Consider a Takagi - Sugeno - Kanga  (TSK) Model consisting of rules of the form :

If x₁ is A_i1 and ... and x_r is A_ir

THEN y = f_i (x₁, x₂, ...., x_r) = b_i0 + b_i1x₁ + b_irx_r

assume, α_i is the matching degree of rule i, then the total output of the model is given by :

The correct answer is option 2.

Key Points

If x1 is Ai1 and ... and xr is Air

THEN y = fi (x1, x2, ...., xr) = bi0 + bi1x1 + birxr

assume, αi is the matching degree of rule i,

 The consequence polynomials deteriorate to constant numbers β0r when the outputs are discrete crisp numbers (to represent symbolic values). Given an input vector (A∗1,..., A∗m), the TSK engine performs inference in the following steps:

1. Determine the firing strength of each rule Rr (r ∈ {1, 2,..., n}) by integrating the similarity degrees between its antecedents and the given inputs.
2. Calculate the sub-output led by each rule Rr based on the given observation (A×1,..., A×m).
3. Generate the final output by integrating all the sub-outputs from all the rules:

\(y = \frac{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}{f_i}\left( {{x_1},\;{x_2}, \ldots .,\;{x_r}} \right)}}{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}}}\)

∴ Hence the correct answer is \(y = \frac{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}{f_i}\left( {{x_1},\;{x_2}, \ldots .,\;{x_r}} \right)}}{{\mathop \sum \nolimits_{i = 1}^L {\alpha _i}}}\)