APPLICATION OF FUZZY REGRESSION METHODOLOGY IN AGRICULTURE USING SAS

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1 APPLICATION OF FUZZY REGRESSION METHODOLOGY IN AGRICULTURE USING SAS Himadri Ghosh and Savita Wadhwa I.A.S.R.I., Library Avenue, Pusa, New Delhi Multiple linear regression modelling is a very powerful technique and is extensively used in agricultural research (Lalitha et al. 1999, Guo and Sun 2001). This technique estimates linear relationship between dependent (response) and independent (explanatory) variables. If X i, i=1,2,,n are explanatory variables and Y is response variable, the model is expressed as : Y b0 b1 X1... bn X n e (1) where b s are parameters and e is the error term assumed to be following a normal distribution. The parameters are generally estimated using method of least squares. A good description of various aspects of multiple linear regression methodology is given in Draper and Smith (1998). One drawback of the above methodology is that the underlying relationship is assumed to be crisp or precise, as its gives a precise value of response for a set of values of explanatory variables. However, in a realistic situation, the underlying relationship is not a crisp function of a given form; it contains some vagueness or impreciseness. So, by assuming a crisp relationship, some vital information may be lost (Slowinski 1998). A very promising technique of fuzzy regression has been developed. This technique can be applied to solve agricultural research problems. A fuzzy regression model corresponding to equation (1) can be written as: Y A A X...A (2) nxn Here explanatory variables X i s, as before, are assumed to be precise. However, as mentioned above, response variable Y is not crisp but is instead fuzzy in nature. This implies that the parameters are also fuzzy in nature. Our aim is to estimate these parameters. In subsequent discussion, it is assumed that A i s are symmetric fuzzy numbers (ie vagueness is expressible as equidistant from the center) and so can be represented by intervals. For example, A i can be expressed as fuzzy set given by: A 1 a a 1 1c, w (3) where a 1 c is centre and belief of regression coefficient around a 1 w is radius or vagueness associated. The above fuzzy set describes a ic in terms of symmetric triangular membership function. It is also to be noted that the methodology is applied when the underlying phenomenon is fuzzy which means that the response variable is fuzzy and the relationship is also considered to be fuzzy. Equation (3) is sometimes also written as: where A 1 [ a 1L, a 1R ] (4) a1l a1c a1w and a1r a1c a1w (Kacprzyk and Fedrizzi 1992)

2 Method of estimation of parameters of equation (2) is different from that of equation (1). In fuzzy regression methodology, parameters are estimated by minimizing total vagueness in the model, ie sum of radii of predicted intervals, From equation (2): Y j A 0 A X 1 ij... A n X nj Using equation (3), y a,a a a x... a,a x y,y, say Thus j 0c 0w 1c, 1w y jc a c a1c x1 j... ancxnj 1j 0 (5a) y jw a0w a1w x1 j... anw xnj (5b) As y jw represents radius and so cannot be negative, therefore on the right-hand side of equation (5b), absolute values of x ij are taken. Suppose there are m data points, each comprising an 1 row vector. Then parameters A i are estimated by minimizing the quantity, which is total vagueness of the model-data set combination, subject to the constraints that each data point must fall within estimated value of response variable. This can be visualized as the following linear programming problem (Tanaka 1987): m Minimize a a x... a x j1 nc 0 w 1w 1j nw nj (6) nw nj jc jw and a 0 iw Subject to n n a0 c aicx ij a0w aiwx ij Y j i1 i1 n n a c aicx ij a w aiwx ij Y 0 0 j i1 i1 To solve the above linear programming problem, Simplex procedure (Taha 1997) is generally employed. ILLUSTRATION 1: Data given in article of Sengupta et al. (2001) are considered. They have studied the effect of sulphur-containing fertilizers on productivity of rainfed greengram (Phaseolus radiatus L.). The response variable is dry-matter accumulation (Y) and the explanatory variables are plant height (X 1 ) and leaf area index (X 2 ). Only the data pertaining to maturity level, i.e. 60 days after sowing (DAS), are considered for data analysis and the same are presented in Table 1 for ready reference.

3 Table 1: Data of dry matter accumulation, plant height and leaf area index for effect of sulphur on growth of greengram crop Dry-matter accumulation Plant Height Leaf area index (g/m 2 ) (cm) The multiple linear regression model and fuzzy regression model are fitted to the above data using SAS, version 9.2 software package and following are the SAS codes and results obtained: Method of Multiple linear regression (MLR) Title 'Method of least square'; ods csv file= resultls.csv ; data plant; input y x1 x2; cards; ; proc reg; model y=x1 x2; output out=all; proc print data=all; run; quit; ods csv closed; Method of Fuzzy regression (FR) (OPTMODEL) Title Linear programming ; data plant; input y x1 x2; datalines;

4 ; run; ods rtf file='result_ex1.rtf'; proc optmodel; set j= 1..8; number y{j}, x1{j}, x2{j}; read data plant into [_n_] y x1 x2; /*Print y x1 x2*/ print y x1 x2; number n init 8; /* Total number of Observations*/ /* Decision Variables*/ var aw{1..3}>=0; /*Theses three variables are bounded*/ var ac{1..3}; /* These three variables are not bounded*/ /* Objective function*/ min z1= aw[1] * n + sum{i in j} x1[i] * aw[2] + sum{i in j} x2[i] * aw[3]; /*Linear Constraints*/ con c{i in 1..n}: ac[1]+x1[i]*ac[2]+x2[i]*ac[3]-aw[1]-x1[i]*aw[2]- x2[i]*aw[3] <= y[i]; con c1{i in 1..n}: ac[1]+x1[i]*ac[2]+x2[i]*ac[3]+aw[1]+x1[i]*aw[2]+x2[i]*aw[3] >= y[i]; expand; /* This provides all equations */ solve; print ac aw; quit; ods rtf close; RESULTS: Partial SAS output: Method of Multiple linear regression (MLR) Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept x x

5 Method of Fuzzy regression (FR) (OPTMODEL) [1] y x1 x Objective Sense Problem Summary Objective Function Objective Type Minimizati on z1 Linear Number of Variables 6 Bounded Above 0 Bounded Below 3 Bounded Below and Above Free 3 Fixed 0 0 Number of Constraints 16 Linear LE (<=) 8 Linear EQ (=) 0 Linear GE (>=) 8 Linear Range 0 Constraint Coefficients 96

6 Solver Objective Function Solution Summary Solution Status Dual Simplex z1 Optimal Objective Value Iterations 7 Primal Infeasibility Dual Infeasibility 0 Bound Infeasibility 0 0 [1] ac aw From the above results ac1= ac2= ac3= aw1= 7.97 aw2=0 aw3=0 The fitted model for MLR is Y= X X 2 (7) Standard Errors (107.63) (2.16) (7.57) The fitted model for FR is Y = <217.08, 7.97 > + <-3.06, 0 > X 1 + < 59.73, 0 > X 2 (8) In order to compare performance of above 2 approaches, viz multiple linear regression methodology and fuzzy regression methodology, width of prediction intervals corresponding to each observed value of response variable is computed. For the former, upper limits of prediction interval are computed from the prediction equation (7) by taking the coefficient as their corresponding estimated values plus standard error, i.e. using the equation Y = ( ) + ( ) X 1 + ( ) X 2 Similarly, lower limits of prediction interval for multiple linear regression models are computed using the equation Y = ( ) + ( ) X 1 + ( ) X 2

7 Further, for fuzzy regression model, the prediction equations for computing upper and lower limits, obtained from equation (8), are respectively and Y = ( ) + ( ) X 1 + ( ) X 2 Y = ( ) + ( ). X 1 + ( ) X 2 The width of prediction intervals in respect of multiple linear regression model and fuzzy regression model corresponding to each set of observed explanatory variables is computed in MS Excel (We can open above SAS results in MS Excel directly) and the results are reported in the following Table 2. From this table, average width for former was found to be , while that for latter was only 15.93, indicating thereby the superiority of fuzzy regression methodology. Table 2: Fitting of MLR and FLR Multiple Linear Regression (MLR) Model Fuzzy Regression (FR) Model Lower limit Upper limit Width Lower limit Upper limit Width Average width Average width In reality, underlying phenomenon is fuzzy; therefore, as emphasized above, correct methodology to obtain relationship between response and explanatory variables is to apply fuzzy regression methodology rather than multiple linear regression methodology. ILLUSTRATION 2: Length (L) weight (W) data of a fish species is given below: Length (mm): Weight (g) : Length (mm): Weight (g) : Assuming the underlying phenomenon to be fuzzy, fit Fuzzy linear regression using the Method of fuzzy least squares (FLS) to the deterministic allometric model W=a L b. For estimating Length-weight relationship statistical form of the above deterministic allometric model is: Log W= log a + b log L +e Also compare the results with those obtained through fitting least squares (LS). (i)

8 Note: Optmodel procedure can also be used for this illustration SAS Codes: /* Method of least squares (LS)*/ data LW; input l logl w logw ; cards; ; Ods rtf file= result.rtf ; proc reg; model logw=logl/p; run; quit; ods rtf close; Partial SAS output : The REG Procedure Model: MODEL1 Dependent Variable: logw Number of Observations Read 16 Number of Observations Used 16 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total

9 Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept <.0001 logl <.0001 Substituting the values of parameter estimates in model (i) of illustration 2 logw = logl Standard Errors (0.38) (0.08) Now we substitute the values of parameters i.e. a=exp(-11.99) and b=2.96 and their corresponding standard errors in the deterministic allometric model W=a L b and calculate the width as follows: Width= exp( )*l ( ) -(exp( )*l ( ) for different values of L(Length) given in the illustration 2. /* Method of fuzzy least squares (FLS) */ proc nlp; min Y; decvar ar br ac bc; bounds ar>=0, br>=0, ac= , bc= 2.96; lincon ac+4.38*bc-ar-4.38*br<=1.12; lincon ac+4.44*bc-ar-4.44*br<=1.12; lincon ac+4.50*bc-ar-4.50*br<=1.30; lincon ac+4.55*bc-ar-4.55*br<=1.52; lincon ac+4.61*bc-ar-4.61*br<=1.55; lincon ac+4.65*bc-ar-4.65*br<=1.81; lincon ac+4.70*bc-ar-4.70*br<=1.89; lincon ac+4.74*bc-ar-4.74*br<=2.03; lincon ac+4.79*bc-ar-4.79*br<=2.21; lincon ac+4.83*bc-ar-4.83*br<=2.32; lincon ac+4.87*bc-ar-4.87*br<=2.34; lincon ac+4.91*bc-ar-4.91*br<=2.56; lincon ac+4.94*bc-ar-4.94*br<=2.67; lincon ac+4.98*bc-ar-4.98*br<=2.73; lincon ac+5.01*bc-ar-5.01*br<=2.85; lincon ac+5.04*bc-ar-5.04*br<=3.04; lincon ac+4.38*bc+ar+4.38*br>=1.12; lincon ac+4.44*bc+ar+4.44*br>=1.12; lincon ac+4.50*bc+ar+4.50*br>=1.30;

10 lincon ac+4.55*bc+ar+4.55*br>=1.52; lincon ac+4.61*bc+ar+4.61*br>=1.55; lincon ac+4.65*bc+ar+4.65*br>=1.81; lincon ac+4.70*bc+ar+4.70*br>=1.89; lincon ac+4.74*bc+ar+4.74*br>=2.03; lincon ac+4.79*bc+ar+4.79*br>=2.21; lincon ac+4.83*bc+ ar+4.83*br>=2.32; lincon ac+4.87*bc+ar+4.87*br>=2.34; lincon ac+4.91*bc+ar+4.91*br>=2.56; lincon ac+4.94*bc+ar+4.94*br>=2.67; lincon ac+4.98*bc+ar+4.98*br>=2.73; lincon ac+5.01*bc+ar+5.01*br>=2.85; lincon ac+5.04*bc+ar+5.04*br>=3.04; Y=16*ar *br; run; Partial SAS output: Newton-Raphson Ridge Optimization Without Parameter Scaling 4 Lower Bounds 4 Upper Bounds 2 Linear Constraints 32 Using Sparse Hessian _ Optimization Start Active Constraints 2 Objective Function Max Abs Gradient Element All parameters are actively constrained. Optimization cannot proceed. PROC NLP: Nonlinear Minimization Optimization Results N Parameter Estimate Gradient Objective Function 1 ar Active Bound Constraint 2 br E Lower BC

11 Optimization Results N Parameter Estimate Gradient Objective Function Active Bound Constraint 3 ac Equal BC 4 bc Equal BC Value of Objective Function = Substituting the values of parameter estimates in model (i) of illustration 2 logw = logl Standard Errors (0.15) (0.00) Now we substitute the values of parameters i.e. a=exp(-11.99) and b=2.96 and their corresponding standard errors in the deterministic allometric model W=a L b and calculate the width as follows: Width = exp( ) *L (exp( )*l 2.96 L(Length) given in the illustration 2. for different values of Table 3: Width of predicted interval by LS and FLS approach Length Weight Estimated weight Width of predicted interval (g) (mm) (g) (g) LS FLS Average width Conclusion: The predicted interval computed using Method of fuzzy least squares have much shorter average width as compared to that obtained using Method of least squares. This implies that former procedure is more efficient than latter. The main

12 message emerging out of this illustration is that correct methodology to determine length-weight relationship in fish is that of Fuzzy least squares rather than ordinary Least squares. ILLUSTRATION 3: Wheat crop and spectral vegetation indices Normalized Difference Vegetation Index (NDVI) and Ratio Vegetation Index (RVI) (95 days a fter sowing) are observed from the experimental study at IARI farms, New Delhi and are produced below: Y (Qtl/Hec.): NDVI: RVI : Use Fuzzy regression Methodology (FRM) to fit the data using linear programming approach available in SAS software package and show its superiority over corresponding Multiple linear regression (MLR) model. Models are same as given in equations (1) and (2) above. Note: Optmodel procedure can also be used for this illustration SAS CODES: Method of Least squares (LS) Data plant; input Y NDVI RVI; cards; ; proc reg; model Y= NDVI ; proc reg; model Y= RVI; proc reg; model Y= NDVI RVI; run; quit;

13 Partial SAS Output: Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept NDVI Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept RVI Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept NDVI RVI So our equations are: i) Y= NDVI Standard Errors (13.46) (27.04) ii) Y= RVI Standard Errors (11.51) (3.82) iii) Y= NDVI RVI Standard Errors (25.04) (238.14) (32.11) Method of Fuzzy linear regression (FLR) /*FOR NDVI*/ Proc nlp; min Y; decvar a0c a0w a1c a1w ; bounds a0w>=0, a1w>=0; lincon a0c+.52*a1c-a0w-.52*a1w<=52.29; lincon a0c+.54*a1c-a0w-.54*a1w<=43.05; lincon a0c+.52*a1c-a0w-.52*a1w<=44.20; lincon a0c+.48*a1c-a0w-.48*a1w<=44.05; lincon a0c+.48*a1c-a0w-.48*a1w<=36.08;

14 lincon a0c+.48*a1c-a0w-.48*a1w<=40.04; lincon a0c+.54*a1c-a0w-.54*a1w<=46.93; lincon a0c+.52*a1c-a0w-.52*a1w<=47.64; lincon a0c+.53*a1c-a0w-.53*a1w<=46.62; lincon a0c+.49*a1c-a0w-.49*a1w<=46.13; lincon a0c+.38*a1c-a0w-.38*a1w<=29.57; lincon a0c+.46*a1cc-a0w-.46*a1w<=45.17; lincon a0c+.52*a1c+a0w+.52*a1w>=52.29; lincon a0c+.54*a1c+a0w+.54*a1w>=43.05; lincon a0c+.52*a1c+a0w+.52*a1w>=44.20; lincon a0c+.48*a1c+a0w+.48*a1w>=44.05; lincon a0c+.48*a1c+a0w+.48*a1w>=36.08; lincon a0c+.48*a1c+a0w+.48*a1w>=40.04; lincon a0c+.54*a1c+a0w+.54*a1w>=46.93; lincon a0c+.52*a1c+a0w+.52*a1w>=47.64; lincon a0c+.53*a1c+a0w+.53*a1w>=46.62; lincon a0c+.49*a1c+a0w+.49*a1w>=46.13; lincon a0c+.38*a1c+a0w+.38*a1w>=29.57; lincon a0c+.46*a1c+a0w+.46*a1w>=45.17; Y=a0w* *a1w; run; Partial SAS output: PROC NLP: Nonlinear Minimization N Parameter Optimization Results Estimate Gradient Objective Function 1 a0c a0w a1c a1w Value of Objective Function = So our estimates are: parameters: a0c a0w a1c a1w estimates: /* For RVI*/ proc nlp; min Y; decvar a0c a0w a2c a2w ; bounds a0w>=0, a2w>=0; lincon a0c+3.18*a2c-a0w-3.18*a2w<=52.29;

15 lincon a0c+3.36*a2c-a0w-3.36*a2w<=43.05; lincon a0c+3.20*a2c-a0w-3.20*a2w<=44.20; lincon a0c+2.87*a2c-a0w-2.87*a2w<=44.05; lincon a0c+2.88*a2c-a0w-2.88*a2w<=36.08; lincon a0c+2.88*a2c-a0w-2.88*a2w<=40.04; lincon a0c+3.35*a2c-a0w-3.35*a2w<=46.93; lincon a0c+3.19*a2c-a0w-3.19*a2w<=47.64; lincon a0c+3.24*a2c-a0w-3.24*a2w<=46.62; lincon a0c+2.95*a2c-a0w-2.95*a2w<=46.13; lincon a0c+2.20*a2c-a0w-2.20*a2w<=29.57; lincon a0c+2.67*a2c-a0w-2.67*a2w<=45.17; lincon a0c+3.18*a2c+a0w+3.18*a2w>=52.29; lincon a0c+3.36*a2c+a0w+3.36*a2w>=43.05; lincon a0c+3.20*a2c+a0w+3.20*a2w>=44.20; lincon a0c+2.87*a2c+a0w+2.87*a2w>=44.05; lincon a0c+2.88*a2c+a0w+2.88*a2w>=36.08; lincon a0c+2.88*a2c+a0w+2.88*a2w>=40.04; lincon a0c+3.35*a2c+a0w+3.35*a2w>=46.93; lincon a0c+3.19*a2c+a0w+3.19*a2w>=47.64; lincon a0c+3.24*a2c+a0w+3.24*a2w>=46.62; lincon a0c+2.95*a2c+a0w+2.95*a2w>=46.13; lincon a0c+2.20*a2c+a0w+2.20*a2w>=29.57; lincon a0c+2.67*a2c+a0w+2.67*a2w>=45.17; Y=a0w* *a2w; run; Partial SAS output: PROC NLP: Nonlinear Minimization N Parameter Optimization Results Estimate Gradient Objective Function 1 a0c a0w a2c Active Bound Constraint 4 a2w E Lower BC Value of Objective Function = So our estimates are: parameters: a0c a0w a2c a2w estimates:

16 /* For NDVI and RVI*/ proc nlp; min Y; decvar a0c a0w a1c a1w a2c a2w ; bounds a0w>=0, a1w>=0, a2w>=0; lincon a0c+.52*a1c+3.18*a2c-a0w-.52*a1w-3.18*a2w<=52.29; lincon a0c+.54*a1c+3.36*a2c-a0w-.54*a1w-3.36*a2w<=43.05; lincon a0c+.52*a1c+3.20*a2c-a0w-.52*a1w-3.20*a2w<=44.20; lincon a0c+.48*a1c+2.87*a2c-a0w-.48*a1w-2.87*a2w<=44.05; lincon a0c+.48*a1c+2.88*a2c-a0w-.48*a1w-2.88*a2w<=36.08; lincon a0c+.48*a1c+2.88*a2c-a0w-.48*a1w-2.88*a2w<=40.04; lincon a0c+.54*a1c+3.35*a2c-a0w-.54*a1w-3.35*a2w<=46.93; lincon a0c+.52*a1c+3.19*a2c-a0w-.52*a1w-3.19*a2w<=47.64; lincon a0c+.53*a1c+3.24*a2c-a0w-.53*a1w-3.24*a2w<=46.62; lincon a0c+.49*a1c+2.95*a2c-a0w-.49*a1w-2.95*a2w<=46.13; lincon a0c+.38*a1c+2.20*a2c-a0w-.38*a1w-2.20*a2w<=29.57; lincon a0c+.46*a1c+2.67*a2c-a0w-.46*a1w-2.67*a2w<=45.17; lincon a0c+.52*a1c+3.18*a2c+a0w+.52*a1w+3.18*a2w>=52.29; lincon a0c+.54*a1c+3.36*a2c+a0w+.54*a1w+3.36*a2w>=43.05; lincon a0c+.52*a1c+3.20*a2c+a0w+.52*a1w+3.20*a2w>=44.20; lincon a0c+.48*a1c+2.87*a2c+a0w+.48*a1w+2.87*a2w>=44.05; lincon a0c+.48*a1c+2.88*a2c+a0w+.48*a1w+2.88*a2w>=36.08; lincon a0c+.48*a1c+2.88*a2c+a0w+.48*a1w+2.88*a2w>=40.04; lincon a0c+.54*a1c+3.35*a2c+a0w+.54*a1w+3.35*a2w>=46.93; lincon a0c+.52*a1c+3.19*a2c+a0w+.52*a1w+3.19*a2w>=47.64; lincon a0c+.53*a1c+3.24*a2c+a0w+.53*a1w+3.24*a2w>=46.62; lincon a0c+.49*a1c+2.95*a2c+a0w+.49*a1w+2.95*a2w>=46.13; lincon a0c+.38*a1c+2.20*a2c+a0w+.38*a1w+2.20*a2w>=29.57; lincon a0c+.46*a1c+2.67*a2c+a0w+.46*a1w+2.67*a2w>=45.17; Y=a0w* *a1w+35.97*a2w; run; Partial SAS output: N Parameter PROC NLP: Nonlinear Minimization Optimization Results Estimate Gradient Objective Function 1 a0c a0w a1c Active Bound Constraint 4 a1w E Lower BC 5 a2c

17 N Parameter Optimization Results Estimate Gradient Objective Function 6 a2w Active Bound Constraint Value of Objective Function = So our estimates are: parameters: a0c a0w a1c a1w a2c a2w estimates: For NDVI : Upper and lower widths of prediction interval for Multiple linear regression models are computed respectably as Y = ( ) + ( ) NDVI (a) Y = ( ) + ( ) NDVI (b) Upper and lower widths of prediction interval for Fuzzy linear regression models are computed respectably as Y = ( ) + ( ) NDVI (c) Y = ( ) + ( ) NDVI (d) By subtracting equation(b) from (a) and then taking average, we can get average width for Multiple linear regression model and by subtracting equation(d) from (c) and then taking average, we can get average width for Fuzzy linear regression model. Similarly we can get average widths for RVI and both NDVI and RVI. The following table shows the average width for the three predictor variables. Predictor Variable Table 4: Average width for fitted regression models Method of Least Squares (LS) Fuzzy Linear Regression (FLR) Model NDVI RVI Both (NDVI & RVI) Conclusion: The above table 4 shows that average widths for linear regression models vis-à-vis their fuzzy counterparts are much higher. Thus Fuzzy regression methodology is more efficient than Multiple linear regression technique. It may also be pointed out that, for fuzzy approach, average widths, when both NDVI and RVI are taken into account, are generally smaller than when only one of these is considered, which is quite logical. This clearly shows that, unlike multiple linear regression technique, fuzzy regression methodology is capable of handling situations in which predictor variables are highly correlated.