(i) ANFIS:

For simplicity, we assume the fuzzy inference system under consideration has two inputs x and y and one output z.

Suppose that the rule base contains two fuzzy it..then rules of Takegi and Sugeno's type:

Rule 1: If x is $A_1$and y is$B_1$ , then

$f_1=p_1x+q_1y+r_1$

Rule 2: If x $A_2$is and y is $B_2$ , then

$f_2=p_2x+q_2y+r_2$

Depending on the two rule ANFIS architecture is as follows:

Layer 1:

Every node i in this layer is a square node with a node function

$O_i^1=\mu_{Ai}(x)$

where x is the input to node i, and Ai is the linguistic label (small,large etc) associated with this node function

Layer 2:

Every node in this layer is a circle node labeled π which multiplies the incoming signals and send the product out. For instance,

$w_i=\mu_{Ai}(x) \times i \mu_{Bi}(y); i=1,2,3..$

Each node output represents the firing strength of a rule.

Layer 3:

Every node in this layer is a circle node labeled N. The i th node calculates the ration of the i th rules firing strength to the sum of all rules firing strengths:

$\bar w_i=\frac{w_i}{w_2+w_2 };i=1,2,..$

For convenience, outputs of this layer will be called "Normalized firing strengths".

Layer 4:

Every node i in this layer is a square node with a node function

$O_i^4=\bar w_if_i=\bar w_i(p_ix+q_iy+r_i)$

where w¯iw¯i  is the output of layer 3 and {pi,qi,ripi,qi,ri} is tha parameter set. Parameters in this layer will be referred to as consequent parameters.

Layer 5:

The single node in this layer is a circle node labeled ∑∑ that computes the overall output as the summation of all incoming signals i.e.

$O_i^5=$ overall output = $\sum_{i} w_if_i=\frac{\sum_{i}w_if_i}{\sum_{i}w_i}$

Thus we have constructed an adaptive network which is functionally equivalent to a Type-3 fuzzy inference system.

(ii) Brain state in box mode:

The "brain-state-in-box" sounds like we have a brain in a box without body.  But the model is defined as follows:

Let W be a symmetric weight matrix whose largest eigenvalues have positive real components.  In addition, W is required to be positive semi-definite, i.e.,  xTWx >= 0 for all x.  Let x(0) denotes the initial state vector.  The BSB algorithm is defined by the pair of equations:

        $y(n) = x(n) + h Wx(n)\\ x(n+1) = f(y(n)).$

Or more concisely, the updating rule of the "brain state" x (a vector) is

      $ x = f(x + h W x)$

where h is a small positive constant called the feedback factor.  The f is a piecewise-linear function of the form

       ( f(x) = +1 \hspace{1cm} if\hspace{1cm} x > 1;\

f(x) = x \hspace{1cm} if \hspace{1cm}-1 < x < 1;\

f(x) = -1 \hspace{1cm} if \hspace{1cm}x < -1.)

When the W is choosing with the required property (positivity of largest eigenvalues), the effect of the algorithm is to drive the system for components of x to binary values +1 or 1 for each of the neuron.   We can view it as a mapping from continuous inputs x(0) to discrete binary outputs.  The final states are of the form (-1,+1,-1,-1,+1,+1, ..., +1).  This represents a corner of cube in an N-dimensional space of linear size 2, centered at origin.  This is the box of the brain-state-in "a box".   The dynamics is such that the state moves to the wall of the box and then drives to the corner of the box.

Application of BSB model

What is a good use of BSB model?  A natural application for the BSB model is clustering.  Such as the classification of radar signals from the source of emitters.  The matrix W has to be (unsupervised) learned using some of the methods discussed in early chapters.