ISYE 7400 Homework 1, Due Feb 10, 2009
http://www2.isye.gatech.edu/ jeffwu/courses/isye7400/
January 28, 2009
Problem 1. Consider the following model for simulating flowrate through a borehole:
y=2πTu(Hu−Hl)
ln(r/rw)n1 + 2LTu
ln(r/rw)r2
wKw+Tu
Tlo,
where the ranges of the interest for the eight variables are: rw∈(0.05,0.15), r∈(100,50000),
Tu∈(63070; 115600), Hu∈(990; 1110), Tl∈(63.1; 116), Hl∈(700; 820), L∈(1120; 1680),
and Kw∈(9855; 12045). Simulate n= 50 observations using the design given in the excel
file Design.xls, in which each variable is scaled in [0,1].
Fit (a) an ordinary kriging model with µ(x) = β0and (b) a universal kriging model
with linear effects for the 8 factors, di.e., µ(x) = β0+P8
i=1 βixi. Use a Gaussian product
correlation function and use MLE for estimating the correlation parameters. Compute the
RM SP E =(1
10000
10000
X
i=1
{ˆyi(xi)−yi}2)1/2
for the two kriging models by randomly generating 10,000 points in the experimental region.
Which model gives better prediction?
Problem 2. Prove the Gauss-Markov Theorem:
(i) Assume Yi=PK
j=1 βjXij +i, for i= 1, . . . , n, where βjare nonrandom and unobservable
parameters, Xij are non-random and observable, iare random. If ifor i= 1, . . . , n satisfy,
(a) E(i) = 0, (b) V ar(i) = σ2, and (c) Cov(i, j) = 0, for i6=j, the ordinary least square
estimator ˆ
β= (XTX)−1XTyis a BLUE of β.
(ii) By changing condition (b) to V ar(i) = σ2
ifor i= 1, . . . , n, the generalized least square
estimator ˆ
β= (XTΣ−1X)−1XTΣ−1yis a BLUE of β, where Σ = diag{σ2
1, . . . , σ2
n}.
1