Specialized Control Charts |
Diagnosing and Modeling Autocorrelation
You can diagnose autocorrelation with an autocorrelation
plot created with the ARIMA procedure.
proc arima data=chemical;
identify var = xt;
run;
Refer to SAS/ETS User's Guide
for details on the ARIMA procedure.
The plot, shown in Figure 49.2,
indicates that the data are highly autocorrelated with
a lag 1 autocorrelation of 0.83.
Autocorrelations |
Lag |
Covariance |
Correlation |
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 |
Std Error |
0 |
48.348400 |
1.00000 |
| |********************| |
0 |
1 |
40.141884 |
0.83026 |
| . |***************** | |
0.100000 |
2 |
34.732168 |
0.71837 |
| . |************** | |
0.154229 |
3 |
29.950852 |
0.61948 |
| . |************ | |
0.184683 |
4 |
24.739536 |
0.51169 |
| . |********** | |
0.204409 |
5 |
20.594420 |
0.42596 |
| . |********* | |
0.216840 |
6 |
18.427704 |
0.38114 |
| . |********. | |
0.225052 |
7 |
17.400188 |
0.35989 |
| . |******* . | |
0.231417 |
8 |
17.621272 |
0.36446 |
| . |******* . | |
0.236948 |
9 |
18.363756 |
0.37982 |
| . |******** . | |
0.242489 |
10 |
16.754040 |
0.34653 |
| . |******* . | |
0.248367 |
11 |
16.844924 |
0.34841 |
| . |******* . | |
0.253156 |
12 |
17.137208 |
0.35445 |
| . |******* . | |
0.257906 |
13 |
16.884092 |
0.34922 |
| . |******* . | |
0.262732 |
14 |
17.927976 |
0.37081 |
| . |******* . | |
0.267334 |
15 |
16.801860 |
0.34752 |
| . |******* . | |
0.272429 |
16 |
17.076544 |
0.35320 |
| . |******* . | |
0.276826 |
17 |
17.815028 |
0.36847 |
| . |******* . | |
0.281296 |
18 |
16.501312 |
0.34130 |
| . |******* . | |
0.286082 |
19 |
14.662196 |
0.30326 |
| . |****** . | |
0.290126 |
20 |
12.612280 |
0.26086 |
| . |***** . | |
0.293278 |
21 |
11.105364 |
0.22969 |
| . |***** . | |
0.295590 |
22 |
8.891648 |
0.18391 |
| . |**** . | |
0.297369 |
23 |
6.794132 |
0.14052 |
| . |*** . | |
0.298504 |
24 |
4.732816 |
0.09789 |
| . |** . | |
0.299165 |
"." marks two standard errors |
Partial Autocorrelations |
Lag |
Correlation |
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 |
1 |
0.83026 |
| . |***************** | |
2 |
0.09346 |
| . |** . | |
3 |
0.00385 |
| . | . | |
4 |
-0.07340 |
| . *| . | |
5 |
-0.00278 |
| . | . | |
6 |
0.09013 |
| . |** . | |
7 |
0.08781 |
| . |** . | |
8 |
0.10327 |
| . |** . | |
9 |
0.07240 |
| . |* . | |
10 |
-0.11637 |
| . **| . | |
11 |
0.08210 |
| . |** . | |
12 |
0.07580 |
| . |** . | |
13 |
0.04429 |
| . |* . | |
14 |
0.11661 |
| . |** . | |
15 |
-0.10446 |
| . **| . | |
16 |
0.07703 |
| . |** . | |
17 |
0.07376 |
| . |* . | |
18 |
-0.07080 |
| . *| . | |
19 |
-0.02814 |
| . *| . | |
20 |
-0.08559 |
| . **| . | |
21 |
0.01962 |
| . | . | |
22 |
-0.04599 |
| . *| . | |
23 |
-0.07878 |
| . **| . | |
24 |
-0.02303 |
| . | . | |
|
Figure 49.2: Autocorrelation Plots for Chemical Data
The partial autocorrelation plot in Figure 49.2
suggests that the data can be modeled with a first-order
autoregressive model, commonly referred to as an AR(1) model.
![\(
\tilde{x}_{t} \equiv x_{t} - \mu =
\phi_{0} + \phi_{1} \tilde{x}_{t-1} + \epsilon_{t}
\)](images/saceq4.gif)
You can fit this model with the ARIMA procedure. The
results in Figure 49.3 show that the equation of
the fitted model is
.
proc arima data=chemical;
identify var=xt;
estimate p=1 method=ml;
run;
Maximum Likelihood Estimation |
Parameter |
Estimate |
Standard Error |
t Value |
Approx Pr > |t| |
Lag |
MU |
85.28375 |
2.32973 |
36.61 |
<.0001 |
0 |
AR1,1 |
0.84694 |
0.05221 |
16.22 |
<.0001 |
1 |
Constant Estimate |
13.05329 |
Variance Estimate |
14.27676 |
Std Error Estimate |
3.77846 |
AIC |
552.8942 |
SBC |
558.1045 |
Number of Residuals |
100 |
|
Figure 49.3: Fitted AR(1) Model
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.