INTERNATIONAL JOURNAL OF ENVIRONMENTAL SCIENCES Volume 2, No 1, 2011 Copyright 2010 All rights reserved Integrated Publishing Association Research article ISSN 0976 4402 Prediction of daily air pollution using wavelet decomposition and Adaptive- Network-based fuzzy inference system School of Basic and Applied Science, Guru Gobind Singh Indrapastha University, Delhi, India rashmib22@gmail.com ABSTRACT In this paper, forecasting of daily air pollution is presented using the method of wavelet decomposition and Adaptive-Network-Based Fuzzy Infer system (ANFIS). Daily averaged value of air pollution parameters Carbon Monoxide (CO) for the year 2007 is used for decomposition using daubechies 8 wavelet at level 3. To obtain the acceptable accuracy of forecast, the wavelet coefficients are predicted using Adaptive-Network-Based Fuzzy Infer system (ANFIS). ANFIS serve as a basis for constructing a set of fuzzy if-then rules with appropriate membership functions to generate the stipulated input-output pairs. On the basis of these predicted values the final forecasting is prepared.the result is compared with actual future data and relative performance of this procedure is investigated. Keywords: Time Series prediction, Air pollution forecasting, ANFIS, Wavelet. 1. Introduction The aim of this work is to construct a forecasting model for the daily averaged future pollution that would be applicable for use by the pollution control authority. The main objective of this paper is to study the prediction of Indian daily averaged air pollution using wavelets and Adaptive-Network-Based fuzzy Inference (ANFIS). Several work using wavelet and other techniques has been done for prediction of time series. A wavelet based prediction procedure is introduced for investigation of market efficiency in future market for oil prices(yousefi et.al, 2005). The use of support vector machine and wavelet decomposition was modeled on the observed data of air pollution and actual meteorological parameters, like wind, temperature, humidity and pressure(stanislaw and Garanty, 2007). We propose here the forecasting model based on Adaptive-Network-Based Fuzzy Infer system (ANFIS) and the wavelet decomposition of the air pollution time series. The forecasting procedure makes use of daily-averaged air pollution parameters CO for the year 2007 measured at ITO- Crossing, of Delhi, India. The forecasting system based directly on these measurements was not able to provide the acceptable accuracy of prediction. To solve this problem we decompose the measured signals into wavelets (Can et.al, 2005 and Daubechies 1992). The prediction is performed for the wavelet coefficients (the detailed coefficients up to some level and the approximated coarse signal corresponding to the last level) in the original resolution. On the basis of these predicted values the reconstruction of the real value of the forecasted pollution for all considered pollutants is performed by simply summing up the predicted decomposition signals. The paper proceeds as follows, in section-ii of this paper we discuss the time series of daily averaged Carbon monoxide measured at ITO-Crossing. Concept of wavelet analysis required for prediction will be discussed in section-iii while elements of neuro fuzzy architecture needed will be descried in section -IV. A prediction procedure using wavelets and neuro fuzzy is discussed in section-v whilie its application to time series of Received on July, 2011 Published on September 2011 185
Carbon Monoxide data is presented in section-vi. Section-VII includes discussion and concluding remarks. 2. Wavelet Analysis Wavelet analysis is a refinement of Fourier analysis [Daubechies 1992, Georgiou and Kumar 1994, Hu and Nitta 1996, Jang 1997, Kumar and Foufoula 1997 Mallat 1998, Siddiqi 2003, 2004 Can et al. 2004, 2005 Furati et al. 2006 Manchanda et al. 2007] which has been used for prediction of time series of oil, meteorological pollution wind speed etc. (Yousefi et.al., 2005, Stanislaw and Garanty, 2007). In this section some important Concepts relevant to our work have been described. The underlying mathematical structure for wavelet bases of a function space is a multi-scale decomposition of a signal, known as multi resolution or multi scale analysis. It is called the heart of wavelet analysis. Let L 2 (R) be the space of all signals with finite energy. A family {Vj} of subspaces of L 2 (R) is called a multi resolution analysis of this space if (i) intersection of all V j, j = 1, 2, 3,... be non-empty, that is (ii)this family is dense in L 2 (R), that is, = L 2 R (iii) f(x) V j if and only if f(2x) V j + 1 (iv) V 1 V 2... V j V j + 1 V φ j j (v) There is a function preferably with compact support of such that translates φ(x k) k Z, span a space V 0. A finer space V j is spanned by the integer translates of the scaled functions for the space V j and we have scaling equation φ( x) = a φ(2x 1) (3.1) k with appropriate coefficient a k, k ε Z. φ is called a scaling function or farther wavelet. The mother wavelet ψ is obtained by building linear combinations of φ. Further more φ and ψ should be orthogonal, that is, < φ(. k) ψ(. l) > = 0, l, k ε Z. (3.2) These two conditions given by (3.1) and (3.2) leads to conditions on coefficients b k which characterize a mother wavelet as a linear combination of the scaled and dilated father wavelets φ : ψ(x) = Σ k ε Z b k φ(2x k) (3.3) Haar, Daubechies and Coefmann are some well known wavelets. It is quite clear that in the higher case the scaled, translated and normalized versions of ψ are denoted by ψ j, k (t) = 2 j/2 ψ(2 j x k) (3.4) With orthogonal wavelet ψ the set {ψ j, k j, k ε Z} is an orthogonal wavelet basis. A function f can be represented as f = Σ Σ c j,k ψ j,k (t), c j,k = <f,ψ j,k > (3.5) j ε Z k ε Z 186
The Discrete Wavelet Transform (DWT) corresponds to the mapping f c j,k. DWT provides a mechanism to represent a data or time series f in terms of coefficients that are associated with particular scales [Stanislaw and Garanty (2007), Siddiqi et al.(2003)] and therefore is regarded as a family of effective instrument for signal analysis. The decomposition of a given signal f into different scales of resolution is obtained by the application of the DWT to f. In real application, we only use a small number of levels j in our decomposition (for instance j = 4 corresponds to a fairly good level wavelet decomposition of f). The first step of DWT corresponds to the mapping f to its wavelet coefficients and from these coefficients two components are received namely a smooth version, named approximation and a second component that corresponds to the deviations or the so-called details of the signal. A decomposition of f into a low frequency part a, and a high frequency part d, is represented by f = a 1 + d 1. The same procedure is performed on a 1 in order to obtain a decomposition in finer scales : a 1 = a 2 + d 2. A recursive decomposition for the low frequency parts follows the directions that are illustrated in the following diagram. f... a 1... a 2... a 3... a n \ \ \ \ d 1 d 2 d 3 d 4... d n The resulting low frequency parts a 1, a 2,... a N are approximations of f, and the high frequency parts d 1, d 2,... d n contain the details of f. This diagram illustrates a wavelet decomposition into N levels and corresponds to f = d 1 + d 2 + d 3 +... + d N 1 + d N + a N. (3.6) In practical applications, such a decomposition is obtained by using a specific wavelet. Several families of wavelets have proven to be especially useful in various applications. They differ with respect to orthogonality, smoothness and other related properties such as vanishing moments or size of the support. 3. Adaptive-Network-Based Fuzzy Inference (ANFIS) Adaptive-Network-based Fuzzy Inference, or simply ANFIS, serve as a basis for constructing a set of fuzzy if-then rules with appropriate membership functions to generate the stipulated input-output pairs. The ANFIS Structure is obtained by embedding the fuzzy inference system into the framework of adaptive network. ANFIS, standing for Adaptive- Network-based Fuzzy Inference a class of adaptive networks which are functionally equivalent to fuzzy inference systems. We have used matlab for training the system using anfis edit to train the wavelet coefficient(jang and Gulley, 1995). ANFIS Architecture According to the Takagi and Sugeno s type (Takagi and Sugeno 1985) the fuzzy inference system has two inputs x & y and one output z. Rule 1 : If x is A 1 & y is B 1 then f 1 = p 1 x + q 1 y + r 1 Rule 2 : If x is A 2 & y is B 2 then f 2 = p 2 x + q 2 y + r 2 187
Figure1: Graphical representation of ANFIS Layer 1 Every node i in this layer is a square node with a node function O i = µa i (x). (4.1) Where x is the input to node i, and A i is the linguistic label (small, large etc.) associated with this node function. In other words, O i 1 is the membership function of A i and it specifies the degree to which the given x satisfies the quantifier A i. Usually we choose µa i (x) to be boil-shaped with maximum equal to 1 & minimum equal to 0, such as the generalization ball function 1 µ A i(x) = 1 + 2 bi (4.2) x O i a i or gaussian function µ A i(x) = exp x c i a i 2 (4.3) where {a i, b i, c i } is the parameter set. As the values of these parameters change, the bell shaped functions vary accordingly thus exhibiting various forms of membership functions on linguistic label A i. In fact, any continuous and piecewise differentiable functions, such as commonly used trapezoidal or triangular-shaped membership functions, are also qualified candidates for node functions in this layer. Parameter in this layer are referred as premise parameter. Layer 2 Every node in this layer is a circle node labeled which multiplies the incoming signals and send the product out. For instance W i = µa i (x) µb i (y). i = 1, 2. (4.4) Each node output represents the firing strength of a rule. Layer 3 Every node in this layer is a circle node labelled N. The i th node calculates the ratio of the ith rule s firing strength to sum of all rule s firing strengths W i =, i = 1, 2. (4.5) 188
For convenience, outputs of this layer will be called normalized firing strength. Layer 4 Every node i in this layer is a square node with a node function O i 4 = W i f i = W i (p i x + q i y + n i ) (4.6) where W i is the output of layer 3 and (p i, q i, n i ) is the parameter set. Parameters in this layer will be referred to as consequent parameters. Layer 5 The single node in this layer is a circle node labeled that computes the overall output as the summation of all incoming signals i.e., O 1 5 = Overall output = i W i f i = S iw i f i S i W i (4.7) Thus, we have constructed an adaptive network which is functionally equivalent to a type 3 fuzzy inference system. For type-1 fuzzy inference systems, the extension is quite straight forward & the type-1 ANFIS is shown in fig. 2, where the output of each rule is induced jointly by the output & the firing strength. 4. Time Series Data Used in Study The present study uses measurement of daily-averaged air pollution parameters Carbon Monoxide(CO) for the year 2007. Data are collected from air pollution monitoring stations ITO-Crossing, of Delhi India. Matlab Time Series for the data is depicted in Fig. 2. Out of 365 data samples, 346 have been used to develop prediction model and 19 sample have been used for the test model. 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 50 100 150 200 250 300 350 400 Figure 2: Time Series of daily averaged air pollution parameter Carbon Monoxide for 2007. 4.1 Predictability Method The present study is based the comparison of two results obtained by applying ANFIS directly and applying ANFIS on the Wavelet coefficients. Daubechies wavelets Db8 have been applied in the decomposition. All signals (the first three levels of wavelet coefficients from d 1 to d 3 and the coarse approximation a 3 ) are illustrated in the original resolution. We observe the substantial difference of variability of the signals at different levels. The higher is the wavelet level, the lower variation of the coefficients and easier prediction of them. Our main idea is to substrate the prediction task of the original time series of high variability by 189
the prediction of its wavelet coefficients on different levels of lower variability s, and then using equation (3.6) for final prediction of the Carbn Monoxide(CO) at any time instant n. Since most of the wavelet coefficients are of lower variability we expect the increase of the total prediction accuracy. The wavelet tool available in Matlab is used for the process of wavelet decomposition of the time series representing daily averaged Carbonmonoxide(CO) measured at ITO-Crossing Delhi for year 2007. This step involves several different families of wavelets and a detailed comparison of their performance. In our case, The Daubechies wavelets of order 8 are performed. A three level wavelet decomposition of the given time series X N = f is performed f = a 3 + d 3 + d 2 + d 1 The smooth part of f is stored in a 3, and details on different levels are captured by d 1, d 2, d 3. Consequently a decomposition of the time series in three different scales is obtained. Fig.3 illustrates the decomposition of the original signals. 10000 Approximation A3 5000 0 0 50 100 150 200 250 300 350 400 Detail D1 5000 0-5000 0 50 100 150 200 250 300 350 400 Detail D2 5000 0-5000 0 50 100 150 200 250 300 350 400 Detail D3 2000 0-2000 0 50 100 150 200 250 300 350 400 Figure 3: Wavelet decomposition of daily averaged CO for 2007. The forecasting procedure methodology explained in section 4 is used to predict the next value. The basic idea is to use the wavelet transforms and predict the data by neuro fuzzy for individual coefficients of wavelet transform represented by a3, d 1, d 2, d 3. The input to the ANFIS architecture to predict the wavelet coefficients is explained in fig. 4. 190
Figure 4: Mechanism for forecasting Procedure The total predicted Carbon Monoxide(CO) at any instant (i) is given by F (i) =f1 (i) +f2 (i) +f3 (i) +f4 (i) (5.1) 5. Results of numerical experiment (A) Result Obtained by Direct method: The forecasting of daily averaged Carbon Monoxide for the year 2007 is done directly using Adaptive-Network-based Fuzzy Inference (ANFIS). First 346(approx. 95%) data are used for training and next 19(approx. 5%) data are used to test the predicted values. Average training error was recorded 1002.8752 and average testing error was recorded 1744.7927. Figure 5: actual(blue) and predicted(red) Carbon Monoxide for training data. Figure 6: Actual(blue) and predicted(red) Carbon Monoxide for testing data. 191
The relative normalized error is given as d y = x100(%) d The relative normalized error obtained for the daily averaged Carbon monoxide(d) and the predicted Carbon Monoxide data (y ) is given in the figure below Figure 7: Relative normalized error by direct method. (B) Result Obtained Using Wavelet Decomposition: Now the numerical experiment of predicting the future air pollution parameter Carbon Monooxide have been performed for the wavelet coefficients of decomposed data. Three level decomposition of Daubechies wavelets Db8 have been applied to the daily averaged Carbon Monoxide(CO) for predicting A3, D1, D2, D3. The results of learning and testing are done on the basis of normalized error. The main experiment have been performed using neuro fuzzy predictor, the introductory experiments were not encouraging. This was the main reason, we proposed the additional step of decomposing the data into the wavelets and used neuro fuzzy for predicting the wavelet coefficient of each decomposition level. Figure 8: Actual(blue) and Predicted(red) Approximate coefficient (A3) for training data. Figure 9: Actual(blue) and Predicted(red) detail coefficient(d1) for training data. 192
Figure 10: Actual(blue) and Predicted(red) detail coefficients(d2) for training data. Figure 11: Actual(blue) and Predicted(red) detail coefficient(d3) for training data. Table 1: Average Training Error for wavelet coefficients Coefficients of wavelets Average Training Error A3 38.99 D1 251.39 D2 228.70 D3 51.57 The trained predicted output is obtained from the decomposed wavelet coefficients by simple summation represented by S(n). S(n) = D 1 + D 2 + D 3 + A 3 The predicted trained signal is represented in the Fig.10. and Fig.11 respectively. The 5% data which was left for testing is applied to the trained system. The actual testing coefficients obtained from wavelet decomposition are given to the ANFIS system individually. Table 2. Average Testing Error for wavelet coefficients Coefficients of wavelets Average Testing Error A3 32.37 D1 763.14 D2 448.07 D3 153.66 193
Figure 12: Actual(blue) and Predicted(red) detail coefficient(a3) for testing data. Figure 13: Actual(blue) and Predicted(red) detail coefficient(d1) for testing data. Figure 14: Actual(blue) and Predicted(red) detail coefficient(d2) for testing data. Figure 15: Actual(blue) and Predicted(red) detail coefficient(d3) for testing data. 194
7000 6000 5000 4000 3000 2000 1000 0 0 2 4 6 8 10 12 14 16 18 20 Figure 16: Actual(blue) output and predicted(red) Time Series of daily average CO for next 19 days. Figure 17: Relative normalized error between actual output and predicted output for next 19 days in percentage. 5.1 Conclusions In this work we have predicted the air pollution parameter Carbon Monoxide of next nineteen days which gives satisfactory results on employing wavelet decomposition with neuro fuzzy. The wavelet transform has given the strength of generalization to neural network and specialization to Tagaki Sugeno inference fuzzy logic for training the non stationary data and predicting the output. The average normalized error for testing is reduced 5 times by using wavelet decomposition. The variability of data was unable to be trained by using only neuro fuzzy. Therefore, the wavelet decomposition and coefficient prediction plays a vital role in the analysis of air pollution data. Acknowledgements We are very thankful to Council of Science and Industrial Research for their financial support. We are also thankful to Central Pollution Control Board for providing us air pollution data. 6. References 1. Can Z., Aslan Z. and Oguz O. (2004). One dimensional wavelet Real analysis of gravity waves, The Arabian journal of scince and engineering, 29, Issue 2c, pp 33-42. 195
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