Applied Mathematical Sciences, Vol. 7, 2013, no. 73, 3629 - 3639 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.35253
Multiple Linear Regression Equation for Estimation of Daily Averages Solar Radiation in Chonburi, Thailand Jatupat Mekparyup, Kidadan Saithanu and Jiraporn Dujjanutat Department of Mathematics Faculty of Science, Burapha University 169 Muang, Chonburi, Thailand
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[email protected] Copyright © 2013 Jatupat Mekparyup et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract The objectives of the present study were to estimate solar radiation in Chonburi, Thailand using regression analysis. For building multiple linear regression equation, the data, solar radiation, maximum temperature, minimum temperature, hours of bright sunshine, sea level pressure, humidity and water vapor pressure, were collected from Atmospheric Ozone and Solar Radiation Monitoring, Meteorological Observation Bureau, Thai Meteorological Department since 2005 to 2009. The results of the present study found that the regression equation for estimation of solar radiation in March provided less accurate estimation and the n′ =−7.71+0.13SUNSHINE+269 HUMIDITY ′ equation was SOLAR +0.16 PRESSURE with adjusted coefficient of determination 0.632 and standard error of estimation 0.295. As the best one was in n =3.53+0.06MAX_TEMP− October and the equation was SOLAR 0.16MIN_TEMP−0.04HUMIDITY+0.11PRESSURE with adjusted coefficient of determination 0.923 and standard error of estimation 0.101. Mathematics Subject Classification: 62J05 Keywords: Multicollinearity, Variance inflation factor
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Jatupat Mekparyup, Kidadan Saithanu and Jiraporn Dujjanutat
1 Introduction Thailand is located near the equator which receives more solar radiation during April and May. The highest average annual solar radiation values, of about 20-24 (MJ/M2/DAY), was recorded in the Northeast. In Thailand, there were 38 monitoring stations of solar radiation. Chonburi is a province with a monitoring station of solar radiation located at the Department of Meteorology, Ban Bueng, using Pyranometer, a World Meteorological Organization First Class Radiometer designed for the measurement of solar and sky radiation. However, the instrumentation was very expensive and somewhat difficult to measure accurately so solar radiation determination was able to do in some areas. Therefore, the estimation of solar radiation will reduce the cost of buying the equipment and simplify to measure solar radiation. Moreover, it will use as a guide in estimation of solar radiation in areas where no monitoring stations of solar radiation provided as well. There were abundance studies conducted estimation of solar radiation, for example, [1-3], [6-9], [11-12], [14-15], [17], [19-20], [22-28], [30], [32-35], [3940]. Nevertheless, there is still scarce study about the estimation of solar radiation in Thailand. Thus, the present study objectifies to determine the multiple regression equation in each month for the estimation of solar radiation in Chonburi, Thailand.
2 Materials and Methods The data, solar radiation, maximum temperature, minimum temperature, hours of bright sunshine, sea level pressure, humidity and water vapor pressure, were collected from Atmospheric Ozone and Solar Radiation Monitoring, Meteorological Observation Bureau, Thai Meteorological Department since 2005 to 2009. For generating multiple linear regression equation to estimate the solar radiation in Chonburi, regression model was used as Equation (1) yij = β0 j + β1 j x1ij + β 2 j x2ij + β3 j x3ij + β 4 j x4ij + β5 j x5ij + β 6 j x6ij + ε ij (1)
The model consists of one dependent variable; yij = daily averages solar radiation, and six independent variables; x1ij = maximum daily temperature, x2ij = maximum daily temperature, x3ij = hours of bright sunshine, x4ij = sea level pressure, x5ij = humidity and x6ij = water vapor pressure, where β j = regression coefficient of jth month and ε j = error of jth month. Simple correlation coefficients were calculated to identify relationship between the dependent variable and the independent variables. All possible linear regression equations were used to consider the best fitted linear regression
Multiple linear regression equation
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2 equation by adjusted coefficients of determination ( Radj ) and standard error of
estimation (S) including the test of the regression equation. After obtained the best fitted linear regression equation, then checking assumptions for multiple regression analysis was proceeded. There are four assumptions to be tested; (I) normality of the error distribution using Anderson-Darling statistic by Equation (2); 2i − 1 [lnF (Yi ) + ln(1 − F (Yn +1−i ))] i =1 n n
AD = −n − ∑
(2)
(II) independence of the errors using Durbin-Watson statistic by Equation (3); n
n
2 DW = ∑ (εli − εn i −1 )
∑ εli
i =1
2
(3)
i=1
(III) homoscedasticity (constant variance) of the errors using Breush-Pagan statistic by Equation (4); BP =
SSR* ⎛ SSE ⎞ ÷⎜ ⎟ 2 ⎝ n ⎠
2
(4)
l where SSR* =Sum of squares in regression between e2j and xij , SSE=Sum of squares in regression error between m y j and xij . (IV) multicollinearity among independent variables using Variance Inflation Factor (VIF) Equation (5). VIF j =
1 1 − R 2j |others
(5)
where R 2j|others =Multiple coefficient of determination between xij and all xi After tested all assumptions, the comparison in each month between the real values and the estimated values of daily averages solar radiation from the obtained multiple linear regression equations was plotted. Then the percentage of measurement error of daily averages solar radiation in each month will be calculated by Equation (6).
εl j
=
m yj − n y Pt j m yj
(6)
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Jatupat Mekparyup, Kidadan Saithanu and Jiraporn Dujjanutat
y j = Estimation where εlj = measurement error given as an absolute fraction, m
of solar radiation at jth month, n yPt j = Pyranometer of solar radiation at jth month.
3 Results The relationship between dependent variable and independent variables was determined by simple correlation coefficients. The correlation coefficients values are shown that there was the positive and significant relationship (P-value
