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class03_22

# class03_22 - 1.017/1.010 Class 22 Linear Regression...

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1.017/1.010 Class 22 Linear Regression Regression Models Fluctuations in measured ( dependent ) variables can often be attributed (in part) to other ( independent ) variables. ANOVA identifies likely independent variables. Regression methods quantify relationship between dependent and independent variables. Consider problem with one random dependent variable y and one independent variable x related by a regression model : y = g ( x, a 1 , a 2 ,…, a m ) + e g (..) = known function (e.g. a polynomial) a 1 , a 2 , …, a m = m unknown regression parameters e = random residual , E [ e ] = 0, Var [ e ] = σ e 2 , CDF = F e ( e ). Data for 1872-1986 ( x i , y i ) Illustrate basic concepts with the following special case, where g (..) is quadratic in x and linear in the a i 's : y ( x ) = g ( x, a 1 , a 2 , a 3 ) + e = a 1 + a 2 x +a 3 x 2 + e Mean of y ( x ) is: E [ y ( x )] = a 1 + a 2 x +a 3 x 2 1

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Objective is to estimate the a i 's from a set of y measurements [ y 1 y 2 .... y n ] taken at different known x values [ x 1 x 2 .... x n ] . The complete set of measurement equations is: n ,..., i x a x a a x i i i i i 1 ; ) ( 2 3 2 1 = + + + = = e y y The residual errors [
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class03_22 - 1.017/1.010 Class 22 Linear Regression...

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