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Newton Raphson Rule Tuning

Newton Raphson Rule Tuning. Rules. Consider a rule base with M rules, r th rule has the form IF x 1 is T r,1 AND … AND x n is T r,n THEN y is y r (or y is y r + other stuff ) TSK fuzzy system has mathematical form. Scalar Newton-Raphson iteration. Example: p 2 -5=0, p=?. Why?

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Newton Raphson Rule Tuning

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  1. Newton Raphson Rule Tuning

  2. Rules Consider a rule base with • M rules, rth rule has the form • IF x1 is Tr,1 AND … AND xn is Tr,n THEN y is yr (or y is yr + other stuff) • TSK fuzzy system has mathematical form

  3. Scalar Newton-Raphson iteration Example: p2-5=0, p=? • Why? • Consider p+dp, ignore dp2, solve for dp • Taylor series expansion.

  4. What is G(p)? • How do we ENGINEER G(p)? • TSK f(x,p) to G(p) • What do we know about p? • Previous value of p • New data about p

  5. New p = Old p New data

  6. Matrix Inverses • Defining relationships • MM-1 = I = M-1M • Primary use • If Mx=b then M-1Mx=M-1b hence x=M-1b • Variety of techniques to compute M-1 • M-1 exists only if M is square and full rank

  7. Pseudo Inverses • Defining relationships • M+MM+=M+; MM+M=M; [MM+]*= MM+; [M+M]*=M+M • Primary use • If Mx=b then x=M+b+[I-M+M]z • Many ways to compute M+ • All matrices have pseudo inverses • “Two Applications of Pseudo Inverses” • pinv function

  8. Modified Newton-Raphson iteration

  9. Example • y=x2 • Tuning the TSK approximation • NRtuning.lib • main()

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