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- 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 202-207, © IAEME
202
LINEAR REGRESSION MODELS FOR ESTIMATING REFERENCE
EVAPOTRANSPIRATION
K. CHANDRASEKHAR REDDY
Professor, Department of Civil Engineering & Principal,
Siddharth Institute of Engineering & Technology, Puttur, Andhra Pradesh, India.
ABSTRACT
In the present study, an attempt is made to develop linear regression models to estimate
reference evapotranspiration(ET0) in Rajendranagar region of Andhra Pradesh, India. ET0 estimated
by the standard FAO-56 Penman-Monteith(PM) method was correlated linearly with the most
influencing climatic parameters such as Temperature(T), Sunshine hours(S), Wind velocity(W) and
Relative Humidity(RH) in the region of the study area for daily, weekly and monthly time steps. The
performance of linear regression models developed was verified based on the evaluation criteria. The
performance indicators such as regression coefficients (slope and intercept of scatter plots), Root
Mean Square Error (RMSE), Coefficient of Determination (R2
) and Efficiency Coefficient (EC) were
used. The regression models performed better at weekly and monthly time steps in ET0 estimation. It
may be due to the fact that the parameters averaged over larger periods generally exhibit the
enhanced effect of linearity among the variables. The simple linear regression models developed may
therefore be adopted in the study region for reasonable estimation of ET0.
Keywords: Reference Evapotranspiration, Linear Regression Model, Multiple and Partial
Correlation Coefficients, Performance Evaluation.
1. INTRODUCTION
Accurate estimation of reference evapotranspiration (ET0) is essential for many studies such
as hydrologic water balance, irrigation system design, and water resources planning and
management. Numerous ET0 equations have been developed and used according to the availability of
historical and current weather data. The FAO-56 Penman-Monteith (PM) equation (Allen et al.,
1998)[1]
is widely used in recent times for ET0 estimation. However, the difficulty in using the
equation, in general, is due to the lack of accurate and complete data. In addition, the parameters in
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING
AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 3, March (2014), pp. 202-207
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2014): 7.8273 (Calculated by GISI)
www.jifactor.com
IJARET
© I A E M E
- 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 202-207, © IAEME
203
the equation potentially introduce certain amount of measurement and/or computational errors,
resulting in cumulative errors in ET0 estimates. Under these circumstances, a simple empirical
equation that requires as few parameters as possible and, producing results comparable with FAO-56
Penman-Monteith method is preferable.
The multiple least-squares regression technique is a popular data analysis and synthesis tool
used in several fields of science and technology. This approach has found wide spread use, even in
agronomic and irrigation studies; most notably in the development of empirical, albeit simple
equations for predicting reference evapotranspiration (ET0) using inputs of measured climatic
variables.
Kotsopoulos and Babajimoponlos (1997)[2]
derived mathematical expressions that describe
the parameters used for the calculation of the Penman reference evapotranspiration through a
nonlinear regression procedure. Comparisons of these expressions to those found in the literature
showed more reliable results.
The present study reports the identification of climatic parameters that influence ET0 estimation
processes the most, development of simple linear regression models for the estimation of daily,
weekly and monthly ET0 for Rajendranagar region of Andhra Pradesh.
2. MATERIALS AND METHODS
Rajendranagar region, located in Rangareddy district of Andhra Pradesh, India, has been
chosen as the study area. The meteorological data at the region for the period 1978-1993 was
collected from IMD, Pune. Data from 1978 to 1988 is used for the purpose of training the model and
that of 1989 to 1993 for testing the model. A brief description of region selected for the present study
is given in Table 1.
Table 1: Brief description of the Rajendranagar region
Longitude Latitude Altitude
Mean daily
relative
humidity
Mean
daily
temperature
Mean daily
wind
velocity
Mean daily
sunshine
hours
Mean daily
vapour
pressure
Mean annual
rainfall
(0
E) (0
N) (m) (%) (0
C) (kmph) (hr) (mm of Hg) (mm)
780
23′
170
19′
536.0 61.8 26.2 7.3 8.0 14.9 920
2.1 Linear regression (LR) model
Linear Regression is a statistical technique that correlates the change in a variable to other
variable(s). The representation of the relationship is called the linear regression model. It is called
linear because the relationship is linearly additive. Below is an example of a linear regression model
Y = C + b1X1 + b2X2+ ………….
Where Y is the dependent variable and X1, X2, ----- are independent variables.
C is the intercept and b1, b2, …… are the regression coefficients.
However, implementation of multiple linear regression considering all the predictor variables
may lead to over-fit and consequent reduction in predictive capability. To overcome this, a step-wise
procedure whose objective is to develop an optimal prediction equation by using statistical criteria to
eliminate superfluous predictor variables, is applied to arrive at the final form of the regression
model involving only those predictor variables that can explain observed variability in the response
variable.
- 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 202-207, © IAEME
204
In the present study, the Linear Regression model is expressed as
ET0 = C + b1 T + b2 S + b3 W + b4 RH
Where ET0 is the reference evapotranspiration (dependent variable) estimated using FAO-56
Penman-Monteith method and T-Temperature, W-Wind velocity, S-Sunshine hours, RH-Relative
Humidity (RH) (independent variables) are the climatic parameters. C is the intercept and b1, b2, b3,
b4 are the regression coefficients.
3. PERFORMANCE EVALUATION CRITERIA
The performance evaluation criteria used in the present study are the Coefficient of
Determination (R2
), the Root Mean Square Error (RMSE) and the Efficiency Coefficient (EC).
3.1 Coefficient of Determination (R2
)
It is the square of the correlation coefficient (R) and the correlation coefficient is expressed as
Where O and P are observed and estimated values, O and P are the means of observed and
estimated values and n is the number of observations. It measures the degree of association between
the observed and estimated values and indicates the relative assessment of the model performance in
dimensionless measure.
3.2 Root Mean Square Error (RMSE)
It yields the residual error in terms of the mean square error and is expressed as (Yu et al.,
1994)[5]
n
op
RMSE ii
n
i
2
1
)( −
=
∑
=
3.3 Efficiency Coefficient (EC)
It is used to assess the performance of different models (Nash and Sutcliffe, 1970)[4]
. It is a
better choice than RMSE statistic when the calibration and verification periods have different lengths
(Liang et al., 1994)[3]
. It measures directly the ability of the model to reproduce the observed values
and is expressed as
( )
( )∑
∑
=
=
−
−
−= n
i
i
n
i
ii
oo
po
EC
1
2
1
2
1
2/1
1
2
1
2
1
)()(
))((
∑ −∑ −
−−∑
=
==
=
n
i
i
n
i
i
ii
n
i
ppoo
ppoo
R
- 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 202-207, © IAEME
205
A value of EC of 90% generally indicates a very satisfactory model performance while a
value in the range 80-90%, a fairly good model. Values of EC in the range 60-80% would indicate an
unsatisfactory model fit.
4. RESULTS AND DISCUSSION
The analysis of multiple linear correlation between FAO-56 Penman-Monteith reference
evapotranspiration (PM ET0) and the climatic parameters was carried out by omitting one of the
climatic factors each time. While carrying out the analysis, the data period was divided into training
and testing periods. The training period data set was used to identify the parameters influencing the
region and, to develop linear ET0 models in terms of these parameters. The verification of
applicability of the models developed was checked using the testing period data set. The multiple
linear correlation coefficients and partial correlation coefficients between PM ET0 and climatic
parameters for the regions selected for the present study were computed for both training and testing
periods are presented in Table 2 and 3.
Table 2: Multiple correlation coefficients
Time step
Multiple correlation coefficient
Independent variable omitted
---- T S W RH VP R
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Daily 0.9643 0.9689 0.8684 0.8533 0.9099 0.9134 0.8865 0.8487 0.9531 0.9657 0.9642 0.9687 0.9643 0.9687
Weekly 0.9755 0.9804 0.8816 0.8584 0.9521 0.9544 0.9161 0.8967 0.9697 0.9783 0.9753 0.9804 0.9754 0.9803
Monthly 0.9900 0.9886 0.9209 0.8884 0.9805 0.9775 0.9509 0.9387 0.9872 0.9869 0.9899 0.9884 0.9900 0.9884
Table 3: Partial correlation coefficients
Time
step
Partial correlation coefficient
T S W RH VP R
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Daily 0.8455 0.8802 0.7697 0.7940 0.8201 0.8838 0.4842 0.3030 0.0524 0.0793 0.0000 0.0793
Weekly 0.8847 0.9233 0.6945 0.7513 0.8360 0.8955 0.4348 0.3095 0.0894 0.0000 0.0634 0.0709
Monthly 0.9322 0.9447 0.6962 0.7003 0.8901 0.8996 0.4665 0.3592 0.0993 0.1309 0.0000 0.1309
From Tables 2 and 3, it may be observed that the influence of T, S, W and RH is relatively
more on ET0 in the region of the study area irrespective of the time step. Further, no significant effect
of Vapour Pressure (VP) and Rainfall (R) on ET0 is found in the region. This may be due to the fact
that the region lies in the semi-arid zone, mostly experienced by high temperature and radiation.
In the regression analysis, the PM ET0 was used as the dependent variable and T, S, W and
RH as independent variables to derive the coefficients in the linear regression models. The linear
regression equations relating daily, weekly and monthly PM ET0 and corresponding climatic
parameters influencing the regions were developed as presented in Table 4. The scatter plots for the
training and testing periods of the ET0 values computed using the relations presented in Table 4 with
those of PM ET0 as shown in Fig.1, depict the closeness of the values for different time steps and
thereby reflect the appropriateness of the analysis.
The performance of linear regression models developed was verified based on the evaluation
criteria. The performance indicators such as regression coefficients (slope and intercept of scatter
plots), R2
, RMSE and EC of the models are presented in Table 5. The slope and intercept of scatter
- 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 202-207, © IAEME
206
plots nearly equal to one and zero respectively, high values of R2
and EC and low values of RMSE
indicate satisfactory performance of the models. The models showed better performance for weekly
or monthly time steps over daily time step in ET0 estimation. It may be due to the fact that the
parameters averaged over larger periods generally exhibit the enhanced effect of linearity between
the variables.
The simple linear regression models developed may therefore be adopted in the study regions
for reasonable estimation of ET0.
Table 4: Linear regression ET0 models
Time step Regression equation
Daily ET0 = – 2.559 + 0.235 T + 0.225 S + 0.141 W – 0.026 RH
Weekly ET0 = – 3.219 + 0.248 T + 0.240 S + 0.140 W – 0.023 RH
Monthly ET0 = – 3.436 + 0.253 T + 0.239 S + 0.146 W – 0.021 RH
A) Training period B) Testing period
Fig.1: Scatter plots of ET0 values estimated using Linear Regression (LR) models with those
estimated by Penman-Monteith (PM) method
- 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 3, March (2014), pp. 202-207, © IAEME
207
Table 5: Performance indices of Linear Regression (LR) models
Time step
Slope of the
scatter plots
Intercept of the
scatter plots
R2 RMSE
(mm)
EC
(%)
Training
Period
Testing
period
Training
period
Testing
period
Training
period
Testing
period
Training
Period
Testing
period
Training
period
Testing
period
Daily 1.0000 0.9310 0.0005 0.2386 0.9299 0.8940 0.45 0.49 92.99 89.40
Weekly 1.0000 0.9166 -0.0004 0.3410 0.9516 0.9127 0.33 0.40 95.16 91.27
Monthly 1.0000 0.9420 0.0003 0.2553 0.9802 0.9357 0.19 0.33 98.02 93.57
5. CONCLUSION
The effect of climatic parameters on ET0 at Rajendranagar region is brought out through
multiple correlation analysis. The sunshine hours, wind velocity, temperature and relative humidity
mostly influenced ET0 in the study area. The linear regression ET0 models comparable with FAO-56
Penman-Monteith method for the region have been developed in terms of the influencing climatic
parameters. The performance of linear regression models developed was verified based on the
evaluation criteria. The slope and intercept of scatter plots nearly equal to one and zero respectively,
high values of R2
and EC and low values of RMSE indicate satisfactory performance of the models.
The models showed better performance for weekly or monthly time steps over daily time step in ET0
estimation. It may be due to the fact that the parameters averaged over larger periods generally
exhibit the enhanced effect of linearity among the variables. Therefore the simple linear regression
models developed may be used in the study area and also the other regions of similar climatic
conditions for satisfactory ET0 estimation.
REFERENCES
[1] Allen, R. G., Pereira, L. S., Raes, D. and Smith, M. (1998), Crop evapotranspiration - Guidelines
for computing crop water requirements. FAO Irrigation and Drainage Paper 56, FAO, Rome.
pp.326.
[2] Kotsopoulos, S. and Babajimopoulos, C. (1997). Analytical estimation of modified Penman
equation parameters. Journal of Irrigation and Drainage Engineering, ASCE, Vol.123, No.4,
pp.253-256.
[3] Liang, G. C., O-Connor, K. M. and Kachroo, R. K. (1994), a multiple-input single-output variable
gain factor model. Journal of Hydrology, Vol.155, No.1-2, pp.185-198.
[4] Nash, J. E. and Sutcliffe, J. V. (1970), River flow forecasting through conceptual models part I: A
discussion of principles. Journal of Hydrology, Vol.10, No.3, pp.282-290.
[5] Yu, P. S., Liu, C. L. and Lee, T. Y. (1994), Application of transfer function model to a storage
runoff process. In Hipel K. W., McLeod A.I. and Panu U.S. (Ed.), Stochastic and Statistical
Methods in Hydrology and Environmental Engineering, Vol.3, pp.87-97.
[6] K. Chandrasekhar Reddy, “Evaluation And Calibration of Temperature Based Methods For
Reference Evapotranspiration Estimation In Tirupati Region” International Journal of Advanced
Research in Engineering & Technology (IJARET), Volume 5, Issue 2, 2014, pp. 87 - 94, ISSN
Print: 0976-6480, ISSN Online: 0976-6499.
[7] Sameer Ul Bashir, Younis Majid and Ubair Muzzaffer Rather, “Effect of Rapidite on Strength of
Concrete in Warm Climates”, International Journal of Civil Engineering & Technology (IJCIET),
Volume 4, Issue 6, 2013, pp. 126 - 133, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.
[8] K. Chandrasekhar Reddy, “Reference Evapotranspiration Estimation By Radiation Based
Methods” International Journal of Civil Engineering & Technology (IJCIET), Volume 5, Issue 2,
2014, pp. 81 - 87, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.