6. Process or Product Monitoring and Control
6.6. Case Studies in Process Monitoring
6.6.2. Aerosol Particle Size

6.6.2.3.

Model Estimation

Dataplot ARMA Output for the AR(2) Model

Based on the differenced data, Dataplot generated the following estimation output for the AR(2) model: ############################################################# # NONLINEAR LEAST SQUARES ESTIMATION FOR THE PARAMETERS OF # # AN ARIMA MODEL USING BACKFORECASTS # ############################################################# SUMMARY OF INITIAL CONDITIONS ------------------------------ MODEL SPECIFICATION FACTOR (P D Q) S 1 2 1 0 1 DEFAULT SCALING USED FOR ALL PARAMETERS. ##STEP SIZE FOR ######PARAMETER ##APPROXIMATING #################PARAMETER DESCRIPTION STARTING VALUES #####DERIVATIVE INDEX #########TYPE ##ORDER ##FIXED ##########(PAR) ##########(STP) 1 AR (FACTOR 1) 1 NO 0.10000000E+00 0.77167549E-06 2 AR (FACTOR 1) 2 NO 0.10000000E+00 0.77168311E-06 3 MU ### NO 0.00000000E+00 0.80630875E-06 NUMBER OF OBSERVATIONS (N) 559 MAXIMUM NUMBER OF ITERATIONS ALLOWED (MIT) 500 MAXIMUM NUMBER OF MODEL SUBROUTINE CALLS ALLOWED 1000 CONVERGENCE CRITERION FOR TEST BASED ON THE FORECASTED RELATIVE CHANGE IN RESIDUAL SUM OF SQUARES (STOPSS) 0.1000E-09 MAXIMUM SCALED RELATIVE CHANGE IN THE PARAMETERS (STOPP) 0.1489E-07 MAXIMUM CHANGE ALLOWED IN THE PARAMETERS AT FIRST ITERATION (DELTA) 100.0 RESIDUAL SUM OF SQUARES FOR INPUT PARAMETER VALUES 138.7 (BACKFORECASTS INCLUDED) RESIDUAL STANDARD DEVIATION FOR INPUT PARAMETER VALUES (RSD) 0.4999 BASED ON DEGREES OF FREEDOM 559 - 1 - 3 = 555 NONDEFAULT VALUES.... AFCTOL.... V(31) = 0.2225074-307 ##### RESIDUAL SUM OF SQUARES CONVERGENCE ##### ESTIMATES FROM LEAST SQUARES FIT (* FOR FIXED PARAMETER) ######################################################## PARAMETER STD DEV OF ###PAR/ ##################APPROXIMATE ESTIMATES ####PARAMETER ####(SD 95 PERCENT CONFIDENCE LIMITS TYPE ORD ###(OF PAR) ####ESTIMATES ##(PAR) #######LOWER ######UPPER FACTOR 1 AR 1 -0.40604575E+00 0.41885445E-01 -9.69 -0.47505616E+00 -0.33703534E+00 AR 2 -0.16414479E+00 0.41836922E-01 -3.92 -0.23307525E+00 -0.95214321E-01 MU ## -0.52091780E-02 0.11972592E-01 -0.44 -0.24935207E-01 0.14516851E-01 NUMBER OF OBSERVATIONS (N) 559 RESIDUAL SUM OF SQUARES 109.2642 (BACKFORECASTS INCLUDED) RESIDUAL STANDARD DEVIATION 0.4437031 BASED ON DEGREES OF FREEDOM 559 - 1 - 3 = 555 APPROXIMATE CONDITION NUMBER 3.498456

Interpretation of Output	The first section of the output identifies the model and shows the starting values for the fit. This output is primarily useful for verifying that the model and starting values were correctly entered. The section labeled "ESTIMATES FROM LEAST SQUARES FIT" gives the parameter estimates, standard errors from the estimates, and 95% confidence limits for the parameters. A confidence interval that contains zero indicates that the parameter is not statistically significant and could probably be dropped from the model. The model for the differenced data, Yt, is an AR(2) model: with 0.44. It is often more convenient to express the model in terms of the original data, Xt, rather than the differenced data. From the definition of the difference, Yt = Xt - Xt-1, we can make the appropriate substitutions into the above equation: to arrive at the model in terms of the original series:
Dataplot ARMA Output for the MA(1) Model	Alternatively, based on the differenced data Dataplot generated the following estimation output for an MA(1) model:
	############################################################# # NONLINEAR LEAST SQUARES ESTIMATION FOR THE PARAMETERS OF # # AN ARIMA MODEL USING BACKFORECASTS # ############################################################# SUMMARY OF INITIAL CONDITIONS ------------------------------ MODEL SPECIFICATION FACTOR (P D Q) S 1 0 1 1 1 DEFAULT SCALING USED FOR ALL PARAMETERS. ##STEP SIZE FOR ######PARAMETER ##APPROXIMATING #################PARAMETER DESCRIPTION STARTING VALUES #####DERIVATIVE INDEX #########TYPE ##ORDER ##FIXED ##########(PAR) ##########(STP) 1 MU ### NO 0.00000000E+00 0.20630657E-05 2 MA (FACTOR 1) 1 NO 0.10000000E+00 0.34498203E-07 NUMBER OF OBSERVATIONS (N) 559 MAXIMUM NUMBER OF ITERATIONS ALLOWED (MIT) 500 MAXIMUM NUMBER OF MODEL SUBROUTINE CALLS ALLOWED 1000 CONVERGENCE CRITERION FOR TEST BASED ON THE FORECASTED RELATIVE CHANGE IN RESIDUAL SUM OF SQUARES (STOPSS) 0.1000E-09 MAXIMUM SCALED RELATIVE CHANGE IN THE PARAMETERS (STOPP) 0.1489E-07 MAXIMUM CHANGE ALLOWED IN THE PARAMETERS AT FIRST ITERATION (DELTA) 100.0 RESIDUAL SUM OF SQUARES FOR INPUT PARAMETER VALUES 120.0 (BACKFORECASTS INCLUDED) RESIDUAL STANDARD DEVIATION FOR INPUT PARAMETER VALUES (RSD) 0.4645 BASED ON DEGREES OF FREEDOM 559 - 1 - 2 = 556 NONDEFAULT VALUES.... AFCTOL.... V(31) = 0.2225074-307 ##### RESIDUAL SUM OF SQUARES CONVERGENCE ##### ESTIMATES FROM LEAST SQUARES FIT (* FOR FIXED PARAMETER) ######################################################## PARAMETER STD DEV OF ###PAR/ ##################APPROXIMATE ESTIMATES ####PARAMETER ####(SD 95 PERCENT CONFIDENCE LIMITS TYPE ORD ###(OF PAR) ####ESTIMATES ##(PAR) #######LOWER ######UPPER FACTOR 1 MU ## -0.51160754E-02 0.11431230E-01 -0.45 -0.23950101E-01 0.13717950E-01 MA 1 0.39275694E+00 0.39028474E-01 10.06 0.32845386E+00 0.45706001E+00 NUMBER OF OBSERVATIONS (N) 559 RESIDUAL SUM OF SQUARES 109.6880 (BACKFORECASTS INCLUDED) RESIDUAL STANDARD DEVIATION 0.4441628 BASED ON DEGREES OF FREEDOM 559 - 1 - 2 = 556 APPROXIMATE CONDITION NUMBER 3.414207

Interpretation of the Output

The model for the differenced data, Yt, is an ARIMA(0,1,1) model: with 0.44. It is often more convenient to express the model in terms of the original data, Xt, rather than the differenced data. Making the appropriate substitutions into the above equation: we arrive at the model in terms of the original series: