6. Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.5. Multivariate Time Series Models
|If each time series observation is a vector of numbers, you can model them using a multivariate form of the Box-Jenkins model||
The multivariate form of the Box-Jenkins univariate models is
sometimes called the ARMAV model, for AutoRegressive Moving Average
Vector or simply vector ARMA process.
The ARMAV model for a stationary multivariate time series, with a zero mean vector, represented by
|Estimation of parameters and covariance matrix difficult||
The estimation of the matrix parameters and
matrix is complicated and very difficult without computer
software. The estimation of the Moving Average matrices is
especially an ordeal. If we opt to ignore the MA component(s) we
are left with the ARV model given by:
The parameter matrices may be estimated by multivariate least squares, but there are other methods such as maximium likelihood estimation.
|Interesting properties of parameter matrices||There are a few interesting properties associated with the phi or AR parameter matrices. Consider the following example for a bivariate series with n =2, p = 2, and q = 0. The ARMAV(2,0) model is:|
Without loss of generality, assume that the X series is input and
the Y series are output and that the mean vector = (0,0).
Therefore, tranform the observation by subtracting their respective averages.
|Diagonal terms of Phi matrix||
The diagonal terms of each Phi matrix are the scalar
estimates for each series, in this case:
1.22, .2.22 for the output series Y.
The lower off-diagonal elements represent the influence of the
input on the output.
This is called the "transfer" mechanism or transfer-function model as discussed by Box and Jenkins in Chapter 11. The terms here correspond to their terms.
The upper off-diagonal terms represent the influence of the output on the input.
This is called "feedback". The presence of feedback can also be seen
as a high value for a coefficient in the correlation matrix of the
residuals. A "true" transfer model exists when there is no feedback.
This can be seen by expressing the matrix form into scalar form:
Finally, delay or "dead' time can be measured by studying the lower
off-diagonal elements again.
If, for example, 1.21 is non-significant, the delay is 1 time period.