Environ Sci Pollut Res (2012) 19:256–268 DOI 10.1007/s11356-011-0554-9
RESEARCH ARTICLE
Wavelet transform-based artificial neural networks (WT-ANN) in PM10 pollution level estimation, based on circular variables Maryam Shekarrizfard & A. Karimi-Jashni & K. Hadad
Received: 4 April 2011 / Accepted: 14 June 2011 / Published online: 7 July 2011 # Springer-Verlag 2011
Abstract Introduction In this paper, a novel method in the estimation and prediction of PM10 is introduced using wavelet transform-based artificial neural networks (WT-ANN). Discussion First, the application of wavelet transform, selected for its temporal shift properties and multiresolution analysis characteristics enabling it to reduce disturbing perturbations in input training set data, is presented. Afterward, the circular statistical indices which are used in this method are formally introduced in order to investigate the relation between PM10 levels and circular meteorological variables. Then, the results of the simulation of PM10 based on WT-ANN by use of MATLAB software are discussed. The results of the above-mentioned simulation show an enhanced accuracy and speed in PM10 estimation/prediction and a high degree of robustness compared with traditional ANN models. Keywords Wavelet transform . Circular variable . PM10 . Pollution . Neural network
Responsible editor: Euripides Stephanou M. Shekarrizfard (*) : A. Karimi-Jashni Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran e-mail:
[email protected] K. Hadad School of Mechanical Engineering, Shiraz, Iran
1 Introduction PM10 has been identified as one of the major air pollutants responsible for various health problems. In fact, recent studies have shown that PM10 is clearly associated with respiratory diseases (Pires et al. 2008; Ostro et al. 1999a, b; Schwartz and Dockery 1992; Dockery et al. 1992; Dockery and Pope 1994; Pope et al. 1995). Many attempts have been made to model PM10 through the use of meteorological data (Papanastasiou et al. 2007; Perez and Reyes 2002; Hooyberghs et al. 2005). Chepil et al. (1962) indicated that wind erosion, one of the major sources of PM10 pollution, could be determined as a function of the third power of wind speed. Considering the fact that wind is distinct from other linear meteorological variables (such as temperature and humidity), it should be investigated through a different computation analysis. Wind direction is a circular variable which can be represented as a point on the circumference of a circle. There have been numerous publications on meteorologicalbased PM10 prediction, e.g., traditional artificial neural network (ANN) modeling and principal component analysis (PCA)-based ANN modeling (Lu et al. 2002; Sousa et al. 2007; Kukkonen et al. 2003; Corani 2005). Perez and Reyes (2002) developed a neural network-based model that uses measured and forecasted values of meteorological variables as input in order to predict the maximum PM10 concentration for a given day in Santiago, Chile. Chaloulakou et al. (2003a) and Hooyberghs et al. (2005) applied NN models to predict daily PM10 concentrations in Athens, Greece. Moreover, Grivas and Chaloulakou (2006) and Hrust et al. (2009) evaluated the accuracy of NN models to make reliable predictions of hourly PM10 concentrations. Their results showed that NN models provide adequate solutions to the issue of PM10 estimation. Paschalidou et al. (2010)
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applied two types of artificial neural network models to predict hourly PM10 levels in Cyprus based on available meteorological data. The models were developed using the multilayer perceptron and the radial basis function techniques. Their results demonstrated that the artificial NN model provides accurate predictions about PM10 concentration levels. A detailed comparison of the pros and cons of traditional ANN modeling and PCA-based ANN modeling has already been done by Hadad et al. (2008) and is beyond the scope of this paper. In brief, as the number and the complexity of data sets increases in traditional ANN, its efficacy in diagnosing and identifying the higher values of PM10 is reduced; consequently, the main goal of PCA-based ANN is to process the input data set in a way that reduces the workload of ANN. One of the primary complexities of data sets in PM10 analysis is the presence of fast transients. PCA-based methods do improve ANN performance; however, when dealing with fast transients (high perturbations), ANN estimations become less reliable (Hadad et al. 2011). Wavelet transfer enables the decomposition of signals into compactly supported oscillating components (Meyer 1994; Safavi and Romagnoli 1997). The removal of noise from data series before being used in ANN modeling is of great importance (Hadad et al. 2011; Guegan 2008; Guegan and Hoummiya 2005). To this end, we present the wavelet transform-based artificial neural network (WT-ANN) application, which helps improve the speed and accuracy of predictions and increases reliability when dealing with highly perturbed PM10 data. For this purpose, circular statistical indices such as mean wind direction and circular and linear–circular correlations were used in order to investigate the relationship between wind features and PM10 levels. The results obtained from these statistical analyses were studied to select the appropriate input data for the WT-ANN model. In the following, the theoretical basis of wavelet analysis is presented. Then, the circular statistical indices and their associations are identified. Finally, the results of a real case study analyzing PM10 in Iran are presented.
2 Theory and method The process of detecting patterns in PM10 data generally becomes more complicated with large data. As a result, the predictions from raw meteorological data are only accurate in the short term. Enhancing the accuracy of predictions made with ANN requires the processing and de-noising of the raw data. Therefore, the observations of the present study were de-noised so as to achieve a higher degree of precision. Though several de-noising techniques have been developed for noisy time series (Farmer and Sidorovich
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1988; Guegan and Tchernig 2001; Hadad et al. 2011), the simplest one is the moving average approach proposed by Grassberger and Schreiber (1991). Alternatively, the wavelet method has also been applied extensively in various cases (Meyer 1994; Safavi and Romagnoli 1997; Hadad et al. 2011). The following is an explanation of the application of PM10 concentration estimation. 2.1 Wavelet method Windowed Fourier transform (WFT) is one way of achieving time and frequency localization when the time and frequency resolutions are both fixed. However, WFT is inadequate in the case of temporal transient signals; instead, WT must be used for time and frequency localization. WT uses short windows at higher frequencies and long windows at lower frequencies. This alternate use of short and long windows enables WT to “zoom in” on singularities, thus making it practical for the analysis of transient signals (Hadad et al. 2011). Furthermore, this method facilitates the decomposition of signals into compactly supported oscillating components. This property allows WT to detect discontinuities or sharp changes in noisy signals. Assuming Yt is the set of observation signals of a specific process, e.g., meteorological data, and by defining the coefficients as follows, Eqs. 1 to 7 can be used to separate the noise from signals using wavelet analysis. Z Wj;K ¼ 2
j=2
Z
Yt y 2 t K dt ¼ j
Yt y j;K dt
ð1Þ
where j is the scale parameter, K is the translation parameter, t is time, and yj,k is known as the Wavelet function. Wj,K can be determined for any N-sample signal Y1,…,YN. Considering Wj,K and for any level of signal approximation, Yt is defined as: X Dj ð2Þ AJ ¼ j>J
where Dj is the details of the signal Yt and is defined as: X Wj;K :y j;K ðtÞ ð3Þ Dj ¼ K2Z
y j,k is called the scaling function, which is defined as: y j;K ðtÞ ¼ 2j=2 y 2j t K ð4Þ Expression 2 can be rewritten as: XX Wj;K :2j=2 y 2j t K AJ ¼
ð5Þ
j>J K2Z
This last expression constitutes the basic notation of the wavelet decomposition of a function. Starting from the
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approximation of the function at, for example, a resolution of j=0, one may rewrite Eq. 5 as: AJ ¼ A0 þ
1 X þ1 X
Wj;K :2j=2 y 2j t K ;
K2Z
ð6Þ
j¼0 K¼1
Cr ¼ 1 R
X
W0;K :f0;K ;
K2Z
ð7Þ
K
where f0,K is a scaled function as defined by Safavi and Romagnoli (1997). In order to model dust pollution levels (PM10) using WTANN, the correlation between circular meteorological data and dust levels needs to be investigated first. For this purpose, different circular statistical indices have been presented below. Afterward, the theoretical foundation underlying neural networks, which are used in this study for the short-term modeling of dust levels on the basis of meteorological parameters, is explained. 2.2 Circular statistical indices and variables
R n
Wind mean direction is estimated by converting angular measurements to points on a circle and computing the resultant vector of unit vectors indicated by the data points. The mean direction is the direction of this resultant vector and is a criterion for expressing the concentration of the circular data. The following equations are used to estimate the mean direction of wind:
R ranges from 0 to 1. A value of zero means that the data are highly concentrated around one direction and a value of 1 denotes a wide dispersion of wind data. 2.2.2 Circular–linear and circular–circular correlations Another statistical index which can be used in the investigation of the correlation between circular and noncircular variables is called the circular–linear correlation. Mardia (1976) proposed the following equation to determine this type of correlation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ r2 2r r r rxc xc xs cs xs r¼ 2 1 rcs
C¼
S¼
i¼1
Xn i¼1
ð8Þ
cos ai
ð9Þ
sin ai
ð10Þ
where αi is the hourly wind direction; additionally, the wind mean direction (a) is determined as: s 8 > arctan ; if c 0 > > < cs a ¼ arctan þ p; if c < 0 > > c > : Undefined: if R ¼ 0
ð14Þ
rxc ¼ corrðx; cos aÞ
ð15Þ
rxs ¼ corrðx; sin aÞ
ð16Þ
rcs ¼ corrðcos a; sin aÞ
ð17Þ
where x is the linear variable and α is the angle of the wind speed. Also, the correlation of two circular variables has been defined by Jammalamadaka and Sarma (1988) and Jammalamadaka and SenGupta (2001) as follows:
where C and S are two parameters calculated as follows: Xn
ð13Þ
where rxc, rxs, and rcs are parameters which are defined as:
2.2.1 Circular mean and variance
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ C2 þ S 2
ð12Þ
where R is determined as follows: R¼
where A0 ¼
Also, a criterion can be obtained for circular dispersion, known as circular variance (Cr):
ð11Þ
n P
rc;n ¼
i¼1
sinðai aÞ sin bi b
2 sin ðai aÞ2 sin bi b
ð18Þ
where αi and βi are the circular variables and a and b are the circular mean directions, calculated using Eq. 11. The closer rc,n is to zero, and the more independent two variables are; on the other hand, if this value is close to one, it can be understood that there is a correlation between our two variables. The above indices are utilized in order to investigate the correlation of PM10 levels with wind direction and wind speed. This is where effective variables come into play: to improve PM10 prediction through the use of the neural network method. Following, the overall structure of neural network model is discussed.
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2.3 Neural networks, a brief description ANN are massive parallel-distributed information processing systems that have certain performance characteristics resembling those of the biological neural network of the human brain. ANN has been developed as a general, concrete representation of the mathematical models utilized by human cognition and neural biology. ANN, with its ability to derive meaning and discern nonlinear relationships from complicated or imprecise data, has proven to be an effective alternative to the more conventional statistical techniques (Schalkoff 1997). In such networks, the available data set is partitioned into three separate groups, one corresponding to training and the other two corresponding to the validation and testing of the model. The purpose of training is to determine the set of connection weights and nodal thresholds that enable the ANN to estimate outputs that are sufficiently close to target values. The training process is halted when no appreciable change is observed in the values associated with the connection weights or some termination criterion is satisfied. The overall performance of the network can be determined by computing the percentage error between the predicted and desired values. A number of data records were selected for the sake of input for the neural network model according to the results obtained from correlational analysis. The proposed neural network model is a two-layer model with 30 neurons in the first layer and a solitary neuron in the second (often referred to as the “hidden layer”). The transformation functions of the first and second layers were sigmoid and linear, respectively. Six input data series, including lag-one PM10 daily records (PM10 at time t−1) and five daily meteorological data series (relative humidity, temperature, wind speed, wind direction, and rainfall at time t) provided the required input for the model. The target data record is the predicted PM10 concentration at time t. Pre-processing is desirable for both input and target data series before starting the learning process; this includes, on the one hand, data series standardizing and, on the other hand, PCA (Hadad et al. 2008; Aminghafari et al. 2006; Camdevyren et al. 2005; Johnson and Wichern 1982; Ouyang 2005). To this end, the data were scaled according to the mean and standard deviation of the series. Additionally, principle component analysis was employed to reduce the size of the data to be entered. In PCA, the input data are ranked on the basis of the change they bring about in the external (target) series, with those causing greater change being ranked higher. Then, the data with the lowest rank (i.e., which have the smallest impact on the target series) are removed from the analysis altogether. In this study, the cutoff level of change for removing an entry series was set at 2%. Two algorithms, i.e., the Levenberg–Marquardt (LM) and the scaled conjugate gradient, were used for the
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learning process. Previous studies (Hadad et al. 2008) have shown that in the presence of highly perturbed data arising from a nonlinear system, the LM algorithm is the most effective when it comes to the learning process in comparison with the scaled conjugate gradient algorithm. The performance function which is used in the network learning phase is the sum of squared errors. The existing data were broken into three subsets. The first set includes the learning data which are used to estimate the weights in the neural network, while the second one is used for validation and the third for testing. 2.4 Study area The Shiraz metropolitan area is located in the south of Iran (52°32′ eastern and 29°37′ northern) with a population of 1,749,926 people. The environmental protection agency of Shiraz was the organization charged with collecting the air pollution data through two air quality monitoring stations in central and southern Shiraz. In this study, the effect of meteorological parameters, e.g., wind speed and direction, on PM10 levels has been investigated. The daily meteorological data from a climatological station located in Shiraz were used to apply in the proposed methodology. The available data were checked with regard to their conformance to established observing practices and their internal consistency. For the sake of the present study, PM10 data from various traffic sites in the main urban center of Shiraz (the downtown district) were collected and used. PM10 monitoring was introduced to Shiraz in 2003. The statistics recorded since then have been a source of serious concern as PM10 levels have regularly exceeded critical limit values (designated as 150 μg/m3). PM10 mass concentrations (measured in micrograms per cubic meter) were recorded using Beta Ray attenuation gauging method (APDA370 Ambient Dust Monitor, HORIBA, Japan). The data consisted of daily mean values gathered during a 1-year time span between Jan 2005 and Dec 2005. Before application, the data set was made uniform by, first, removing days with errors or missing data (approximately 2.5% of the data set) and, then, standardizing the values between −1 and +1 in order to improve network sensitivity.
3 Results and discussions 3.1 Monthly PM10 and meteorological data variations Meteorological data have been frequently used to predict the level of dust in wind erosion equations and models. One of the better-known wind erosion equations was proposed by Skidmore and Woodruff (1968), which uses rainfall, evapo-
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ration, temperature, and wind speed as the main criteria in dust modeling. In this study, several meteorological parameters, namely wind speed and direction, monthly temperature, monthly rainfall, and air moisture, were selected for the purpose of dust modeling. To investigate the possibility of using circular variables in modeling (e.g., wind direction), the circular means of the monthly wind directions were calculated using Eqs. 8 to 13. The results of these analyses have been presented in Table 1. In general, PM10 levels fluctuated proportional to wind speed, denoting a positive correlation between these two variables. As can be seen from Table 1, PM10 levels increased during months 2, 3, and 4 due to decreasing rainfall and relative humidity levels. The highest PM10 levels were observed in months 6, 7, and 8. It is notable that during these 3 months, the lowest rainfall and/or humidity values and the highest wind speeds of the year were recorded, which is in accordance with the general observation mentioned previously. Looking at these results, we can come to the conclusion that there seems to be a relationship between monthly changes of PM10 and meteorological parameters such as temperature, moisture, rainfall, and wind speed and direction. To verify these findings, statistical analyses such as linear and circular correlation and principal component analysis were run to calculate the levels of significance of the relationships among these parameters. Following this, the neural network model was utilized for the purpose of predicting PM10 concentration levels.
was higher during the dry months of the year. In this study, statistical methods were also used in the correlational analysis of PM10 and meteorological data series during the dry season. In this regard, a 10-mm rainfall depth was considered as the cutoff point in the classification of dry and non-dry months. In the following section, the results of these analyses have been presented. 3.3 Wind direction—PM10 correlation Rose diagrams are one of the most effective ways of depicting the relationship between wind and PM10 data series. As such, a number of wind rose diagrams were developed to illustrate the various levels of dust quantiles (70%, 80%, and 90%). In the making of these rose diagrams, the usage of wind data was restricted to days when the dust level was above the selected quantile values (e.g., 150 μg/m3 is the restriction level for the 70% quantile). The values of quantiles at the different levels of probability determined for this study (70%, 80%, and 90%) were 150, 172.75, and 216.95 μg/m3, respectively. Figure 1a shows a rose diagram of daily wind directions as well as dominant wind directions. Figure 1b is the rose diagram for wind data with a dust level of 150 μg/m3 and below (quantile 71% and below). As can be understood from the diagram, the dust pollution levels for the eastern winds were the lowest of the researched winds. Figures 1c, d demonstrate dust pollution levels pertaining to quantiles above 70 and 80%. In contrast to the previous group, the southeastern winds had the highest level of frequency in both of these cases. Moreover, according to Fig. 1e, the highest frequency of wind also belonged to the southeastern winds when the level of
3.2 Correlation analysis of PM10 levels and meteorological data series Firstly, statistical analyses were applied to all of the available data. As shown in Table 1, PM10 concentration
Table 1 Mean monthly variation of meteorological data and the PM10 level Variable
Month Ann
Sample size PM10 level (μg/m3)
Mean SD Temp (°C) Mean SD Wind speed (m/s) Mean SD Wind direction (degrees) circ mean (ά) circ disp (1−ά) Rain (mm) Humidity (%)
Mean SD
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
353 31 28 19 30 31 30 31 31 30 31 135 86 86 104 114 175 179 239 244 137 144 71 27 40 43 24 105 84 206 226 19 23 18 5.9 7.9 12.9 18.1 21.5 27.1 29.7 28.7 24.7 18.7 2.2 2.1 3.3 1.8 2.6 2.4 2.1 1.8 1.1 2.2 2.1 9.8 7.7 11.8 11.0 13.0 11.7 10.5 10.8 10.7 9.1 7.3 3.7 4.6 5.4 4.1 5.6 4.9 2.9 3.0 2.7 2.1 2.0 105.9 105.3 110.3 126.9 130.2 112.2 107.4 94.1 99.7 104.4 94.5 0.4 0.4 326.5 108.1 42.0 64.6 8.6 10.8
0.4 36.5 58.2 14.4
0.4 31.5 51.5 15.5
SD standard deviation, circ mean circular mean, circ disp circular dispersion
0.4 2.3 38.6 8.8
0.4 0.1 29.0 7.1
0.4 0.0 21.0 3.4
0.5 4.0 26.5 6.4
0.4 0.0 25.5 4.2
0.4 0.0 28.7 4.1
Nov 30 67 21 12.2 2.9 7.2 3.4 92.3
Dec 31 45 28 9.5 2.1 6.5 3.2 94.1
0.4 0.3 0.3 0.0 111.5 32.5 33.3 63.3 64.5 3.9 18.9 6.0
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Fig. 1 a–e Wind rose for different levels of dust quantiles
pollution is increased to 90% of the dust quantiles. As a result, it seems that the level of dust pollution generally increases with winds coming from the southeast (120°), and there is a correlation between wind direction and the amount of air pollution. Linear–circular correlation was used to show the relationship between monthly wind direction and monthly PM10 levels using Eqs. 14 to 17. The linear–circular correlation value obtained from the calculations is 0.62, which shows that dust level is correlated with wind direction at a level of significance of 99%. Furthermore, the levels of PM10 are presented for different wind directions in Fig. 2a. According to Fig. 2a, the highest dust levels were recorded for winds coming
from the southeast (120°). Also, a new formula was developed to estimate the mean directional pollution index based on the wind frequency of each direction. Directional Pollution Index ðDPIÞ ¼ directional wind frequency dust pollution level
ð19Þ
In Eq. 19, directional wind frequency is formally defined as the number of winds observed in a specific direction divided by the total number of observed values (the length
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a
b 40.000
80
30
60
35.000
DPI (µg/ g/ m3)
100
330
300
60
40
30.000 20 103.86
25.000
113.9 90
270
20.000
120.45 141.87
15.000
134.71
10.000
185.4 120
240
5.000 150
210
G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20
0.000
180
Average PM10 level by daily wind direction
Wind direction
Fig. 2 a, b DPI index results for different wind directions
of each petal in the rose diagram). The estimated values of DPI are presented in Fig. 2b. This figure shows that the highest levels of pollution are related to the winds blowing from the southeastern direction. G1 to G20 in Fig. 2b is pertaining to wind directions 0° to 360°, respectively. 3.4 Relation between PM10 and linear meteorological data Spearman’s correlation coefficients were used in order to investigate the correlation between wind speed and PM10 concentration levels. In this case, the value of correlation coefficients was equal to −0.79, which was found to be significant at the level of 99%. Although wind speed has universally been considered an important factor in dust modeling in previous studies, other parameters such as moisture, rainfall, and relative moisture are of great importance as well (Giri et al. 2008; Berlyand 1991; Chaloulakou et al. 2003a, b, c; Papanastasiou and Melas 2008). Therefore, the correlation among these parameters and dust levels was also investigated in the present study. For this purpose, the Pearson linear correlation coefficient was used. The results appear in Table 2. Furthermore, correlation coefficients are provided in the case of using monthly averages of meteorological data and dust levels.
For this purpose, the 3-day, 7-day, and monthly moving average of data were used to estimate the values of linear– linear, linear–circular, and circular–circular correlations (Table 2). Regarding the results depicted in Table 2, it can be concluded that there is a significant correlation between the different levels of dust and the determined meteorological parameters, especially with respect to the long-term average data series. As a result, using meteorological data to predict dust levels through on ANN analysis is justifiable. 3.5 The results of WT-ANN modeling A data set was considered for ANN simulation using WTANN and traditional ANN. The models make use of meteorological data gathered from previous days as an input vector. The second model uses WT to de-noise the observed data. In the sections below, the results of traditional ANN analysis (for chaotic observations) are presented, followed by a discussion of the results of wavelet-based ANN analysis. To compare the results, three different statistical indices, namely the index of agreement (IA), fractional bias (FB), and Rsquared, were used. The properties of these statistical indices have already been discussed by Willmott (1981). The IA and
Table 2 Correlation values between PM10 level and meteorological parameters Variables
Monthly PM10 Temperature (°C) Humidity % Rain (mm) Wind speed (m/s) Wind direction (radian)
Meteorological parameters Monthly PM10
Temperature (°C)
Humidity %
Rain (mm)
Wind speed (m/s)
Wind direction (radian)
1
0.90 1
−0.87 −0.94 1
−0.62 −0.72 0.82 1
0.43 0.35 −0.45 −0.50 1
−0.12 −0.13 −0.04 −0.20 0.71 1
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263
1
10
0
10
Mean Squared Error (mse)
FB indices are determined based on observation (O) and prediction (P) series as follows:
Train Validation Test Best
ðP O Þ2 IA ¼ 1 2 abs P O þ abs P O
ð20Þ
-1
10
FB ¼
-2
10
PO 0:5 P þ O
ð21Þ
The best model has a lower FB value and a higher IA value. The ANN and WT-ANN models are compared using Eqs. 20 and 21 and the value of R-squared in the subsequent section.
-3
10
-4
10
0
2
4
6
8
10
12
14
16
Epoch
3.6 Traditional ANN modeling (chaotic observations)
Fig. 3 Mean squared errors for training, validation, and test process for chaotic data
Fig. 4 Regression analysis values as a result of network training, validation, and testing for chaotic data
The meteorological data were divided into three sets: 70% for training, 15% for validation, and 15% for testing. After 11 epochs, the training algorithm was stopped. In Fig. 3, the training process and error values for the validation, testing,
Training: R=0.96953
Validation: R=0.77533 0.4
0.5
Y=T Fit Data
0.4
Output~=0.76*Target+0.039
Output~=0.93*Target+0.0084
Y=T Fit Data
0.3
0.2
0.1
0
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
Target Test: R=0.69255 Y=T Fit Data
0.4
0.3
0.2
0.1
0 0
0.1
0.2
0.3 Target
0
0.1
0.2 Target
0.5
Output~=0.34*Target+0.072
0.3
0.4
0.5
0.3
0.4
264
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3.7 PM10 simulation using the WT-ANN model The de-noised data sets were used for ANN modeling. Once more, the data were divided into three groups in the same manner: 70% for training, 15% for validation, and 15% for testing. The error values for the validation, testing, and training processes were also estimated to be RMSE= 0.15, which is evidently less than the corresponding value in the noisy data analysis (Fig. 3). The regression analysis values obtained through wavelet analysis pertaining to network training, validation, and testing for the denoised data are shown in Fig. 6. Once more, it can be seen that the R values for the processes of training, testing, and validation compare well, and again, there was no occurrence of over-fitting. A comparison of Figs. 4 and 6 reveals that the values of R correlation coefficients regarding the results of training, testing, and validation increased after using wavelet analysis. Another comparison, this time of the observed and predicted values of PM10, was done, the results appearing in Fig. 7. This latter comparison demonstrates that the values were not underestimated. Furthermore, the values of IA and FB were 0.96 and −0.0067, respectively. These values are higher than the corresponding values obtained from the simple ANN model. The aforementioned R value, IA, and FB values prove that the wavelet approach possesses great potential for improving the predictive accuracy of PM10 levels.
Fig. 5 Observation values and prediction levels of PM10 as resulted from ANN model based on chaotic data
and training processes are illustrated. According to Fig. 3, the resulted error values for the training, testing, and validation processes compare well and over-fitting did not occur. In addition, the errors on the training, validation, and test sets and the performance of the trained network were evaluated by performing a regression analysis between the network response and the corresponding targets. Figure 4 shows the results of network training, validation, and testing obtained through observation and prediction comparison. In general, the R values indicate that the networks have reliable performance and can predict PM10 concentration levels with a high degree of accuracy. Therefore, there is no need for further training; in other words, the network’s structure in its current state is suitable for modeling daily PM10 levels. Moving on, the forecasted levels of PM10 concentration and the observed values have been compared in Fig. 5. As illustrated in this figure, PM10 levels were underestimated in some cases. It should be mentioned that the values of IA and FB were determined using Eqs. 20 and 21 and stood at 0.92 and −0.0105, respectively.
3.8 Model application: prediction of a typical dust-storm episode The suitability of the developed WT-ANN model for operational use (especially in terms of its ability to predict extreme concentrations) was evaluated according to daily PM10 concentration time series during 50 days of the year 2005 in Shiraz. This extra-analysis on the proposed model’s predictive ability is presented in Fig. 8. The figure illustrates a more-than-acceptable agreement between the predicted and observed PM10 values, though there were a few cases where the highest measured values were substantially overestimated by the proposed model. The proposed WT-ANN model also showed good accuracy in terms of predicting extreme concentrations. The values of the correlation coefficients (R), FB, and IA are all shown in the figures. The results of various other models developed worldwide as well as the results obtained during the course of this paper clearly indicate that the WT-ANN model outperformed both the ANN and linear regression models in terms of overall precision in making predictions regarding PM10 concentration levels. Table 3 provides a brief comparison of the WT-ANN model and its alternatives. Although the predictive ability of the standard
Environ Sci Pollut Res (2012) 19:256–268 Validation: R=0.98135
Training: R=0.96894
0.4
1 Y=T Fit Data
0.9 0.8
Output~=1.1*Target+-0.0073
Output~=0.93*Target+0.0085
Fig. 6 Regression analysis values as a result of network training, validation, and testing for de-noised data using wavelet analysis
265
0.7 0.6 0.5 0.4 0.3
Y=T Fit Data
0.3
0.2
0.1
0.2 0.1 0
0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Target
0
1
0.1
0.2 Target
0.3
0.4
Test: R=0.84612
Output~=1.1*Target+-0.0065
0.4
Y=T Fit Data
0.3
0.2
0.1
0 0
0.1
0.2 Target
artificial neural network model is very high and gives a highly accurate prediction with respect to regression
0.4
models (Pao 2008, Gardner and Dorling 2000, Chaloulakou et al. 2003c), it can still be improved upon through the
260
Standardized values of dust pollution ( g/ m3)
Fig. 7 Observation values and prediction levels of PM10 by ANN model based on de-noised data from wavelet analysis
0.3
Observed Predicted
240
220
200
180
160
140
120 0
50
100
150
200
Time (days)
250
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350
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Environ Sci Pollut Res (2012) 19:256–268
Fig. 8 Extra-analysis on the proposed WT-ANN model to show the predictive ability (a typical dust-storm episode from June 1 to July 20)
1400 FB=-4.4e-4 IA=0.9834 R=0.51
Simulated PM10 ( g/m3)
1200
Observed Simulated
1000 800 600 400 200 0 0
10
20
30
40
50
60
70
Time (days)
use of wavelet analysis as it is very robust when dealing with noisy or unexpected data. Regarding regression and ANN model analysis (Grivas and Chaloulakou 2006; Paschalidou et al. 2010), it can be concluded that regression-based predicted values of PM 10 are less accurate with respect to the alternative models (i.e., ANN). This is mainly due to the underlying assumptions of linearity, normality, etc. in regression-based models which may be violated when using real data. The artificial neural network model is not founded on these assumptions, and recognizing the pattern formed by the data is done only on the basis of a large amount of data. In contrast to regression models, which provide easy and simple explanations on the estimated parameters, ANN-based models have been described as a “black box” in previous studies done in the field. Although WT-ANN does reduce the computational load in comparison with the ANN model, training time and the determination of the optimal number of neurons can take quite some time and can be very complicated and exhaustive in both of these models. The regression method, by contrast, is simple to apply using available statistical computer software.
4 Summary and conclusion In this study, it was shown that PM10 levels can be estimated using both circular variables (e.g., wind direction) and non-circular variables (e.g., wind direction, wind speed, and relative humidity). For this purpose, a number of statistical indices such as mean direction, circular variance, linear–circular correlation, and circular–circular correlation were used. The results of the application of circular statistical indices demonstrated a meaningful correlation between the direction and levels (density) of PM10 and wind speed. Furthermore, a wavelet-based neural network model was developed in order to model the relationship between PM10 levels and meteorological data according to several parameters, such as wind speed, wind direction, moisture, and temperature. This wavelet model was also used to reduce the noise inherent in the meteorological data. According to the results of this study, it was found that the use of wavelet-based ANN led to a noticeable increase in the accuracy of predictions about PM10 levels in comparison with conventional ANN methods. In general, it can be concluded that the proposed
Table 3 Comparing different PM10 modeling methods Model
Properties Predictive ability
Underlying assumptions
Computational effort Applicability
Regression analysis Moderate
Linearity, normality Low
ANN
highly accurate-
No
High
WT-ANN
Highly accurate-robust No with noisy
High
Explanation on the parameters
Very simple to apply Easy and simple explanation on the estimated parameters Complicated and Black box exhaustive Complicated and Black box exhaustive
Environ Sci Pollut Res (2012) 19:256–268
WT-ANN model is an effective method for PM10 level prediction.
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