Evaporation and Production Efficiency Modelling Using Fuzzy Linear Recurrence

International journal of Rural Development, Environment and Health Research(IJREH) [Vol-2, Issue-4, Jul-Aug, 2018]
https://dx.doi.org/10.22161/ijreh.2.4.3 ISSN: 2456-8678
www.aipublications.com/ijreh Page | 20
Evaporation and Production Efficiency Modelling
Using Fuzzy Linear Recurrence
Saeid Eslamian1, Fatemeh Sorousha1, Morteza Soltani2
, Kaveh Ostad-Ali-
Askari3*, Shahide Dehghan4, Mohsen Ghane5, , Vijay P. Singh6, Nicolas R.
Dalezios7
1Department of Water Engineering, Isfahan University of Technology, Isfahan, Iran.
2Department of Architectural Engineering, Shahinshahr Branch, Islamic Azad University, Shahinshahr, Iran
3*Department of Civil Engineering, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran. Email Corresponding
author: Koa.askari@khuisf.ac.ir
4Department of Geography, Najafabad Branch, Islamic Azad University, Najafabad, Iran
5Civil Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Iran
6Department of Biological and Agricultural Engineering & Zachry Department of Civil Engineering, Texas A and M University,
321 Scoates Hall, 2117 TAMU, College Station, Texas 77843-2117, U.S.A.
7Laboratory of Hydrology, Department of Civil Engineering, University of Thessaly, Volos, Greece & Department of Natural
Resources Development and Agricultural Engineering, Agricultural University of Athens, Athens, Greece.
*Corresponding Author:
Dr. Kaveh Ostad-Ali-Askari, Department of Civil Engineering, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan,
Iran. Email: Koa.askari@khuisf.ac.ir
Address: Islamic Azad University, Khorasgan Branch, University Blvd, Arqavanieh, Jey Street, Isfahan, Iran. P.O. Box:81595-
158, Phone: (+98)-31-35354001-9, Fax: (+98)-31-35354060, Mobile: +98912247143.
Abstract— The relationship between crop production and
amount of evapotranspiration is very important to
agronomists, engineers, economists, and water resources
planners. These relationships are often determined using
classical least square regression (LSR). However, one
needs high amount of samples to determine probability
distribution function. Linear regression also requires so
many measurements to obtain the valid estimates of crop
production function coefficients. In addition, deriving ET-
yield regression for each crop and each district is usually
expensive, since lysimetric experiments should be repeated
for several years for each crop. The object of this study is to
introduce a fuzzy linear regression as an alternative
approach to statistical regression analysis in determining
coefficients of ET- yield relations for each crop and each
district with minimum data. The application of possibilistic
regression has been examined with a case study. Two data
set for winter wheat in Loss Plateau of China and North
China Plain have been used. The current finding shows
capability of possibilistic regression in estimation of crop
yield in data shortage conditions.
Keywords— Data shortage; evapotranspiration; fuzzy
regression; grain yield; production function.
I. INTRODUCTION
Water shortage is the major constraint to agricultural
production. The relationships between crop yield and water
use have been a major focus of agricultural research in the
arid and semi-arid regions (Zhang and Oweis, 1999). Water
management is very important in these regions. Many
researchers have studied the effect of deficit irrigation on
crop production as a solution (Zhang et al., 1999 and Kang
et al., 2002).
In agriculture water management, the adequate
representation of production or crop yield functions is
crucial for modeling purposes in environmental economic
analyses. The discussion and estimation of different
functional forms have therefore gained much attention in
agronomic and agricultural economics literature (Finger and
Hediger, 2007). Various functional forms have been
considered so far, but less attention has been given to the
estimation techniques. In general, crop yield is estimated by
least square regression. Classical linear or non-linear
regression assumes that the measurement errors are
normally distributed and independent of each other. Since
one needs so many samples to determine a probability
distribution, linear or nonlinear regression require at least 8

to 30 measurements or observations to obtain valid estimate
of parameters (Eslamian et al. 2012, Cheng Si and
Bodhinayake, 2005).
Measurement of some parameters such as
evapotranspiration in yield function is expensive and time
consuming. Therefore, it is difficult and sometimes
impossible to obtain a simple yield function for regions with
same climate. Moreover, evapotranspiration determination
is subjected to different kind of uncertainties. These arise
from measurement errors due to human and assumptions on
deep percolation and uniformity of soil distribution. In these
circumstances, classical regression may not give valid
estimation for yield. In particular, confidence interval
estimated with a few data points is very wide and may not
provide suitable information that is usual for predictive
purpose (Eslamian et al. 2001, Cheng Si and Bodhinayake,
2005).
Fuzzy sets theory can quantitatively deal with
uncertainty in experimental data or ambiguity in human
perception, and so it has been applied to various fields in
which uncertainty and/or ambiguity have a serious
influence. The theory does not need strict assumptions of
probability functions as in the statistical methods, such as
the normal distribution described above, and it can deal with
the uncertainty more easily and more flexibly (Shimosaka et
al., 1996). The objective of this study is to investigate
whether fuzzy linear regression (Tanaka et al., 1982) would
predict crop production and to provide a method for yield
forecasting with less observation than least square
regression.
II. THEORY
Water use-yield relationship:
Crops consume water in the process of transpiration, and
water evaporates from the soil. These processes are defined
collectively as evapotranspiration (Thornyhwaite, 1948).
The relationship between crop production and the amount of
water applied to crop is important. This importance is
currently considered due to declining in water resources and
competition among users.
Crop production models with resource and management
inputs have been widely used, particularly by agricultural
economist, and called production function (vaux 1983,
Ostad-Ali-Askari et al. 2015). Hanks et al. (1969) reported
that dry matter is linearly related to evapotranspiration for
wheat, millet, oat and grain sorghum in both lysimetric and
field plots. Cole and Mathews (1923) and Mathews and
Brown (1938) investigated grain yield for winter wheat and
sorghum. They used linear regression techniques to evaluate
the yield- evapotranspiration as follows:
bETaY  (1)
Where Y is grain yield (kg ha-1), ET is the growing
season evapotranspiration (mm) and a (kg ha-1 mm-1) and b
(kg ha-1), regression coefficients.
ET is usually calculated using the soil water balance
equation for growing season as given:
fRDSgPIWET  (2)
Where ET is actual evapotranspiration, ΔW the change in
soil water storage between two soil moisture content
measurements, I the irrigation, P the rainfall, Sg the
capillary rise from the lower soil layer to the crop root zone,
D the deep percolation fromthe crop root zone, and Rf is the
surface runoff (Kang et al. 2002). When the groundwater
table is lower than 4 m below the ground surface, Sg is
usually negligible (Zhang et al., 1999). It is usually assumed
that soil infiltration rate is larger than rainfall and irrigation
density.
Some studies had shown that the empirical relation
between crop yield and seasonalevapotranspiration can take
different forms and that the empirical coefficients in the
relations vary with climate, crop type and variety, irrigation
method, soil texture, fertilizer and tillage methods. These
differences relate to regional variability in environment and
agronomic practices, Information specific to a region is
needed to define production function (Eslamian et al. 2015,
Kang et al., 2002, Ostad-Ali-Askari et al. 2016). So,
derivation of production functions for each region would be
expensive and obtaining adequate data for linear regression
would be difficult.
Fuzzy linear regression method
Fuzzy regression analysis was first proposed by Tanaka
et al. (1982). Since membership functions of fuzzy sets are
often described as possibility distributions, this approach is
usually called possibilistic regression analysis (Tanaka et
al., 1982). The basic concept of fuzzy theory of fuzzy
regression is that the residuals between estimators and
observations are not produced by measurement errors, but
rather by the parameter uncertainty in the model, and the
possibility distribution is used to deal with real observations
(Tseng et al., 1999, Eslamian et al. 2016). This method
provides the means by which the goodness of a relationship
between two variables, y and x, may be evaluated on the
basis of a small sample size. In this approach, the regression
coefficients are assumed to be fuzzy number (Sahin and
Hall, 1996, Ostad-Ali-Askari et al. 2017).
The fuzzy linear regression (FLR) model can be expressed
as:

iinni xAxAxAAY
~~~~~
110   (3)
Where  ini xxxxi ,,, 10  is a vector of independent
variables in the ith data mi ,,1 ; ]
~
,,
~
[
~
0 nAAA  is a
vector of fuzzy parameters exhibited in the form of
symmetric triangular fuzzy numbers denoted
by njcpA jjj ,,1),,(
~
 , with its membership function
depicted as (4) bellow where jp is its central value and
jc is its half width (See Figure 1).
A fuzzy linear relationship can be represented by a band
(the bold lines having membership=0) with a centre line
(the dashed line having a membership=1) as in Figure 2.









.,0
,,1
)(~
otherwise
cpacp
c
pa
a jjjjj
j
jj
jjA
 (4)
Therefore, Eq. (3) can be written as:
.),(),(),(
~
11100 innni xcpxcpcpY   (5)
Fig.1: Triangular representation of fuzzy numbers
Fig.2: A Fuzzy linear relationship
Since the regression coefficients are fuzzy numbers, the
estimated dependent variable Y
~
is a fuzzy number.
Finally, the method uses the criterion of minimizing the
total vagueness, S, defined as the sumof individual spreads
of the fuzzy parameters of the model.
 

m
i
n
j
jii xcmcSMinimize
1 1
0 (6)
The fuzzy coefficients are determined such that the
estimated fuzzy output Y
~
has the minimum fuzzy width jc ,
while satisfying a target degree of belief h. The term h can
be viewed as a measure of goodness of fit or a measure of
compatibility between the regression model and data. Each
of the observed data sets, must fall within the estimated Y
~
at h levels (Figure 3). The value of h is between 0 and 1 and
h=0 indicates that the assumed model is extremely
compatible with the data, while h=1 illustrated the assumed
model is extremely incompatible with the data. h is chosen
by the decision maker. A choice of the h-level value
influences the widths c of the fuzzy parameters:
.,,2,1,)(~ mihyiY
 (7)
Taheri et al. (2006) purposed a method of sensitivity
analysis based on credible level h. Their results showed that
as the credible level h, increases, the mean of predictive
capability (MPC) increases, too. On the other hand, by
increasing h, the total vagueness of model, S, increases as
well. For selecting a suitable h we would analyze the
variations of S and h. Variations of S is gradual from h
equal zero up to optimal h, after optimal h, increasing of h
makes an abrupt variation in S value.
The problem of finding the fuzzy regression parameters
was formulated by Tanaka et al. (1982) as a linear
programming problem:
 

m
i
n
j
jii xcmcSMinimize
1 1
0
Subject to:
j
n
j
iji
n
j
ijj yxcchxpp 








   1
0
1
0 )1(
j
n
j
iji
n
j
ijj yxcchxpp 








   1
0
1
0 )1( (8)

Eq. (8) is linear, thereby allowing the optimization problem
to be solved by means of linear programming.
Fig.3: Triangular membership function of fuzzy output
III. APPROACH
The evapotranspiration (ET)-wheat yield (Yield) data
presented in Kang et al. (2002) and Zhang et al. (1999) was
used in this study.
One of our data bases is consist of experimental
irrigation data, grain yield, seasonal ET, water use
efficiency and climatic data summary during growing
season winter wheat at four locations in the piedmont and
lowland of the North China Plain (Zhang et al., 1999). The
locations are divided into two groups that represented
different geographic characteristics in the regions based on
the groundwater table and geography. Luacheng and
Gaocheng are located in the piedmont of the Taihang
Mountains, and Linxi and Nanpi are located in the lowland
of the Haihe floodplain. The irrigation treatments are ranged
from no irrigation (rain-fed: I0) to a maximum of seven
irrigations (I1, I2, I3, I4, I5, I6, and I7) where subscript
represents the number of irrigations during the crop-
growing season in Gaocheng and Linxi, and to a maximum
of five irrigations in Luancheng and Nanpi. The amount of
water applied was about 45–75 mm each irrigation. Grain
yield and seasonal evapotranspiration are listed in Table 1.
Another data base (Kang et al., 2002) is consist of
dataset form a lysimeter experiment that has been conducted
for winter wheat (Triticum aestivum L.) during the period
1995-1998 to evaluate the effects of limited irrigation on
grain yield on the Loess Plateau of China. Kang et al.
(2002) applied a controlled soil water deficit, either mild or
severe, at different stages of crop growth. The average
values of evapotranspiration and grain yield for different
treatments in 1995-1998 are given in Table 2.
Table.1: Grain yield and seasonal evapotranspiration for four locations in North China (Zhang et al., 1999)
Gaocheng Linxi Luancheng Nanpi
Irrigation
treatment
ET
(mm)
Yield
(Kg/ha)
ET
(mm)
Yield
(Kg/ha)
ET
(mm)
Yield
(Kg/ha)
ET
(mm)
Yield
(Kg/ha)
I0 242 2580 247 2610 264 3220 281 2800
I1 305 3600 277 3740 356 4770 355 3010
I2 365 4960 358 4670 379 5250 420 4060
I3 407 5230 414 4990 377 5250 418 4940
I4 437 5280 428 5120 439 5100 443 4750
I5 437 4240 426 4890 453 4790 456 5160
I6 419 4360 478 4940
I7 423 4950 489 4440
In current study, linear fuzzy regression (Tanaka et al.,
1982) are employed and Evapotranspiration- Yield fuzzy
relationships for Luancheng, Napai (Zhang et al., 1999) and
Loess Plateau of China (Kang et al., 2002) were obtained.
For this purpose, complete dataset of Luancheng and
Nanpi are applied. Zhang et al. (1999) has mixed Luacheng
– Gaocheng datasets and presented a least square regression
model for piedmont. In addition, the least square model for
Linxi - Nanpi was reported as lowland. In this study, fuzzy
regression model is obtained for Luancheng and Nanpi and
Gaocheng and Linxi datasets are used for validation of

fuzzy regression models which are derived from Luancheng
and Nanpi datasets, respectively.
Moreover, the dataset of eight different soil water
content treatments (1, 3, 5, 7, 9, 11, 13, 15) in 1995-1996
(Table. 2) is used to obtain ET-Yield fuzzy regression
model in the Loess Plateau of China. Finally, for model
validation, yield estimation of fuzzy model for water
content treatments: 2, 4, 6, 10, 12 and 14 evaluated with
observation data.
In these cases, (having only 5 or 8 observation), it is
impossible to satisfy the basic assumption of statistical
regression analysis (such as normality of error,
independence of errors, and so on). So fuzzy regression can
be used as an alternative approach.
Value of total vagueness (S) calculated for h = 0-0.95
with 0.05 intervals and acceptable value of h was
determined.
Table.2: Total evapotranspiration and grain yield in three growing seasons in the Loess Plateau of China (Kang et al., 2002).
1995-1996 1996-1997 1997-1998
Treatments
ET
(mm)
Yield
(Kg/ha)
ET
(mm)
Yield
(Kg/ha)
ET
(mm)
Yield
(Kg/ha)
1 267 2493 213 1750 220 1612
2 308 3520 300 3180 277 3060
3 304 3089 278 3375 231 2039
4 310 3533 385 3905 232 1771
5 301 3060 359 3570 310 4079
6 339 3506 291 3505 235 2040
7 356 3441 338 3870 296 3060
8 370 3659 387 4020 285 2788
9 362 3672 323 4080 254 3076
10 305 3680 389 4230 285 3852
11 292 3294 403 4245 227 2045
12 399 4233 519 4200 358 4060
13 354 4325 420 4600 330 4749
14 367 4485 383 4775 340 4811
15 370 4553 390 4920 329 4792
IV. RESULTS
In applying fuzzy linear regression, grain yield(Kg/ha) is
employed as the dependent variable and evapotranspiration,
ET(mm) is assumed as independent variable. All the Yield
and ET values are assumed to be crisp. The symmetric
triangular form of the membership function is chosen for
representing the regression parameters. According to Figure
4, it is obvious that by taking large value for h, amount of S
increase quickly. So, it seems that the values around 0.7 for
h, are suitable values for h and this is in an agreement with
Bardossy et al. (1990). According to Bardossy et al. (1990),
the level of credibility is generally chosen so
that 7.05.0  h .
The fuzzy model with symmetric triangular fuzzy
coefficients for crop production modeling of winter wheat
in three locations in China, as a function of growing season
evapotranspiration, can be stated as follows:
ETcpcpY ),(),(
~
1100 
Based on 6 data in Table 1, for Nanpi region, and
adapting relation (8), the objective function is:
10 23735 ccSMinimize 
In addition, constrains (12 constrains) related to
observations (6 observations) must be formulated, based on
relation (8). For example, two constrains corresponding to
the first observation, with h=0.7, are:

Fig.4: The variation of the total vagueness (S), based on different amounts for h.
2800)281(3.0281
2800)281(3.0281
1010
1010


ccpp
ccpp
By minimizing the objective function S subject to 12
constrains, with linear programming methods, the
coefficients of the model are as follows:
)44.4,29.14(
~
,)00.0,34.1589(
~
10  AA
Therefore, the possibility regression model for Nanpi region
is:
ETY )44.4,29.14()00.0,34.1589(
~

In addition, the coefficients of the possibilistic
regression model were calculated for Luancheng and the
Loess Plateau of China. The results are shown in Table 3.
The results of fuzzy regression model for simulation
data are shown in Figure 5. An estimation area at the high
evapotranspiration is wider than low evapotranspiration
(Figure 5).
Table.3: The possibilistic regression models for three sample area with h=0.7.
Location Model
Total
vagueness (S)
Nanpi ETY )44.4,29.14()00.0,34.1589(
~
 10538
Luancheng ETY )82.4,75.9()00.0,98.1026(
~
 10942
Loess Plateau of
China
ETY )35.4,95.11()00.0,00.351(
~
 11302

Fig.5: Fuzzy regression relationships between winter wheat yields and ET in three locations in China.
The variation of estimation area illustrates that
uncertainly of simulation data, along the ET axis changes.
From the simulation results, it can be understood that the
estimation area can well express the degree of dispersion at
each evapotranspiration more practically than the
conventional regression method can, and therefore the area
not only represents the relation between ET and grain yield
but also has information on reliability, while the
conventional crop production function represents only the
relations between ET and yield.
The uncertainty in field data is caused by variation in the
climate of region (drought, wind and frost) and offense of
insects and pests, etc.
Interestingly, the half-width for the intercept is
optimized to a value of zero during the minimization of the
vagueness criterion in three locations (Nanpi, Luancheng
and Loess Plateau of China), (Table. 3). Hence, the
intercept of the fuzzy regression model is a crisp number
and all of the fuzziness in the model arises from the slop
being a fuzzy quantity.
Figure 6 shows a representation of fitness of fuzzy
regression. Validation of fuzzy regression models for
estimation of coefficients of crop production functions in
these regions is evaluated with test data. Figure 6 (a) shows
position of ET-Yield data of Linxi district in possibilistic
regression model for Nanpi region.

Fig.6: Representation of fitness of fuzzy model, using testing data.
According to Zhang et al. (1999), Linxi and Nanpi are
located in the lowland of the Haihe floodplain and they
represented same geographic characteristics in the region
based on the groundwater table and geography. So, the
estimated model for Nanpi should be applicable in Linxi.
Figure 6(a) shows that Linxi data is in a good agreement
with derived linear regression model for Napai. The derived
Luancheng regression model is verified with Gaocheng data
(Figure 6(b)).
Also, the fuzzy regression model for Loess Plateau of
China evaluated with 37 ET-Yield data in this region (Table
2.). Figure 6(c) illustrates capability of fuzzy linear
egression in estimation of production function despite of
deficit data.
V. CONCLUSION
A fuzzy linear regression is used to estimate coefficients
of crop production function. For this purpose,
evapotranspiration- yield measurements of winter wheat are
used for three districts in China. Crop yield is a sensitive
parameter and climate, soil, water and crop alter the
predicted yield. Evapotranspiration is the most important
factor in yield estimation. Having crop production function
in each district is necessary for estimation of yield
condition, but, there should be many data estimation of crop
production function with classical least square regression.
As received from this study, fuzzy linear regression
provides a convenient alternative to characterize crop yield
in deficit data condition. The degree of believe is
determined by Taheri et al. (2006) method. Validation of
model is done by test data. However, this approach is
suitable for crop yield predicting by few data.

REFERENCES
[1] Bardossy, A. (1990). Note on fuzzy regression. Fuzzy
sets System. 37, 65-75.
[2] Cheng Si, B., & Bodhinayake, W. (2005). Determining
soil hydraulic properties from tension infiltrometer
measurements: Fuzzy regression. Soil Science Society
of American Journal. 69,1922–1930.
[3] Cole, J.S., & Mathews, O.R. (1923). Use of water by
spring wheat on the Great Plains. USDA Bureau of
Plant Industry, Bulletin No. 1004.
[4] Finger, R., Hediger, W. (2007). The Application of
robust regression to a production function comparison
– the example of Swiss corn. Available at:
http://mpra.ub.uni-muenchen.de/9097/ MPRA Paper
No. 9097, posted 16. June 2008 / 08:09.
[5] Hanks, R. J., H. R. Gardner, & Florian R. L. (1969).
Plant growth- evapotranspiration relations for several
crops in the central Great Plains. Agronomy Journal.
61, 30-34.
[6] Kang, S., Zhang, L., Liang, Y., Hu, X., Cai, H., & Gu,
B. (2002). Effect of limited irrigation on yield and
water use efficiency of winter wheat in the Loess
Plateau of China. Agricultural Water Management. 55,
203-216.
[7] Mathews, O. R., & Brown, L. A. (1938). Winter wheat
and sorghum production in the southern Great Plains
under limited rainfall. USDA Circular 477. U.S.
Government Printing Office, Washington, DC.
[8] Sahin, V. & Hall, M. J. (1996). The effects of
afforestation and deforestation on water yields. Journal
of Hydrology. 178, 293-309.
[9] Shimosaka, T., Kitamori, T., Harata, A., & Sawada, T.
(1996). Fuzzy linear regression method in and
analytical interpretation ultratrace analysis. Analytical
Sciences. 12: 385-391.
[10] Taheri, S. M., Tavanai, H., & Nasiri, M. (2006). Fuzzy
versus statistical rRegression in false- twist texturing.
WSEAS Transactions on Mathematics. 10(5): 1109-
2769.
[11] Tanaka, H., Uejima, S., & Asima, K. (1982). Linear
regression analysis with fuzzy model. IEEE
Transactions on Systems, Man, and Cybernetics. 12,
903-907.
[12] Thornthwaite, C.W. (1948). An approach towards a
rational classification of climate. Geographical
Review. 38, 55-94.
[13] Tseng, F. M., Tzeng, G. H., & Yu, H. C. (1999). Fuzzy
seasonal time series for forecasting the production
value of the mechanical industry in Taiwan.
Technological Forecasting and Social Change. 60,
263–273.
[14] Vaux, H.J., Jr., & Pruitt, W.O. (1983). Crop-water
production function. In D. Hillel (ed.) Advances in
irrigation. Vol. 2. Academic Press, New York, p. 61-
97
[15] Zhang, H., & Oweis, T. (1999). Water-yield relations
and optimal irrigation scheduling of wheat in the
Mediterranean region. Agricultural Water
Management. 38-195-211.
[16] Zhang, H., Wang, X., You, M., & Liu, C. (1999).
Water – yield relations and water-use efficiency of
winter wheat in the North China Plain. Irrigation
Science. 19: 37–45.

Evaporation and Production Efficiency Modelling Using Fuzzy Linear Recurrence

More Related Content

What's hot (20)

Similar to Evaporation and Production Efficiency Modelling Using Fuzzy Linear Recurrence (20)

More from AI Publications (20)

Recently uploaded (20)

Evaporation and Production Efficiency Modelling Using Fuzzy Linear Recurrence