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Chapter 7 - Non-linear Models
Lab Solution
1 Problem 9
set.seed (1) library (MASS) attach (Boston)
(a). Use the poly() function to fit a cubic polynomial regression to predict nox using dis. Report the regression output, and plot the resulting data and polynomial fits.
lm.fit = lm (nox ~ poly (dis, 3), data = Boston) summary (lm.fit)
Call:
lm(formula = nox ~ poly(dis, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-0.121130 -0.040619 -0.009738 0.023385 0.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.554695 0.002759 201.021 < 2e-16 ***
poly(dis, 3)1 -2.003096 0.062071 -32.271 < 2e-16 ***
poly(dis, 3)2 0.856330 0.062071 13.796 < 2e-16 ***
poly(dis, 3)3 -0.318049 0.062071 -5.124 4.27e-07 ***
---
Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.06207 on 502 degrees of freedom
Multiple R-squared: 0.7148,Adjusted R-squared: 0.
F-statistic: 419.3 on 3 and 502 DF, p-value: < 2.2e-
dislim = range (dis) dis.grid = seq (from = dislim[1], to = dislim[2], by = 0.1) lm.pred = predict (lm.fit, list (dis = dis.grid)) plot (nox ~ dis, data = Boston, col = "darkgrey") lines (dis.grid, lm.pred, col = "red", lwd = 2)
dis
nox
(b). Plot the polynomial fits for a range of different polynomial degrees (say, from 1 to 10), and report the associated residual sum of squares.
all.rss = rep (NA, 10) for (i in 1:10) { lm.fit = lm (nox ~ poly (dis, i), data = Boston) all.rss[i] = sum (lm.fit$residuals^2) } all.rss
## [1] 2.768563 2.035262 1.934107 1.932981 1.915290 1.878257 1.849484 1.835630 1.
## [10] 1.
(c). Use the bs() function to fit a regression spline to predict nox using dis. Report the output for the fit using four degrees of freedom. How did you choose the knots? Plot the resulting fit.
library (splines) sp.fit = lm (nox ~ bs (dis, df = 4, knots = c (4, 7, 11)), data = Boston) summary (sp.fit)
Call:
dis
nox
(d). Now fit a regression spline for a range of degrees of freedom, and plot the resulting fits and report the resulting RSS. Describe the results obtained.
all.cv = rep (NA, 16) for (i in 3:16) { lm.fit = lm (nox ~ bs (dis, df = i), data = Boston) all.cv[i] = sum (lm.fit$residuals^2) } all.cv[- c (1, 2)]
## [1] 1.934107 1.922775 1.840173 1.833966 1.829884 1.816995 1.825653 1.792535 1.
## [10] 1.788999 1.782350 1.781838 1.782798 1.
lm.fit = lm (nox ~ bs (dis, df = 14), data = Boston) lm.pred = predict (lm.fit, list (dis = dis.grid)) plot (nox ~ dis, data = Boston, col = "darkgrey") lines (dis.grid, lm.pred, col = "red", lwd = 2) lm.fit = lm (nox ~ bs (dis, df = 5), data = Boston) lm.pred = predict (lm.fit, list (dis = dis.grid)) lines (dis.grid, lm.pred, col = "blue", lwd = 2)
