Let yt represent the global temperature series (gtemp) shown in Figure 1.2.

(a) Fit a smoothing spline using gcv (the default) to yt and plot the result superimposed on the data. Repeat the fit using spar=.7; the gcv method yields spar=.5 approximately. (Example 2.14 on page 75 may help. Also in R, see the help file ?smooth.spline.)

(b) Write the model yt = xt + vt with ∇2xt = wt, in state-space form. [Hint:

The state will be a 2 × 1 vector, say, xt = (xt, xt−1)0

.] Assume wt and vt are independent Gaussian white noise processes, both independent of x0. Fit this state-space model to yt, and exhibit a time plot the estimated smoother, xbn t and the corresponding error limits, xbn t ±2

√Pbn t superimposed on the data.

(c) Superimpose all the fits from parts

(a) and

(b) [include the error bounds]

on the data and briefly compare and contrast the results.