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Unformatted text preview: STA 414/2104 Mar 9, 2010 Notes I Sample test questions posted I Review and/or questions on Thursday this week I Test will have 3 questions: one from Sample test, one specific to 414/2104 I Extra Office Hour Monday, March 15, 34 I Watch web site for late breaking announcement re MidTerm 1 / 20 STA 414/2104 Mar 9, 2010 Neural Networks I “feed forward single layer neural network” I Y k = g k { β k + M X m = 1 β km σ ( α m + p X ‘ = 1 α ‘ m X ‘ ) } = f k ( X ‘ ) I σ ( x ) = 1 1 + e x tanh ( x ) = e x e x e x + e x , maps to ( 1 , + 1 ) 2 / 20 STA 414/2104 Mar 9, 2010 ... neural networks I Y k = g k { β k + M X m = 1 β km σ ( α m + p X ‘ = 1 α ‘ m X ‘ ) } = f k ( X ‘ ) I θ = ( α m ,α m ,β k ,β k ) I R ( θ ) = ∑ N i = 1 ∑ K k = 1 { y ik f k ( x i ) } 2 , or I R ( θ ) = ∑ N i = 1 ∑ K k = 1 y ik log f k ( x i ) I dim ( θ ) = M ( p + 1 ) + K ( M + 1 )→ regularization/shrinkage, also called weight decay I minimize R ( θ ) + λ J ( θ ) = R ( θ ) + λ X km β 2 km + X m ‘ α 2 m ‘ ! I standardize inputs to mean 0, variance 1 for regularization I backfitting algorithm for minimizing R ( θ ) described in § 11.4; extension to R ( θ ) + λ J ( θ ) in § 11.5.2 3 / 20 STA 414/2104 Mar 9, 2010 ... neural networks I nnet in MASS library: recommend λ ∈ ( 10 4 , 10 2 ) for squared error loss; λ ∈ ( . 01 ,. 1 ) for loglikelihood I compare Figure 11.4 top/bottom I results very sensitive to starting values: R ( θ ) has many local maxima I recommendation (Ripley): take average predictions over several nnet fits I weight decay seems to be more important than number of hidden units I See § 11.7, 8, 9 for interesting examples where neural nets work well 4 / 20 STA 414/2104 Mar 9, 2010...
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 Spring '09
 neural network, MAR, αi yi, yi yj xiT

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