Traveling systems regression model development with restricted wind speed and variables

3.6.1. Variables and data description

The data used in the analysis in section 3.5 was then divided into two parts depending on the system type, traveling systems or stationary systems, in order to investigate whether the model fit better for the separated data than the combined data in section 3.5. The analysis and results of traveling systems are given in section 3.6 and the analysis and results of stationary systems are given in section 3.7. The goals of the analysis were to select a reliable model and build tables of CU for field use. The candidate variables used in this section were the same as those in section 3.5 except for “system type”. They were, “nozzle diameter”, “nozzle type”, “gun model”, “nozzle pressure”, “sprinkler spacing in percent of wetted diameter”, and “wetted diameter”.

3.6.2. Main effects selection

Main effects selection was conducted using PROC GLMSELECT (SAS, 2003) with Stepwise, Forward, and Backward Selection methods. All three selection methods selected only one variable, “sprinkler spacing in percent of wetted diameter” and it was not considered to be reasonable. Therefore, it was determined to skip the main effect selection step and use all six candidate variables for further analysis. However, there were collinearity problems with the six candidate variables because that some VIF values were greater than ten. Since the detailed output of Backward main effects selection showed that the “nozzle diameter” was the first variable chosen to leave the model and the variable “nozzle diameter” also had a high VIF value, the variable “nozzle diameter” was dropped from the analysis in order to avoid collinearity problems. However, the remaining five variables still had slight collinearity with a VIF value greater than ten. Considering there was some relationship between WD (wetted diameter) and Per_Over (sprinkler spacing in percent of wetted diameter). The variable WD was chosen to be dropped. Subsequently, after checking the VIF

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values of the remaining four variables, there were no collinearity problems. Therefore, the variables used for further analysis were “nozzle type”, “gun model”, “nozzle pressure”, and “sprinkler spacing in percent of wetted diameter”.

3.6.3. Select interaction terms and quadratic terms

The Stepwise, Forward, and Backward Selection were conducted to evaluate the interaction and quadratic terms selection at five different significance levels, 0.1, 0.15, 0.2, 0.25, and 0.3. The fifteen resultant models are shown in table 3-10. Since some of the results were the same, there were six different models to compare. They were from Stepwise1, Stepwise2, Stepwise4, Backward1, Backward4, and Backward5 Selection and are listed in table 3-11. The six candidate models had similar criteria values including R2 and adjusted R2. The model from Stepwise4 was selected after considering the Press residual, R2, and the number of terms in the model.

Table 3-10 Selected variab les fro m d ifferent selection methods

effects SL

Stepwise1 Intercept Per_Over*gunm Per_Over*Per_Over

nozp*nozp SLE=SLS=0.1

Stepwise2 Intercept gunm Per_Over*gunm Per_Over*Per_Over

nozp*nozp SLE=SLS=0.15

Stepwise3 Intercept gunm Per_Over*gunm Per_Over*Per_Over

nozp*nozp SLE=SLS=0.2

Stepwise4 Intercept gunm Per_Over*gunm nozp*gunm

Per_Over*Per_Over SLE=SLS=0.25

Stepwise5 Intercept gunm Per_Over*gunm nozp*gunm

Per_Over*Per_Over SLE=SLS=0.3

Forward1 Intercept Per_Over*gunm Per_Over*Per_Over

nozp*nozp SLE=0.1

Forward2 Intercept gunm Per_Over*gunm Per_Over*Per_Over _nozp*nozp SLE=0.15 Forward3 Intercept gunm Per_Over*gunm Per_Over*Per_Over

78 Tab le 3-10 continued

effects SL

Forward4 Intercept gunm Per_Over*gunm Per_Over*Per_Over

nozp*nozp SLE=0.25

Forward5 Intercept gunm Per_Over*gunm Per_Over*Per_Over

nozp*nozp SLE=0.2

Backward1 Intercept gunm Per_Over nozp nozp*gunm

Per_Over*Per_Over SLS=0.1

Backward2 Intercept gunm Per_Over nozp nozp*gunm

Per_Over*Per_Over SLS=0.15

Backward3 Intercept gunm Per_Over nozp nozp*gunm

Per_Over*Per_Over SLS=0.2

Backward4 Intercept gunm Per_Over Per_Over*gunm nozp

nozp*gunm Per_Over*nozp Per_Over*Per_Over SLS=0.25 Backward5

Intercept gunm Per_Over Per_Over*gunm nozp nozp*gunm Per_Over*nozp Per_Over*Per_Over

nozp*nozp

SLS=0.3

Tab le 3-11 Criteria of d ifferent models

Stepwise1 Stepwise2 Stepwise4 Backward1 Backward4 Backward5 Root MSE 8.33027 8.27516 8.24001 8.26904 8.22945 8.22675 Dependent Mean 80.65701 80.65701 80.65701 80.65701 80.65701 80.65701 R-Square 0.1488 0.1705 0.1878 0.1717 0.1951 0.2008 Adj R-Sq 0.1224 0.1339 0.1413 0.1352 0.1435 0.144 AIC 713.93484 713.63046 714.09451 713.38331 714.59925 715.41568 AICC 5.29125 5.29208 5.29819 5.2906 5.30315 5.31015 PRESS 12019 11979 11974 11895 12094 12150 SBC 732.64281 738.57441 745.27445 738.32726 748.89719 752.8316

3.6.4. Final model determination

The selected model above (bold in tables 3-10 and 3-11) was analyzed using PRO GLM (SAS, 2003). All the terms were significant with p-values lower than 0.05, thus it was determined to be the final model and was expressed as:

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2

0 1 2 3 4

( ) _ _

E CU gunm Per Over gunm nozp gunm Per Over

( 3.5) where, E(CU)=expected value of CU

gunm=gun model nozp=nozzle pressure

Per_Over=sprinkler spacing in percent of wetted diameter 3.6.5. Model evaluation

Compared to the traveling model in section 3.3, the model did not provide a better fit as it had a lower R2 and adjusted R2 and wider CLM and CLI intervals. The reason might be that this model did not have the important variable “wind speed”. Compared with the combined model in section 3.5, the model in this section also had lower R2 and adjusted R2 and wider CLM and CLI intervals (the average CLM and CLI intervals of traveling observations of the combined model in section 3.5 were 6 and 29).

3.6.6. Conclusions

1) Only one variable “sprinkler spacing in percent of wetted diameter” was selected as the candidate variables in the main effect selection for further analysis. However, since it was not reasonable that there was only one variable in the model, the main effect selection step was skipped. Interaction and quadratic terms selections were conducted among all the variables after fixing the collinearity problems.

2) The final model was expressed in equation 3.5. It contained main effects, interaction terms and quadratic terms.

3) The final model R2 was 0.1878, the adjusted R2 was 0.1413, the root MSE was 8.24001, the maximum ranges of CLM and CLI were 19 and 37, the minimum

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ranges of CLM and CLI were 4 and 33, the average range of CLM is 7, and the average range of CLI is 34.

4) The residuals of the model were approximately normally distributed (Table A5-9, figure A5-1, figure A5-2).

3.7. Stationary systems regression model development with restricted wind speed and