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Solar Energy 85 (2011) 808–817

Prediction of hourly solar radiation using a novel hybrid modelof ARMA and TDNN

Ji Wu ⇑, Chee Keong Chan

Nanyang Technological University, School of Electrical and Electronics Engineering, Singapore

Received 23 August 2010; received in revised form 26 January 2011; accepted 26 January 2011Available online 24 February 2011

Communicated by: Associate Editor Frank Vignola

Abstract

In this work, a new approach that contains two phases is used to predict the hourly solar radiation series. In the detrending phase,several models are applied to remove the non-stationary trend lying in the solar radiation series. To judge the goodness of different detr-ending models, the Augmented Dickey–Fuller method is applied to test the stationarity of the residual. The optimal model is used todetrend the solar radiation series. In the prediction phase, the Autoregressive and Moving Average (ARMA) model is used to predictthe stationary residual series. Furthermore, the controversial Time Delay Neural Network (TDNN) is applied to do the prediction.Because ARMA and TDNN have their own strength respectively, a novel hybrid model that combines both the ARMA and TDNN,is applied to produce better prediction. The simulation result shows that this hybrid model can take the advantages of both ARMAand TDNN and give excellent result.� 2011 Elsevier Ltd. All rights reserved.

Keywords: Solar radiation prediction; ARMA; TDNN; Hybrid model

1. Introduction

Solar radiation prediction is of great importance formany applications such as generation of electricity andproviding portable clean water (Sozen et al., 2004; Saylanet al., 2003; Dincer et al., 1996; Kaplanis, 2006; Rahmanand Chowdhury, 1988; Dinelli, 1995). Accurate predictioncan greatly improve the performance of these systems(Kaygusuz and Sari, 2003; Acock and Pachepsky Ya,2000; Fujioka, 1995; Tugay and Yilmaz, 2004). The solarradiation sequence can be treated as a time series, whichmeans that we can use mathematical models to fit theunderlying random process and predict the next values(Cao and Cao, 2006; Sfetsos and Coonick, 2000; Chenaet al., 2007; Cucumo et al., 2007; Aguiar and Collares-Pereira, 1992; Kaplanis and Kaplani, 2007). Many different

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doi:10.1016/j.solener.2011.01.013

⇑ Corresponding author.E-mail address: [email protected] (J. Wu).

models are used to predict the solar radiation time serieslike the classic Auto-Regression, Auto-Regression andMoving Average (Box and Jenkins, 1970) and MarkovChain. Furthermore, adaptive methods such as the TimeDelay Neural Network (TDNN), which has been provento be reliable in predicting the future trend of a time series,can also be used to solve this problem.

One of the most popular and widely used time series mod-els is the Autoregressive and Moving Average (ARMA)model ( McKenzie, 1984). The popularity of the ARMAmodel is its ability to extract useful statistical propertiesand the adoption of the well-known Box–Jenkins methodol-ogy (Box and Jenkins, 1970). ARMA models are very flexiblesince they can represent several different types of time seriesby using different order. It has been proved to be competentin prediction when there is an underlying linear correlationstructure lying in the time series.

One major requirement for ARMA model is that thetime series must be stationary. However, from a stationary

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J. Wu, C.K. Chan / Solar Energy 85 (2011) 808–817 809

test using Augmented Dickey–Fuller (ADF) test (Dickeyand Fuller, 1981; Harris, 1992), the solar radiation serieswas found to be non-stationary. Thus a phase of detrend-ing is needed to obtain the stationary series. In the first sec-tion, we used different models to detrend the solarradiation. After detrending, the Augmented Dickey–Fullermethod is used to check the stationarity (Dickey andFuller, 1981;Harris, 1992). Also, auto correlation and partialcorrelation of the residual series and Akaike informationcriterion (AIC) of different orders are checked to configurethe optimal ARMA model in prediction (Akaike, 1974).

Time Delay Neural Network (TDNN) has been keenlyexplored and applied in predicting time series (Winyoftand Cander, 1999). The strength of this algorithm is itsability to model nonlinear series. With TDNN, there isno need to specify a particular model form, since the modelis adaptively formed based on the features presented by thedata. This data driven algorithm is suitable for many timeseries where no theoretical model is available. In this paper,TDNN is applied to predict the solar radiation as well.

The use of hybrid models has gained popularity as ittakes advantage of different models Makridakis et al.,1982). There are many approaches to combine differentmodels ( Reid, 1968; Bates and Granger, 1969; Clemen,1989). The basic idea of the model combination in forecast-ing is to use each model’s unique feature to capture differ-ent patterns in the data. Both theoretical and empiricalfinding suggests that combining different models can bean efficient way to improve the forecast performance. Wehave chosen to use a hybrid model of ARMA and TDNNto improve the prediction accuracy.

The rest of the paper is organized as follows. In Sections2 and 3, we tried different detrending models on the solarradiation series to find the optimal one to get the stationaryseries. In Sections 4 and 5, we simulated ARMA, TDNNand the hybrid model in the prediction of the solar radia-tion series and compared their performance.

2. The detrend models

As mentioned earlier, the solar radiation is non-station-ary and we need to detrend it. Because of the unpredictablenoise, it is not easy to find the trend in a specific day’s ser-ies. Instead, we explored the trend in a much more stableseries namely the monthly average series. Below are somemodels that model the general distribution of hourly solarseries:

2.1. Jain’s model

Jain’s model (Baig et al., 1991) tries to fit the solar radi-ation series with a Gaussian function. That is:

rh ¼1

rffiffiffiffiffiffi2pp exp �ðh�mÞ2

2r2

!ð1Þ

rh is the solar radiation of different time. h is time, r is thestandard deviation of the Gaussian curve. It is a parameterthat should be decided by the actual data. The parameter m

represents the peak hour of a day.

2.2. Baig’s model

Baig’s model (Bevington, 1969) is developed based onthe Jain’s model. Baig modified Jain’s model to better fitthe record data at the start and end of series. The model is:

rh ¼1

rffiffiffiffiffiffi2pp exp �ðh� mÞ2

2r2

!þ cos 180

ðh� mÞ2

So � 1

" #( )

ð2Þ

So is the sunshine hour of the day at a site with latitude uand sun’s declination could be calculated by:

S0 ¼2

15cos�1 � tan u tan dð Þ ð3Þ

d is the angle between the rays of the Sun and the plane ofthe Earth’s equator.

2.3. S. Kaplanis’ model

Another interesting model is proposed by Kaplanis(2006). That is:

rh ¼ aþ b cos2pðh� mÞ

24

� �ð4Þ

The a and b in the equation should be decided according tothe actual situation of different area. m is the peak hour ofsolar radiation in this area.

2.4. Al-Sadah’s model

Al-Sadah found that high order polynomial model isquite good in fitting hourly global radiation on a horizontalsurface (Al-Sadah et al., 1990). This model is:

rh ¼ a1 þ a2hþ a2h2 ð5Þ

3. The simulation of detrending models

In this section, these detrending models are simulatedwith actual data. The data collected is from the observationstation set up at Nanyang Technological University,Singapore. The sampling interval is 10 min. We ignoredthe data between 10:00 pm and 7:00 am as there is consis-tently no solar energy received during this period. The unitof solar radiation energy is watt-hour per square meter(W h/m2). The monthly mean series of February 2009 isshown in Fig. 1. Different models are applied successivelyto fit the monthly mean series.

Using Jain’s model to fit the monthly mean series, toestimate the parameter r and m, the method of least squaremethod was applied to train the model to find those

Fig. 1. The monthly mean series of February 2009.Fig. 3. The comparison of Baig’s model and actual series of February2009.

Fig. 4. The comparison of S. Kaplanis’ model and actual series of

810 J. Wu, C.K. Chan / Solar Energy 85 (2011) 808–817

parameters. The comparison of Jain’s model and actualmonthly mean series is presented in Fig. 2.

The solid line is the actual series and dot line is the esti-mated series of Jain’s model.

We can see that Jain’s model generally fits the mid-rangeof the monthly mean series. But it fails badly at the begin-ning and ending of the model.

Baig’s model aims to improve the accuracy at the begin-ning and ending of the actual series, as shown in Fig. 3.

The solid line is the actual series and the dot line isBaig’s model. Since the Baig’s model is based on Jain’smodel, the general profile is the same. However, the modelprovides a much better fit at the beginning and endingstage of the actual series (see Fig. 4).

S. Kaplanis’ model is different from the previous twomodels. This model is based on cosine function. The exper-imental data provides the estimate for the parameter a andb in the function.

Fig. 2. The comparison of Jain’s model and actual series of February2009.

February 2009.

The solid line the actual series and dot line is theS. Kaplanis’ model.

Using Al-Sadah’s model also requires us using actualdata to train the model to get the value of these unknowncoefficients. We used the least square method and theactual data as input. Fig. 5 is the result achieved aftertraining.

To evaluate the stationarity of the solar radiation afterdetrending by these models, we use the AugmentedDickey–Fuller (ADF) test (Dickey and Fuller, 1981; Harris,1992). The ADF test is a test for unit root in a time series.If there is a unit root in time series, the time series is notstationary; otherwise, it should be stationary.

The model of ADF test is:

@Y t ¼ lþ bt þ qY t�1 þ @1Y t�1 þ � � � þ @pY t�p þ et ð6Þ

The l in this equation is a constant, b represent the trend.And p is the order of autoregressive process. {et} is a

Fig. 5. The comparison between Al-Sadah’s model and the actual series ofFebruary 2009.


sequence of independent normal random variables withmean zero and variance r2 = 1.

ADF test applies Eq. (6) to construct a statistics analo-gous to the regression t statistics for the test of hypothesisthat the series has unit root (or q = 1).

Fig. 6 shows the residual of monthly mean solar radiationseries after detrending by the models introduced above. Thetest is carried out on the residual series under the null hypoth-esis that the true underlying process contains a unit root. Ifthe test result is above the critical value, that means weshould accept the null hypothesis and the time series as notstationary; otherwise we should reject the null hypothesisand the time series as stationary. After this testing procedure,we can attain the statistical power, which is the probabilitythat the test rejects a false null hypothesis.

In the test procedure, we use the same significance level0.05 for all the four models (see Figs. 7 and 8).

Fig. 6. The residual series after detren

From the table we can see that the test results for allthese models are below the critical value. Hence, we cansurmise that the residual series of all these models can beregarded as stationary series. The statistical power ofAl-Sadah’s model is the highest one, which means theresidual series detrended by the Al-Sadah’s model has thelowest probability to contain a unit root.

Another key factor that should be taken into consider-ation is the accuracy of fitting the actual series. To evaluatethis accuracy, we use the indicator: root mean square error(RMSE) and the normalized root mean square error(NRMSE):

RMSE ¼ 1

n

XN

1

ei � mið Þ2 !1

2

ð7Þ

NRMSE ¼1N

P1N ei � mið Þ2

� �12

1N

PN1 mi

ð8Þ

In the above two equations, {ei} is the model fitted seriesand {mi} is the actual series (see Table 1).

The RMSE and NRMSE of different models are shownin Table 2.

From Table 2 we see that Al-Sadah’s model yields thebest result in fitting accuracy.

Since Al-Sadah’s model renders the best performance inboth detrending and fitting, we chose it to detrend the solarradiation series for further prediction.

4. The prediction model

4.1. ARMA model

The Autoregressive Moving Average (ARMA) model isusually applied to auto correlated time series data (Box andJenkins, 1970; McKenzie, 1984). This model is a great tool

ding by the four different models.

Fig. 7. The partial correlation of the residual series.

Fig. 8. The auto correlation of the residual series.

Table 1The ADF test for the detrending models.

Detrendingmodel

Statisticalpower

Significantlevel

Testresult

Criticalvalue

Jain’s model 0.9961 0.05 �3.9002 �2.8994Baig’s model 0.9941 0.05 �3.7176 �2.8994S. Kaplanis’ model 0.9961 0.05 �3.8936 �2.8994Al-Sadah’s model 0.9990 0.05 �6.3815 �2.8994

Table 2The RMSE and NRMSE of different models.

Model Error (RMSE) Error (NRMSE)

Jain’s model 0.0459 0.1407Baig’s model 0.0434 0.1330S. Kaplanis’ model 0.0444 0.1360Al-Sadah’s model 0.0231 0.0706


for understanding and predicting the future value of a spec-ified time series. ARMA is based on two parts: autoregres-sive (AR) part and moving average (MA) part. Also, thismodel is usually referred as ARMA (p, q). In which p

and q are the order of AR and MA respectively.Autoregressive (AR) models are based on the idea that

the current value of the series, xt can be explained as afunction of p past values: xt�1; xt�2; . . . ; xt�p where p deter-

mines the number of steps into the past needed to predictthe current value. An autoregressive model of order p,abbreviated AR (p), is of the form:

xt ¼ b1xt�1 þ b2xt�2 þ � � � þ bqxt�q þ wt ð9Þ

where xt is stationary. b1; . . . ; bq are constants (q – 0). wt isa Gaussian white noise series with mean zero.

The moving average (MA) model is an alternative to theautoregressive representation in which the xt on the left-hand side of the equation are assumed to be combinedlinearly, while the moving average model of order q,abbreviated as MA (q), assumes the white noise wt on theright-hand side of the defining equation are combinedlinearly to form the observed data.

The moving average model of order q, or MA (q) model,is defined to be:

xt ¼ wt þ h1wt�1 þ h2wt�2 þ � � � þ hqwt�q ð10ÞNow we mix the Autoregressive and Moving Average to

get the ARMA model. The definition for the model is asfollow.

A time series {xt; t = 0, ±1, ±2. . .} is ARMA (p, q) if itis stationary and:

xt ¼ b1zt1 þ � � � þ bqztq þ wt þ h1wt�1 þ � � � þ hqwt�q ð11Þ

The parameters p and q are called the autoregressive andthe moving average orders, respectively. {wt; t = 0, ±1,±2. . .} is a Gaussian white noise sequence.

To determine the order of ARMA model, we need tocalculate partial correlation and auto correlation of the ser-ies. The plot for partial correlation and auto correlationwith different lags are as follows.

The figures above apparently show that both the partialcorrelation and auto correlation decrease sharply after 1lag. Therefore, the p and q order for ARMA should bothbe 1.

The Akaike information criterion (AIC) can also beapplied to decide the order of ARMA model. AIC is a mea-sure of the goodness of fitting an estimated model. It isbased on the concept of entropy. Entropy is a measure ofthe information lost when a mathematical model is usedto describe the actual data. AIC is a powerful tool formodel selection. The model with the lowest AIC has thebest performance.

The AIC is defined by the following equation:

AIC ¼ log V þ 2dN

ð12Þ

V is the loss function, d is the number of estimated parame-ters, and N is the number of values in the estimation data set.

The loss function V is:

V ¼ det1

NRN

1 ðeðt; hN ÞÞ eðt; hNÞð ÞT� �

ðHere T means transpose of the matrixÞ ð13Þ

hN represents the estimated parameters.


Fig. 9 is the AIC of ARMA model with different order.From the figure, we see that the result is the same as

what was found in the partial correlation and auto cor-relation plot. ARMA (1, 1) has the lowest AIC. Thus,that is the best order for ARMA model (see Figs. 10and 11).

Fig. 10. The architecture of TDNN.

4.2. Time Delay Neural Network

Time Delay Neural Network (TDNN) is developedfrom general feed forward neural network to obtain therelationship between the input and output position in timeseries (Winyoft and Cander, 1999; Fatih et al., 2008). Theconventional neurons of a neural network provide theirresponse to the weighted sum of the current inputs. Butfor TDNN, it extends the sum to a finite number of pastinputs. In this way, the output provided by a given layerdepends on the output of the previous layers computedbased on the temporal domain of input values. Becauseof the very similar structure of the TDNN and the generalMLP, back-propagation with some modifications can beapplied to train the TDNN.

As in the classical NN, the TDNN also has a trainingphase. There are basically two different methods of trainingand updating the weights. In the first method, weights areupdated for each of the input patterns using an iterationmethod. In the second method, an overall error for allthe input output patterns of training sets is calculated oncethe weights and biases are obtained.

There are several methods to update the weights. Forexample, the conjugate gradient-based minimization algo-rithms’ update is along the conjugate direction where theerror function decreases fastest. Other algorithms, whichcan produce faster convergence, are the Newton’s methodor the Levenberg–Marquard algorithm, which considersthe second derivative of the error function. The methodwhich can be used to train NN is vast, but 4 of the mostpopular methods are: (1) Basc back Propagation algorithm(BP), (2) Quasi-Newton algorithm (QN), (3) Levenberg–Marquard algorithm (LM), (4) Conjugate Gradient algo-rithm (CG).

Fig. 9. The AIC of ARMA with different order.

4.3. The hybrid model

Both ARMA and TDNN can be predicting time series.But none of them is a universal model that addresses bothlinear and nonlinear problems. ARMA model has beenproven to be suitable for linear problems, but be inade-quate for complex nonlinear problems. On the other hand,using TDNN to solve linear problem may yield mixedresults. Some cases showed that if there are outliers inthe data, the networks could easily out-perform linearregression models. Some other cases also found that theperformance of TDNN for linear regression problemsdepends on the sample size and noise level. Therefore, itis not wise to use TDNN blindly to any type of data, sinceit is difficult to know all the characteristics of the data in anactual problem. A hybrid model that combines both thesemodels would then be able to capture both the linear andnonlinear aspects.

We assume that the daily solar radiation series is com-posed by linear and nonlinear component (Zheng, 2003).

Then, it should be:

Y t ¼ Lt þ Nt: ð14Þ

Lt denotes linear component and Nt denotes nonlinearcomponent.

First of all, we used the ARMA model to fit the linearcomponent. This means the residual series should containnonlinear component only. Then we used the TDNNmodel to find the nonlinear pattern lying in the residual.This hybrid model has the potential to harness the uniquefeature and strength of both models,

5. The simulation of predicting models

The solar radiation data we use in the simulation isobtained from the observation station in Nanyang Techno-logical University. Interested readers can obtain the datafrom the website: http://nwsp.ntu.edu.sg/weather/new_-portal_intro.htm. We have used the data of 2010 to evalu-ate the performance of different prediction model. Asample is taken every 10 min.

During our investigations, we found that the solar radi-ation received from 22:00 to 07:00 is constantly zero in

http://www.nwsp.ntu.edu.sg/weather/new_portal_intro.htm

http://www.nwsp.ntu.edu.sg/weather/new_portal_intro.htm

Fig. 11. The solar radiation of February 2010.

Fig. 12. The actual and predicted series by ARMA (1, 1).

Fig. 13. The actual and predicted series by TDNN.


Singapore. Thus we only took the sample between 07:00and 20:00 into consideration.

Our goal in the simulation is to explore the optimalmodel to predict the hourly solar radiation of a specifiedday. Hence, we used all the prediction models to predictthe solar radiation 10 min ahead in a specified day.

First, the ARMA model is applied to do the predictionand the Al-Sarah’s model to detrend the solar radiationseries. Then, the data of 1st February 2010 is used to trainthe ARMA (1, 1) model. After training, the ARMA modelis used to do prediction of the data of 2nd February 2010.The prediction figure is in Fig. 12.

In the figure, the dot line is the predicted series; the solidline is the actual series. We can see that the ARMA modelcan generally predict the trend of the solar series. But thereis some “lag” in the prediction.

Next, the Time Delay Neural Network is used in thesimulation. In all the TDNN structure, the sigmoid func-tion was finally adopted and all the popular training meth-ods were tested with various time delay and neuron sizes. Itwas experimentally found that the fastest convergence withthe smallest prediction error was obtained by using the LMmethod in the training phase 3-step time delay and fiveneurons in the hidden layer.

The comparison of prediction by TDNN and the actualdata is shown in Fig. 13. The training data is 1st February2010, and the prediction data is 2nd February 2010.

In Fig. 13, the dot line is predicted series and solid line isactual series. The figure shows that TDNN does not haveobvious lag as ARMA. But it cannot catch the peak ofthe solar radiation when it fluctuates.

Finally, we used the hybrid model of ARMA andTDNN to do the prediction, using the same set of dataof 1st February 2010 to train the model and the data of2nd February 2010 to verify the prediction performance.The comparison of actual and predicted series is presentedin Fig. 14. The dot line is the predicted series and the solidline is the actual series.

The prediction of hybrid model is more accurate thanusing the ARMA or TDNN model separately.

The result of the experiment with just a day’s data is notenough. To better verify the performance of these models,we applied 10 months (January, 2010–October, 2010) datato do the prediction and to confirm the result. These dataare first detrended by Al-Sadah’s model, then three differ-ent prediction models are applied the residual data. ARMA(1, 1) is chosen as the proper ARMA model. The data ofthe previous day was used to train the ARMA (1, 1) model.The trained model was used to predict the next day’s data.

Fig. 14. The actual and predicted series by hybrid model.


For TDNN and hybrid model, we also use the sameapproach; the previous day’s solar energy data was usedto train the model, and then applied the trained model topredict the series of the day after.

Still, RMSE and NRMSE are used to measure thegoodness of prediction. The RMSE and NRMSE ofARMA, TDNN and Hybrid model are calculated. Tocompare the RMSE and NRMSE of different modelsmore intuitively, we plot the result of 10 months (tofacilitate visualize, we take the interval of 2 days) which

Fig. 15. The RMSE of the prediction of differen

Fig. 16. The NRMSE of the prediction of differe

spread over a year in the same coordinate system. TheRMSE and NRMSE of these models are as shown fromFigs. 15–18.

It is easily observed that although the overall perfor-mance of TDNN is better than ARMA, however, it isnot stable. Sometimes, the prediction is close to the actualsituation but at other times a huge error can occur. Thereason for this phenomenon might be attributed to the sen-sitivity of the TDNN model to the training data. Thiswould explain why TDNN is able to capture the trend oftime series without lagging, yet when the training data setis not properly pre-processed, it would affect the predictionperformance. On the other hand, the ARMA model is acomparatively stable algorithm. The prediction accuracyof ARMA for everyday is very close. However, in mostcases, the ARMA model produces huge error. In compar-ison, the hybrid model’s performance is the best in mostcases. It may not be as stable as the ARMA model, butthe stability of hybrid model is much better than theTDNN. Thus, we can see that the hybrid model has har-nessed the advantages of both the TDNN and the ARMAmodel. Although the hybrid model doesn’t always have thebest performance in prediction, it can maintain the stabilityand accuracy of its performance at a high level. See Fig. 19.

To explore the effect of weather condition on the predic-tion models, we plot the NRMSE of different models against

t models on first half year’s solar radiation.

nt models on first half year’s solar radiation.

Fig. 17. The RMSE of the prediction of different models on second half year’s solar radiation.

Fig. 18. The NRMSE of the prediction of different models on second half year’s solar radiation.


different level of daily integral solar radiation. The day withhigh energy level can be regarded as a clear day. Day with lowenergy level usually indicates bad weather condition like acloudy or rainy day. There is clearly a down trend of theNRMSE as the daily integral solar energy increases. TheNRMSE is extremely high when the daily integral solarenergy falls below 6000 W h/m2. Thus we come to the con-

Fig. 19. The NRMSE on different leve

clusion that the prediction models perform very well in cleardays, but when weather condition is not ideal, the accuracyof prediction models decrease. Typically, for a clear day,the daily integral falls in the range of 15,000–30,000 W h/m2 with acceptable NRMSE. Improving the accuracy of pre-diction models under bad weather condition will be regardedas one of our major task in the future.

l of daily integral of solar energy.


6. Conclusion

In this work, we tested several models’ performance indetrending the hourly solar radiation. To judge the good-ness of different detrending models, ADF test is used tomeasure the stationarity of the detrended series. Also wehave used RMSE and NRMSE to measure the goodnessof curve fitting of these different models. Coincidently, wefound that the Al-Sadah’s model’s performance is the bestin both aspects.

After the detrending phase, we applied classical modelARMA to the stationary series, and checked its orderaccording to auto correlation and partial correlation, andconcluded that the best order for it is ARMA (1, 1). Aninspection with AIC provides the same result. Therefore,(1, 1) is the best order for the series. The TDNN modelwas also used to predict the trend series, and found to bemuch more sensitive than the ARMA model, but not asstable. To capture both the advantages of ARMA andTDNN, we used a hybrid model that combined both.The hybrid model uses ARMA model to predict the linearcomponent of the series and the TDNN model to predictthe nonlinear component.

The same data set is applied to the hybrid model to eval-uate its performance. Our conclusion is that the predictionof the hybrid model is quite good since it is stable and accu-rate. This is due to the fact that the solar radiation seriescontain both linear and nonlinear component.

Further works will be focus on exploring other potentialhybrid models and developing a novel method to furtherimprove the accuracy of the prediction. And as we statedat the end of last section, how to improve the accuracyof prediction models in bad weather is also an importanttask.

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