
384 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
Matrix Φ is symmetric and has an inverse. Therefore, we can solve for coefcients a = Φ
−1
ρ.
For illustration, when AR(1) there is only one equation with obvious solution
()1
1
a (11.35)
For example, consider the series in Figure 11.16, the autocorrelation at lag 1 is ρ(1) = −0.48 and
relatively smaller for higher lags. Thus, we could model the series as AR(1), then solving Equation
11.35 we have a
1
= ρ(1) = −0.48.
When AR(2) we have two equations
()
() ()
12
aa
(11.36)
in matrix form
ρ
ρ
ρ
ρ
()
()
()
()
1
2
11
11
1
2
=
a
a
(11.37)
Lag 1