Diagram of measurement series elements deviation from local linear approximations

Diagram of m easurement series element s deviation from local linear approximations Lande D.V. (dwl@visti.net), Snarskii А . А . (asnarskii@gmai l.com), National Technical University of Ukraine “Kyiv Polytechnic Institute” For the detection and visualization of trends, periodicities, local peculiarities in measurement s eries, the methods of fractal and wavelet-a nal ysis are of consi derable current use. One of such methods - DFA (Detrended fluctuation analysis), [1, 2], is used to re veal statistical self-similarity of sign als. The essence o f this method i s as follows. Let there be a measurement series , 1 , ..., t x t N ∈ . Denote the average value of this measurement series 1 1 N k k x x N = = ∑ . From the source series, the accumulation series is built: ( ) 1 t t k k X x x = = − ∑ . Then t X s eries is divid ed in to time windows of length L , l inear approximation ( , j L L ) is built ac cording to , , k j L X value s fr om , j L X i nside each window (in its turn, , j L X i s s ubset t X , 1 , ..., j J = , / J N L = - the number of obse rvation windows) and a deviation of a ccumulation series points from linear approximation is calculated: ( ) 2 2 , , , , , , 1 1 1 1 ( , ) | | L L k j L k j L k j L k k E j L X L L L = = = − = ∆ ∑ ∑ , where , , k j L L is the value of local linear approximation at point ( 1 ) t j L k = − + . Here , , | | k j L ∆ is absolute deviation of element , , k j L X from local linear approximation. Then the average value is calculated: 1 1 ( ) ( , ) J j F L E j L J = = ∑ , following whi ch, i n case ( ) F L L α ∝ , wher e α i s certain constant, conclusions are made on the availability of statistical self-similarity and behaviour of measurement series under study. Of interest is behaviour of absolute deviation of accumulation series poin ts from linear approximation , , | | k j L ∆ (let us call it L ∆ -method) for t he real proce sses, for ex ample, reflecting the intensity of publications on this subject in the Internet. Most commonl y , time series corresponding to themati c information flows pos sess t he properties of statis tic self-similarity [3], which is confirm ed, in particular, by DFA method. Visualiz ation of parameters , , | | k j L ∆ as a function of ( 1 ) t j L k = − + and L as a “relief” diagram i s of certain int erest for stud ying local peculiarities of the process corresponding to the source measurement series. Note that di vision of the source r ange of val ues 1 , ..., t N ∈ into J nonoverlapping observation windows results in some “inequ ality” of points inside these windows, which is not principal in case of summation and subsequent approximate estimation, but essential in the analysis of local v alues and visualiz ation. The refore, withou t giving up the ide a of li near approximation, it is proposed to choose for each point t such observation window of length L , that this point appears to be in its centre (or with a shift at 1 in case of even L ). Undoubtedl y, with regard to this correction, the calculation speed is decelerated , , | | k j L ∆ , which is largely compensated by the simplicit y of a lgorithm. 2 As a tim e series under study, on which basis the method opportunit ies will be considered, we shall use a s eries o f daily number of published works on certain su bjects in the Internet during a y ear (Fig. 1). This series was obtained by m eans of InfoStream content-monitoring system, regularly scanning over 3000 on-line Russian and Ukrainian mass media [4] . “Relief “ diagrams obtained as a result of proposed method (thi s diagram is exemplified in Fig. 2, wh ere li ghter tones correspond to larger valu es , , | | k j L ∆ ), res emble scalegrams obtained as a result of continuous wavelet-transformations. The fact that dark stripes in the centre of m any li ght co loured areas testify to “stabilization” of large values of series under consideration on a high level is noteworth y. Fig. 1. Time series of intensity of publications on gi ven subjects (abscissa axis –days of the year, ordinate axis –number of publications) Fig.2. L ∆ -diagram of time series of intensity of subject publications (abscissa axis –days of the year, ordinate axis –size of measurement window ) Note that L ∆ -method proves to be rather effic ient fo r revealing harmonic com ponents of series under study. Fig. 3 sh ows L ∆ -diagram of the series corresponding to sinusoid ( ( ) sin ( / 7 ), 1 , ..., 366 y i i i π = = ). Application of L ∆ -method to the series composed of the number of publications scanned b y I nfoStream s ystem from I nternet without considering subject division, has a pronounced harmonic component (total number of publications depends on the day of the week), t hat ca n be seen in Fig. 4. Besides, in this diagram one can see deviations from general dynamics of publication volumes on holida ys. Fig.3. L ∆ -diagram of sinusoid 3 Fig.4. L ∆ -diagram of series of the number of publications scanned daily by InfoStream system in 2008 L ∆ -diagrams are similar in appearance to scale grams obtained as a result of wavelet- analysis of measurement series. The basic i dea of wavelet-transformations lies i n the fact that some numerical series, like in the above-mentioned method, is divided into “observation windows”, and on e ach of them there is g enerated a set of c oefficients that are functions of two variables: time and frequenc y, and thus can be also represented as “ relief” diagrams, the so- called sc alegrams. In t heir nature, wavelet-coefficients represent a certain d egree of p roximit y of measurement series under study to certain special function called wavelet [31, 32] . Continuous wavelet-transformation for function ( ) f t is constructed b y means of continuous scale transfo rmations and t ransfers of wavelet ( ) t ψ with arbitrary values of scale coefficient a and shift parameter b : ( ) ( ) ( ) * 1 , ( ), ( ) t b W a b f t t f t dt a a ψ ψ ∞ −∞ −   = =     ∫ . Fig. 5 shows a scalegram – the result of continuous wavelet-analysis (Gauss wavelet) of time series corresponding to process under stud y. Fig. 5. S calegram o f time series u nder stu dy (Gaus s wavel et) Proposed method for visualization of absolute deviations L ∆ , like the wavelet- transformations method, allows (not worse, as it is shown in the example) detecting single and irregular “bursts”, drastic changes in the values of quantit y fi gures in different time p eriods. Note that wavelet-transformations method can be employ ed with the use of various wavelets. In particular, the us e of Haar wavelet ( Fig. 6), apparently, is m ore suitable for an alysis of se quenc e under consideration. However, even the use of Haar wavelet did not allow identifying the peculiarity (local maxim um) of the sourc e me asurement series durin g the last days of y ear 2008, at least, this peculiarity is not shown as skeleton in Fig. 6 б . 4 а ) Б ) Fig. 6. S calegram o f time series u nder stu dy (Haar wavelet ): а ) scalegram (absci ssa axis – day of the year, or din ate axis - f requency ); б ) local maximum lines o f scal egram L ∆ -method that is much easier to realize, nevertheless allowed determi ning this anomaly. Besides, select ion of suitable wavelet f or anal ysis is always a complicated task that need not be solved in case of using L ∆ -method. Proposed method is sufficientl y easy in program realization and, as experience suggests, can be efficientl y us ed in t he analysis of time series in such fields as economics and sociology. References [1] Peng, C.K. et al. (1994) Mosaic organization of DNA nucleotides, Ph ys R ev E, 49 (2) 1685-1689. URL: http://prola.aps.org/pdf/PRE/v49/i2/p1685_1 [2] P eng C.-K., Havlin S ., Stanle y H.E., Goldberger A. L. Quantification of scaling exponents and crossover phenomena i n nonstationary h eartbeat time series // Chaos. - Vol. 5. - 1995. - pp. 82. [3] Dodonov A.G., L ande D.V. Self-similarity of arra ys of network publications on computer virology // R ey estr atsiya, Zberigann ya i Obrobka Danyh, 2007, V. 9, - N 2. - P. 53-60. [4] InfoStream content-monitoring system. URL: http://i nfostream.ua/ [5] Astaf ye va N.M. W avelet-analysis: theoretical basis and applic ation examples // Uspekhi Fizicheskih Nauk. -1996. - V. 166. - № 11. –P . 1145-1170. [6] Buckhe it J ., Donoho D. Wavelab and reproducible research // Stanford Universit y Technical Report 474: Wavelets and Statistics Lecture Notes, 1995. -27 p. 5 Abstract Diagram of m easurement series element s deviation from local linear approximations Lande D. V. (dwl@ visti.n et), Snars kii А . А . (asnars kii@ gmail.co m), Natio nal Techn ical Uni versity of Ukraine “ Kyiv Polytechni c Institut e” Method for dete ction a nd visualiz ation of trends, periodicities, local peculiarities in measurement s eries ( L ∆ -method) based on D FA te chnolog y (Detrended fluctuation anal ysis) is proposed. The e ssence of the method lies in reflecting the values of absolute deviation of measurement accumulation series points from th e respective values of line ar approximation. I t is shown t hat L ∆ -method in some cases allows better determination of local peculiarities than wavelet-analysis. Easy-to-realize approach that is proposed can be used in the analysis of time series in such fields as economics and sociology. Key words: measurement series, linear approxim ation, wavelet-analysis, Detrended fluctuation analysis, L ∆ -method