2. To combine these, we require a membership function that relates the actual reaction rate to the fuzzy descriptors "low" and "high" (Figure 8.16). 

Membership functions to convert between experimental reaction rates and fuzzy sets. 

The center of gravity is a reliable and well-used method. Its principal disadvantage is that it is computationally slow if the membership functions are complex (though it will be clear that if the membership functions are as simple as the ones we have used here, the computation is quite straightforward). Other methods for determining the nonfuzzy output include center-of-largest-area and first-of-maxima. 

---