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In-Depth Information
2.5.6 Multivariate Time Series Models
The observation values of some time series are multivariate, made up of
components that themselves are observations of some time series. Such
multivariate values are presented as vector values
, and the entire
set of multiple values as a matrix made up of individual observation vectors
[, . T
n
x
xx
x
12
ª
º
x
x
...
x
11
12
1
n
«
»
x
x
x
«
...
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x
«
21
22
2
n
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¬
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12
n
Multivariate time series are processed using multivariate analysis , which is the
statistical methodology for processing of multidimensional data.
Model building of multivariate time series is required when the values of one
variable of an individual time series are dependent on the values of variables in
other related time series. For better modelling and for more accurate analysis, all
values concerned should be taken into account simultaneously. For instance, the
corresponding joint observations of two mutually dependent variables have to be
modeled under consideration of the components of a two-dimensional observation
vector
, for i =1, 2, 3, … etc . Thus, a bivariate time series has to be
modeled based on two-dimensional observation vectors of the interdependent
univate time series. But, before building the model it should be checked whether
x
(
yz
,
)
i
i
i
x the two time series (represented by y and z values) mutually correlate, in
which case only the correlation analysis has to be carried out, or
x the two series are causally related, in which case the time series model
should be built.
In practice, the number of time series to be considered simultaneously can be larger
than two, so that multivariate time series have to be built using the observation
vectors and the related observation matrix. Using this presentation approach, the
great majority of basic theory of univariate time series can formally be extended to
the multivariate time series. For instance, in an analogous way the equivalent
ARMA model for a stationary multivariate time series, with zero mean vector, can
be written as
x
DD
x
x
...
DH EH
x
EH
...
EH
n
i
11
t
2 2
t
QQ
t
t
11
t
2 2
t
n
t
n
where x and H are n -dimensional column vectors, H being the multivariate
white noise, and D and E are the elements of the corresponding [ n u n ] matrix of
ARMA model parameters
DD
{}
, for j = 1, 2, …, Q
j
j kk
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