Vector-Valued Series–Notation
• Studies of time series data often involve p > 1 series.
• E.g. Southern Oscillation Index and recruitment in a fish
population (p = 2).
• Treated as a p × 1 column vector:


xt = 


xt,1
xt,2
...
xt,p





1
Mean Vector
• Assume jointly weakly stationary.
• mean vector:

µ



= E (xt) = 



E xt,1
E xt,2
...
E xt,p






=



µ1
µ2
...
µp





2
Autocovariance Matrix
• Autocovariance matrix contains individual autocovariances
on the diagonal and cross-covariances off the diagonal:
Γ(h) = E



=

h
xt+h − µ (xt − µ)
γ1,1(h) γ1,2(h)
γ2,1(h) γ2,2(h)
...
...
γp,1(h) γp,2(h)
0
i

. . . γ1,p(h)
. . . γ2,p(h) 


...
...

. . . γp,p(h)
3
Sample mean and autocovariances
• sample mean:
n
1 X
xt
x̄ =
n t=1
• sample autocovariance:
X 1 n−h
Γ̂(h) =
xt+h − x̄ (xt − x̄)0
n t=1
for h ≥ 0, and Γ̂(−h) = Γ̂(h)0.
4
Multidimensional Series (Spatial Statistics)
• Some studies involve data indexed by more than one variable.
• E.g. soil surface temperatures in a field
• Notation: xs is the observed value at location s (s for spatial).
5
Soil temperatures
10
6
60
40
s
4
row
ature
Temper
8
30
colu 20
mns
20
10
6
Autocovariance and Variogram
• Stationary : E (xs) and cov xs+h, xs do not depend on s.
• For a stationary process, the autocovariance function is
γ(h) = cov xs+h, xs = E
h
xs+h − µ
xs − µ
i
• Intrinsic: E xs+h − xs and var xs+h − xs do not depend on
s.
• For an intrinsic process, the (semi-)variogram is
1
Vx(h) = var xs+h − xs
2
7
• A stationary process is intrinsic (see Problem 1.26), but an
intrinsic process is not necessarily stationary.
• In one dimension, the random walk is intrinsic but not stationary.
• When stationary, Vx(h) = γ(0) − γ(h).
• Isotropic: an intrinsic process is isotropic if the variogram is
a function only of |h|, the Euclidean distance between s + h
and s.
8