Membership value

The distance D between the analyte and the reference was found as the normed sum of intersections of the corresponding membership values, and the similarity was estimated as 5=1- DID. ... 

Conventional set theory distinguishes between those elements that are members of a set and those that are not, there being very clear, or crisp boundaries. Figure 10.2 shows the crisp set medium temperature . Temperatures between 20 and 30 °C he within the crisp set, and have a membership value of one. 

In equation (10.2) / is a delimiter. Hence the numerator of each term is the membership value in fuzzy set M associated with the element of the universe indicated in the denominator. When = 11, equation (10.2) can be written as... 

Fuzzifieation is the proeess of mapping inputs to the FLC into fuzzy set membership values in the various input universes of diseourse. Deeisions need to be made regarding... 

Fuzzy inference is therefore the process of mapping membership values from the input windows, through the rulebase, to the output window(s). 

The assignment of the membership values is done by an optimization procedure. Of the many criteria that have been described, probably the best known is that of Ruspini [32] ... 

One of the disadvantages of the method is that one must determine the smoothing parameter by optimisation. When the smoothing parameter is too small (Fig. 33.16a) many potential functions of a learning class do not overlap with each other, so that the continuous surface of Fig. 33.15 is not obtained. A new object u may then have a low membership value for a class (here class K) although it clearly belongs to that class. An excessive smoothing parameter leads to a too flat surface (Fig. 33.16b), so that discrimination becomes less clear. The major task of the... 

Fig. 2.19. Various sets of analytical data (A) Hard reference data set, mR(x). (B) Hard test data set, mT(x), which is slightly shifted compared with (A). (C) Fuzzy set of test data, mT(x) = exp (—(x — a)2/b2). (D) Intersection mT R(x) of test data and reference data which is empty in this case. (E) Intersection of fuzzed test data and reference data with a membership value of about 0.8 in this case...

The degree of membership or the membership value of an object in a set measures the extent to which the object belongs in that set. For crisp sets, the only possible membership values are 0 and 1 (Figure 8.1). A membership of 1 tells us that the object is a member of that set alone, while a membership of 0 shows that it has no membership in the set. 

Fuzzy logic gets around this difficulty by replacing hard boundaries between sets with soft divisions. Objects are allocated to fuzzy sets, which are sets with fuzzy boundaries. The membership value of an object within a fuzzy set can lie anywhere within the real number range of 0.0 to 1.0 (Figure 8.4). 

Fuzzy and Crisp Membership Values in the "Volatile" Set for Some Liquids... 

Table 8.1 compares the fuzzy membership and crisp membership values in the volatile set for a few liquids (Figure 8.5). 

Although it is the purpose of fuzzy systems to handle ill-defined information, this does not mean that we can get away with uncertainty in the allocation of membership values. If some of the membership values for liquids in a database were proposed by one person and the rest by a second person, the two groups of memberships could well be inconsistent unless both people used the same recipe for determining membership. Any deductions of the fuzzy system would then be open to doubt. In fact, even the membership values determined by just one person might be unreliable unless they had used a properly defined method to set membership values. The hold-a-wet-finger-in-the-air style of finding a membership value is not supportable. 

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,"... 

In a conventional expert system, the only rules to fire are those for which the condition is met. In a fuzzy system, all of the rules fire because all are expressed in terms of membership, not the Boolean values of true and false. Some rules may involve membership values only of zero, so have no effect, but they must still be inspected. Implicitly, we assume an or between every pair of rules, so the whole rule base is... 

Once the input data have been used to find fuzzy memberships, in the second step we compute the membership value for each part in the condition or antecedent of a rule. These are then combined to determine a value for the conclusion or consequent of that rule. If the antecedent is true to some degree, the consequent will be true to the same degree. 

This is easily done. Rather than averaging the values of all consequents that predict membership of the same fuzzy set, we include separately the membership values for every rule in the calculation of equation (8.6). As Figure 8.18 shows, there are now three areas to consider, so the numerator in equation (8.6) becomes ... 

Developing a classification rule This step requires the known class membership values for all calibration samples. Classification rules vary widely, but they essentially contain two components ... 

The final step in the first iteration is to compute new membership values u j using Equation 6 and the distance definition of Equation 4. 

The computation of the shared membership values u j by Equation 6 usually results in relatively small values being assigned to outliers or noisy measurement vectors. It is not difficult to locate these values in the final membership matrix, and the corresponding data vectors can be singled out for deletion, or closer examination. 

First, a list of unique scaffolds was derived and sorted by complexity. The complexity was calculated from four structural descriptors, namely number of rings in the smallest set of smallest rings, number of heavy atoms, number of bonds and the sum of heavy atomic numbers in the scaffold. Each scaffold or class center in the list was assigned an ID that corresponded to its position in the list. How much a molecule resembled its class center was determined by the number of side-chains attached to the scaffold. Fewer side-chains will give a closer resemblance to the class center. The similarity of a drug with the class center was reflected in the membership value. The membership value was based on the sum of heavy atomic numbers, the number of rotating bonds, the number of one and two nodes and the number of double and triple bonds in a molecule compared with its scaffold. Since the membership value indicated the contribution of rings in the class center for a certain... 

The average operator simultaneously takes all membership values into account. But, no guarantee can be made for any... 

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