High Frequency Time Series Analysis using Wavelets

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1 POSTER 015, PRAGUE MAY 14 1 High Frequency Time Series Analysis using Wavelets Jaroslav SCHÜRRER Masaryk Institute of Advanced Studies, Czech Technical University, Kolejní 637/a, Praha, Czech Republic jaroslav.schurrer@muvs.cvut.cz Abstract. This paper proposes a new method how to analyze high frequency time series with Wavelets analysis. Our motivation is to introduce a new method, which can be used during decomposition of tick-by-tick stock data for example intraday stock data. Each tick is one logical unit of information. High frequency time series are represented by a discrete set of values and usually include tens of thousands of samples per day. In this situation we are dealing with the necessity to reduce input data for further analysis. This paper covers description of high frequency time series, wavelet analysis and method definition for model preparation for analysis Keywords Wavelets, Wavelet decomposition, High-Frequency Data, Tick-by-tick Data, WHA method. 1. Preface The fundamentals of Wavelets theory lie in the field of pure mathematics and applications of Wavelet analysis are found in signal processing, scientific calculation and image processing. Wavelets are used in wide range of scientific fields for their time scale properties. These properties are suitable for decomposition of financial time series where we need to analyze their multiscale relationships. Let s imagine a large number of investors who participate in the stock market and make decisions over different time scales. These participants can be seen as a diverse group consisting of intraday traders, portfolio managers, bank brokers, corporations, government institutions etc. Each such subgroup uses a different time horizon for their decision making therefore relationships between variables will vary over different time scales. Increased availability of highfrequency financial data has introduced new challenges for their analysis. Both financial time series and financial theory include an important element of uncertainty. In the past the adoption of this element of uncertainty has led to extensive development of statistical methods. Financial theory embraces various definitions for asset volatility, stock return series and so on thus by using those accordingly we have at our disposal many different statistical models. This statistical approach to time series spans from simple linear models like ARMA, unit root nonstationarity over Conditional Heteroscedastic Models like GARCH and all the way nonlinear models for example TAR, STAR or neural networks. Common feature of statistical models is the sequence: theoretical model and then application of this model to empirical data plus analysis of the model s data. Our approach is little bit different, because we focus on empirical data and their direct transformation to new entity, which is consequently analyzed. We think about time series as a function and we are looking for their analytical expression. Also we are dealing with high-frequency data, which brings problems with computational complexity of massive data files.. High-Frequency Data In finance time series are considered as a sequence of observed values associated with time (ordered and usually independent). The simplest case is represented by stationary time series where the data are time-independent. In most real situations we deal with non-stationary time series where mean, variance and covariance are variable in time. High- Frequency data are observations taken at fine time intervals. Usually we mean observations on daily or finer time scale. Regularly spaced time intervals are generally taken as an assumption for many models. In limit case orders initiated by market participants arrive at random times. Today s high frequency financial data reflect transactions that take place in matter of seconds or milliseconds, due to massive usage of computer trading. As stated by (TSAY 010) we can characterize following properties of high-frequency data: Unequally spaced time intervals Discrete value prices Existence of daily periodic pattern Multiple transactions within a single second For subsequent definitions we define signal in following manner and we identify time series with signal. A signal is defined as a sequence of numbers x n, n Z satisfying ( Z x n < It is clear that a signal has to be bounded if we want to think about convergent series. This means that there exists some number M > 0 such that x n M, n Z.

2 J. SCHÜRRER, HIGH FREQUENCY TIME SERIES ANALYSIS USING WAVELETS If this sequence satisfies condition x n. < ( then we speak about a signal with finite energy. This property is crucial for our method, as without it the proposed method cannot be applied. By the energy of signal we mean the sum of the values squares: E 0 = x. + x..,, x (.. We also need to define maximum time interval between neighbor s values in signal as: max x n + 1 x n, n Z Later on we will use this value in preprocessing phase. Last defined signal characteristics is the distance between minimum and maximum values in signal x n. Minimum value in signal x ;<( = min x n, n Z and as well as maximum value x ;?0 = max x n, n Z. So distance between min and max value in input signal is x ;?0 x ;<(. 3. Discrete Wavelet Analysis Wavelets are wave-like functions that can be translated and scaled. Usually the main property of a Wavelet is compact support and finite energy. Wavelet transform represents an analyzed signal as translated and scaled Wavelets. This means that we can analyze features on different scales independently. Because the energy of the signal is finite, not all values of decomposition are needed to exactly reconstruct the original signal, if we use Wavelet satisfying admissibility condition. In these cases Continuous Wavelet Transformation is redundant and Discrete Wavelet Transformation (DWT) is sufficient. The Discrete Wavelet transform transforms input signal into time and frequency domain. It means that a time series is decomposed into high and low frequency components. These two parts holds completely different characteristics of the original time series. Low frequency part matches longer time intervals and high frequency part matches shorter time intervals. Low frequency Wavelet component includes coarse structure long-term trends. High frequency Wavelet part captures singularities and discontinuities. Because Wavelets are often used in signal processing domain, we will present our theory in this spirit. We also bear in mind reality that high frequency financial data have discreet manner. 3.1 Haar Transform We will explain concept of Discrete Wavelet transform on Haar Transformation. Notation and definition used in this part are based on [7]. Earlier we defined discrete signal as sequence of numbers x n. So we can use form x = (x, x.,, x ( ) where n is even positive integer. Wavelet transforms decompose signal to two sub signals. The first is trend and the second fluctuation. Every decomposition is marked by number, where decomposition 1 means first level of decomposition and level means decomposition based on the level 1 and so on. First trend a = (a, a.,, a(. ) is defined by equation a ; = (x.;e + x.; ) where m = 1,,, n. In similar manner we define first fluctuation d = (d, d.,, d(. ) with formula d ; = (x.;e x.; ) This definition assumes that signal has even number of data points. In numerical computation we can use several techniques sometimes called signal extension modes: zero-padding - signal is extended by adding zero samples constant-padding - border values are replicated symmetric-padding - signal is extended by mirroring samples periodic-padding - signal is treated as a periodic one Haar transform 1- level H mapping is denoted by x H K a d ) We should notice that Haar level 1 transform generates the same number of coefficients as the original signal. There is of course inverse mapping, so for level 1 we get x = ( a 1 + d, a 1 d,, a n + d (., a n d (. ) One very important property of Haar transform is the energy conservation of the transformed signal. This property is expressed by equation E x = E a 1 d 1 ) Another important feature of Haar transform is energy compaction where the first trend a include large percentage of energy of the transformed signal a d ). A fixed amount of energy cannot be localized into an arbitrarily small time interval. This is due to Heisenberg s Uncertainty Principle (for reference see []). To compute the second level Haar transform we compute the second trend a. and second fluctuation d. for the first trend a only. Haar transform of level two is then a. d. d and for level 3 a O d O d. d and so on for other levels. 3. Haar Wavelets First level Haar Wavelets are defined as W = 1, 1, 0,,0 W. = (0,0, W (. = 0,0,, 1, 1, 0,,0) 1, 1 and values of the first fluctuation of the signal can be expressed as d ; = x. W ; for m = 1,,, n.

3 POSTER 015, PRAGUE MAY 14 3 Each has energy of 1 and consist of fluctuation between two values, E with average equal to zero. Similarly the first.. level scaling signals are defined V = 1,, 0,,0 1 V. = (0,0, V (. = 0,0,, 1, 1, 0,,0) 1, 1 and again the values for the first trend are a ; = x. V ; where m = 1,,, n. We can see that Haar scaling signals are similar to Haar Wavelets and that all have energy equal to 1 and their support includes two sequential values. Equations described above can be extended to every level. For the second level a = (x. V., x. V...,, x. V ( T ) d = (x. W., x. W...,, x. W ( T ) Original signal x can be expressed in natural expansion in term of natural basis due to addition of two signals x = (x V x. V x ( V n 0 ) where V 1 0 = (1,0,,0) and V n 0 = (0,0,,1). 3.3 Haar MRA Multi resolution analysis (MRA) expresses signal x as a sum of Haar wavelets and scaling signals of certain level. For first level of Haar MRA we get expansion x = ( a 1, a 1,, a n, a n ) +( d 1, d 1,, d n, d n ) This equation describe signal x as a sum of first averaged signal plus sum detail signal x = A 1 + D 1 where A 1 = ( a 1, a 1,, a n, a n ) D 1 = ( d 1, d 1,, d n, d n ) When we use Haar scaling functions and Wavelets we get A 1 = a V a. V a (. V n D 1 = d W d. W d (. W n These two equations can we rewritten to following equations A 1 = (x. V 1 1 ). V (x. V n 1 ). V n s D 1 = (x. W 1 1 ). W (x. W n ). W n Second level of Haar MRA is than expressed by equation x = A + D + D 1 where we can see A 1 = A + D. For signal x where the number of values is divisible k times by we get x = A k + D k + + D + D Discrete Wavelet Families Discrete Wavelet families usually include a subset of the following Wavelets 1 : HaarWavelet (edge detection) DaubechiesWavelet (time series prediction) BattleLemarieWavelet BiorthogonalSplineWavelet CDFWavelet CoifletWavelet (currency pair rates) MeyerWavelet ReverseBiorthogonalSplineWavelet ShannonWavelet SymletWavelet The Difference between these Wavelet families is how compactly the basis functions are localized in space and how smooth the Wavelets are. Some Wavelets are distinguished within their family by number of coefficients and level of iteration, e.g. Daubechies Wavelets DB, DB0. 4. WHA method Wavelet Based Method for High Frequency Data Analysis uses three main phases: (1) original high-frequency data preprocessing; () optimized discrete wavelet transform (DWT) to produce a characteristic D feature vector; (3) characteristic D feature vector analysis. 4.1 Data preprocessing This part depends on the type of financial time series and is not required in all cases, but nevertheless recommended. Mandatory checks includes verification that input signal has high-frequency nature and signal size is in terms of mega/samples. First of all we calculate maximum time interval between neighbor s values from input signal to get evidence about time deltas in input signal. Maximum time interval has to be under the estimated value related to average time interval. If time delta is acceptable we can continue to next step. In opposite case we create equally spaced series with time delta Δ where time delta Δ is calculated from maximum time interval between neighbor s values from input signal. If there is no observed transaction/value at time nδ, n N then we put the last value before this interval. 1 List based on availability in Mathematica 10.

4 4 J. SCHÜRRER, HIGH FREQUENCY TIME SERIES ANALYSIS USING WAVELETS Next step involves elimination of outliers and in case of need zero values, which are replaced with last value before zero and distance calculation between min and max value of input signal. Distance between minimum and maximum value is important only from numerical standpoint of view, where we need to use proper data type during computation phase. Because of the nature of financial time series we do not expect problems with this and typically floating-point types are ok. Next phase stems from the fact that we have massive data base with equally spaced observations. It means that we can use interesting properties of Wavelets like small fluctuation feature and energy compaction/conservation. 4. Optimized DWT Optimized Wavelet transform is represented by an algorithm with an optimization rule to select the optimal Wavelet family and the level of decomposition complying with the cost criterion. Output is represented by a Characteristics D feature vector, analytically describing the input signal in our case high-frequency financial time series. We can also use cost function to calculate difference between original signal and signal derived by inverse DWT to compare performance of different Wavelets. Advantage of using Wavelets in this method is that we can simultaneously perform decomposition and denoising of signal based on Wavelet thresholding. Any suitable function can be used as a cost function. In our case we use very simple cost function Root Means Square Error (RMS) which is defined Cost function = (x y ). + + (x ( y ( ). where y ( is inverse DWT of certain Wavelet family. 4.3 D Feature Vector Analysis D Feature Vector represent optimized output from the WHA method. Specifically include best trend and detail coefficients of DWT on certain level and selected Wavelet family, which fulfil the best energy conservation and original signal approximation. D feature vector can be used for further analysis in two different ways: Direct analysis of trend and fluctuation part Analysis of the new signal created by inverse transform of the D feature vector. 4.4 Assumptions and constrains The main assumption was mentioned earlier. We assume that a signal has finite energy. This property is used to fine-tune the model and the signal energy represents the optimization criterion. We also do not assume any quantization, which means that it is not necessary to take into account the finite precision of numerical data handled n by digital methods. The reason is that financial data values are discrete by nature. 4.5 High Level Algorithm Picture The proposed method involves several steps, which are repeated in a loop until a threshold condition is true. Output from algorithm is best approximation of input signal based on Wavelet families used in algorithm. High-level structure of WHA method is depicted below. START Data preprocessing Select first Wavelet family from pool of Wavelets Loop 1 END Do one level decomposition w/ or w/o tresholding Calculate energy of decomposed signal If CostFunctionOfEnergy < threshold then Goto Loop 1 Else Save DFeatureVector If possible to select next Wavelet from pool Then select next Wavelet family from pool of Wavelets Goto Loop 1 Select from DFeatureVectors the vector with lowest CostFunction and return DFeatureVector. The method s overall s characteristics is a set of basic properties that allow for comparison of different method runs based on variants of the cost function and data preprocessing and, thus letting us compare the distinct characteristic D feature vectors. 5. Example of WHA method For demonstration of WHA method we use generated Fractional Brownian motion process with drift 0, volatility 1, and Hurst index 1 and twenty millions of data points. This process represents effective market realization in terms of effective market theory. Computation is done in high-level, high-performance dynamic programming language Python with Open Source wavelet transform software for Python - PyWavelets. We used 1D forward and inverse Discrete Wavelet Transform with signal extrapolation set to periodization, which is similar to periodic-padding where signal is treated as a periodic one, but generates smallest possible number of coefficients. It is necessary to note that every Wavelet package implements advanced algorithms to ensure best performance and also high-level interface.

5 POSTER 015, PRAGUE MAY 14 5 In such cases we may have to deal with some discrepancies between theory and practical results. The PyWavelets specific feature is automatically maintaining of the signal energy during its decomposition to ensure the best signal reconstruction. For simulation we use random-generated process with twenty millions of points and we do not use in this case cost function for energy. Maximum level of decomposition was used by function from PyWavelets. RMS error cost function was used for D feature vector selection. The following figure denotes the analyzed signal progression. The first noticeable result is depicted in the following picture fig.. where we can see the dramatically decreasing length of trend coefficients (number of values for ca coefficient. Total signal length is twice as long, because during IDWT we need ca and cd coefficients (length of cd is same as ca). At minimum decomposition level 15 we need 611 values with minimum RMS error equal to For comparison at first decomposition level we need 10,000,000 values with minimum RMS error equal to nearly zero for Haar Wavelet. 10,000,000" 1,000,000" 100,000" 10,000" 1,000" 100" 10" 1" Level"1" Level"5" Level"6" Level"7" Level"8" Level"10" Level"1" Level"15" Fig.. The number of coefficient s versus decomposition level (logarithmic scale on vertical axis) Fig. 1. The random process with 0M samples. The initial simulation setup parameters are recorded in following table tab.1. For input data the maximum decomposition level is 4 for Harr Wavelet and for other Wavelets it is between 18 and depending on Wavelet support length. We choose decomposition level 15. Data type Number of samples Wavelet families Brownian Motion 0 M samples Haar, Daubechies, Symlets, Coiflets, Biorthogonal, Reverse biorthogonal, Discrete FIR approximation of Meyer wavelet (dmey) Maximum decomposition level 15 Cost function for D Feature Vector Tab. 1. Setup for simulation RMS Error Functions used for decomposition and reconstruction: pywt.wavedec(data, wavelet, mode= per ) pywt.upcoef(part, coeffs, wavelet) with following Wavelets: haar, db, db15, db0, sym6, sym9, sym19, coif1, coif3, coif5, sym4, sym14, sym0, bior1.3, bior.4, rbio6.8, dmey. During simulation we have identified that the best Wavelet for decomposition level 5, 6, 8 is Biorthogonal 1.3, which is symmetric, not orthogonal, and biorthogonal. Haar Wavelet is sufficient in other cases. The last graph shows the growth of RMS error with increasing level of decomposition where the number of coefficients is significantly lower but the approximation of the analyzed signal is worse. Maximum reasonable level for further analysis is # 4.00# 3.00#.00# 1.00# 0.00# Fig. 3. Level#1# Level#5# Level#6# Level#7# Level#8# Level#10# Level#1# Level#15# RMS err1or in dependence on decomposition level In conclusion we can state that the simulation verified our assumptions about the proposed method even despite some discrepancies in energy preservation due to the algorithms implemented in PyWavelet for Python. We also verified that numeric calculations with Python s 53-bit precision float type provide sufficient accuracy. RMS error also shows very good approximation of original signal up to level 8.

6 6 J. SCHÜRRER, HIGH FREQUENCY TIME SERIES ANALYSIS USING WAVELETS 6. Conclusion This paper presented the basics of high-frequency financial data (based on [6]) and properties of Discrete Wavelet Transformation (based on [1], [], [3], [7]) as a tool for non-stationary time series analysis. We proposed a new method how to analyze tic-by-tick data for further analysis and show overall complexity of such analysis. We are also mentioning the existing limitations due to Heisenberg s principle of uncertainty. At the end we present illustrative example of using this method to analyze a random process representing effective market realization of financial data. About Author Jaroslav SCHÜRRER is a student of combined study course in Computational Economics and Finances on Masaryk Institute of Advanced Studies, Czech Technical University. 7. Conventions and notations R N Z DWT IDWT MRA WHA The real axis Time delta Natural numbers Integral numbers Discrete Wavelet Transformation Inverse DWT Multi Resolution Analysis Wavelet Based Method for High Frequency Data Analysis Acknowledgements Research described in the paper was supervised by Prof. M. Vošvrda, Institute of Information Theory and Automation of the ASCR References [1] CHRISTENSEN, O., CHRISTENSEN, K.L. Approximation Theory: From Taylor Polynomials to Wavelets. Boston: Birkhäuser, 004. [] VIDAKOVIC, B. Statistical Modeling by Wavelets. New York: Wiley Series in Probability and Statistics, [3] VALNUT, D.L. Introduction to Wavelet Analysis. Boston: Birkhäuser, 00. [4] RAMSEY, J. The Contribution of Wavelets to the Analysis of Economic and Financial Data. Wavelets: the key to intermittent information, OUP, 1 36, 000. [5] MEYER, Y. Wavelets and Operators. New York: Cambridge University Press, [6] TSAY, R. S. Analysis of Financial Time Series. Hoboken: John Wiley & Sons, 010. [7] WALKER, J. S. A Primer On Wavelets and Their Scientific Applications. NW: Chapman and Hall/CRC; edition, 008. [8] Python, available at [9] PyWavelets - Discrete Wavelet Transform in Python, available at