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VALIDATION OF TECHNIQUE TO ESTIMATE LOGIST IC MODEL PARAMETERS FROM LINEAR PLATEAU KARL MAXWELL WALLACE FALL 2014 SUMMA CUM LAUDE BACHELOR OF SCIENCE IN AGRICULTURAL AND BIOLOGICAL ENGINEERING
A BSTRACT The extended logistic model is useful for describing plant responses to applied nutrients, but determining model parameters requires complicated and time consuming nonlinear regression techniques. This study aims to validate estimate of logistic parameters from the readily applied linear plateau model In this analysis, logistic parameters were determined using both nonlinear regression and estimates from the linea r plateau model for tall fescue ( Festuca arundinacea Schreb.) and dallisgrass ( Paspalum dilatatum Poir) responses to applied nitrogen Standard error was calculated for each parameter and 95% confidence contours were generated using the parameters obtaine d by nonlinear regression. Finding that the estimated parameters fell close to or within the confidence contours, it was determined that the estimation technique has potential as a practical alternative to nonlinear regression. From a management perspect ive, both sets of logistic parameters found o ptimum seasonal nitrogen application rates to be 158 kg ha 1 for tall fescue and 267 kg ha 1 for dallisgrass grown in their respective study sites. This work may be applicable to other logistic models such as those describing population growth, chemical reactions, and economic trends. Further justification of the logistic model and validation of the parameter estimation technique should be investigated but are beyond the scope of this work.
C ONTENTS Introduction ................................ ................................ ................................ ................................ .............................. 1 Methods ................................ ................................ ................................ ................................ ................................ ... 1 Obtaining Model Parameters ................................ ................................ ................................ ................................ 1 Standard Error ................................ ................................ ................................ ................................ ..................... 3 Confidence Contours ................................ ................................ ................................ ................................ ........... 3 Results ................................ ................................ ................................ ................................ ................................ ...... 4 Discussion ................................ ................................ ................................ ................................ ................................ 7 Acknowledgements ................................ ................................ ................................ ................................ .................. 8 References ................................ ................................ ................................ ................................ ................................ 8 Appendix A: Nitrogen Uptake and Yield Data ................................ ................................ ................................ ........ 10 Appendix B: Nonlinear Regression ................................ ................................ ................................ ......................... 12 Appendix C: Hessian Elements ................................ ................................ ................................ ............................... 14 Appendix D: Logistic Parameters from Linear Plateau ................................ ................................ ........................... 15 Appenidx E: VBA Macros ................................ ................................ ................................ ................................ ...... 17
1 1. I NTRODUCTION Effective nutrient management is central to sustainable agriculture as over fertilization of crops is economically infeasible with the potential for environmental pollution. Mathematical models for plant nutrient uptake and biomass production responses to applied nutrients have been used to determine optimum fertilizer rates ( Willcutts et al., 1998 ). Overman et al. (1990) first proposed the logistic model to describe forage grass responses to nitrogen. Nitrogen uptake, N u is related to total seasonal applied nitrogen, N by the simple logistic equation (1) where A N is the maximum plant N uptake at high N, kg ha 1 ; b N is the dimensionless N uptake shifting parameter for plant N uptake at N = 0; and c N is the applied N response parameter, ha 1 kg. Overman et al. (1994) extended t he logistic model to account for biomass yield Y (2) where A is the relative maximum seasonal biomass production, Mg ha 1 ; b is the dimensionless biomass yield shifting parameter; and c is the applied N response parameter, ha 1 kg. A common c is assumed, c = c N to produce a model with five parameters: A N A b N b and c The significance of these parameters is discussed by Scholtz and Overman (2014) Determining these parameters involves complicated and time consuming optimization techniques. A simpler alternative is the linear plateau model given by Willcutts et al. (1998) and Scholtz and Wallace ( 2014 ) as (3) where A N is the relative maximum plant N uptake at high N, kg ha 1 ; N 1 is the zero uptake threshold, kg ha 1 ; is the difference between N 3 and N 1 kg h a 1 ; and N 3 is the peak uptake threshold, kg ha 1 Likewise, the linear plateau can be extended to biomass by (4) where A is relative maximum seasonal biomass production, Mg ha 1 ; N 0 is the zero yield threshold, kg ha 1 ; is the difference between N 2 and N 0 kg ha 1 ; and N 2 is the peak yield threshold, kg ha 1 In order to compare the linear plateau model with the logistic model, it is assumed that (5) Fitting the linear plateau model to data can be as simple as performing two separate linear regressions, which can be readily performed by available computer graphing applications, or in some cases even approximated by visual inspection. The major disadvantage of using the linear plateau model are that it cannot be used to determine instantaneous N uptake efficiency used to establish optimum fertilizer application rates. The derivative of a straight line is a constant which incorrectly suggests tha t N uptake and yield are independent of applied N between the minimum and maximum threshold values. Additionally, since N concentration in the plant N c is defined as (6) the linear plateau is undefined for N concentrations below the minimum threshold value. In comparison, the logistic model is more mathematically robust and consistent with observations but less practical to apply than the linear plateau ( Willcutts et al., 1998 ; Overman et al., 2003 ). The solution proposed by Scholtz and Wallace ( 2014 ) is to approximate the parameters for the logistic model from th e linear plateau model. This approximation is based on the observation that the logistic model approximates a straight line at 50% response ( Willcutts et al., 1998 ) and that both models exhibit rotational symmetry abo ut that point and both models are bounded by a minimum and maximum response. The purpose of this study is to validate a technique to estimate logistic model parameters from the linear plateau model. This work does not claim that the logistic model is the best model for describing crop responses to applied N, but rather investigates whether it is statistically acceptable to convert one model into a more favorable one. The ultimate goal is to help make the logistic model more practical. 2. M ETHODS 2.1 Obtaining Model Parameters Both the logistic and linear plateau models were fit to data for dallisgrass ( Paspalum dilatatum Poir; Robinson et al., 1988 ) and tall fescue ( Festuca arundinacea Schreb.; Overman and Wilkinson, 1993 ). Both data sets can be found in Appendix A. Optimum values for each parameter in equation 1 were obtained using nonlinear regression. The technique for nonlinear regression of the logistic model follows Overman
2 and Scholtz (2002) and are included in Appendix B. The applied N response parameter c was then held constant in the regression of equation 2 to find the remaining parameters A and b The parameters for equation 3 were obtained using the Solver Add In for Microsoft Excel to minimize the error sum of squares between observed N uptake and the three portions of the piecewise function by changing the two inflection points and the relative m aximum. This process was repeated holding constant and using the biomass yield data with equation 4. The relative maximum N uptake, A N and biomass yield, A found from the linear plateau are immediately usable in equations 1 and 2, respectively. The remaining three parameters are derived from the linear plateau using a technique by Scholtz and Wallace ( 2014 ) described below. From Reck and Overman (2006) the logistic equatio n can be written in the form ( 7 ) where is a characteristic N defined as ( 8 ) and N 1/2 is the applied N to reach 50% of the relative maximum biomass yield, which can also be expressed as ( 9 ) The dimensionless form of the logistic equation is given by Overman (1995) as ( 10 ) where is the dependent dimensionless variable and is the independent dimensionless variable. The variables are defined so that ( 11 ) and (1 2 ) The dimensionless form of the linear plateau equation is given by Scholtz and Wallace ( 2014 ) as (1 3 ) where is the independent dimensionless variable and is the transitional parameter defined so that ( 14 ) and ( 15 ) Subtracting equation 1 4 from equation 1 5 and solving for c gives ( 16 ) Substituting equation 1 6 into equation 9 and solving for b gives ( 17 ) The challenge is finding the value of that minimizes the total error between the two models defined as ( 18 ) Scholtz and Wallace ( 2014 ) determined using methods summarized in Appendix D that the minimum total error occurs when ( 19 ) Substituting equation 19 into equations 1 6 and 1 7 gives ( 20 ) and ( 21 ) This same derivation can be performed for N uptake by replacing N 0 with N 1 and N 2 with N 3 Doing so will find b N as ( 22 )
3 The parameters found for the linear plateau model were converted to the parameters for the logistic model using equations 20 through 22 and the direct values for A N and A 2.2 Standard Error The standard error was calculated for each parameter following the procedure described by Overman and Scholtz (2002) Analysis was completed separately for the logistic model parameters obtained from nonlinear regres sion and from the linear plateau model. The procedure is illustrated here using biomass yield but is identical for nutrient uptake. For the logistic model, the error sum of squares E is ( 23 ) where Y is the o bserved biomass yield and is the biomass yield estimated by the logistic model. The Hessian matrix for the logistic model is then ( 24 ) where each element is given by ( 25 ) Variance of the estimate is defined as ( 26 ) where n is the number of observations. The variance covariance matrix is then ( 27 ) The standard error, j for parameter a j is ( 28 ) The elements of the inverted Hessian matrix needed to calculate the standard error for A b and c are respectively given by ( 29 ) ( 30 ) ( 31 ) 2.3 Confidence Contours Validation of the linear plateau estimating technique requires that the logistic parameters estimated from the linear plateau and their standard error fall within the confidence contours generated from the logistic model parameters obtained b y nonlinear regression. Equal probability contours were obtained for the 95% probability using the procedures described by Draper and Smith (1981) Ratkowsky (1983) Overman et al. (1990) and Overman and Scholtz (2002) Again, the procedure is illustrated here using biomass yield but is identical for nutrient uptake. The error sum of squares E q at a particular probability level q is related to the minimum error sum of squares E 0 by ( 32 ) where p is the number of nonlinear parameters, n is the number of observations, and F (p, n p, q ) is the F statistic. The minimum error sum of squares is calculated using the optimum values for A, b, and c in equation 23 N uptake and biomass yield are treated separately so that for each p = 2. q, p, n p function in Microsoft Excel. Combining Eq. 22 and Eq. 31 gives ( 33 ) where E q is written to emphasize dependence on combinations of A and b A and c and b and c Three confidence contours were created by fixing one parameter at a time to its optimum value and finding combinations of the remaining two parameters that satisfy Eq. 32. This last step was accomplished using the Solver Add In for Microsoft Excel. A large number of points were need to create smooth curves. A VBA macro was created to expedite the process by selecting the objective and variable cells based on the current cell selection. The VBA code for this macro is included in Appendix E.
4 3. R ESULT S Table 1 summarizes the linear plateau parameters for dallisgrass and tall fescue. Table 2 and Table 3 contain the logistic model parameters obtained from nonlinear regression and conversion of the linear plateau parameters for tall fescue Table 1: Linear plateau parameters for tall fescue and dallisgrass Parameter Tall Fescue Dallisgrass A (Mg ha 1 ) 9.29 15.33 A N (kg ha 1 ) 238.24 419.56 N (kg ha 1 ) 438.27 841.36 N 0 (kg ha 1 ) 86.10 310.57 N 1 (kg ha 1 ) 59.12 156.06 Table 2 : Logistic model parameters, standard error, and error sum of squares at 95% confidence for tall fescue Nonlinear Regression Linear Plateau N Uptake A N (kg ha 1 ) 239.79 13.8 238.24 16.5 b N 1.66 0.20 1.90 0.29 c (ha kg 1 ) 0.0105 0.0017 0.0119 0.0026 E 95% 3329.73 Biomass Yield A (Mg ha 1 ) 9.31 0.34 9.29 0.40 b 1.35 0.13 1.58 0.18 c (ha kg 1 ) 0.0105 0.0012 0.0119 0.0018 E 95% 2.5 Table 3 : Logistic model parameters, standard error, and error sum of squares at 95% confidence for dallisgrass Nonlinear Regression Linear Plateau N Uptake A N (kg ha 1 ) 427.62 5.41 419.56 12.8 b N 1.47 0.03 1.64 0.10 c (ha kg 1 ) 0.0055 0.0002 0.0062 0.0005 E 95% 432.17 Biomass Yield A (Mg ha 1 ) 15.51 0.24 15.33 0.28 b 0.59 0.04 0.68 0.05 c (ha kg 1 ) 0.0055 0.0003 0.0062 0.0005 E 95% 1.16 and dallisgrass, respectively. The error sum of squares required for 95% confidence and the standard error for each parameter are also included. Comparison of the linear plateau, logistic, and estimated logistic is presented graphically in Figure 1. Visual inspection s hows agreement between the two sets of logistic Figure 1. Dependence of annual biomass yield ( Y ), plant N uptake ( N u ), and plant N concentration ( N c ) on applied nitrogen ( N ) for tall fescue grown at Watkinsville, GA. Data adapted from Overman and Wilkinson (1993) C urves drawn from parameters in Eq. 3, 4, and 6 and parameters from T able 1 fo r the linear plateau model and Eq. 1, 2, and 6 and parameters from T able 2 for the logistic model. 0 5 10 15 0 200 400 600 Annual Yield, Mg ha 1 Applied Nitrogen, kg ha 1 0 10 20 30 Nitrogen Concentration, g kg 1 Observed Linear-Plateau Logistic Estimated Logistic 0 0 0 100 200 300 Nitrogen Uptake, kg ha 1 0 0
5 Figure 2. Confidence contours for logistic model parameters at 95% confidence level for tall fescue grown at Watkinsville, GA. Also shown are best estimates and standard error. Data adapted from Overman and Wilkinson (1993) Best estimates are from Table 2. Standard error calculated from Eq. 26 through 31. Confidence con tours generated from Eq. 33. 160 310 460 0.5 1.5 2.5 3.5 A N kg ha 1 b N 160 310 460 0 0.01 0.02 0.03 A N kg ha 1 c, ha kg 1 0 0.01 0.02 0.03 0.5 1.5 2.5 3.5 c, ha kg 1 b N 7.5 9.5 11.5 0.5 1.5 2.5 A, Mg ha 1 b 7.5 9.5 11.5 0 0.01 0.02 A, Mg ha 1 c, ha kg 1 0 0.01 0.02 0.5 1.5 2.5 c, ha kg 1 b
6 Figure 3 Confidence contours for logistic model parameters at 95% confidence level for dallisgrass grown at Baton Rouge, LA Also shown are best estimates and standard error. Data adapted from Robinson et al. (1988) Best estimates are from Table 3. Standard error calculated from Eq. 26 through 31. Confidence contours generated from Eq. 33. 310 410 510 1.2 1.5 1.8 A N kg ha 1 b N 310 410 510 0.004 0.007 0.01 A N kg ha 1 c, ha kg 1 0.004 0.007 0.01 1.2 1.5 1.8 c, ha kg 1 b N 14 15 16 17 0.35 0.65 0.95 A, Mg ha 1 b 14 15 16 17 0.004 0.006 0.008 A, Mg ha 1 c, ha kg 1 0.004 0.006 0.008 0.35 0.65 0.95 c, ha kg 1 b
7 parameters This agreement is supported by the confidence contour plots presented in Figure 2. All estimated parameters and their standard error are enveloped by the 95% confidence contours generated from the parameters obtained by nonlinear regression. This close agreement clearly supports the use of the estimation technique. Interpretation of the confidence contours created for dallisgrass (Fig. 3) is less straightforward. Only the cont ours for b against c and A against c enclose the estimated parameters for biomass yield. In general, combinations of A ( A N ) and b (b N ) show the weakest agreement, followed by A ( A N ) and c and finally b (b N ) and c The poorer agreement with combinations of b or c with A is explained by the nature of the linear plateau model to underestimate the relative maximum N uptake or biomass yield. This observation may suggest that estimating logistic parameters from linear plateau is limited to b b N and c Onc e these parameters are known, the A and A N parameters can be solved for directly (see Appendix B) Failure to satisfy the confidence contours for dallisgrass is attributed to the small error sum of squares at 95% confidence: 432. 1 7 kg 2 ha 2 compared to 3329 .73 kg 2 ha 2 for tall fescue N uptake and 1.16 kg 2 ha 2 compared to 2.5 Mg 2 ha 2 for tall fescue biomass yield. The relatively small allowable error sum of squares is a result of the closeness of fit of the logisti c model to the dallisgrass data Int erestingly, the linear plateau (R = 1.000) fits the data better than the logistic model (R=0.999). The observations fall almost directly on the linear plateau estimates. In this context, the parameters and their confidence intervals are shifted away from the truer values given by the linear plateau rather than the linear plateau shifting away from the nonlinear regression parameters. Additional summary statistics illustrating the differences between the two models are included in Appendix A. Despite fail ure to satisfy the 95% confidence contours, the estimated logistic parameters did not perform poorly. When the ellipses represented by the standard error bars are included, all estimated parameters intersect the confidence contours Additionally, visual inspection of the N c N u and Y plots for dallisgrass (Fig. 4) demonstrates similar performance of the logistic parameters obtained by nonlinear regression and those estimated from the linear plateau. 4. D ISCUSSION From the confidence contours and response curves generated in this analysis, it is concluded that logistic model parameters can be successfully estimated from the linear plateau model. In some cases, the estimated parameters could be within 95% confidence of the parameters obtained f rom non linear regression. One of the major advantages the logistic model has over the linear plateau model is that the instantaneous Figure 4 Dependence of annual biomass yield ( Y ), plant N uptake ( N u ), and plant N concentration ( N c ) on applied nitrogen ( N ) for dallisgrass grown at Baton Rouge, LA Data adapted from Robinson et al. (1988) C urves drawn from parameters in Eq. 3, 4, and 6 and parameters from T able 1 fo r the linear p lateau model and Eq. 1, 2, and 6 and parameters from T able 3 for the logistic model. nitrogen uptake efficiency, dN u /dN can be calculated by differentiation such that ( 34 ) The instantaneous nitrogen use efficiency is plotted over applied N for tall fescue and dallisgrass in F igures 5 through 6. In contrast, the linear plateau can only be used to estimate the average rate of change in N uptake over applied N. 0 5 10 15 20 0 200 400 600 800 1000 Annual Yield, Mg ha 1 Applied Nitrogen, kg ha 1 0 100 200 300 400 500 Nitrogen Uptake, kg ha 1 0 10 20 30 Nitrogen Concentration, g kg 1 Observed Linear-Plateau Logistic Estimated Logistic 0 0
8 Figure 5. Instantaneous nitrogen uptake efficiency for tall fescue grown at Watkinsville, GA. Data adapted from Overman and Wilkinson (1993) Curves drawn using Eq. 34 and parameters from Table 2 for logistic model and the slope of the linear plateau model. Figure 6. Instantaneous nitrogen uptake efficiency for dallisgrass grown at Baton Rouge, LA. Data adapted from Robinson et al. (1988) Curves drawn using Eq. 34 and parameters from Table 2 for logistic model and the slope of the linear plateau model. The instan tan eous nitrogen uptake efficiency improves upon the agronomic definition of fertilizer efficiency , which is given by ( 34 ) where Y N is the yield with nitrogen application at rate N and Y is the yield without nitrogen application. Using equation 34 and Figures 5 through 6, the optimum seasonal nitrogen application rate was found to be 158 kg ha 1 for tall fescue and 267 kg ha 1 for dallisgrass grown at their respective study sites and con ditions Although the logistic model with estimated parameters overestimates the nitrogen uptake efficiency, the maximum efficiency occurs at the same application rate as determined from the logistic model with parameters obtained from nonlinear regressio n. From a management perspective, the performance of the estimated parameters is identical to those obtained from nonlinear regression. Th e simplified approach to obtaining logistic parameters examined in this study should be validated on additional se ts of data to fully identify its limitations, but this initial attempt has demonstrated an alternative to nonlinear regression to make the logistic model quicker and easier to apply. The significance of the estimation technique validated here is not limit ed to crop modeling, as logistic functions have been applied to population growth, growth of tumors, chemical reactions, and economics. 5. A CKNOWLEDGEMENTS The author would like to thank Dr. Scholtz for his academic and professional mentorship. This work would not have been possible without his guidance. 6. R EFERENCES Abramowitz, M., and I. A. Stegun. 1965. Handbook of Mathematical Functions. New York NY: Dover Publications Adby, P. R. and M. A. H. Dempster. 1974. Introduction to Optimization Methods 1st ed. New York, NY: John Wiley & Sons. Draper, N. R. and H. Smith. 1981. Applied Regression Analysis 2nd ed. New York, NY: John Wiley & Sons. Overman, A. R. 1995. Rational basis for the logistic model for forage grasses. J. Plant Nutrition 18(5): 995 1012. Overman, A. R., F. G. Martin, and S. R. Wilkinson. 1990. A logistic equation for yield response of forage grass to nitrogen. Comm. Soil Sci. Plant Anal. 21: 5 95 609. Overman, A. R., and R. V. Scholtz III. 2002. Chapter 6: Nonlinear regr ession for mathematical models. In Mathematical Models of Crop Growth and Yield 305 319 New York, NY: Marcel Dekker, Inc.. Overman, A. R., R. V. Scholtz III, F. G. Martin. 2003. In defense of the extended logistic model of crop production. Comm. Soil Sci. Plant Anal. 34(5 6): 0.0 0.5 1.0 0 100 200 300 400 500 600 Nitrogen Uptake Efficiency Applied nitrogen, kg ha 1 Linear-Plateau Logistic Estimated Logistic 0.0 0.5 1.0 0 200 400 600 800 1000 Nitrogen Uptake Efficiency Applied nitrogen, kg ha 1 Linear-Plateau Logistic Estimated Logistic
9 851 864. Overman, A. R. and S. R. Wilkinson. 1993. Modeling tall fescue cultivar responses to applied nitrogen. Agron. J. 85: 1156 1158. Overman, A. R., S. R. Wilkinson, and D. M. Wilson. 1994. An extended model of forage grass response to applied nitrogen. Agron. J. 86: 617 620. Ratkowsky, D. A. 1983. Nonlinear Regression Modeling: A Unified Practical Approach New York, NY: Marcel Dekker, Inc.. Reck, W. R. and A. R. Overman. 2006. Modeling effect of residual nitrogen on response of corn to applied nitrogen. Comm. Soil Sci. Plant Anal. 37(11 12): 1651 1662. Robinson, D. L., K. G. Wheat, N. L. Hubbert, M. S. Henders on, and H. J. Savoy Jr.. 1988. Dallisgrass yield, quality and nitrogen recovery responses to nitrogen and phosphorus fertilizers. Comm. Soil Sci. Plant Anal. 19: 529 542. Scholtz III, R. V. and K. M. Wallace 2014 A comparison of extended linear plateau and extended logistic models for biomass production and N uptake estimation [white paper] unpublished. Scholtz III, R. V. and A. R. Overman. 2014. Estimating seasonal nitrogen removal and biomass yield by annual s with the extended logistic model. PLOS ONE 9(4): 1 9. Willcutts, J. F., A. R. Overman, G. J. Hochmuth, D. J. Cantliffe, and P. Soundy. 1998. A comparison of three mathematical models of response to applied nitrogen: A case study using lettuce. HortSci ence 33(5): 833 836.
10 A PPENDIX A: N ITROGEN U PTAKE AND Y IELD D ATA Table A 1 : Seasonal plant N removal, dry matter yield, and plant N concentration for tall fescue grown at Watkinsville, GA Applied N N Uptake Biomass Yield N Concentration N N u Y N c (kg ha 1 ) (kg ha 1 ) (Mg ha 1 ) (g kg 1 ) 0 28.7 1.61 17.8 67 65.8 3.09 21.3 134 119 5.21 22.8 268 170 7.33 23.2 536 238 9.21 25.8 Data adapted from Overman and Wilkinson (1993) Table A 2 : Seasonal plant N removal, dry matter yield, and plant N concentration for dallisgrass grown at Baton Rouge, LA Applied N N Uptake Biomass Yield N Concentration N N u Y N c (kg ha 1 ) (kg ha 1 ) (Mg ha 1 ) (g kg 1 ) 0 77 5.33 14.4 56 103 6.56 15.7 112 129 7.97 16.2 224 194 10.53 18.4 448 305 13.21 23.1 896 417 15.34 27.2 Data adapted from Robinson et al. (1988)
11 Table A 3 : Comparison of mathematical models describing tall fescue response to applied nitrogen Extended Linear Plateau Extended Logistic Logistic from Linear Plateau Applied N N Uptake Biomass N Cont N Uptake Biomass N Cont N Uptake Biomass N Cont N N u Y N c N u Y N c N u Y N c (kg ha 1 ) (kg ha 1 ) (Mg ha 1 ) (g kg 1 ) (kg ha 1 ) (Mg ha 1 ) g kg 1 (kg ha 1 ) (Mg ha 1 ) g kg 1 0 32 1.82 17.6 38 1.91 20.0 31 1.59 19.5 67 69 3.24 21.1 66 3.19 20.8 59 2.91 20.4 134 105 4.67 22.5 105 4.78 21.9 101 4.67 21.6 268 178 7.51 23.7 182 7.56 24.1 186 7.73 24.1 536 238 9.29 25.6 235 9.19 25.6 236 9.21 25.6 CNESS* 0.0095679 05 0.0117766 65 E 0 277.24 0.40 0.48 451.91 0.34 6.68 652.32 0.48 6.26 RMS 3.33 0.13 0.14 4.25 0.12 0.52 5.11 0.14 0.50 R 2 0.990 0.989 0.986 0.984 0.991 0.807 0.976 0.987 0.819 R 0.995 0.995 0.993 0.992 0.996 0.898 0.988 0.994 0.905 Table A 4 : Comparison of mathematical models describing dallisgrass response to applied nitrogen Extended Linear Plateau Extended Logistic Logistic from Linear Plateau Applied N N Uptake Biomass N Cont N Uptake Biomass N Cont N Uptake Biomass N Cont N N u Y N c N u Y N c N u Y N c (kg ha 1 ) (kg ha 1 ) (Mg ha 1 ) (g kg 1 ) (kg ha 1 ) (Mg ha 1 ) g kg 1 (kg ha 1 ) (Mg ha 1 ) g kg 1 0 78 5.66 13.8 80 5.55 14.4 68 5.15 13.3 56 106 6.68 15.8 102 6.68 15.3 91 6.40 14.2 112 134 7.70 17.4 128 7.87 16.2 118 7.71 15.3 224 190 9.74 19.5 189 10.17 18.5 184 10.26 17.9 448 301 13.82 21.8 312 13.45 23.2 317 13.64 23.3 896 420 15.33 27.4 414 15.31 27.1 411 15.21 27.0 CNESS* 0.005481 0.001605 E 0 70.91 1.19 4.66 96.64 0.26 0.23 651.30 0.40 4.91 RMS 1.20 0.16 0.31 1.40 0.07 0.07 3.65 0.09 0.32 R 2 0.999 0.984 0.962 0.999 0.997 0.998 0.993 0.995 0.960 R 1.000 0.992 0.981 0.999 0.998 0.999 0.996 0.997 0.980 *CNESS is the combined normalized error sum of squares calculated using the method described in Scholtz and Overman (2014)
12 A PPENDIX B: N ONLINEAR R EGRESSION The method for nonlinear regression used to find optimum values for the logistic model is described by Overman and Scholtz (2002) and is presented here: Error sum of square E between the data and the model is defined by (B.1) where Y is measured yield and is estimated yield. To find optimum values of A, b, and c then simultaneously (B.2) The partial derivative of E with respect to A is (B.3) which can be solved explicitly for A as (B.4) From equation B.4, A can be calculated if b and c are assumed or known. The partial derivatives of E with respect to b and c are implicit so an iterative technique is needed. The second order Newton Raphson method described by Adby and Dempster (1974) uses a truncated Taylor series: (B.5) and (B.6) Equations B.5 and B.6 can be simplified to (B.7)
13 and (B.8) This system of equations can be rewritten in matrix form as (B.9) where the elements of the Hessian matrix are (B.10) and the elements of the Jacobian vector are (B.11) equation B.9 gives (B.12) and (B.13) where the determinants are given by (B.14) (B.15) (B.16) The equations used to determine the Hessian elements are include in Appendix C. Corrections for estimates of b and c are found using (B.17) and (B.18) Iteration is continued until (B.19) 3 10 5
14 A PPENDIX C: H ESSIAN E LEMENTS (C.1) (C.2) (C.3) (C.4) (C.5) (C.6)
15 A PPENDIX D: L OGISTIC P ARAMETERS FROM L INEAR P LATEAU Scholtz and Wallace ( 2014 ) proposed the following method to obtain the linear plateau transitional parameter needed for equations 15 and 16 to determine logistic model parameters from the linear plateau: The total error, E between the dimensionless forms of the logistic, and l inear plateau, models is defined as (D.1) Using equations 9 and 12, equation D.1 can be expanded to (D.2) The optimal value of the linear plateau transitional parameter is determined by setting the derivative of the total error with respect to the transitional parameter equal to zero. From the Leibniz Rule ( Abramowitz a nd Stegun, 1965 ) defined as (D.3) which simplifies equation D.2 to (D.4) The derivative of the dilogarithm is defined (D.5) ( Abramowitz and Stegun, 1965 ) written as the sum (D.6) The integral portion of equation D.4 can be found as (D.7) From which equation D.4 becomes (D.8) One of the functional properties of the dilogarithm is given by (D.9) And using the property that z =
16 (D.10) Setting equation D.10 equal to zero, the optimal value of with(RootFinding): evalf_8(NextZero(x ln((1+exp(x))*(1+exp( x))) 5/6 2/x*polylog(2,( exp( x))),1); which finally gives (D.1 1 ) The result is shown graphically in Figure D.1 Figure D.1: Dimensionless forms of the logistic and linear plateau models 0 0.5 1 Linear-Plateau Logistic 0
17 A PPENDIX E: VBA M ACROS Sub StandardError() 'calculates standard error for A, b, and c given their optimum values and a data set and Scholtz (2002) Dim rng As Range 'user selected range (applied N in left column and yield in right column) Dim myArray As Variant 'intermediate array populated with user inputs Dim N() As Double 'applied nutrient Dim Y() As Double 'dry matter yield Dim number As Integer Dim A As Double 'maximum dry matter yield at high N Dim A_num As Double 'numerator of Eq. 6.5 Dim A_den As Double 'denominator of Eq. 6.5 Dim b As Double 'intercept parameter Dim c As Double 'nutrient response coefficient Dim delta_b As Double Dim delta_c As Double Dim H_bb As Double Dim H_bc As Double Dim H_cc As Double Dim J_b As Double Dim J_c As Double Dim D As Double Dim D_b As Double Dim D_c As Double Dim epsilon As Double 'stopping criterion Dim i_pre As Integer 'index for preliminary matrices Dim i_A As Integer 'index for calculation of A Dim i As Integer 'index for calculating elements of Hessian and Jacobian matrices Dim address As String 'populate arrays with measured inputs Set rng = Application.InputBox("Select target range with the mouse", Type:=8) myArray = rng number = UBound(my Array, 1) ReDim N(1 To number) As Double ReDim Y(1 To number) As Double For i_pre = 1 To UBound(myArray, 1) Step 1 N(i_pre) = myArray(i_pre, 1) Y(i_pre) = myArray(i_pre, 2) Next i_pre A = Application.InputBox("Input A", Type:=8) b = Application.InputBox("Input b", Type:=8) c = Application.InputBox("Input c", Type:=8) epsilon = 10 ^ 5 'may change to machine epsilon
18 'reset sums for matrix calculations H_bb = 0 H_cc = 0 H_bc = 0 J_b = 0 J_c = 0 For i = 1 To UBound(N) Step 1 'calculate elements of Hessian matrix H_bb = H_bb + 2 A (Y(i) M(b, c, N(i)) / K(b, c, N(i)) ^ 2 2 Y(i) W(b, c, N(i)) / K(b, c, N(i)) ^ 3 A M(b, c, N(i)) / (K(b, c, N(i)))^3 + 3 A W(b, c, N(i)) / K(b, c, N(i)) ^ 4) H_cc = H_cc + 2 A (Y(i) N(i) ^ 2 M(b, c, N(i)) / K(b, c, N(i)) ^ 2 2 Y(i) N(i) ^ 2 W(b, c, N(i)) / K(b, c, N(i)) ^ 3 A N(i) ^ 2 M(b, c, N(i)) / K(b, c, N(i)) ^ 3 + 3 A N(i) ^ 2 W(b, c, N(i)) / K(b, c, N(i)) ^ 4) H_bc = H_bc 2 A (Y(i) N(i) M(b, c, N(i)) / K(b, c, N(i)) ^ 2 2 Y(i) N(i) W(b, c, N(i) ) / K(b, c, N(i)) ^ 3 A N(i) M(b, c, N(i)) / K(b, c, N(i)) ^ 3 + 3 A N(i) W(b, c, N(i)) / K(b, c, N(i)) ^ 4) 'calulate elements of Jacobian vector J_b = J_b + 2 A (Y(i) M(b, c, N(i)) / K(b, c, N(i)) ^ 2 A M (b, c, N(i)) / K(b, c, N(i))^3) J_c = J_c 2 (Y(i) N(i) M(b, c, N(i)) / K(b, c, N(i)) ^ 2 A N(i) M(b, c, N(i)) / K(b, c, N(i)) ^ 3) Next i 'calculate determinants D = H_bb H_cc H_bc ^ 2 D_b = J_b H_cc + J_c H_bc D_c = J_c H_bb + J_b H_bc 'Calculated Y^(i) ARRAY Dim Y_hat() ReDim Y_hat(1 To number) Dim iy As Integer For iy = 1 To UBound(N) Step 1 Y_hat(iy) = ELM(A, b, c, N(iy)) ad dress = rng.Cells(1).address(0, 0) Range(address).Offset( 1 + iy, 2) = Format(Y_hat(iy), "###.##") Next iy ''''Standard error Dim H_AA As Double Dim H_Ab As Double Dim H_Ac As Double Dim i_se As Integer Dim n_data As I nteger Dim p As Integer Dim ESS_0 As Double 'minimum Error sum of Squares Dim variance As Double Dim inv_AA As Double Dim inv_bb As Double Dim inv_cc As Double
19 Dim se_A As Double Dim se_b As Double Dim se_c As Double n_data = number p = 3 ESS_0 = 0 For i_se = 1 To UBound(N) Step 1 ESS_0 = ESS_0 + (Y(i_se) Y_hat(i_se)) ^ 2 H_AA = H_AA + 2 / (K(b, c, N(i_se))) ^ 2 'Eq. 6.33 H_Ab = H_Ab + 2 (Y(i_se) M(b, c, N(i_se)) / K(b, c, N(i_se)) ^ 2 2 A M(b, c, N(i_se)) / K(b, c, N(i_se)) ^ 3) H_Ac = H_Ac 2 (Y(i_se) N(i_se) M(b, c, N(i_se)) / K(b, c, N(i_se)) ^ 2 2 A N(i_se) M(b, c, N(i_se)) / K(b, c, N(i_se)) ^ 3) Next i_se variance = ESS_0 / (n_ data p) inv_AA = (H_bb H_cc (H_bc) ^ 2) / invDenom(H_AA, H_bb, H_cc, H_Ab, H_Ac, H_bc) inv_bb = (H_AA H_cc (H_Ac) ^ 2) / invDenom(H_AA, H_bb, H_cc, H_Ab, H_Ac, H_bc) inv_cc = (H_AA H_bb (H_Ab) ^ 2) / invDenom(H_AA, H_bb, H_cc, H_Ab, H_Ac, H_bc) se_A = Abs(Sqr(variance inv_AA)) se_b = Abs(Sqr(variance inv_bb)) se_c = Abs(Sqr(variance inv_cc)) ''''Confidence Contours Dim ESS As Double 'Error sum of sqrs at probability level q Dim p_cc As Double 'number of nonlinear parameters for confidence contour Dim q As Single 'Desired probability level (decimal) p_cc = 2 q = 0.95 ESS = 0 ESS = ESS_0 (1 + p_cc WorksheetFunction.F_Inv(q, p_cc, n_data p_cc) / (n_data p_cc)) 'Print to worksheet Range(address).Offset( 8, 1) = A Range(address).Offset( 7, 1) = b Range(address).Offset( 6, 1) = c Range(address).Offset( 8, 2) = se_A Range(address).Offset( 7, 2) = se_b Range(address).Offset( 6, 2) = se _c Range(address).Offset( 5, 1) = ESS End Sub 'function to simplify calculations and debugging Function K(ByVal b As Double, ByVal c As Double, ByVal N As Double) K = 1 + Exp(b c N) End Function
20 'function to simplify calculations and debugging Function M(ByVal b As Double, ByVal c As Double, ByVal N As Double) M = Exp(b c N) End Function 'function to simplify calculations and debugging Function W(ByVal b As Double, ByVal c As Double, ByVal N As Do uble) W = Exp(2 (b c N)) End Function 'function to simplify matrix inversion Function invDenom(ByVal H_AA As Double, ByVal H_bb As Double, ByVal H_cc As Double, ByVal H_Ab As Double, ByVal H_Ac As Double, ByVal H_bc As Double) invDenom = H_A A H_bb H_cc + 2 H_Ab H_bc H_Ac H_AA (H_bc) ^ 2 H_bb (H_Ac) ^ 2 H_cc (H_Ab) ^ 2 End Function 'extended logistic model Function ELM(ByVal A As Double, ByVal b As Double, ByVal c As Double, ByVal N As Double) ELM = A / (1 + Exp(b c N)) End Function Sub Solver() 'macro to automate solver to find points for confidence contours 'assign to a shape as a button SolverReset SolverOk SetCell:=ActiveCell.Offset(9, 0), MaxMinVal:=3, ValueOf:=0, ByChange:= ActiveCell.address, Engine:=1, EngineDesc:="GRG Nonlinear" SolverSolve End Sub