Application of fuzzy regression methodology in agriculture*iasri.res.in/sscnars/socialsci/18-APPLICATION OF FUZZY REGRESSION... · Application of Fuzzy Regression Methodology in Agriculture

APPLICATION OF FUZZY REGRESSION METHODOLOGY

IN AGRICULTURE USING SAS

Himadri Ghosh and Savita Wadhwa

I.A.S.R.I., Library Avenue, Pusa, New Delhi – 110012

[email protected], [email protected]

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 Xi, i=1,2,…,n are explanatory variables and Y is response variable, the model is expressed as :

eXbXbbY nn ...110 (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:

nn110 XA...XAAY (2)

Here explanatory variables Xi’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 Ai ‘s are symmetric fuzzy numbers (ie vagueness is expressible as equidistant from the center) and so can be represented by intervals. For example, Ai can be

expressed as fuzzy set given by:

wc aaA 111 , (3)

where ca1 is centre and wa1 is radius or vagueness associated. The above fuzzy set describes

belief of regression coefficient around ica 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:

],[ 111 RL aaA (4)

where wcwcL aaandaaa 111R 111 a (Kacprzyk and Fedrizzi 1992)

mailto:[email protected]

Application of Fuzzy Regression Methodology in Agriculture Using SAS

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):

njnij10j XA...XAAY

Using equation (3),

,y,yxa,a...xaaa,a y jwjcnjnwncj1w1,c1w0c0j say

Thus

njncjccjc xaxaay ...110 (5a)

||...|| 110 njnwjwwjw xaxaay (5b)

As jwy represents radius and so cannot be negative, therefore on the right-hand side of

equation (5b), absolute values of ijx are taken. Suppose there are m data points, each

comprising rowna 1 vector. Then parameters Ai 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):

Minimize

m

jnjnwjww xaxaa

1110 ||...|| (6)

Subject to

10

10 j

n

iijiww

n

iijicc Yxaaxaa

j

n

iijiww

n

iijicc Yxaaxaa

10

10

and 0iwa

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 (X1) and leaf area index (X2). 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.


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

(g/m2)

Plant Height

(cm)

Leaf area index

247.32 60.41 3.74

324.52 61.08 4.80

364.56 64.98 5.71

328.44 64.16 5.27

349.48 62.99 5.45

339.92 65.20 5.34

320.48 63.24 5.11

357.16 67.19 5.66

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;

247.32 60.41 3.74

324.52 61.08 4.80 364.56 64.98 5.71

328.44 64.16 5.27 349.48 62.99 5.45 339.92 65.20 5.34

320.48 63.24 5.11 357.16 67.19 5.66

; 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;

247.32 60.41 3.74 324.52 61.08 4.80

364.56 64.98 5.71 328.44 64.16 5.27 349.48 62.99 5.45


339.92 65.20 5.34 320.48 63.24 5.11

357.16 67.19 5.66 ;

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 Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 186.10038 107.62844 1.73 0.1444

x1 1 -3.01638 2.16085 -1.40 0.2216 x2 1 65.21831 7.57036 8.61 0.0003


Method of Fuzzy regression (FR) (OPTMODEL)

[1] y x1 x2

1 247.32 60.41 3.74

2 324.52 61.08 4.80

3 364.56 64.98 5.71

4 328.44 64.16 5.27

5 349.48 62.99 5.45

6 339.92 65.20 5.34

7 320.48 63.24 5.11

8 357.16 67.19 5.66

Problem Summary

Objective Sense Minimization

Objective Function z1

Objective Type Linear

Number of Variables 6

Bounded Above 0

Bounded Below 3

Bounded Below and

Above

0

Free 3

Fixed 0

Number of Constraints 16

Linear LE (<=) 8

Linear EQ (=) 0

Linear GE (>=) 8

Linear Range 0

Constraint Coefficients 96


Solution Summary

Solver Dual Simplex

Objective

Function

z1

Solution Status Optimal

Objective Value 63.742782934

Iterations 7

Primal

Infeasibility

0

Dual Infeasibility 0

Bound

Infeasibility

0

[1] ac aw

1 217.0811 7.9678

2 -3.0657 0.0000

3 59.7343 0.0000

From the above results

ac1= 217.08 ac2= -3.06 ac3= 59.73 aw1= 7.97 aw2=0 aw3=0

The fitted model for MLR is Y=186.10 – 3.02 X1 + 65.22 X2 (7) Standard Errors (107.63) (2.16) (7.57)

The fitted model for FR is Y = <217.08, 7.97 > + <-3.06, 0 > X1 + < 59.73, 0 > X2 (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 = (186.10 + 107.63) + (-3.02 + 2.16) X1 + (65.22 + 7.57) X2

Similarly, lower limits of prediction interval for multiple linear regression models are

computed using the equation

Y = (186.10 - 107.63) + (-3.02 - 2.16) X1 + (65.22 - 7.57) X2


Further, for fuzzy regression model, the prediction equations for computing upper and lower limits, obtained from equation (8), are respectively

Y = (217.08 +7.97) + (-3.06 + 0) X1 + (59.73 + 0) X2 and Y = (217.08 -7.97) + (-3.06 - 0). X1 + (59.73 - 0) X2

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 568.00, 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 Lower limit Upper limit Width

Fuzzy Regression (FR) Model Lower limit Upper limit Width

-18.84 514.01 532.85 247.65 263.58 15.93

38.80 590.59 551.80 308.91 324.85 15.93

71.06 653.48 582.42 351.33 367.27 15.93

49.94 622.16 572.22 327.56 343.49 15.93

66.37 636.26 569.89 341.89 357.83 15.93

48.59 626.36 577.77 328.56 344.49 15.93

45.48 611.30 565.82 320.82 336.75 15.93

56.72 647.94 591.21 341.58 357.52 15.93

Average width 568.00 Average width 15.93

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): 80 85 90 95 100 105 110 115

Weight (g) : 3.05 3.07 3.68 4.56 4.72 6.10 6.65 7.65 Length (mm): 120 125 130 135 140 145 150 155

Weight (g) : 9.16 10.14 10.43 12.99 14.48 15.32 17.35 20.90

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 Lb

.

For estimating Length-weight relationship statistical form of the above deterministic

allometric model is:

Log W= log a + b log L +e (i)

Also compare the results with those obtained through fitting least squares (LS).


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;

80 4.38 3.05 1.12

85 4.44 3.07 1.12 90 4.5 3.68 1.3

95 4.55 4.56 1.52 100 4.61 4.72 1.55 105 4.65 6.1 1.81

110 4.7 6.65 1.89 115 4.74 7.65 2.03

120 4.79 9.16 2.21 125 4.83 10.14 2.32 130 4.87 10.43 2.34

135 4.91 12.99 2.56 140 4.94 14.48 2.67

145 4.98 15.32 2.73 150 5.01 17.35 2.85

155 5.04 20.90 3.04

; 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 1 5.75209 5.75209 1340.15 <.0001

Error 14 0.06009 0.00429

Corrected Total 15 5.81218


Root MSE 0.06551 R-Square 0.9897

Dependent Mean 2.06625 Adj R-Sq 0.9889

Coeff Var 3.17068

Parameter Estimates

Variable DF Parameter

Estimate

Standard

Error

t Value Pr > |t|

Intercept 1 -11.98624 0.38421 -31.20 <.0001

logl 1 2.96076 0.08088 36.61 <.0001

Substituting the values of parameter estimates in model (i) of illustration 2 logw = -11.99 + 2.96 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 Lb and calculate the width as follows:

Width= exp(-11.99+0.38)*L(2.96+0.08)-(exp(-11.99-0.38)*L(2.96-0.08) 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= -11.99, 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+4.83*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+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;




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+4.94*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+75.94396*br;

run;

Partial SAS output:

Newton-Raphson Ridge Optimization

Without Parameter Scaling

Parameter Estimates 4

Lower Bounds 4

Upper Bounds 2

Linear Constraints 32

Using Sparse Hessian _

Optimization Start

Active Constraints 2 Objective Function 30.221322328

Max Abs Gradient Element 75.94396

All parameters are actively constrained. Optimization cannot proceed.

PROC NLP: Nonlinear Minimization

Optimization Results

Parameter Estimates

N Parameter Estimate Gradient

Objective

Function

Active

Bound

Constraint

1 ar 0.145200 16.000000

2 br -6.34258E-18 75.943960 Lower BC



Parameter Estimates


Objective

Function

Active

Bound

Constraint

3 ac -11.990000 0 Equal BC

4 bc 2.960000 0 Equal BC

Value of Objective Function = 2.3232

Substituting the values of parameter estimates in model (i) of illustration 2 logw = -11.99 + 2.96 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 Lb and calculate the

width as follows:

Width = exp(-11.99+0.15) *L2.96-(exp(-11.99-0.15)*L2.96 for different values of

L(Length) given in the illustration 2.

Table 3: Width of predicted interval by LS and FLS approach

Length

(mm)

Weight

(g)

Estimated weight

(g)

Width of predicted interval (g)

LS FLS

80 3.05 2.67 4.50 0.65

85 3.07 3.20 5.43 0.77

90 3.68 3.79 6.48 0.91

95 4.56 4.45 7.66 1.06

100 4.72 5.19 8.98 1.24

105 6.10 6.00 10.44 1.42

110 6.65 6.89 12.06 1.63

115 7.65 7.87 13.84 1.86

120 9.16 8.93 15.79 2.10

125 10.14 10.09 17.91 2.37

130 10.43 11.34 20.22 2.65

135 12.99 12.69 22.73 2.96

140 14.48 14.14 25.43 3.29

145 15.32 15.70 28.35 3.64

150 17.35 17.37 31.48 4.02

155 20.90 19.15 34.84 4.42

Average width 16.63 2.19

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


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 (Qt l/Hec.): 52.29 43.05 44.20 44.05 36.08 40.04 46.93 47.64 46.62 46.13 29.57 45.17

NDVI: 0.52 0.54 0.52 0.48 0.48 0.48 0.54 0.52 0.53 0.49 0.38 0.46

RVI : 3.18 3.36 3.20 2.87 2.88 2.88 3.35 3.19 3.24 2.95 2.20 2.67

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;

52.29 0.52 3.18 43.05 0.54 3.36 44.2 0.52 3.2

44.05 0.48 2.87 36.08 0.48 2.88

40.04 0.49 2.88 46.93 0.54 3.35 47.64 0.52 3.19

46.62 0.53 3.24 46.13 0.49 2.95

29.57 0.38 2.2 45.17 0.46 2.67

;

proc reg;

model Y= NDVI ; proc reg; model Y= RVI;

proc reg; model Y= NDVI RVI;

run; quit;


Partial SAS Output:

Parameter Estimates


Estimate

Standard

Error

t Value Pr > |t|

Intercept 1 -6.68100 13.45930 -0.50 0.6304

NDVI 1 101.16673 27.04375 3.74 0.0038

Parameter Estimates


Estimate

Standard

Error

t Value Pr > |t|

Intercept 1 4.03703 11.51212 0.35 0.7331

RVI 1 13.15890 3.81911 3.45 0.0063

Parameter Estimates


Estimate

Standard

Error

t Value Pr > |t|

Intercept 1 -26.36657 25.03924 -1.05 0.3198

NDVI 1 322.31429 238.14074 1.35 0.2089

RVI 1 -30.01393 32.10847 -0.93 0.3743

So our equations are:

i) Y= -6.68 + 101.17 NDVI Standard Errors (13.46) (27.04)

ii) Y= 4.04 + 13.16 RVI Standard Errors (11.51) (3.82)

iii) Y= -26.37+ 322.31 NDVI - 30.01 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+.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+.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;


Y=a0w*12+5.95*a1w; run;

Partial SAS output: PROC NLP: Nonlinear Minimization


Parameter Estimates


Objective

Function

1 a0c -14.548333 0

2 a0w 5.131667 12.000000

3 a1c 117.416667 0

4 a1w 1.250000 5.950000


So our estimates are: parameters: a0c a0w a1c a1w

estimates: -14.55 5.13 117.42 1.25

/* 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;


lincon a0c+3.36*a2c-a0w-3.36*a2w<=43.05; lincon a0c+3.20*a2c-a0w-3.20*a2w<=44.20;


lincon a0c+2.88*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+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*12+35.97*a2w;

run; Partial SAS output:

PROC NLP: Nonlinear Minimization


Parameter Estimates


Objective

Function

Active

Bound

Constraint

1 a0c 1.883824 0

2 a0w 6.010882 12.000000

3 a2c 13.960784 0

4 a2w -4.55365E-18 35.970000 Lower BC


So our estimates are:

parameters: a0c a0w a2c a2w estimates: 1.88 6.01 13.96 0


/* 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*12+5.95*a1w+35.97*a2w; run;

Partial SAS output: PROC NLP: Nonlinear Minimization


Parameter Estimates


Objective

Function

Active

Bound

Constraint

1 a0c -47.558016 0

2 a0w 2.082937 12.000000

3 a1c 365.841270 0

4 a1w 6.045511E-16 5.950000 Lower BC

5 a2c -30.144345 0



Parameter Estimates


Objective

Function

Active

Bound

Constraint

6 a2w 1.064980 35.970000


So our estimates are: parameters: a0c a0w a1c a1w a2c a2w

estimates: -47.56 2.08 365.84 0 -30.14 1.06

For NDVI :

Upper and lower widths of prediction interval for Multiple linear regression models are computed respectably as Y = (-6.68+13.46) + (101.17+27.04) NDVI (a)

Y = (-6.68-13.46) + (101.17-27.04) NDVI (b)

Upper and lower widths of prediction interval for Fuzzy linear regression models are computed respectably as

Y = (-14.55+5.13) + (117.42+1.25) NDVI (c) Y = (-14.55-5.13) + (117.42-1.25) 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.

Table 4: Average width for fitted regression models

Predictor Variable Method of Least Squares

(LS)

Fuzzy Linear

Regression (FLR)

Model

NDVI 53.73 11.50

RVI 45.92 12.02

Both (NDVI & RVI) 478.73 10.51

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.

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