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# note10 - STAT5044 Regression and Anova Inyoung Kim Outline...

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STAT5044: Regression and Anova Inyoung Kim

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Outline 1 Polynomial Regression
Polynomial Regression (nonlinearity) Using taylor series approximate polynomial function m th order polynomial, y i = β 0 + β 1 x i + β 2 x 2 i + ··· + β m x m i + ε i , ε i N ( 0 , σ 2 ) . The number of parameter p = m + 1 y = 1 x 1 x 2 1 ··· x m 1 1 x 2 x 2 2 ··· x m 2 . . . . . . . . . ··· . . . 1 x n x 2 n ··· x m n β 0 . . . β m + ε = X [ m ] β + ε Polynomial regression is still a linear model (linear model of parameter)

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General Anova H 0 : there is no “polynomial type of relationship” between x and y . ⇐⇒ β 1 = β 2 = ... = β m = 0 H 1 : at least one β i ( i 1 ) does not equal to zero. Full model (general model) Y i = β 0 + β 1 x i + β 2 x 2 i + ... + β m x m i + ε i Reduced model (corespond to H 0 ) Y i = β 0 + ε i
General Anova Residual sum of square RSS ( β 0 , β 1 ,..., β m ) = n i = 1 ( y i - ˆ β 0 - ˆ β 1 x i -···- ˆ β m x m i ) 2 RSS ( β 0 ) = n

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note10 - STAT5044 Regression and Anova Inyoung Kim Outline...

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