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

6.4.5.1.

Example of Multivariate Time Series Analysis

A multivariate Box-Jenkins example

As an example, we will analyze the gas furnace data from the Box-Jenkins textbook. In this gas furnace, air and methane were combined in order to obtain a mixture of gases which contained CO₂ (carbon dioxide). The methane gas feedrate constituted the input series and followed the process

Methane Gas Input Feed = .60 - .04 X(t)

the CO₂ concentration was the output, Y(t). In this experiment 296 successive pairs of observations (X_t, Y_t) were read off from the continuous records at 9-second intervals. For the example described below, the first 60 pairs were used. It was decided to fit a bivariate model as described in the previous section and to study the results.

Plots of input and output series

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

From a suitable Box-Jenkins software package, we select the routine for multivariate time series analysis. Typical output information and prompts for input information will look as follows:

SEMPLOT output

MULTIVARIATE AUTOREGRESSION   Enter FILESPEC       GAS.BJ                                                      Explanation of Input   How many series? :  2      the input and the output series   Which order?        :  2      this means that we consider times                                           t-1 and t-2 in the model , which is                                            a special case of the general ARV                                            model     SERIES     MAX           MIN            MEAN      VARIANCE        1            56.8000        45.6000        50.8650         9.0375                   2              2.8340        -1.5200           0.7673         1.0565                  NUMBER OF OBSERVATIONS:  60 .    THESE WILL BE MEAN CORRECTED.  so we don't have to    fit the means  -------------------------------------------------------------------------------    OPTION TO TRANSFORM DATA    Transformations?    :  y/N  -------------------------------------------------------------------------------    OPTION TO DETREND DATA    Seasonal adjusting? :  y/N  -------------------------------------------------------------------------------    FITTING ORDER:  2    OUTPUT SECTION     the notation of the output follows the notation of the previous    section                                     MATRIX FORM OF ESTIMATES         1       1.2265     0.2295      -0.0755     1.6823         2      -0.4095    -0.8057       0.0442    -0.8589    Estimate Std. Err t value Prob(t) Con 1 -0.0337 0.0154 -2.1884 0.9673 Con 2 0.003 0.0342 0.0914 0.0725 1.11 1.2265 0.0417 29.4033 > .9999 1.12 0.2295 0.0530 4.3306 0.9999 1.21 -0.0755 0.0926 -0.8150 0.5816 1.22 1.6823 0.1177 14.2963 > .9999 2.11 -0.4095 0.0354 -11.5633 > .9999 2.12 -0.8057 0.0714 -11.2891 > .9999 2.21 0.0442 0.0786 0.5617 0.4235 2.22 -0.8589 0.1585 -5.4194 > .9999         -------------------------------------------------------------------------------    Statistics on the Residuals               MEANS      -0.0000     0.0000    COVARIANCE MATRIX         0.01307      -0.00118        -0.00118       0.06444    CORRELATION MATRIX       1.0000    -0.0407      -0.0407     1.0000  ----------------------------------------------------------------------  SERIES   ORIGINAL      RESIDUAL    COEFFICIENT OF                    VARIANCE     VARIANCE   DETERMINATION    1                9.03746           0.01307           99.85542                             2                1.05651           0.06444           93.90084                           This illustrates excellent univariate fits for the individual series.  ---------------------------------------------------------------------  This portion of the computer output lists the results of testing for independence (randomness) of each of the series.    Theoretical Chi-Square Value:    The 95th percentile       =   35.16595    for degrees of freedom  =   23    Test on randomness of Residuals for Series:  1    The Box-Ljung  value    =   20.7039      Both Box-Ljung and Box-Pierce    The Box-Pierce value    =   16.7785       tests for randomness of residuals    Hypothesis of randomness accepted.      using the chi-square test on the                                                                  sum of the squared residuals.    Test on randomness of Residuals for Series:  2    The Box-Ljung  value    =   16.9871       For example, 16.98 < 35.17    The Box-Pierce value    =   13.3958       and 13.40 < 35.17    Hypothesis of randomness accepted.         --------------------------------------------------------                           FORECASTING SECTION       --------------------------------------------------------  The forecasting method is an extension of the model and follows the theory outlined in the previous section. Based on the estimated variances and number  of forecasts we can compute the forecasts and their confidence limits. The user, in this software, is able to choose how many forecasts to obtain, and at what confidence levels.    Defaults are obtained by pressing the enter key, without input.    Default for number of periods ahead from last period = 6.    Default for the confidence band around the forecast  = 90%.   How many periods ahead to forecast?     6    Enter confidence level for the forecast limits : .90:                               SERIES:  1                        90 Percent Confidence limits      Next Period      Lower      Forecast         Upper           61            51.0534       51.2415       51.4295           62            50.9955       51.3053       51.6151           63            50.5882       50.9641       51.3400           64            49.8146       50.4561       51.0976           65            48.7431       49.9886       51.2341           66            47.6727       49.6864       51.7001                               SERIES:  2                        90 Percent Confidence limits      Next Period      Lower      Forecast         Upper           61              0.8142        1.2319        1.6495           62              0.4777        1.2957        2.1136           63              0.0868        1.2437        2.4005           64             -0.2661        1.1300        2.5260           65             -0.5321        1.0066        2.5453           66             -0.7010        0.9096        2.5202