nonparametric

nonparametric - Nonparametric Techniques Nonparametric...

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Nonparametric Techniques Nonparametric Techniques
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2 Nonparametric Techniques x w/o assuming any particular distribution b the underlying function may not be known (e.g. multi-modal densities) b too many parameters x Estimating density distribution directly x Transform into a lower-dimensional space where parametric techniques may apply (more on this later on dimension reduction)
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3 Example x Estimate the population growth, annual rainfall, etc. in the US x p(x,y)dxdy is the probability of rain fall in [x,x+dx,y,y+dy]
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4 Example (cont.) x A simple parametric model for p(x,y) probably does not exist x In stead b partition the area into a lattice b At each (x,y), count the amount of rain r(x,y) b Do that for a whole year b Normalize Σ r (x,y) = 1
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5 Density estimation probability value (x) p x ( ) probability value (x) P p x dx x x i j = ( ) x i x j
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6 x From equation x From observation x Hence P p x dx p x x x x x j i i j = - ( ) ( )( ) P k n = p x k n x x k n V j i ( ) / ( ) / - = Density estimation
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7 Comparison x In Reality: x The number of training samples is limited x if V is too small, k becomes erratic b What does 0 mean? x if V is too large, is not representative x In theory: x If n becomes infinitely large, k/n approaches the probability, p(x) = (k/n)/V is then only a space average x Hence, V must be allowed to go to zero as n goes to infinity p x ( ) p x k n x x k n V j i ( ) / ( ) / - =
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nonparametric - Nonparametric Techniques Nonparametric...

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