20120140503022
Upcoming SlideShare
Loading in...5
×
 

20120140503022

on

  • 90 views

 

Statistics

Views

Total Views
90
Views on SlideShare
90
Embed Views
0

Actions

Likes
0
Downloads
0
Comments
0

0 Embeds 0

No embeds

Accessibility

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

20120140503022 20120140503022 Document Transcript

  • 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
  • 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.
  • 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
  • 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
  • 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
  • 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.