Traveling gun system regression model development

Sections 3.3 and 3.4 discuss the separated regression analysis of traveling gun systems and stationary systems with subsets of the data. The data used in section 3.2 was split into two subsets based on the system type. The subset of data collected from traveling gun systems is discussed in section 3.3, and the subset of data collected from stationary systems is discussed in section 3.4. The reasons of analyzing the data by system type were 1) from the analysis of the combined data in section 3.2, it was evident that different systems (traveling or stationary) perform differently in application uniformity; 2) to investigate whether the model would better fit the data for each specific system possibly using different variables in either models; 3) to determine which system (traveling or stationary) introduced more variance into the combined model in section 3.2. The candidate variables used in this analysis were the same as those in section 3.2 except for the variable “system type”, since system type was limited to traveling gun systems. Therefore, the candidate variables were “nozzle diameter”, “nozzle type”, “gun model”, “nozzle pressure”, “sprinkler pacing in percent of wetted diameter”, “wetted diameter”, and “site type”.

3.3.2. Main effects selection

Variable selection was done as described in section 3.2 using PROC GLMSELECT (SAS, 2003) with Stepwise, Forward, and Backward Selections to choose the main effects. The SLE and SLS were set at the default values. All three selection methods chose the same three variables which were “sprinkler spacing in percent of wetted diameter”, “wind speed”, and “site type”. From a physical perspective, it was not reasonable that nozzle pressure was not selected as the nozzle pressure had been proven to be a very important factor affecting the application uniformity. It had been determined that variable “nozzle pressure” did not create any collinearity problem with the other three selected variables. Therefore, in later analysis

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for interaction terms and quadratic terms, the variable “nozzle pressure” was forced into the model.

3.3.3. Inte raction terms and quadratic te rms selection

Stepwise, Forward, and Backward Selections were conducted to select the interaction terms and quadratic terms at five different SL (significance levels), 0.1, 0.15, 0.2, 0.25, and 0.3, resulting in fifteen candidate models (table 3-4). Since some of the resultant models were the same, there were only six candidate models to compare (table 3-5) and they were from Stepwise1, Stepwise2, Forward1, Forward2, Backward1, and Backward4. Comparing the six models, the one from Backward1 was selected for the reasons: 1) it had the lowest values of AIC, AICC, PRESS residual, and SBC ; 2) although the adjusted R2 of the model was not the highest, it was only a little bit lower than the highest one (0.4334 compared to 0.435); 3) and it had fewer terms than the model of the highest R2 and adjusted R2.

Tab le 3-4 Selected variab les fro m d ifferent selection methods

effects SL

stepwise1 Intercept Per_Over*windsd windsd*site SLE=SLS=0.1 stepwise2 Intercept Per_Over*windsd site Per_Over*site windsd*site

Per_Over*Per_Over windsd*windsd SLE=SLS=0.15 stepwise3 Intercept Per_Over*windsd site Per_Over*site windsd*site

Per_Over*Per_Over windsd*windsd SLE=SLS=0.2 stepwise4 Intercept Per_Over*windsd site Per_Over*site windsd*site

Per_Over*Per_Over windsd*windsd SLE=SLS=0.25 stepwise5 Intercept Per_Over*windsd site Per_Over*site windsd*site

Per_Over*Per_Over windsd*windsd SLE=SLS=0.3 forward1 Intercept Per_Over Per_Over*windsd windsd*site

Per_Over*Per_Over SLE=0.1

forward2 Intercept Per_Over Per_Over*windsd site Per_Over*site

windsd*site Per_Over*Per_Over windsd*windsd SLE=0.15 forward3 Intercept Per_Over Per_Over*windsd site Per_Over*site

64 Table 3-4 continued

effects SL

forward4 Intercept Per_Over Per_Over*windsd site Per_Over*site

windsd*site Per_Over*Per_Over windsd*windsd SLE=0.25 forward5 Intercept Per_Over Per_Over*windsd site Per_Over*site

windsd*site Per_Over*Per_Over windsd*windsd SLE=0.3

backward1

Intercept Per_Over windsd Per_Over*windsd site Per_Over*site windsd*site nozp Per_Over*nozp windsd*nozp Per_Over*Per_Over windsd*windsd

SLS=0.1

backward2

Intercept Per_Over windsd Per_Over*windsd site Per_Over*site windsd*site nozp Per_Over*nozp windsd*nozp Per_Over*Per_Over windsd*windsd

SLS=0.15

backward3

Intercept Per_Over windsd Per_Over*windsd site Per_Over*site windsd*site nozp Per_Over*nozp windsd*nozp Per_Over*Per_Over windsd*windsd

SLS=0.2

backward4

Intercept Per_Over windsd Per_Over*windsd site Per_Over*site windsd*site nozp Per_Over*nozp windsd*nozp Per_Over*Per_Over windsd*windsd

nozp*nozp

SLS=0.25

backward5

Intercept Per_Over windsd Per_Over*windsd site Per_Over*site windsd*site nozp Per_Over*nozp windsd*nozp Per_Over*Per_Over windsd*windsd

nozp*nozp

SLS=0.3

Table 3-5 Criteria of d ifferent models

stepwise1 stepwise2 forward1 forward2 backward1 backward4 Root MSE 8.0915 7.34743 7.97839 7.34743 7.09087 7.08068 Dependent Mean 80.71746 80.71746 80.71746 80.71746 80.71746 80.71746 R-Square 0.2719 0.413 0.2984 0.413 0.4607 0.4647 Adj R-Sq 0.2622 0.3916 0.2827 0.3916 0.4334 0.435 AIC 961.5575 922.231 957.0652 922.231 908.80819 909.09159 AICC 5.20885 5.04035 5.19027 5.04035 4.98472 4.98714 PRESS 15302 13411 15143 13411 12543 12584 SBC 975.2924 953.1345 977.6676 953.1345 950.01285 953.72997

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3.3.4. Final model determination

The selected model in the above section (bold in tables 3-4 and 3-5) was then analyzed in PROC GLM (SAS, 2003). All the terms in the selected model in section 3.3.3 were proven to be significant with p- values less than 0.05. Therefore, this model was chosen to be the final model and was expressed as:

0 1 2 3 4 5 6 7 8 2 2 9 10 11 ( ) _ _ _ _ _

E CU Per Over windsd Per Over windsd site Per Over site windsd site nozp Per Over nozp windsd nozp Per Over windsd

(3.2)

where, E(CU)=expected value of CU site=site type

nozp=nozzle pressure

Per_Over=sprinkler spacing in percent of wetted diameter windsd=wind speed

3.3.5. Model evaluation

The R2 and adjusted R2 of the selected traveling model were slightly higher than the combined model in section 3.2 and the CLM and CLI intervals were slightly wider. The average CLM and CLI intervals of the traveling observations from the combined model in section 3.2 were 5 and 27, and from the traveling model in this section were 6 and 28.

3.3.6. Conclusions

1) Variables “sprinkler spacing in percent of wetted diameter”, “wind speed”, and “site type” were chosen as the candidate variables for further analysis in the main effect selection step. Compared with the combined model in section 3.2, the traveling model had fewer candidate variables selected in the main effect selection step, though the

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variable “nozzle pressure” was forced into the model. The variable “site type” was included which meant that producer farms and at LATDU had different abilities in achieving CU, while it was not the case for the combined model.

2) The final model was expressed in equation 3.2. The number of terms in the model was twelve, including the main effects, interaction terms, and quadratic terms. The two quadratic terms were “sprinkler spacing in percent of wetted diameter” and “wind speed” instead of “sprinkler spacing in percent of wetted diameter” and “nozzle diameter” in the combined model in section 3.2. It may reflect the difference between the data collected from traveling gun systems and the combined data. The loss of the quadratic term of “nozzle diameter” might result from the different nozzle size ranges for traveling gun systems and stationary systems. When only traveling systems were analyzed, there were no small nozzle sizes included.

3) The final model R2 was 0.461, the adjusted R2 was 0.433, the PRESS residual was 12543, the root MSE is 7.091, the maximum ranges of CLM and CLI were 16 and 32, the minimum ranges of CLM and CLI were 3 and 28, the average range of CLM was 6, and the average range of CLI was 28.

4) The residuals of the model were approximately normally distributed (Table A2-9, figure A2-1, figure A 2-2)

3.4. Stationary systems regression model development