At present, the numerical prediction models fail to predict effectively due to the lack of basic data of pollutant concentration in a short term in China. Therefore, it is necessary to study the statistical prediction methods based on historical data. The traditional Back Propagation Neural Network (BPNN) has been used to predict the pollutant concentration. The missing data also has an impact on modeling, and how to use historical data effectively of multiple monitoring stations in a city should be concerned. In this study, the Improved Newton Interpolation (INI) algorithm has been adopted to solve the problem of missing data, and assigning weight (AW) method has been proposed to enrich data of per station. The Neighbor-Principal Component Analysis (Neighbor-PCA) algorithm has been employed to reduce the dimension of data in order to avoid overfit- ting caused by high dimension and linear correlation of multiple factors. The strategy of early stopping and gradient descent algorithm have been utilized to avoid the slow convergence speed and overfitting by the traditional BPNN. The methods (INI, AW, Neighbor-PCA) have been integrated as a prediction model named NNP-BPNN. Forecasting experiments of PM 2.5 have shown that the NNP-BPNN model can improve the accuracy and generalization ability of the traditional BPNN model. Specifically, the average root mean square error (RMSE) has been reduced by 24% and the average correlation relevancy has been increased by 9.4%. It took 20 s to implement BPNN model, it took 170 s to implement NN-BPNN model and it took 47 s to implement NNP-BPNN model. The time used by NNP-BPNN model is reduced by 72% than that of NN-BPNN model. Keywords Newton interpolation · Improved Newton interpolation algorithm · Assigning weight method · Neighbor-PCA · BPNN · Pollutant concentration prediction 1 Introduction research area in the future (Li et al. 2017; Liu et al. 2017; Li et al. 2017; Cui et al. 2016). However, the NAP can not be Currently, environmental pollution poses a serious threat adopted among many cities in China due to the lack of the to human health (Zheng et al. 2016). The air quality pre- basic data of pollutant concentration in a short time (Huchao diction model mainly adopts the classic regression statisti- et al. 2015; Niska et al. 2005). Hence, statistical prediction cal model and the Numerical Air Prediction model (NAP) methods based on historical data should be studied in order (Demuzere et al. 2009). The NAP uses mathematical models to obtain a high prediction performance. of the atmosphere and oceans to predict the air quality based Shi et al. (2012) have proved that Artificial Neural on current atmosphere conditions and the pollutant sources Network (ANN) can provide better results than the tradi- (Zhang et al. 2014). Using cloud computing to implement tional multiple linear regression models about pollutant high performance of computing of NAP would be the key concentration prediction based on historical monitoring data. The combined model of ARMA and BPNN based on historical monitoring data studied by Zhu and Lu (2016) * Yi Wang has a smaller prediction error than that of the traditional wynk@mail.nankai.edu.cn BPNN. The time series data is decomposed into wavelet Hong Zhao coefficients, and the prediction experiments based on three zhaoh@nankai.edu.cn types of neural network models (Multi-layer Perceptron, Elman and Support Vector Machine) based on historical Nankai University, Weijin Road, Tianjin, China data have shown that the improved models have provided Nankai University, Tongyan Road, Tianjin, China Vol.:(0123456789) 1 3 H. Zhao et al. low RMSE and mean absolute error in comparison to the original models, which is developed by Feng et al. (2013). 2 f(.) Grivas and Chaloulakou (2006) has employed a novel ANN which selected factors of historical monitoring data via genetic algorithm, proving the model has produced Input neurons Output smaller error than that of the regression model in the pre- diction of PM concentration. Shi et al. (2012) has used Fig. 1 The neuron map of the BPNN Feed-forward BPNN with the hyperbolic tangent sigmoid activation function and the Levenberg–Marquardt opti- mization method to predict PM concentration in each station. However, there is no explanation that the interac- tion between central station and neighbor stations in their 2 studies. The monitoring data that is often lost from multiple stations in a city. Nejadkoorki and Baroutian (2011) have Input Hidden removed the missing data, resulting in the waste of data. The Newton Interpolation algorithm has good inter- Fig. 2 The structure of BPNN polation effect when the data loss ratio is not too high (Breu et al. 2016). Thus, in this study, the missing data is effectively interpolated by INI algorithm. In order to 2 Related algorithm enlarge the data, we have taken neighbor stations data into consideration. Because different distance makes dif- In this section, the basic principles of BPNN, Newton inter- ferent effects on central station, weight has been assigned polation algorithm and PCA algorithm are respectively to each stations data. The oversize of the dimension of introduced. input data would affect the generalization ability of the BPNN, therefore, the PCA algorithm is utilized to extract 2.1 BPNN principles the eigenvector from neighbor stations effectively in the study (Skrobot et al. 2016). The extracted eigenvector from BPNN is composed by a series of simple unit connected neighbor stations and the eigenvectors of the central sta- with each other densely. In data mining , neural network has tion are regarded as input to the BPNN, which can not been employed by Chang and Yang (2017). Each unit has a only keep information as much as possible, but also ensure certain amount of input and output. The structure of neuron that the dimension of the input to the BPNN is suitable, is shown in Fig. 1 and the structure of BPNN is shown in increasing the generalization ability of the BPNN. Finally, Fig. 2. early stop strategy (Yang and Zhang 2017) has been per- The stimulation delivered by neuron is called x . The con- formed during training. That is, when the error reached the nection weight is called w . This accumulation is called a . i j value set by program, the training should be terminated Sigmoid(x) is put as the activation function. The error is in order to avoid overfitting. The learning rate gradient utilized to modify the connection weights for feedback to descent strategy (Huang and Lin 2009) has been employed complete the learning process, which is the feedback mecha- to avoid the slow convergence speed. nism of the BPNN (Guan et al. 2016). The inner of neuron NEXT, INI algorithm would be proved in detail for also interferes with the output result, so an extra bias named dealing with missing data. Multiple stations data and AW as b is brought. So we get the formula (1). method have enlarge data for modeling. However, this affects the speed and efficiency of BPNN. Thus, Neighbor-PCA n algorithm has been employed to reduce the dimension of a = f (x × w )+ b (1) j i j data in order to avoid overfitting. BP model and NN_BPNN model are used for contrast. The effectiveness of NN_BPNN But in practice, the traditional BPNN has some shortcom- model has showed that the integration of INI algorithm and ings, such as the long training time, the slow convergence AW method proved to be persuasive and the effectiveness speed, local minimum and the poor stability. It is hard to of NNP_BPNN model has showed that the combination of adjust the initial weight and learning rate parameters (Xiao INI algorithm, AW method and Neighbor-PCA algorithm et al. 2012). If the dimension of the input is high enough, proved to be persuasive. there is a greater challenge for training. 1 3 The pollutant concentration prediction model of NNP‑BPNN based on the INI algorithm, AW method… Table 1 The steps of PCA algorithm 2.2 Newton interpolation algorithm Input data ={X X ...X } low dimensional: k 1 2 P Interpolation function has many different types. Li et al. 1 i=1 Steps 1. Centralize all the data: X − X has introduced multiple methods of interpolation (Li et al. i p i 2. Calculate the covariance matrix for the dataset: X X 2017). Using basis function to get the Lagrange interpola- tion polynomial is common in the theoretical analysis. The 3. Eigen value decomposition for the matrix X X basis function would be changed with the change of nodes, 4. Take the eigenvectors of the largest k eigenvalues: v ,v ,...,v which results in the change of formula. Newton interpola- 2 k Output Projection matrix: V = (v ,v ,...,v ) 1 2 k tion algorithm can overcome this shortcoming (Varsamis and Karampetakis 2012). Newton interpolation algorithm function is determined by independent variables and dependent variables. The first-order function,second-order 3.1 INI algorithm function,kth-order function are shown in formulas (2), (3) and (4) respectively. Newton interpolation formula defined There will be some outliers or even missed value in the in formula (5) is deducted by formulas (2), (3) and (4). data because of all kinds of reasons. If we use the New- ton interpolation algorithm directly, large error would be f (x )− f (x ) j i brought. The INI algorithm is seen as formula (6). f [x , x ]= (2) i j x − x j i f (x)−f (x) newton f (x) ≤ (I) newton f (x) f (x)= ⎨ (6) k f (x)−f (x) newton f (x)= x p >(II) f [x , x ]− f [x , x ] i i j k i j 1 f (x) f [x , x , x ]= (3) i j k x − x k i f[x , … , x , x ]− f[x , x , … , x ] 0 k−2 k 0 1 k−1 f[x , x , … , x ]= 0 1 k f (x) = f (x ) (7) x − x k k−1 k i=1 (4) f (x) is the interpolation results of formula (5). f (x) is newton f (x)= f (x )+ f [x , x ](x − x )+…+ f [x , x ](x − x )…(x − x ) 0 0 1 0 0 n 0 n−1 the expectation of k samples. x ( i = 1, 2, 3,… , k − 1 ) i s + f [x, x , … , x ](x − x )…(x − x )= N (x)+ R (x) 0 n 0 n n n independent variables from the nearest k hours. The data of (5) kth hour is the dependent variable. The p is the probability Whereas, the instability of the interpolation results is 1 and it is a constant of . affected by the high power of formula (5 ) during the process If the formula I is true, then the samples of (k - 1) is of interpolation (Hlbach 1979). used as the independent variables and the kth sample is used as the dependent variable, getting the f (x). The newton 2.3 PCA algorithm expectation value of ( k − 1 ) samples is as the input to f (x). If the formula II is true, then f(x) is calculated newton Mapping high dimensional data to low dimensional data by formula (7).The data before interpolation is named as through PCA algorithm (Nejadkoorki and Baroutian 2011). D . The data after the interpolation is named of D. orinal It seems convenient to introduce the following steps of PCA algorithm (Table 1). 3.2 Multiple stations data and the dimension reduction 3 NNP‑BPNN model In this part, we mainly introduce the data composition from multiple stations, Neighbor-PCA algorithm and the In order to improve the data utilization and prediction training process of BPNN model. accuracy, the NNP-BPNN model is proposed for pollut- ant concentration prediction. In this paper, the INI algo- 3.2.1 Multiple stations data and AW method rithm has been utilized to handle missing data. The AW method is calculated based on geographical location. The Due to different distance, neighbor stations have a different Neighbor-PCA algorithm is used to deal with the data of impact on the central station. The air pollutant concentration the neighbor stations. 1 3 H. Zhao et al. of all monitoring stations is recalculated by AW method et al. 2017). The relevancy, error and RMSE between the defined as formula (8 ). prediction and actual value are respectively defined as for - mulas (16), (17) and (18). ij k = 1 − ∑ V =(v , v , … , v ) ij (8) (13) n 1 2 m ij j=1 =[D , V] (14) The k is the weight between the ith station and the jth sta- ij √ N = m + q + a (15) tion. The d is the distance between the ith station and the jth ij station. It is calculated based on the latitude and longitude and defined in formula (9 ). (90 − C)× PI (90 − F)× PI B × PI E × PI (9) d = 637814 × ACOS 1 − sin × cos − sin × cos ij 180 180 180 180 The C is the latitude of the ith station, and The B is the (x − x ̄)(y − y ̄) longitude of the ith station, The F is the latitude of the jth R = ∑ ∑ (16) 2 2 (x − x ̄) (y − y ̄) station, and The E is the longitude of the jth station. The closer distance is, the greater the impact on the cen- tral station. The k from formula (8) is the minus function. ij error =(y − y ̂) (17) The neighbor stations that are closer from central station can get greater weight from k , enhancing the ability of BPNN ∑ ij n error model. (18) RMSE = The D is the data of ith station and its composition is seen as formula (10) . The D is the collection of data of k stations and is seen as formula (11). is the data of D i i 4 Modeling methods ×k where n ranges from 0 to n and is seen as formula (12). ij is the data of D × k where n ranges from 1 to n. neighbor i ij In order to verify the effectiveness of the NNP-BP model, D =(F , F ) i pollutant neighbor (10) three models have been established, including the BPNN model, the NN-BPNN model with INI algorithm and AW D =[D , … , D , … , D ] (11) 1 i n method and the NNP-BPNN model with INI algorithm, AW =[D × 1, D × k , … , D × k , … , D × k ] method and Neighbor-PCA algorithm. i 0 1 i1 i ij n in (12) Modeling is performed in Matlab7.0, and data D is as input Compared with the traditional methods, we have consid- to the models. We choose traingda function as gradient descent ered the neighbor stations based on geographic information, function and set 100 as interval. Learning rate is 0.1. We set which improves the accuracy of BPNN model. 1000 as the largest number of training and set 0.0001 as target error. The input nodes is 15, the number of output nodes is 1, and a is 8. So, the hidden layer node is 12 according to the 3.2.2 Dimension reduction process formula (15). One half of data has been applied to train and another half of data has been treated to evaluate the model. If the dimension is too high or the sample size is too small, The learning rate (lr) valued between 0 and 1, which deter- the BPNN would be unable to learn the general rule (Meng mines the step size for updating in each iteration. If lr is too and Meng 2010). In this paper, the PCA algorithm has been used to handle data . neighbor The extracted eigenvector from data using PCA neighbor Table 2 The steps of BPNN model algorithm is named as V defined as formula (13). defined in formula (14) has been used as the input to BPNN. A three- Input Original data D orinal layer structure BPNN mentioned above has been applied Steps 1. Calculate the number of hidden layer nodes in this paper. The number of nodes of the hidden layer are of BPNN according to formula (15) and set determined by the formula (15). The m is the number of parameters nodes of the input layer and the q is the number of nodes of 2. Training the BPNN model the output layer. The a is the constant from 1 to 10 (Wang Output The sum of error and RMSE 1 3 The pollutant concentration prediction model of NNP‑BPNN based on the INI algorithm, AW method… Table 3 The steps of NN-BPNN Input Original data D orinal model Steps 1. The missing value is calculated to obtain the data D according to formulas (6) and (7). 2. Calculate the distance d of each station according to the formula (9) ij 3. Calculate the weight k according to d and formula (10) ij ij 4. We can get the data according to the k and formula (11) ij 5. The data is used as the input for BPNN model 6. Calculate the number of hidden layer nodes of BPNN according to formula (15) and set other parameters 7. Training the NN-BPNN model Output The sum of error and RMSE Table 4 The steps of NNP- Input Original data D orinal BPNN model Steps 1. Use PCA algorithm to deal with data to obtain data V neighbor 2. Get the input to BPNN according to the data V and formulas (13) and (14) 3. Calculate the number of hidden layer nodes of BPNN according to for- mula (15) and set other parameters 4. Training the NN-BPNN model Output The sum of error and RMSE big, it is easy to oscillate, and if lr is too small, it converges 5.1 Data description too slowly. The gradient descent method can not only solve the problem, to a certain extent, but also make results closer to the 16 stations data in one city, including pollutant concen- global minimum (Table 2). tration and meteorological parameters of 254 days from July 30, 2015 to April 10, 2016, have been employed in 4.1 Modeling of BPNN model this paper. The basic information is consist of stations name, time, longitude, latitude. Pollutants are consist of CO, SO , O , NO, NO , NOX, PM .5 , P M 0 , VOC. The 2 3 2 2 1 4.2 Modeling of NN‑BPNN model meteorological parameters are consist of relative humidity, wind direction, wind speed, temperature, pressure, visibil- Assuming data of the ith hour is missed, we take the data ity, total rainfall. Among them, the longitude and latitude named x from the (i−4)th to the (i−2)th and the (i−1)th data are used to calculate the distance. A continuous 24 h data is named as y, which is used to calculate Newton interpola- is lost among 16 stations. tion polynomial. The value of f ( x ̄ ) can be calculated. newton The ranged from 0 to 1. The missing data can be obtained by 5.2 Results and discussions formulas (6) and (7). Different weights have been assigned for each station to get the data (Table 3). Three models have been established, including BPNN model, NN-BPNN model, NNP-BPNN model, which the 4.3 Modeling of NNP‑BPNN model same data and parameters are used. Here, we discuss the effectiveness of NN-BPNN model with INI algorithm and Throughout this section, The PCA algorithm is performed to the AW method. Then we discuss the validity of NNP-BPNN deal with data to obtain data V and train NNP-BPNN model with INI algorithm, AW method and Neighbor-PCA neighbor model as follows (Table 4). algorithm. The RMES has been regarded as the evaluation criterion shown in formula (18). 5 Experiments and analysis 5.2.1 The effectiveness of NN‑BPNN model Here, we mainly talk about data in experiment and discus- Figure 3 is the comparison of RMES between BPNN model sions about results. and NN-BPNN model. In the neural network, the local 1 3 H. Zhao et al. BPNN Model NN_BPNN Model Table 5 The value of relevancy 0.15 ID BPNN NN-BPNN NNP-BPNN 0.1 1 0.75 0.77 0.79 0.05 2 0.61 0.70 0.81 3 0.73 0.76 0.82 123456789 10 4 0.70 0.76 0.78 5 0.71 0.61 0.76 Fig. 3 The RMES between BPNN model and NN-BPNN model 6 0.77 0.76 0.81 7 0.78 0.78 0.80 8 0.81 0.71 0.80 BPNN Model NN_BPNN Model NNP_BPNN Model 0.12 9 0.73 0.77 0.79 0.1 0.08 10 0.68 0.78 0.80 0.06 0.04 0.02 12345678 910 and it took 47 s to implement NNP-BPNN model. The time used by NNP-BPNN model is reduced by 72% than that of Fig. 4 The RMES of three models NN-BPNN model. Empirically, the NNP-BPNN model of statistical forecasting method based on historical monitoring minimum value of the error function is not the global mini- data is simple and practical. mum. Ten times of experiments have been performed to prove that the NN-BPNN model has a smaller value of sum of RMES than that of BPNN model, as shown in Fig. 5. 6 Conclusions The RMSE of three models is shown in Fig. 4. Meanwhile, experimental results on the concentration of PM show that 2.5 In this paper, we study the problem that how to handle the sum of RMES of NNP-BPNN model is relatively mini- missing data and how to utilize historical data effectively mal, as shown in Fig. 5. The relevancy of the three models for prediction with BPNN. There are some achievements is shown in Table 5. in this paper. Firstly, the INI algorithm is adopted to deal In ten times of experiments,the area of RMSE produced with missing data, so as to avoid the waste of data. Sec- by BPNN model is the largest among that of three mod- ondly, the AW method not only enriches the experimen- els, the average RMSE of NN-BPNN model is reduced by tal data, improving the utilization rate of data, but also 18% than that of BPNN model, the average RMSE of NNP- reduces the RMSE. The average RMSE of NN-BPNN BPNN model is reduced by 24% than that of BPNN model model is reduced by 18% compared with BPNN model. and the average RMSE of NNP-BPNN model is reduced by Thirdly, the NNP-BPNN model with INI algorithm, AW 7% than that of NN-BPNN model. It took 20 s to implement method and Neighbor-PCA algorithm has improved the BPNN model, it took 170 s to implement NN-BPNN model prediction accuracy and relevancy. The average RMSE of NNP-BPNN model is reduced by 7% compared with NN- BPNN Model BPNN model. The average RMSE of NNP-BPNN model is reduced by 24% compared with BPNN model. NN_BPNN Model There are still some limitation in this research. More NNP_BPNN Model methods of dimension reduction should be studied in the future work. The research of NNP-BPNN model to find 10 2 out the global minimum error. Multiple machine learning models should be explored in air pollutant concentration 9 3 prediction area. Open Access This article is distributed under the terms of the Crea- 8 4 tive Commons Attribution 4.0 International License (http://creat iveco mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- tion, and reproduction in any medium, provided you give appropriate 7 5 credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Fig. 5 The RMES of three models 1 3 The pollutant concentration prediction model of NNP‑BPNN based on the INI algorithm, AW method… networks and intelligent systems. 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Published: Jun 1, 2018
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