HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.38450
Denoising the Temperature Data Using Wavelet Transform
Samsul Ariffin Abdul Karim
Fundamental and Applied Sciences Department Universiti Teknologi PETRONAS
Bandar Seri Iskandar
31750 Tronoh, Perak Darul Ridzuan, Malaysia [email protected]
Mohd Tahir Ismail
Pusat Pengajian Sains Matematik, Universiti Sains Malaysia 11800 Minden, Pulau Pinang, Malaysia
[email protected] Mohammad Khatim Hasan
Jabatan Komputeran Industri, Universiti Kebangsaan Malaysia 43600 UKM Bangi, Selangor, Malaysia
[email protected] Jumat Sulaiman
Program Matematik dengan Ekonomi, Universiti Malaysia Sabah Beg Berkunci 2073, 88999 Kota Kinabalu, Sabah, Malaysia.
Copyright © 2013 Samsul Ariffin Abdul Karim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Wavelets transform are effectively used in data compression and denoising such as in signal and image compression and denoising. One of the advantages of wavelets method is there exist fast algorithm in order to use wavelet for various applications.
In this paper we will apply Discrete Wavelet Transform (DWT) to denoise the temperature data using symlet 16 with 32 corresponding filters (low-pass and high-pass).
We apply various thresholding approaches e.g., Heuristic SURE, SURE, Minimax and Fixed-Form method. We utilized temperature data in Kuala Lumpur from January 1948 until July 2010. We also discuss the advantages of wavelet as compared with Fast Fourier Transform (FFT). Several numerical results will be presented by using Matlab.
Keywords: symlet; Wavelet Transform; temperature data; data denoising; thresholding
I. INTRODUCTION
Wavelets are relatively new in pure and applied mathematics field of research; they have, with respect to theory and applications, strong relations with Fourier Transform.
Wavelets have emerged in the last twenty years as a synthesis of ideas from the fields such as electrical engineering, statistics, physics, computer science, economy, finance and mathematics. Wavelet transform have beautiful and deep mathematical properties, making them well-adapted tool for a wide range of functional spaces, or equivalently, for very different types of data. On the other hand, they can be implemented via fast algorithms, essential to convert their mathematical efficiency into truly practical tool ([1], [2], and [3]).
Temperature, water level or closed market prices over a period of time are examples of time series data. In its descriptive form, time series data may be defined as a set of data collected or arranged in a sequence of order over a successive equal increment of time. Noise is an unwanted modulation of the carrier whose presence interferes with the detection of the desired signal [4]. Noise is extraneous information in a signal that can be filtered out via the computation of averaging and detailing coefficients in the wavelet transformation. In fact many statistical phenomena have wavelet structure. Often small bursts of high frequency wavelets are followed by lower frequency waves or vice versa.
The theory of wavelets reconstruction helps to localize and identify such accumulations of small waves and helps thus to better understand reason for these phenomena. In addition, wavelet theory is different from Fourier analysis and spectral theory since it is based on a local frequency representation [5].
The studies of climate changes using wavelet analysis have received much consideration. Reference [17] discussed the applications of wavelets to detect the climate signal. In this paper, the authors used Morlet wavelets (continuous wavelet transform CWT) and concluded that WT provides understanding of the importance of local versus global climate signals via time-frequency localization of WT. While, reference [18] utilized WT for visual exploration of climate variability changes at seasonally averaged northern hemisphere winter and northern hemisphere summer. In addition, [19] explained in details of wavelet analysis and its applications in atmospheric and oceanic.
In this paper we will utilize wavelet transform which is symlet 16 in order to denoise the temperature data. Firstly we decomposed the original time series (in our cases it is temperature data) and later we study the characteristics of the temperature data then we
apply thresholding method and finally we denoised the original signal with suitable choices of thresholding. The results indicate that symlet 16 gives us good denoising signal with SURE method and Minimax method of thresholding gives higher SNR values.
The remainder of the paper is organized as follows. A brief discussion about wavelet analysis is provided in Section 2. An application of DWT using temperature data is presented in Section 3. Some concluding remarks are made in the final section.
II. WAVELET ANALYSIS
Wavelet analysis is a mathematical model that transforms the original signal (especially with time domain) into a different domain for analysis and processing. This model is very suitable with the non-stationary data, i.e. mean and autocorrelation of the signal are not constant over time. There exist various choices of wavelet basis functions such as Haar, Daubechies, Symlet, Meyer, biorthogonal wavelet and etc. Basically, we define wavelet directly from its counterpart that is scaling function also known as father wavelet and wavelet function also known as mother wavelet ([1]; [2]; [6]). Following [13], suppose that there exists a function φ( )t ∈L2( )R such that the family of functions
( )t =2j2
(
2jt−k)
, j,k∈Z.φ (1) is an orthonormal basis.
We can define the wavelet series as follow:
( )x ( )x ( )x
f
j k
jk jk k
k
k
∑∑
∑
∞=
+
=
0
0 β ψ
ϕ
α (2) where αk,βjk are coefficients defined Eq (4), and
{ }
ψ jk ,k∈Ζ is a basis for Wj. The relation in (2) is called a multiresolution expansion of f. To turn (2) into wavelet expansion we use the following expression( )x j
(
jx k)
jk =22ψ 2 −
ψ , j,k∈Ζ. (3) Basically the function ϕjk( )x and ψjk( )x are called the scaling function (father wavelet) and the mother wavelet respectively. Meanwhile
( ) ( )
∫
= f x k xdx
k ϕ0
α ,βjk =
∫
f( ) ( )
xψjk xdx (4) Where αk are called approximation/coarser coefficients and βjk are called detail coefficients. Since the development of multiresolution analysis by [8], [2] has constructed various wavelets function e.g. Daubechies, symlets and coiflets wavelets.All these wavelet functions do not have a specific formula as the Haar wavelet function but they differ from others only by translation along the time axis and the changes of
scale. However, in this paper we make use of the symlet 16 wavelet (16 filter coefficient). to show the power of DWT. We apply the symlet 16 to denoise time series data up to five level of decomposition.
Figure 1 shows the example of symlet 16 scaling function and its corresponding wavelet function. Symlet 16 has 15 vanishing moments (VM). Due to the shape of the temperature data, symlet 16 is highly suitable to denoise the original time series. Figure 2 shows the original time series (temperature data) and the number of data are 751 (monthly data started from January 1948 and end July 2010). Meanwhile Figure 3 shows the wavelet decomposition for original temperature data. We can reconstruct the signal by adding details from level 1 until level 5 and approximation at level 5. Overall level 5 of the approximations (a5) and detail (d5) indicates an increasing trend in the temperature. This is due to the global warming because of the Green House effect which is getting worse every year. This is why we need to sustain our environment.
Figure 1. Symlet 16 scaling function and wavelet function
Figure 2. Original series
III. DENOISING TEMPERATURE DATA
In order to denoise the data, we need to apply thresholding method. Basically in the literature there exist various thresholding methods such as hard thresholding, soft thresholding, Garrote thresholding, firm thresholding etc. In statistics literature, thresholding is called as shrinkage approach. Refer to [12] for more detailed on various thresholding methods. In data denoising, one of the main objectives is to cut-off the data at certain values and then reconstruct the original signal or image with an acceptable Signal-to Noise-Ratio (SNR) and Root Mean Square Error (RMSE). We can use either hard thresholding method or soft thresholding. Below, we list algorithm that could be used to perform data denosing for any one dimension (1D) problem:
0 100 200 300 400 500 600 700 800
25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 30
(a) (b) Figure 3: Wavelet decomposition (a) detail (b) approximation using symlet 16
A. Denoising algorithm
1. Input noise signal with length N (the temperature data already consist noise- but if the data has no noise we can just add Gaussian white noise)
2. Apply DWT by using symlet 16 to the data (perform wavelet transform of the data)
3. We find thresholding values. Here we may choose hard or soft and global value or by-level dependent value. In practice the best way to find the threshold value is by doing experiment and experienced from the user.
4. Keep only values that are non-zero or significant obtained from the transformation in step 3. Then apply wavelet denoising to the original signal with the threshold values from Step 3.
5. Finally, perform the inverse discrete wavelet transform (IDWT) for the data in step 4. This denoising step produces an approximate of the original data (an example can be seen in Figure 5).
Please refer to the works by [9] and [10], [11] and [12], [5], [15] and [16], [21], [6] and [13] for more detailed on denoising techniques.
In this paper, we apply various thresholding methods i.e. SURE, Heuristic SURE, Fixed-Form and Minimax. Fixed-Form threshold are the classical thresholding approach (VisuShrink threshold) which is given by εj =σ 2lnN (for j=1,...J and all level have same value).
Detail inspection of Figure 4 indicates that thresholding methods, Heuristics SURE and Minimax give us better result in denoising the temperature data. This is because they produced a better smooth series as compared to the other two methods, Fixed Form Threshold and Minimax. This results are in line with the statistical results analysis given in Table 1 below. The SURE method has the highest SNR with 21.37 and Fixed-Form is the lowest one with 17.83.
Table 1: Statistical Results for Denoising using various thresholding method THRESHOLDING RMSE SNR
Minimax 0.0109 19.63
Fixed-Form 0.0165 17.83
Heuristic SURE 0.013 18.86
SURE 0.0073 21.37
As we mentioned in section 1, fast fourier transform (FFT) only tells us about the frequency of the event but not the time that the event occurred. This is because the basis of FFT is cosines and sines which is smooth in nature. Furthermore from the original signal the nature of the data is fractal like shape. So definitely FFT will not achieve best result as compared to Symlet 16 wavelets. Since in FFT we smooth all the data where all of the spike and anomaly in the data will be removed. On the other hand, this will not happen when we denoise using DWT. Since DWT will preserve all the spike and etc.. In order to justify this matter, Figure 5 exhibits the periodogram or the plot of the estimation power spectrum versus frequency. It appears from Figure 5 that fast fourier transform (FFT) is not sufficient to capture the behavior of the series since it represents the data as a function of position. Moreover, a plot of the FFT (Figure 5) of this signals show nothing particular interesting. But with wavelets we can do a lot of analysis as compared with FFT.
Figure 4. Denoised original signal using DWT with various thresholding approach:
from top to bottom: Original time series, Heuristics SURE, SURE, Fixed Form threshold and Minimax
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0
0.5 1 1.5 2 2.5 3 3.5 4
4.5x 108 Periodogram
Figure 5: The periodogram or the Power Spectrum Estimation
CONCLUSIONS
In this paper we have discussed the application of wavelet transform with symlet 16 to denoising temperature data. We apply 4 types of thresholding methods. From the results we conclude that SURE and Minimax give us better results in terms of SNR and RMSE values as compared to Heuristic SURE and Fixed-form method. Future work will be focusing on applying another thresholding method such as BayesShrink by [17] and do the numerical comparison between all thresholding methods and various choices of wavelet filters. This method could be applied to the geographical and geological problems. We will report the related findings in our forthcoming papers.
ACKNOWLEDGMENT
The authors would like to acknowledge Universiti Teknologi PETRONAS (UTP) for the financial support received in the form of a research grant: Short Term Internal Research Funding (STIRF) No. 76/10.11 and 35/2012.
REFERENCES
[1] C. K. Chui, An Introduction to Wavelets. Academic Press, New York, 1992.
[2] I. Daubechies, “Ten Lectures on Wavelets,” CBMS-NSF Reg. Con. Ser. Appl. Math., Society for Industrial Applied Maths (SIAM), Philadelphia, PA, vol. 61, 1992.
[3] S. Mallat, A Wavelet Tour of Signal Processing, San Diego: Academic Press, 1998.
[4] H. L. Resnikoff and R. O. Wells, Wavelets Analysis: A Scalable Structure of Information. Springer-Verlag, New York., 1998.
[5] W. Hardle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation and Statistical Applications, Lecture Notes in Statistics, vol. 129, Springer, New York, 1998.
[6] P. J. Van Fleet, Discrete Wavelet Transformation: An Elementary Approach with Application, New Jersey, John Wiley & Sons, 2008.
[7] Y. Meyer, Wavelet and Operators, Cambridge University, Cambridge, 1992.
[8] S. Mallat, “A theory of multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Learn, vol. 11(9), pp. 674-693, 1989.
[9] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelets shrinkage,”
Biometrika, vol. 81, pp. 425-455, 1994.
[10] D. L. Donoho and I. M. Johnstone, “Adapting to unknown smootnees via wavelet shrinkage, “ Journal of American Statistical Society, vol. 90, pp. 1200-1224, 1995.
[11] A. Antoniadis, “Wavelets in statistics: A review,” Journal of Italian Statistical Society, vol. 6, pp 1-34, 1997.
[12] A. Antoniadis, “Wavelets methods in statistics: some recent developments and their applications,” Statistics Surveys, vol. 1, pp. 16-55, 2007.
[13] S. A. A. Karim, M. T. Ismail and B. A. Karim, “Denoising non-stationary time series using Daubechies wavelets,” Proceedings of Seminar Kebangsaan Matematik
& Masyarakat (SKMM), UMT Terengganu 13-14 February 2008, Grand Continental Hotel, Kuala Terengganu (In CD), 2008.
[14] S. A. A. Karim and M. T. Ismail, “Wavelet method in statistics,” Proceedings of the Sixteenth National Symposium on Mathematical Sciences, 3-5 Jun 2008 at Hotel Renaissance, Kota Bharu, Malaysia, 2008.
[15] S.A.A. Karim, B.A. Karim, M.T. Ismail, M.K Hasan, and J. Sulaiman, Applications of Wavelet Method in Stock Exchange Problem. Journal of Applied Sciences, 11 (8): 1331-1335.
[16] Karim, S.A.A, Ismail, M.T., Karim , B.A., Hasan, M.K. and Sulaiman, J.
Compression KLCI Time Series Data Using Wavelet Transform. World Engineering Congress 2010, 2nd – 5th August 2010b, Kuching, Sarawak, Malaysia Conference on Engineering and Technology Education. In CD, 2010.
[17] S.G. Chang, B. Yu, and M. Vetterli, Adaptive wavelet thresholding for Image denoising and compression. IEEE Transactions on Image Processing, vol. 9(9), pp.
1532-1546., 2000.
[18] H. Janicke, M. Bottinger, U. Mikolajewicz, and G. Scheuermann, Visual Exploration of Climate Variability Changes Using Wavelet Analysis. IEEE Transactions on Visualization and Computer Graphics. 15(6), pp 1384-1391, 2009.
[19] K.-M. Lau, and H. Weng, Climate Signal Detection Using Wavelet Trasnform:
How to Make a Time Series Sing. Bulletin of the American Meterological Society, 76,pp. 2391-2402, 1995.
[20] C. Torrence, and G.P. Compo, A Practical Guide to Wavelet Analysis. Bulletin of the American Meterological Society 79,pp. 61-78, 1998.
[21] G. Strang, and T. Nguyen, Wavelets and Filter Banks. Wellesly-Cambridge Press, Massachussets, Wellesley, 1996.
Received: August 5, 2013
