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AMDA-2008-Session 20 - Non-Parametric Regression Time...

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Non-Parametric Regression & Time Series Analysis
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Nadaraya-Watson Kernel Estimator kernel. boxcar the is K where estimator kernel a of example an is estimator average local The ) ( by given are ) ( weights the and kernel a is K where ) ( ) ( r ˆ by defined is estimator kernel Watson - Nadaraya The bandwidth. the called number, positive a be 0 h Let 1 1 n - - = = = = n j j i i i n i i i h x x K h x x K x l x l Y x l x
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An Interesting Fact estimate. regression kernel the is which ) ( ) ( ) ( r ˆ is solution that the find we calculus Using . ) )( ( w squares of sum weighted the minimize to ) ( ˆ a choose want to We . ) ( Let w Fix x. 1 1 n n 1 i 2 i i = = = = - - = n i i n i i i i n i x w Y x w x a Y x x r h x x K x
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Generalization ( 29 2 1 1 2 1 1 1 2 1 0 0 0 n 1 0 1 2 1 0 1 0 1 0 ) )( ( ) )( ( ) )( ( ) )( ( ) )( ( ) ( ) ( a ˆ that see can one algebra little a and calculus Using ) ( ˆ ˆ ) ( r ˆ get we , particular In ). ( ˆ a ˆ (u) r ˆ get Then we . ) ( ) ( squares of sums ighted locally we the minimizing by , a estimate We ). ( a r(u) i.e. function linear a by r(u) function regression the e approximat can we x to close u of for values Now, r(x). value the estimate want to We Fix x. - - - - - - - = = - + - - - - + = = = = = = = n i i i n i i i n i i
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