Evaporation and Production Efficiency Modelling Using Fuzzy Linear Recurrence

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 ETyield 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 ETyield 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.


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 andKang 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 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 andBodhinayake, 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 us ers.
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: 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: 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 from the 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 seasonal evapotranspiration 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 Where Subject to: Eq. ( 8) is linear, thereby allowing the optimization problem to be solved by means of linear programming.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 cropgrowing 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  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 = 0-0.95with 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

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

 
Based on 6 data in Table 1, for Nanpi region, and adapting relation (8), the objective function is:  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).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   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.

Fig. 3 :
Fig.3: Triangular membership function of fuzzy output 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:

Fig. 4 :
Fig.4: The variation of the total vagueness (S), based on different amounts for h.

Fig. 5 :
Fig.5: Fuzzy regression relationships between winter wheat yields and ET in three locations in China.
) shows position of ET-Yield data of Linxi district in possibilistic regression model for Nanpi region.

Fig. 6 :
Fig.6: Representation of fitness of fuzzy model, using testing data.

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)).

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
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. 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 .aipublications.com/ijrehPage | 24

Table . 3
: The possibilistic regression models for three sample area with h=0.7.