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531f10NR - STAT 531 Newton-Raphson Algorithm HM Kim...

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STAT 531: Newton-Raphson Algorithm HM Kim Department of Mathematics and Statistics University of Calgary Fall 2010 1/21
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Problems Find the extrema of a function h ( θ ) over a domain Θ Find the solution to an implicit equation g ( θ ) = 0 over a domain Θ Focus on the maximization problem max θ Θ h ( θ ) Fall 2010 2/21
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numerical optimization methods performance depends on the analytical properties of the target function (such as convexity, boundedness, and smoothness) - optimise in R (golden section search and successive parabolic interpolation) - nlm in R ( Newton-Raphson algorithm ) Fall 2010 3/21
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optimise , Example 1 : Find x to maximize g ( x ) = log( x ) 1+ x . > z=seq(2.63,6,len=200) > gz = log(z)/(z+1) > plot(z,gz,type="l",xlab="x",ylab="g(x)") Fall 2010 4/21
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> gz = function(z) log(z)/(z+1) > curve(gz,2.63,4, type="l",xlab="x",ylab="g(x)") > optimise(gz,interval=c(0,10),maximum=T)$max [1] 3.591116 > optimise(gz,interval=c(0,10),maximum=T)$objective [1] 0.2784645 Fall 2010 5/21
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, Example 2 : Considering maximizing the likelihood of a Cauchy ( θ , 1) l ( θ | x 1 , ··· , x n ) = n i =1 1 1+( x i - θ ) 2 . mean: not defined variance: not defined Fall 2010 6/21
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, The sequence of maxima is converging to θ * = 0 as n . > ref=rcauchy(400) # n=1,..., 400 > # log-likelihood > f=function(theta){-sum(log(1+(x-theta)^2))} > mi=NULL > for (i in 1:400){ > x=ref[1:i] > aut=optimise(f,interval=c(-10,10), maximum=T) > mi=c(mi,aut$max)} > plot(mi,ty="l",lwd=2,xlab="",ylab="arg") > #likelihood > f=function(theta){prod(1/(1+(x-theta)^2))} > mip=NULL > for (i in 1:400){ > x=ref[1:i] > aut=optimise(f,interval=c(-10,10), maximum=T) > mip=c(mip,aut$max)} > lines(mip,col="sienna",lwd=2) Fall 2010 7/21
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Fall 2010 8/21
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