6.
Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis 6.4.5. Multivariate Time Series Models


Bivariate Gas Furance Example 
The gas furnace data from
Box, Jenkins, and Reinsel, 1994 is used to illustrate the
analysis of a bivariate time series. Inside the gas furnace,
air and methane were combined in order to obtain a mixture
of gases containing CO\(_2\)
(carbon dioxide).
The input series \(x_t\) is the methane gas feedrate and
the CO\(_2\) concentration is the output series \(y_t\).
In this experiment 296 successive pairs of observations \((x_t, \, y_t)\) were collected from continuous records at 9second intervals. For the analysis described here, only the first 60 pairs were used. We fit an ARV(2) model as described in 6.4.5. 

Plots of input and output series 
The plots of the input and output series are displayed below.


Model Fitting 
The scalar form of the ARV(2) model is the following.
$$ \begin{eqnarray}
x_t & = & \phi_{1.11}x_{t1} + \phi_{2.11}x_{t2} +
\phi_{1.12}y_{t1} + \phi_{2.12}y_{t2} + a_{1t} \\
& & \\
y_t & = & \phi_{1.22}y_{t1} + \phi_{2.22}y_{t2} +
\phi_{1.21}x_{t1} + \phi_{2.21}x_{t2} + a_{2t}
\end{eqnarray} $$
The equation for \(x_t\)
corresponds to gas rate while the equation for \(y_t\)
corresponds to CO\(_2\)
concentration.
The parameter estimates for the equation associated with gas rate are the following.
Estimate Std. Err. t value Pr(>t) a_{1t } 0.003063 0.035769 0.086 0.932 φ_{1.11} 1.683225 0.123128 13.671 < 2e16 φ_{2.11} 0.860205 0.165886 5.186 3.44e06 φ_{1.12} 0.076224 0.096947 0.786 0.435 φ_{2.12} 0.044774 0.082285 0.544 0.589 Residual standard error: 0.2654 based on 53 degrees of freedom Multiple RSquared: 0.9387 Adjusted Rsquared: 0.9341 Fstatistic: 203.1 based on 4 and 53 degrees of freedom pvalue: < 2.2e16 The parameter estimates for the equation associated with CO\(_2\) concentration are the following.
Estimate Std. Err. t value Pr(>t) a_{2t } 0.03372 0.01615 2.088 0.041641 φ_{1.22} 1.22630 0.04378 28.013 < 2e16 φ_{2.22} 0.40927 0.03716 11.015 2.57e15 φ_{1.21} 0.22898 0.05560 4.118 0.000134 φ_{2.21} 0.80532 0.07491 10.751 6.29e15 Residual standard error: 0.1198 based on 53 degrees of freedom Multiple RSquared: 0.9985 Adjusted Rsquared: 0.9984 Fstatistic: 8978 based on 4 and 53 degrees of freedom pvalue: < 2.2e16 BoxLjung tests performed for each series to test the randomness of the first 24 residuals were not significant. The \(p\)values for the tests using CO\(_2\) concentration residuals and gas rate residuals were 0.4 and 0.6, respectively. 

Forecasting 
The forecasting method is an extension of the model and follows the theory outlined in the previous section. The forecasted values of the next six observations (6166) and the associated 90 % confidence limits are shown below for each series. 90% Lower Concentration 90% Upper Observation Limit Forecast Limit     61 51.0 51.2 51.4 62 51.0 51.3 51.6 63 50.6 51.0 51.4 64 49.8 50.5 51.1 65 48.7 50.0 51.3 66 47.6 49.7 51.8 90% Lower Rate 90% Upper Observation Limit Forecast Limit     61 0.795 1.231 1.668 62 0.439 1.295 2.150 63 0.032 1.242 2.452 64 0.332 1.128 2.588 65 0.605 1.005 2.614 66 0.776 0.908 2.593 