Neural Fuzzy Inference Network

The main classifier in our proposed system is a particular fuzzy neural network which is the so-called self-constructing neural fuzzy inference network (SONFIN). The SONFIN is a general connectionist model of a fuzzy logic system, which can find its optimal structure and parameters automatically. There are no rules initially in the SONFIN, and they are created and adapted as on-line learning proceeds via simultaneous structure and parameter learning. The SONFIN can always find itself an economic network size, and the learning speed as well as the modeling ability is all superior to normal neural networks.

2.3.1 Desired Output

The desired output is defined as the membership-value which describes how this pattern belongs to a specified class [29]. Here, a formula to evaluate the membership-value would be given. Let’s consider an l-class problem domain, and there should be also l nodes in the output layer. Let the n-dimensional vectors Ok and Vk denote the mean and standard deviation respectively of the numerical training data for the kth class. A weighted distance could be obtained to represent the normalized distance between this pattern and the class. The weighted distance of the training pattern Fi from the kth class is defined as

∑

where Fij is the value of the jth component of the ith pattern point, and Ck

is the kth class. The weight 1/vkj is used to take care of the variance of the classes so that a feature with higher variance has less weight (significance) in characterizing a class. Note that when all the feature values of a class are the same, then the standard deviation will be zero. In that case, we consider vkj = 1 such that the weighting coefficient becomes 1. this is obvious because any feature occurring with identical magnitudes in all members of a training set is certainly an important feature of the set.

Hence its contribution to the membership function should not be reduced.

Therefore, the desired output (dk) of the kth output node for the ith input pattern, is defined as

fe ith pattern in class Ck, zik is the weighted distance of the training pattern from and the positive constants fd and fe are the denominational and exponential fuzzy generators controlling the amount of fuzziness in this class-membership set. They influence the amount of overlapping among the output classes. Note that, here we have used a (nonlinguistic) definition of the output nodes which indicates the degree of belongingness of a pattern to a class. However, this definition may be suitably modified in other application areas to include linguistic definitions. Obviously µk (Fi) lies in the interval [0, 1]. Here (2. 19) is

such that the higher the distance of a pattern from a class, the lower its membership value to that class. It is to be noted that when the distance is 0, the membership value is 1 (maximum) and when the distance is infinite, the membership value is 0 (minimum).

2.3.2 SONFIN

The structure of the SONFIN is shown in Fig. 4-2. This 6-layered network realizes a fuzzy model of the following form:

Rule i: IF x1 is Ai1 and … and xn is Ain

THEN y is m0i + ajixj + … (2. 20)

where Aij is a fuzzy set, m0i is the center of a symmetric membership function on y, and aji is a consequent parameter. It is noted that unlike the traditional TSK model where all the input variables are used in the output linear equation, only the significant ones are used in the SONFIN; i.e., some ajis in the above fuzzy rules are zero. We shall next describe the functions of the nodes in each of the six layers of the SONFIN.

Each node in Layer 1, which corresponds to one input variable, only transmits input values to the next layer directly. Each node in Layer 2 corresponds to one linguistic label (small, large, etc.) of one of the input variables in Layer 1. In other words, the membership value that specifies the degree how an input value belongs to a fuzzy set is calculated in Layer 2. A node in Layer 3 represents one fuzzy logic rule and performs precondition matching of a rule. The number of nodes in layer 4 is equal

normalized in this layer. Layer 5 is called the consequent layer. Two types of nodes are used in this layer, and they are denoted as blank and shaded circles in Fig. 9, respectively. The node denoted by a blank circle (blank node) is the essential node representing a fuzzy set of the output variable.

The shaded node is generated only when necessary. One of the inputs to a shaded node is the output delivered from Layer 4, and the other possible inputs (terms) are the selected significant input variables from Layer 1.

Combining these two types of nodes in Layer 5, we obtain the whole function performed by this layer as the linear equation on the THEN part of the fuzzy logic rule in (2. 20). Each node in Layer 6 corresponds to one output variable. The node integrates all the actions recommended by Layer 5 and acts as a defuzzifier to produce the final inferred output.

Two types of learning, structure and parameter learning are used concurrently for constructing the SONFIN. The structure learning includes both the precondition and consequent structure identification of a fuzzy if-then rule. For the parameter learning, based upon supervised learning algorithms, the parameters of the linear equations in the consequent parts are adjusted to minimize a given cost function. The SONFIN can be used for normal operation at any time during the learning process without repeated training on the input-output patterns when on-line operation is required. There are no rules in the SONFIN initially, and they are created dynamically as learning proceeds upon receiving on-line incoming training data by performing the following learning processes simultaneously,

(B) Construction of fuzzy rules,

(C) Optimal consequent structure identification, (D) Parameter identification.

Processes A, B, and C belong to the structure learning phase and process D belongs to the parameter learning phase.

Fig. 9 network structure of SONFIN