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Forecasting solar irradiation based on influencing factors determined by linear correlation and stepwise regression

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Abstract

Determination of the factors influencing solar irradiation plays an important role in the performance of a prediction model based on machine learning techniques. This study compared the results obtained using a linear correlation analysis and stepwise regression method to identify the key meteorological, weather, and radiation factors that significantly affect the solar irradiation on the following day, as well as the difference between the quantities of solar irradiation on the current day and following day. These factors were used to establish prediction models for the Qinghai–Tibet Plateau, which has a complex topography and weather caprices. The results indicated that the stepwise regression method was capable of screening more effective influencing factors and the corresponding predictive accuracy was acceptable even under the weather conditions of the plateau.

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Abbreviations

w :

weather condition

E :

evaporation (mm)

ST avg :

average surface temperature (°C)

ST max :

daily max surface temperature (°C)

ST min :

daily min surface temperature (°C)

P 20–8 :

20–8 h rainfall (mm)

P 8–20 :

8–20 h rainfall (mm)

P 20–20 :

20–20 h rainfall (mm)

P avg :

daily average station pressure (hPa)

SP max :

daily maximum station pressure (hPa)

SP min :

daily minimum station pressure (hPa)

RH avg :

average relative humidity (%)

RH min :

minimum relative humidity (%)

S :

sunshine hours (h)

T avg :

average temperature (°C)

T max :

daily maximum temperature (°C)

T min :

daily minimum temperature (°C)

W avg :

average wind speed (m/s)

W max :

maximum wind speed (m/s)

W g :

great wind speed (m/s)

H :

daily solar irradiation (MJ m−2 day−1)

H 0 :

daily extra-terrestrial irradiation (MJ m−2 day−1)

H sc :

solar constant (0.082 MJ m−2 min−1)

ΔH :

daily solar irradiation difference (MJ m−2 day−1)

Ht − 1 Ht :

current and next day’s solar irradiation (MJ m−2 day−1)

d :

inverse relative earth–sun distance

ω s :

sunset hour angle (degrees)

φ :

site latitude (degrees)

δ :

solar declination angle (degrees)

j :

day of the year, starting from January 1

N :

total number of datasets

i :

ith data point of data

t-1, t :

current day and next day

x t − 1, i :

input parameter of the current day

y t, i :

daily solar irradiation for the next day (MJ m−2 day−1)

\( \overline{x} \) :

average of input parameters

\( \overline{y} \) :

average of the daily irradiation (MJ m−2 day−1)

R :

correlation coefficients between the influencing factors and H

R’ :

correlation coefficients between the influencing factors and ΔH

\( {y}_{t,i}^o \) :

measured value of daily solar irradiation (MJ m−2 day−1)

\( {y}_{t,i}^f \) :

predicted value of daily solar irradiation (MJ m−2 day−1)

\( \overline{y^o} \) :

average of the measured value of daily solar irradiation (MJ m−2 day−1)

x t, ij :

ith sample value building the regression equation

b j :

coefficient of the regression equation

Qm, Qm + 1 :

sum of residual squares with m and m + 1 factors

Adj-R 2 :

adjustable coefficient of determination

x k :

new factor of multivariate regression equation

V k :

variance contribution of factor xk

R m − p :

Pearson correlation coefficient between the measured value and predicted value

NSE :

Nash–Sutcliffe efficiency coefficient

RMSE :

root–mean–square error

MAE :

mean absolute error

MAPE :

mean absolute percentage error

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Funding

This research was supported by the National Natural Science Foundation of China (Grant No. 91847302, 51879137, 51979276).

Author information

Authors and Affiliations

  1. State Key Laboratory of Hydroscience & Engineering, Tsinghua University, Beijing, 100084, China

    Jun Qiu

  2. State Key Laboratory of Plateau Ecology and Agriculture, Qinghai University, Xining, 810016, China

    Jun Qiu

  3. College of Water Resources & Civil Engineering, China Agricultural University, Beijing, 100083, China

    Xiao-Jie An & Fang-Fang Li

  4. Qinghai Huanghe Electric Power Technology Co., Ltd, Xining, 810016, China

    Zhi-Gang Wu

Authors
  1. Jun Qiu
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  2. Xiao-Jie An
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  3. Zhi-Gang Wu
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  4. Fang-Fang Li
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Correspondence to Fang-Fang Li.

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Appendix

Appendix

  1. I.

    Linear correlation analysis method

LCA is a statistical method used to study the correlation between two or more random variables. In this study, LCA was used to determine the factors influencing the solar irradiation by calculating the Pearson correlation coefficient, as shown in Eq. (10). A correlation coefficient that is closer to + 1 or − 1 indicates a stronger correlation; conversely, a correlation coefficient that is closer to zero shows a weaker correlation:

$$ R=\frac{\sum_{i=1}^N\left({x}_{t-1,i}-\overline{x}\right)\left({y}_{t,i}-\overline{y}\right)}{\sum_{i=1}^N{\left({x}_{t-1,i}-\overline{x}\right)}^2{\sum}_{i=1}^N{\left({y}_{t,i}-\overline{y}\right)}^2}, $$
(10)

where N is the total number of data points, i is the ith data point in all the data, xt − 1, i is the input parameter, yt, i is the daily solar irradiation, t-1 is the day before t,\( \overline{x} \)is the average of the input parameters, and \( \overline{y\ } \)is the average of the daily irradiation.

Table 8 Correspondence between correlation coefficient and correlation degree (http://zh.wikipedia.org/zh-cn)
Full size table
  1. II.

    Stepwise regression method

The basic concept of the SRM is to introduce parameters into the model one at a time and apply the F-test with every additional parameter to verify the performance of the newly established model. In addition, the significance of all the parameters is also investigated every time by applying the T-test. When some of the original influencing factors are no longer significant as a result of the introduction of a particular parameter, they are omitted from the set to ensure that only significant parameters are included in the regression equation before each new parameter is introduced. This process is repeated until no significant parameter needs to be introduced into the regression equation, and no parameter of significance needs to be eliminated from the equation. Essentially, the SRM is a method used to establish the best multiple linear regression equation by introducing and eliminating factors one at a time based on its variance contribution. The variance contribution of a factor is the difference between the sum of the residual squares before and after introducing the factor into the regression equation. The sum of the residual squares with m factors in the equation is shown in Eq. (11):

$$ {Q}^m={\sum}_{i=1}^N{\left({y}_{t,i}-{b}_0-{\sum}_{j=1}^m{b}_j{x}_{t, ij}\right)}^2, $$
(11)

where bj is the coefficient of the regression equation, yt, i is the observed value, and xt, ij is the ith sample value.

Adding factor xk (k > m) and using (m + 1) factors to establish a new multivariate regression equation, the sum of the residual squares can be obtained as shown in Eq. (12):

$$ {Q}^{m+1}={\sum}_{i=1}^N{\left({y}_{t,i}-{b}_0^{\ast }-{\sum}_{j=1}^m{b}_j^{\ast }{x}_{t, ij}-{b}_k^{\ast }{x}_{t, ik}\right)}^2 $$
(12)

The variance contribution of factor xk is

$$ {V}_k={Q}^m-{Q}^{m+1} $$
(13)

The variance contribution is taken as the measurement of the importance of a factor to the equation. The main process of the SRM is illustrated below.

  1. (1).

    Divide all of the factors into two sets: one composed of the factors introduced into regression equation S and the other containing the factors not introduced into the equation, denoted as \( \overline{S} \). At the beginning, all of the factors are in set S, and \( \overline{S} \)is empty.

  2. (2).

    Calculate the variance contribution of all the factors in S by Eqs. (11) to (13) to determine the factor with the greatest variance contribution. If it can pass the F-test as an introduced factor, it will be brought in the regression equation.

  3. (3).

    Calculate the variance contribution of each of the factors in \( \overline{S} \) by Eqs. (11) to (13) and determine the factor with the smallest variance contribution. If it can pass the F-test as an eliminated factor, it will be omitted in the regression equation.

  4. (4).

    Repeat steps (2) and (3) until no factor can be introduced or eliminated.

    Using the SRM, the effective influencing factors can be selected from various factors affecting solar irradiation to establish a multiple regression equation.

  1. III.

    Back-propagation neural network

A BPNN is a multilayer feedforward network trained by the error back-propagation algorithm. It is one of the most widely used neural network models with high prediction accuracy. A BPNN can learn and store a large number of input and output mapping relationships without a mathematical equation describing the mapping relationship beforehand. Its learning rule is to use the steepest descent method to adjust the weights and thresholds of the network through back propagation, to minimise the sum of the squares of errors of the network. The procedure of the BPNN mainly includes the following rules.

  1. (1).

    In forward propagation, input samples are transferred from the input layer to the output layer after being processed by hidden layers. If the actual output of the output layer does not match the expected value, it goes to the backward propagation stage.

  2. (2).

    In back-propagation, the output is propagated layer by layer from the hidden layer to the input layer, and the errors are allocated to all of the neuros in each layer so that the error signals of the neuros in each layer can be obtained. These signals can be used as the basis for correcting the weights of the neuros.

As an ANN model, a BPNN is also composed of an input layer, an output layer, and hidden layers. Although there is no limitation on the hidden layers in theory, some scholars have proved that a neural network with one hidden layer can realise the mapping of continuous functions with arbitrary precision(Goh 1996). Practice has shown that more hidden layers lead to a more complex network, which affects the generalisation ability of the network. In this study, a three-layer BPNN was adopted with the tansig function in Eq. (14) from the input layer to the hidden layer and the logsig function in Eq. (15) from the hidden layer to the output layer.

$$ tansig(x)=\frac{2}{1+{e}^{-2x}}-1 $$
(14)
$$ logsig(x)=\frac{1}{1+{e}^{-x}} $$
(15)

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Qiu, J., An, XJ., Wu, ZG. et al. Forecasting solar irradiation based on influencing factors determined by linear correlation and stepwise regression. Theor Appl Climatol 140, 253–269 (2020). https://doi.org/10.1007/s00704-019-03072-8

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  • DOI: https://doi.org/10.1007/s00704-019-03072-8

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