18 To summarize Rs lmY X function I finds the coefficients b and b 1

# 18 to summarize rs lmy x function i finds the

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To summarize: R’s lm(Y X) function I finds the coefficients b 0 and b 1 characterizing the the “least squares” line ˆ Y = b 0 + b 1 X . I That is it minimizes n i =1 ( Y i - ˆ Y i ) 2 = n i =1 e 2 i . The least squares formulas are b 1 = r xy s y s x and b 0 = ¯ Y - b 1 ¯ X . 19
Exercise (on your own): Take the partial derivatives of the sum of squares of the residuals w.r.t. β 0 and β 1 , set these equal to zero, and solve the series of equations to obtain the least squares estimates. 20
Properties of the least squares fit Developing techniques for model validation and criticism requires a deeper understanding of the least squares line. The fitted values ( ˆ Y i ) and “residuals” ( e i ) obtained from the least squares line have some special properties. I From now on “obtained from the least squares line” will be implied (and therefore not repeated) whenever we talk about ˆ Y i and e i . Lets look at the housing data analysis to figure out what some of these properties are ... 21
The fitted values are perfectly correlated with the inputs. > plot(size, reg\$fitted, pch=20, xlab="X", + ylab="Fitted Values") > text(x=3, y=80, col=2, cex=1.5, + paste("corr(y.hat, x) =", cor(size, reg\$fitted))) 1.0 1.5 2.0 2.5 3.0 3.5 80 100 120 140 160 X Fitted Values corr(y.hat, x) = 1 22
The residuals are “stripped of all linearity”. > plot(size, reg\$fitted-price, pch=20, xlab="X", ylab="Residuals") > text(x=3.1, y=26, col=2, cex=1.5, + paste("corr(e, x) =", round(cor(size, reg\$fitted-price),2))) > text(x=3.1, y=19, col=4, cex=1.5, + paste("mean(e) =", round(mean(reg\$fitted-price),0))) > abline(h=0, col=8, lty=2) 1.0 1.5 2.0 2.5 3.0 3.5 -20 -10 0 10 20 30 X Residuals corr(e, x) = 0 mean(e) = 0 23
What is the intuition for the relationship between ˆ Y , e , and X ? I Lets consider some “crazy” alternative line: 1.0 1.5 2.0 2.5 3.0 3.5 60 80 100 120 140 160 X Y LS line: 38.9 + 35.4 X Crazy line: 10 + 50 X 24
This is a bad fit! We are underestimating the value of small houses and overestimating the value of big houses. 1.0 1.5 2.0 2.5 3.0 3.5 -20 -10 0 10 20 30 X Crazy Residuals corr(e, x) = -0.7 mean(e) = 1.8 I Clearly, we have left some predictive ability on the table! 25
As long as the correlation between e and X is non-zero, we could always adjust our prediction rule to do better. We need to exploit all of the predictive power in the X values and put this into ˆ Y , I leaving no “ Xness ” in the residuals. In Summary: Y = ˆ Y + e where: I ˆ Y is “made from X ”; corr( X , ˆ Y ) = 1 ; I e is unrelated to X ; corr( X , e ) = 0 .

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