7.6 Exercise 7.5 can also be extended to unequally spaced time points t1 < ··· < tn. Show that in this case B is a tri-diagonal matrix with diagonal elements bii =

⎧

⎪

⎨

⎪

⎩

1 2(t2−t1)

, i = 1, 1 2(ti−ti−1) + 1 2(ti+1−ti)

, i = 2, ..., n − 1, 1 2(tn−tn−1)

, i = n, and with super- and sub-diagonal elements bi,i+1 = bi+1,i = − 1 2(ti+1 − ti

)

, i = 1, ..., n − 1.

Further, C = 1 2

(tn − t1).

Hence, deduce that the kriging predictor becomes the piecewise linear interpolator û(t0) =

⎧

⎪

⎨

⎪

⎩

x1, t0 < t1, xi + t0−ti ti+1−ti

(xi+1 − xi

), ti ≤ t0 ≤ ti+1, xn, t0 ≥ tn.

Further, show that the kriging variance is given by

????2 K(t0) =

⎧

⎪

⎨

⎪

⎩

2(t1 − t0), t0 < t1, 2(ti+1 − t0)(t0 − ti

)∕(ti+1 − ti

), ti ≤ t0 ≤ ti+1, 2(t0 − tn), t0 > tn.

Note the kriging variance is quadratic in t between the data sites and linear outside them.

Hint: Adapt the proof of Exercise 7.5.