FCMs are more applicable when the data in the first place is an unsupervised one. The FCMs work on the opinion of experts. FCMs model the world classes and causal relations between classes. FCMs are fuzzy signed directed graphs with feedback. The directed edge from causal concept to concept measures how much causes . The time-varying concept function measures the non-negative occurrence of some fuzzy event, perhaps the strength of political sentiment, historical trend. The edges take values in the fuzzy causal interval [-1, 1]. indicates no causality, indicates causal increase, increases as increases (or decreases as decreases). indicates a causal decrease or negative causality. decreases as increases (or increases as decreases), simple FCMs have edge values in {-1, 0, 1}.
FCMs can successfully represent knowledge, and human experience introduces concepts to represent the essential elements and the cause-and-effect relationship among the concepts to model the behavior of any system. It is a very convenient, simple, and powerful tool used in numerous fields such as social, economic, educational, political, medical, and technical. The FCMs work on the opinion of experts. FCMs model the world as a collection of classes and causal relationships between classes. FCMs are signed and directed graphs with feedback. The directed edge Cij from causal concept Ci to concept Cj measures how much Ci causes Cj. Here Ci, ….. Cn are concepts or nodes of FCMs. The edge weights or causalities Cij are taken from the set {- 1, 0, 1}. The directed graph is drawn using edge weights eij. The Matrix E is defined by E = (eij), where eij is the weight of the directed edge CiCj. E is called the adjacency matrix of FCM. It will work as a dynamical system to identify the joint effects of all the interacting fuzzy knowledge from Experts.
Cognitive maps are techniques: that attempt to depict and analyze the cognitive process of human thinking and human behavior in specific domains by creating models. These models are represented by assigned directed graphs of concepts and by the various casual relationships that exist between the concepts.
Crisp membership function.
Fuzzy-membership.
In 1965, Lotfi Zedah
Uncertain causal knowledge is stored in FCMs. FCMs are fuzzy signed digraphs with feedback. The sign (+ or -) of FCM edges indicates a causal increase or decrease. The fuzzy degree of causality is indicated by a number in [−1, 1]. FCMs learn by modifying their causal connections in sign and magnitude, structurally analogous to how neural networks learn. An appropriate causal learning law for inductively inferring FCMs from time-series data is the differential Hebbian law, which modifies causal connections by correlating time derivatives of FCM node outputs. The differential Hebbian law contrasts with the Hebbian output-correlation learning laws of adaptive neural networks.
FCM nodes represent variable phenomena or fuzzy sets. An FCM node nonlinearly transforms weighted summed inputs into numerical output, again in analogy to a model neuron. Unlike expert systems, which are feed-forward search trees, FCMs are nonlinear dynamical systems. FCM resonant states are limit cycles or time-varying patterns. An FCM limit cycle or hidden pattern is an FCM inference. Experts construct FCMs by drawing causal pictures or digraphs. The corresponding connection matrices are used for inference. By additively combining augmented connection matrices, any number of FCMs can be naturally combined into a single knowledge network. The credibility w in [0, 1] of the ith expert is included in this learning process by multiplying the ith expert's augmented FCM connection matrix by w. Combining connection matrices is a simple type of adaptive inference. An unsupervised learning law, such as the differential Hebbian learning law, generally modifies connection matrices. Under special conditions, differential Hebbian dynamical systems are proved globally stable: they resonate on fixed-point attractors.
FCMs are more applicable when the data in the first place is an unsupervised one. The FCMs work on the opinion of experts.FCM has a major role, mainly when the data concerned is unsupervised. Further, this method is the simplest and most effective one as it can analyze the data by directed graphs and connection matrices. FCMs model the world as a collection of classes and casual relations between classes, which was introduced by Bart Kosko
*Let C1,C2…….Cn be causal concepts and let eij=e(Ci,Cj) be the causal edge function value, the amount of causality Ci imparts to Cj. For a Cognitive map,eij=0,1 or -1.
*Letb E=(eij)1≤i,j≤n, represents the matrix of causal edge values for the given FCM.
Suppose the FCM is a cognitive cognitive map. Then, E is an adjacency matrix. It lists all one-edge paths on the cognitive map.
*E2=[eij2]=ExE lists all two-edge paths on the cognitive map. is non-zero if there is a k such that eik and ekj are non-zero. Similarly E3, E4…..En-1 list the effect of summing all three-edge, four-edge……n-1 edge indirect effects.
*Then the total-effect matrix T is the sum of the powered matrices Ei 4 *If the above process is repeated with E replaced E bar, the matrix of absolute values of E, then ebar (k) is
ij
non-zero if and only if there are ebar (k) many K-edge paths from C to C. Such information is useful when
ij i j
searching for forward and backward chains.
Definition 1. Fuzzy logic is a method of describing and processing ambiguous data. The more traditional propositional logic, such as 'it will rain tomorrow', must be either true or false. However, many facts humans use about the world have some ambiguity. Fuzzy- Not clear, distinct, or precise; blurred.
Definition 2. If X is a universal set and xX, then a fuzzy set is defined as a collection of ordered pairs, 8 Where (x) is called the membership function that maps X to the membership space M.
Definition 3. A fuzzy set defined on the set R of real numbers is called a fuzzy number whose membership function is of the form Ã: R→ [0,1] under certain conditions.
I. Ã is normal.
II. Ã is convex.
III. Ã is piecewise continuous.
Definition 4. An FCM is a directed graph with concepts like policies, events, nodes, and causalities as edges. It represents a causal relationship between concepts. They are called fuzzy nodes.
Definition 5. FCMs with edge weights or causalities from the set {-1, 0, 1} are called simple FCMs.
Definition 6. Consider the nodes/concepts of the FCM. Suppose the directed graph is drawn using edge weight . The matrix E be defined by
where is the weight of the directed edge . E is called the adjacency matrix of the FCM, also known as the connection matrix of the FCM.
It is important to note that all matrices associated with an FCM are always square matrices with diagonal entries as zero.
Definition 7. A finite number of FCMs can be combined to produce the joint effect of all the FCMs. Let be the adjacency matrices of the FCMs with nodes then the combined FCM is got by adding all the adjacency matrices .
We denote the combined FCM adjacency matrix by .
Suppose is a vector that is passed into a dynamical system E. Then after thresholding and updating the vector suppose we get we denote that by . Thus, the symbol ' ' means the resultant vector has been thresholded and updated.
Definition 8. Let be the nodes of an FCM. Let be the edges of the FCM ( ). Then, the edges form a directed cycle. An FCM is said to be cyclic if it possesses a directed cycle. An FCM is said to be acyclic if it does not possess any Directed cycle.
Definition 9. Let be a cycle when is switched on, and if the causality flows through the edges of a cycle and if it again causes , we say that the dynamical system goes round and round. This is true for any node , for i = 1, 2 ….n. The equilibrium state for this dynamical system is called the Hidden pattern.
Definition 10. If the equilibrium state of a dynamical system is a unique state vector, it is called a fixed point. Consider an FCM with as nodes. For example, let us start the dynamical system by switching on . Let us assume that the FCM settles down with and on, i.e., the state vector remains as (1, 0, 0, ….,0, 1) this state vector (1, 0, 0,…,0, 1) is called the fixed point.
Definition 11. If the FCM settles down with a state vector repeating in the form . Then, this equilibrium is called the Limit cycle.
Definition 12. A finite number of FCMs can be combined together to produce the joint effect of all the FCMs. Let E ,E ,....E be adjacency matrices of the FCMs with nodes C ,C ,....C . Then, the combined
1 2 p 1 2 n
FCM is got by adding all the adjacency matrices E ,E ,....E . We denote the combined FCM adjacency matrix
1 2 p
by E = E + E +....+ E .
1 2 p
Definition 13. Advantages and disadvantages of FCM The main advantage of this method is that it is simple. It functions on an expert's opinion. When the data is unsupervised, the FCM comes in handy. This is the only known fuzzy technique that gives the hidden pattern of the situation, as we have a very well-known theory that states that the strength of the data depends on the number of experts' opinions; we can use combined FCMs with several experts' opinions.
At the same time, the disadvantage of the combined FCM is when the weights are1 and -1 for the same Ci Cj, we have the sum adding to zero; thus, at all times, the connection matrices E1, E2……… , Ek may not be conformable for addition.
Combined conflicting opinions tend to cancel out and are assisted by the strong law of large numbers. A consensus emerges as the sample opinion approximates the underlying population opinion. This problem will be easily overcome if the FCM entries are only 0 and 1. The knowledge of guidelines (instead of experts' knowledge) can be easily converted to a number of if-then rules, which are used to construct the FCM model.
Modeling-knowledge representation, Decision making, Enterprise Resource Management, Socio-economic systems, Engineering & Technology Management, Adaptation and Learning, Classification tasks, Robots and control, Political and Social Fields, Military planning, Production Systems, Prediction capabilities, Ecology and environmental
Definition 14. Let C ,C ,...C be n distinct attributes of a problem n very large and a non-prime. If we divide
1 2 n
n into k equal classes. i.e., k/n and if n / k = t , which are disjoint, and if we find the directed graph of each of their classes of attributes with t attributes each, then their corresponding connection matrices are formed, and these connection matrices are joined as blocks to form a n n matrix.
The n n connection matrix forms the combined disjoint block FCM of equal classes. If the classes are not divided to have equal attributes but if they are disjoint classes, we have n n a connection matrix called the combined disjoint block FCM of unequal classes/size.
Here we approach the problem through attributes using Combined Disjoint Block Fcms (CDBFCMs) that are basically matrices that predict the feelings of all the attributes under certain conditions. Before we apply combined o disjoint block FCMs (CDBFCMs) to this problem, we define a set of 20 attributes given by experts. We work with analyzing by using a directed graph and its connection matrices.