Example of Multivariate Time Series Analysis

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 CO2 (carbon dioxide). The input series xt is the methane gas feedrate and the CO2 concentration is the output series yt.

In this experiment 296 successive pairs of observations (xt,yt) were collected from continuous records at 9-second 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. This data set is available as a text file.

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.

xt = ϕ1.11xt−1+ϕ2.11xt−2+ϕ1.12yt−1+ϕ2.12yt−2+a1t
yt = ϕ1.22yt−1+ϕ2.22yt−2+ϕ1.21xt−1+ϕ2.21xt−2+a2t

The equation for xt corresponds to gas rate while the equation for yt corresponds to CO2 concentration.

The parameter estimates for the equation associated with gas rate are the following.

Residual standard error: 0.2654 based on 53 degrees of freedom
Multiple R-Squared: 0.9387
Adjusted R-squared: 0.9341
F-statistic: 203.1 based on 4 and 53 degrees of freedom

Residual standard error: 0.1198 based on 53 degrees of freedom
Multiple R-Squared: 0.9985
Adjusted R-squared: 0.9984
F-statistic: 8978 based on 4 and 53 degrees of freedom
p-value: 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 (61-66) and the associated 90 % confidence limits are shown below for each series.