Center-of-gravity method

The rule base contains all the information required to relate the inputs and outputs. In this case, the independent variables of the membership functions of input and output are different, so the result will be two dimensional. Hence the minimum of the input and the output membership function is performed as per the fuzzy if-then rules. Subsequently, the inference engine operates with the min-max operator to generate the output responses. The output responses are then de-fuzzifled to produce a crisp output using the center of gravity method. Now the minimum of the two membership function is given by ... 

The rule set of the FLC contain 9 rules, which govern the input-output relationship of the FLC and this adopts the Mamdani-style inference engine. We use the center of gravity method to realize defuzzification procedure. In Table 5.3, we present the rule set whose format is established as follows ... 

Height defuzzification. The elements with membership degrees lower than a certain threshold are completely discarded, and the Center-of-gravity method is applied to the remainder of the fuzzy set. 

Center of gravity method ( Centroid name is used in Matlab software) was used in output variables defuzzification process (because all active rules are used in defuzzification process). It improved model sensitivity on input parameters change. Disadvantage of center of gravity method is large amount of calculation (because irregular shaped... 

Figure 10.3. Maximum uncertainty in the band maximum using the center-of-gravity method versus/ when the band has aFWHH of 10 cm and the SNR is 1000. (Recalculated ftom equation 17 in [11], by permission of the Society for Apphed Spectroscopy copyright 1982. It should be noted that the original figure in [11] exhibited a shght error.)...

For the defuzzification the center of gravity method was applied. The authors could achieve three goals by that kind of modeling. They (i) optimized the pHi put to get a minimal CODoutput. (ii) estimated the average efficiency of the anaerobic bioreactor, and (iii) studied the dynamics of the system to draw conclusions for an optimal loading regime of the bioreactor. 

The most popular recipe for defuzzification, although not the fastest, is the centroid method. This finds the vertical line, which would divide the aggregated set determined in the previous step, into two equal portions. The center of gravity (cog) is defined by ... 

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. 

At present the X-ray diffraction method is used mostly to determine atomic positions in a crystal. However, X-ray scattering amplitudes depend directly on the electron-density distribution in a crystal, from which atomic positions can be derived on the assumption of coincidence of the nuclear positions and the center of gravities of total electron densities around atomic nuclei. 

Since at this point we have no method of experimentally identifying a given absorption band in solution in terms of its description in the SLJ couphng scheme, we rely on the similarity in band energy with that established for Pr in various crystal matrices. Figure 2 shows the experimentally determined positions of the center of gravity of the levels of Pr in LaCls (17), LaFs (5) and the levels found in Pr vapor 7,18). 

The method outlined above is strictly applicable only to monodis-perse copolymer samples, unless a viscometer is used for detection of the ThFFF elution profile. This arises because D in equation 3 is an average diffusion coefficient corresponding to a molecular weight My, which is not necessarily equal to the molecular weight at the ThFFF peak maximum. With a viscometer as the ThFFF detector, this limitation could be overcome by using the retention ratio corresponding to the center of gravity of the elution profile. 

There are several experimental methods to measure sY. Macromolecules can be represented as N segments (monomers, groups) of mass m and distance from the center of gravity. The gyration radius is... 

Two typically employed average particle sizes are the arithmetic average particle sizexi o = o [e.g., for a number distribution (r=0) obtained by counting methods], and the weighted average particle size Xi r = Ml r [e.g., tor a volume distribution (r = 3) obtained oy sieve analysis], where Xi r represents the center of gravity on the abscissa of the qr(x) distribution. 

Another method of analysis of elution peaks is based on the statistical moments of the curve and was first proposed by MoQuarrie ( ). The first moment, %, is the center of gravity of... 

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