x t + h 1 , 1 ( h ) 1 , 2 ( h ) 1 ,p ( h ) . . . 2 , 1 ( h - - PowerPoint PPT Presentation

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
• n the diagonal and cross-covariances off the diagonal:

Γ(h) = E

• xt+h − µ
• (xt − µ)′

=

    

γ1,1(h) γ1,2(h) . . . γ1,p(h) γ2,1(h) γ2,2(h) . . . γ2,p(h) . . . . . . ... . . . γp,1(h) γp,2(h) . . . γp,p(h)

    

3

Sample mean and autocovariances

• sample mean:

¯

x = 1

n

n

• t=1

xt

• sample autocovariance:

ˆ Γ(h) = 1 n

n−h

• t=1
• xt+h − ¯

x

• (xt − ¯

x)′

for h ≥ 0, and ˆ Γ(−h) = ˆ Γ(h)′.

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

rows 20 40 60 c

• l

u m n s 10 20 30 Temperature 4 6 8 10

Soil temperatures 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
• xs+h − µ
• xs − µ
• Intrinsic: E
• xs+h − xs
• and var
• xs+h − xs
• do not depend on

s.

• For an intrinsic process, the (semi-)variogram is

Vx(h) = 1 2var

• xs+h − xs
• 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 sta-

tionary.

• 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