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This paper presents a novel computational approach for estimating fuzzy measures directly from Gaussian mixtures model (GMM). The mixture components of GMM provide the membership functions for the input-output fuzzy sets. By treating consequent part as a function of fuzzy measures, we derived its coefficients from the covariance matrices found directly from GMM and the defuzzified output constructed from both the premise and consequent parts of the nonadditive fuzzy rules that takes the form of Choquet integral. The computational burden involved with the solution of

Generalized fuzzy model (GFM) [

The use of GMM in GFM has provided a generalized framework for additive fuzzy systems. A few existing additive fuzzy systems urging our attention are due to fuzzy models given in Kosko [

Incorporation of the nonadditive property into the fuzzy sets is done so that the corresponding output of a non-additive fuzzy system is explored here. Let us throw some light on the additive and non-additive fuzzy systems [

Some important applications of fuzzy measures are now mentioned. A random generation of fuzzy measures have been introduced and some subfamilies of fuzzy measures tackled in [

Fascinated by the ever growing importance of the fuzzy measure theory [

Fuzzy integrals have come into vogue for the information fusion as they aggregate information from several sources. Out of all fuzzy integrals [

The main theme of the present work is the estimation of fuzzy densities, and hence fuzzy measures for the non-additive fuzzy model are derived from GFM; thus the fuzzy densities can be calculated straightaway from the covariance matrices used in GMM [

This paper is organized as follows. Section

The fuzzy measure or fuzzy capacity is a subjective evaluation introduced by Choquet in 1953 and defined by Sugeno in 1974 for fuzzy integrals. Fuzzy measure includes a number of special class of measures like Sugeno’s

Let us denote

If

for all

Let

The

If

for all

Let

The GFM [

The fuzzy set

the input fuzzy sets

the input sets

the output function is additive in fuzzy measures.

We will now show that the above conditions are necessary to represent the defuzzified output of the nonadditive GFM in the Choquet integral form as follows:

with

When a fuzzy measure is available on a finite set

Let

The cluster-wise breakup of the estimated output of the additive fuzzy system [

Equation (

Equation (

As we have already proved that the Choquet integral is the functionality of non-additive GFM, It is now easy to extend to this functionality the GMM case from the fact that the output is Gaussian in the nonadditive case too as per the expression

The algorithm has the following steps.

Normalize the input-output data, so that the data values lie in between 0 and 1:

Find the premise model parameters of GMM using the EM Algorithm.

Determine fuzzy densities

Choose initial values of

Compute

Using (

Compute the estimated output using (

Learn the model. This requires the following two subtasks:

set up an objective function [

Update the values of

Repeat Step

Calculate

Here, we take up the gas furnace data [

Estimated parameters of A 4-component GMM using the EM algorithm.

Cluster or rule no. | Weight of the rule | Mean | Covariance |
---|---|---|---|

Cluster or rule no. 1 | |||

Cluster or rule no. 2 | |||

Cluster or rule no. 3 | |||

Cluster or rule no. 4 |

Fuzzy densities and estimated fuzzy measures of the model are given in Tables

Fuzzy densities using GMM.

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.2473 | 0.7760 | 0.0005 | 0.0000 |

Rule no. 2 | 0.5699 | 0.8204 | 0.0028 | 0.0003 |

Rule no. 3 | 0.6284 | 0.8671 | 0.0070 | 0.0001 |

Rule no. 4 | 0.4844 | 0.7330 | 0.0042 | 0.0012 |

Estimated lambda (

Rule 1 | Rule 2 | Rule 3 | Rule 4 |
---|---|---|---|

−0.1241 | −0.8364 | −0.9108 | −0.6197 |

Estimated fuzzy measures.

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 1.0000 | 0.7923 | 0.0021 | 0.0009 |

Rule no. 2 | 0.9997 | 0.8214 | 0.0032 | 0.0003 |

Rule no. 3 | 0.9999 | 0.8686 | 0.0071 | 0.0001 |

Rule no. 4 | 0.9994 | 0.7359 | 0.0053 | 0.0012 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.2077 | 0.7902 | 0.0012 | 0.0009 |

Rule no. 2 | 0.1783 | 0.8182 | 0.0029 | 0.0003 |

Rule no. 3 | 0.1313 | 0.8615 | 0.0070 | 0.0001 |

Rule no. 4 | 0.2635 | 0.7306 | 0.0041 | 0.0012 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 1.0000 | 0.7823 | 0.0013 | 0.0000 |

Rule no. 2 | 1.0000 | 0.8541 | 0.0635 | 0.0001 |

Rule no. 3 | 1.0000 | 0.9175 | 0.0863 | 0.0003 |

Rule no. 4 | 1.0000 | 0.7230 | 0.0169 | 0.0002 |

For the training dataset containing the first 250 data vectors,

Plot of actual output and model output for training data set (uppermost plot) and plot of actual output and predicted output and the corresponding prediction error for Box and Jenkins’s gas furnace data.

Lattice of fuzzy measure values of the constituent singleton sets [

(a) Membership function for input one. (b) Membership functions for input two. (c) Membership function for input three. (d) Membership function for input four. (e) Membership function for output.

The first rule

the weight of the rule; 0.2991,

the membership functions of the premise parts

The consequent part of a rule can be found using Table

Similarly other rules can also be formed.

In this, we consider the industrial dryer data obtained from the

fuel flow rate,

hot gas exhaust-fan speed,

rate of flow of raw material,

moisture content of the raw material after drying which is

Sampling period of 10 seconds is considered to generate a total of 867 samples. Input and output data are normalized using (

The number of clusters is found to be 4 using EM clustering, which yields the parameters of GMM in Table

Estimated parameters of A 4-component GMM using the EM algorithm.

Cluster or rule no. | Weight of the rule | Mean | Covariance |
---|---|---|---|

Cluster or rule no. 1 | |||

Cluster or rule no. 2 | |||

Cluster or rule no. 3 | |||

Cluster or rule no. 4 |

The estimated fuzzy densities, fuzzy measures,

Fuzzy densities using GMM.

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.4982 | 0.4295 | 0.3881 | 0.7989 |

Rule no. 2 | 0.4996 | 0.7368 | 0.6158 | 0.9027 |

Rule no. 3 | 0.4696 | 0.7543 | 0.3704 | 0.5916 |

Rule no. 4 | 0.4844 | 0.7330 | 0.0042 | 0.0012 |

Estimated lambda (

Rule 1 | Rule 2 | Rule 3 | Rule 4 |
---|---|---|---|

−0.9970 | −1.0000 | −1.0000 | −0.9947 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.9669 | 0.9315 | 0.8779 | 0.7989 |

Rule no. 2 | 0.9027 | 0.8828 | 0.8758 | 0.8749 |

Rule no. 3 | 0.9500 | 0.6529 | 0.6244 | 0.5918 |

Rule no. 4 | 1.0000 | 1.0000 | 1.0000 | 0.9981 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.0354 | 0.0536 | 0.0790 | 0.7989 |

Rule no. 2 | 0.0199 | 0.0070 | 0.0009 | 0.8749 |

Rule no. 3 | 0.2971 | 0.0285 | 0.0326 | 0.5918 |

Rule no. 4 | 0.0000 | 0.0000 | 0.0019 | 0.9981 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 1.0000 | 0.7823 | 0.0013 | 0.0000 |

Rule no. 2 | 1.0000 | 0.8541 | 0.0635 | 0.0001 |

Rule no. 3 | 1.0000 | 0.9175 | 0.0863 | 0.0003 |

Rule no. 4 | 1.0000 | 0.7230 | 0.0169 | 0.0002 |

The training data set yields

Plot of actual output and model output for training data set (uppermost plot) and plot of actual output and predicted output and the corresponding prediction error for industrial dryer data.

Lattice of fuzzy measure values of the constituent singleton sets of rule no.1 of Table

In this, we consider the real time hourly electric load data, that is, hourly peak electric load consumption with uncertain weather information like hourly average temperature, humidity, and wind speed for the month of April 2003. The variables are designated as

hourly average Temperature,

hourly average Humidity,

hourly average Wind speed,

hourly peak electric load consumption which is

A number of 720 samples of Input and output data are normalized using (

The number of clusters in this example also is found to be 4 using the EM clustering after discarding a few datasets as outliers. Application of the EM algorithm yields the parameters of GMM in Table

Estimated parameters of A 4-component GMM using the EM algorithm (load).

Cluster or rule no. | Weight of the rule | Mean | Covariance |
---|---|---|---|

Cluster or rule no. 1 | |||

Cluster or rule no. 2 | |||

Cluster or rule no. 3 | |||

Cluster or rule no. 4 |

Fuzzy densities using GMM.

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.8479 | 0.6488 | 0.5817 | 0.6735 |

Rule no. 2 | 0.0196 | 0.0633 | 0.0001 | 0.2634 |

Rule no. 3 | 0.0264 | 0.0100 | 0.0725 | 0.0255 |

Rule no. 4 | 0.0001 | 0.0469 | 0.0001 | 0.1708 |

Estimated lambda (

Rule 1 | Rule 2 | Rule 3 | Rule 4 |
---|---|---|---|

29.3843 | −0.0776 | −0.9995 | 0.2543 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 1.0000 | 0.0877 | 0.0439 | 0.6735 |

Rule no. 2 | 1.0000 | 0.1200 | 0.0569 | 0.2634 |

Rule no. 3 | 1.0000 | 0.0910 | 0.0818 | 0.0255 |

Rule no. 4 | 1.0000 | 0.4718 | 0.1974 | 0.1708 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 1.0000 | 0.0291 | 0.0145 | 0.0033 |

Rule no. 2 | 1.0000 | 0.1569 | 0.0744 | 0.0614 |

Rule no. 3 | 1.0000 | 0.1468 | 0.1320 | 0.0161 |

Rule no. 4 | 1.0000 | 0.3891 | 0.1628 | 0.1409 |

Estimated

Rule | ||||
---|---|---|---|---|

Rule no. 1 | 0.9709 | 0.0146 | 0.0112 | 0.0033 |

Rule no. 2 | 0.8431 | 0.0825 | 0.0130 | 0.0614 |

Rule no. 3 | 0.8532 | 0.0148 | 0.1159 | 0.0161 |

Rule no. 4 | 0.6109 | 0.2263 | 0.0219 | 0.1409 |

The training data set is found to give

Plot of actual output and model output for training data set (uppermost plot) and plot of actual output and predicted output and the corresponding prediction error for hourly electric load data.

Lattice of

The performance measures of the applications A and B are given in Table

Comparison of

Data | WM [ | Gan et al. [ | Non-additive (using | Proposed algorithm (using fuzzy measure estimation) |
---|---|---|---|---|

Box & Jenkins’s Gas Furnace data | 0.0057 | 0.0032 | 0.0021 | 0.00208 |

Industrial Dryer’s data | NA | 0.0027 | 0.0019 | 0.00189 |

This paper deals with an important issue of computing fuzzy measures directly from covariance matrices found in GMM. Use of EM algorithm in GMM provides us with the input clusters and their Gaussian memberships. The consequent parts of the fuzzy rules, that is, the output function, of GFM are altered for incorporating the non-additive property. It has to be noted that we have been still using the multiplicative T-norm in the premise part but the additive property of S-norm in the consequent part. For this the output function is defined as a linear function of fuzzy measures serving as coefficients. Computation complexity is reduced by replacing Sugeno’s

The defuzzified output of the non-additive GFM rules is then shown to be in the form of Choquet integral. By showing the Choquet fuzzy integral as the functionality of non-additive GFM, we have enhanced its capability to tackle a variety of real-life problems. We have also proved the corresponding non-additive functionality for GMM. This has been accomplished by showing that

The non-additive fuzzy model is applied on the three benchmark applications, namely, Box and Jenkins’s, Dryers’ data and on the real time electric load demand data. The results demonstrate the superiority of this model over the additive fuzzy model of Gan et al. [

In this section we will find the derivatives required for the learning laws. To keep the thread going, we recall the relevant equations from Section

For

The above polynomial and their derivatives in (