{"title": "Wavelet Models for Video Time-Series", "book": "Advances in Neural Information Processing Systems", "page_first": 915, "page_last": 921, "abstract": null, "full_text": "Wavelet Models for Video Time-Series \n\nSheng Ma and Chuanyi Ji \n\nDepartment of Electrical, Computer, and Systems Engineering \n\nRensselaer Polytechnic Institute, Troy, NY 12180 \ne-mail: shengm@ecse.rpi.edu, chuanyi@ecse.rpi.edu \n\nAbstract \n\nIn this work, we tackle the problem of time-series modeling of video \ntraffic. Different from the existing methods which model the time(cid:173)\nseries in the time domain, we model the wavelet coefficients in the \nwavelet domain. The strength of the wavelet model includes (1) a \nunified approach to model both the long-range and the short-range \ndependence in the video traffic simultaneously, (2) a computation(cid:173)\nally efficient method on developing the model and generating high \nquality video traffic, and (3) feasibility of performance analysis us(cid:173)\ning the model. \n\n1 \n\nIntroduction \n\nAs multi-media (compressed Variable Bit Rate (VBR) video, data and voice) traffic \nis expected to be the main loading component in future communication networks, \naccurate modeling of the multi-media traffic is crucial to many important appli(cid:173)\ncations such as video-conferencing and video-on-demand. From modeling stand(cid:173)\npoint, multi-media traffic can be regarded as a time-series, which can in principle \nbe modeled by techniques in time-seres modeling. Modeling such a time-series, how(cid:173)\never, turns out to be difficult, since it has been found recently that real-time video \nand Ethernet traffic possesses the complicated temporal behavior which fails to be \nmodeled by conventional methods[3] [4]. One of the significant statistical properties \nfound recently on VBR video traffic is the co-existence of the long-range (LRD) and \nthe short-range (SRD) dependence (see for example [4][6] and references therein). \nIntuitively, this property results from scene changes, and suggests a complex behav(cid:173)\nior of video traffic in the time domain[7]. This complex temporal behavior makes \naccurate modeling of video traffic a challenging task. The goal of this work is to de(cid:173)\nvelop a unified and computationally efficient method to model both the long-range \nand the short-range dependence in real video sources. \nIdeally, a good traffic model needs to be (a) accurate enough to characterize perti(cid:173)\nnent statistical properties in the traffic, (b) computationally efficient, and (c) fea-\n\n\f916 \n\ns. Ma and C. Jj \n\nsible for the analysis needed for network design. The existing models developed \nto capture both the long-range and the short-range dependence include Fractional \nAuto-regressive Integrated Moving Average (FARIMA) models[4]' a model based \non Hosking's procedure[6], Transform-Expand-Sample (TES) model[9] and scene(cid:173)\nbased models[7]. All these methods model both LRD and SRD in the time domain. \nThe scene-based modeling[7] provides a physically interpretable model feasible for \nanalysis but difficult to be made very accurate. TES method is reasonably fast but \ntoo complex for the analysis. The rest of the methods suffer from computational \ncomplexity too high to be used to generate a large volume of synthesized video \ntraffic. \n\nTo circumvent these problems, we will model the video traffic in the wavelet domain \nrather than in the time domain. Motivated by the previous work on wavelet rep(cid:173)\nresentations of (the LRD alone) Fractional Gaussian Noise (FGN) process (see [2] \nand references therein), we will show in this paper simple wavelet models can simul(cid:173)\ntaneously capture the short-range and the long-rage dependence through modeling \ntwo video traces. Intuitively, this is due to the fact that the (deterministic) similar \nstructure of wavelets provides a natural match to the (statistical) self-similarity of \nthe long-range dependence. Then wavelet coefficients at each time scale is modeled \nbased on simple statistics. Since wavelet transforms and inverse transforms is in \nthe order of O(N) , our approach will be able to attain the lowest computational \ncomplexity to generate wavelet models. Furthermore, through our theoretical anal(cid:173)\nysis on the buffer loss rate, we will also demonstrate the feasibility of using wavelet \nmodels for theoretical analysis. \n\n1.1 Wavelet Transforms \n\nIn L2(R) space, discrete wavelets \u00a2j(t)'s are ortho-normal basis which can be rep(cid:173)\nresented as \u00a2j(t) = 2-j / 2\u00a2(2-i t - m), for t E [0,2 K - 1] with K ~ 1 being an \ninteger. \u00a2(t) is the so-called mother wavelet. 1 ~ j ~ K and 0 ~ m ~ 2K -j - 1 \nrepresent the time-scale and the time-shift, respectively. Since wavelets are the di(cid:173)\nlation and shift of a mother wavelet, they possess a deterministic similar structure \nat different time scales. For simplicity, the mother wavelet in this work is chosen \nto be the Haar wavelet, where \u00a2(t) is 1 for 0 ~ t < 1/2, -1 for 1/2 ~ t < 1 and 0 \notherwise. \n\n-\n\nThen dj can be obtained \n\nLet dj's be wavelet coefficients of a discrete-time process x(t) (t E [0,2 K \ntransform dj = \n1]) . \nK L:;=O-l x(t)\u00a2j(t). x(t) can be represented through the inverse wavelet transform \nX t) = L:j=l L:m=O - dj\u00a2j(t) + \u00a2o, where \u00a2o is equal to the average of x(t). \n\n2K - , 1 \n\nthrough \n\nthe wavelet \n\n( \n\nK \n\n2 Wavelet Modeling of Video Traffic \n\n2.1 The Video Sources \n\nTwo video sources are used to test our wavelet models: (1) \"Star Wars\" [4]' where \neach frame is encoded by JPEG-like encoder, and (2) MPEG coded videos at Group \nof Pictures (GOP) level[7][ll] called \"MPEG GOP\" in the rest of the paper. The \nmodeling is done at either the frame level or the GOP level. \n\n\fWavelet Models for Video Time-Series \n\n917 \n\n31 \n\n31 \n\n34 \n\n32 \n9 \n;30 \n\n> \n\ni21 \ni 21 \nu \n20 . \n\n\u2022 \n\n22 \n\n\" \n\n0 \n\n& \n\n\u2022 .. .+ \n.. -.' \n,. . \n\u2022 \u2022 \n, \n\n. :. + \n\n\u2022 \n\u2022 \n\n\u2022 \n\n\u2022 \n\n\u2022 \n\n20 \n\n.1 \n\n~.O \n\n~ \n\ni \njs \n\n':QOP \n\n. :Si90Soutlt \n\n\u2022 \n\n- I \n\n10 \n\n12 \n\n14 \n\n.. \n\n0 \n\n.,\\ \n0 \n\n\u2022 AMIA('.0.4.o) \n\ndR{' ) \n\n~ AR1IIA{O,Q.4.o) \n\n.0 \n\n.. \n\n\u2022 \u2022 \n.-.. \n\n. , .\u2022 :.: .. \n\n~ \n\n~ \n\n.. \n\" \n. \n+. .. \n.. a --\n\n. . \n.. \n\n8 \nTWoScoIoI \n\n.0 \n\n.2 \n\nFigure 1: Log 2 of Variance of dJ versus \nthe time scale j \n\nFigure 2: Log 2 of Variance of dJ versus \nthe time scale j \n\n- : StarW.,. \n\n.. :GOP \n\n0 .\u2022 \n\n0.8 \n\nj J 0 \u2022\u2022 \n\n0.2 \n\n0 \n\n-0.20 \n\n2 \n\n8 \n\n8 \n\n10 \nLag \n\n12 \n\n1. \n\n18 \n\n18 \n\n20 \n\nFigure 3: The sample auto correlations of ds. \n\n2.2 The Variances and Auto-correlation of Wavelet Coefficients \n\nAs the first step to understand how wavelets capture the LRD and SRD, we plot in \nFigure (1) the variance of the wavelet coefficients dj's at different time scales for \nboth sources. To understand what the curves mean, we also plot in Figure (2) the \nvariances of wavelet coefficients for three well-known processes: FARIMA(O, 0.4, 0), \nFARIMA(l, 0.4, 0), and AR(l). FARIMA(O, 0.4,0) is a long-range dependent pto(cid:173)\ncess with Hurst parameter H = 0.9. AR(l) is a short-range dependent process, and \nFARIMA(l, 0.4,0) is a mixture of the long-range and the short-range dependent \nprocess. \n\nAs observed, for FARIMA(O, 0.4, 0) process (LRD alone), the variance increases \nwith j exponentially for all j. For AR(l) (SRD alone), the variance increases at \nan even faster rate than that of FARIMA(O, 0.4, 0) when j is small but saturates \nwhen j is large. For FARIMA(l, 0.4,0), the variance shows the mixed properties \nfrom both AR(l) and FARIMA(O, 0.4, 0). The variance of the video sources behaves \nsimilarly to that of FARIMA(l, 0.4,0), and thus demonstrate the co-existence of the \nSRD and LRD in the video sources in the wavelet domain. \nFigure 3 gives the sample auto-correlation of ds in terms of m's. The auto(cid:173)\ncorrelation function of the wavelet coefficients approaches zero very rapidly, and \n\n\f918 \n\ns. Ma and C. Ji \n\n. . , \n\nI \n\n~ \n\n~ \nm \n\n. \n; \n\n-... \n\n-2 \n\n0 \n\nso \n\n... \n\nQuantll._ of Stand.rd Norm \u2022\u2022 \n\nFigure 4: Quantile-Quantile of d';' for j = 3. Left: Star Wars. Right: GOP. \n\nthus indicates the short-range dependence in the wavelet domain. This suggests \nthat although the autocorrelation of the video traffic is complex in the time-domain, \nmodeling wavelet coefficients may be done using simple statistics within each time \nscale. Similar auto-correlations have been observed for the other j's. \n\n2.3 Marginal Probability Density Functions \n\nIs variance sufficient for modeling wavelet coefficients? Figure (4) plots the Q - Q \nplots for the wavelet coefficients of the two sources at j = 31 . The figure shows that \nthe sample marginal density functions of wavelet coefficients for both the \"Star \nWars\" and the MPEG GOP source at the given time scale have a much heavier tail \nthan that of the normal distribution. Therefore, the variance alone is only sufficient \nwhen the marginal density function is normal, and in general a marginal density \nfunction should be considered as another pertinent statistical property. \n\nIt should be noted that correlation among wavelet coefficients at different time \nscales is neglected in this work for simplicity. We will show both empirically and \ntheoretically that good performance in terms of sample auto-correlation and sample \nbuffer loss probability can be obtained by a corresponding simple algorithm. More \ncareful treatment can be found in [8]. \n\n2.4 An Algorithm for Generating Wavelet Models \n\nThe algorithm we derive include three main steps: (a) obtain sample variances \nof wavelet coefficients at each time scale, (b) generate wavelet coefficients inde(cid:173)\npendently from the normal marginal density function using the sample mean and \nvariance 2, and (c) perform a transformation on the wavelet coefficients so that the \n\nISimilar behaviors have been observed at the other time scales. A Q - Q plot is a \nstandard statistical tool to measure the deviation of a marginal density function from a \nnormal density. The Q - Q plots of a process with a normal marginal is a straight line. \nThe deviation from the line indicates the deviation from the normal density. See [4] and \nreferences therein for more details. \n\n2The mean of the wavelet coefficients can be shown to be zero for stationary processes. \n\n\fWavelet Models for Video Time-Series \n\n919 \n\nresulting wavelet coefficients have a marginal density function required by the traf(cid:173)\nfic. The obtained wavelet coefficients form a wavelet model from which synthesized \nvideo traffic can be generated. The algorithm can be summarized as follows. \nLet x(t) be the video trace oflength N. \nAlgorithm \n\n1. Obtain wavelet coefficients from x(t) through the wavelet transform. \n2. Compute the sample variance Uj of wavelet coefficients at each time scale \n\nj. \n\n3. Generate new wavelet coefficients dj's for all j and m independently \nthrough Gaussian distributions with variances Uj 's obtained at the previous \nstep. \n\n4. Perform a transformation on the wavelet coefficients so that the marginal \ndensity function of wavelet coefficients is consistent with that determined \nby the video traffic ( see [6] for details on the transformation). \n\n5. Do inverse wavelet transform using the wavelet coefficients obtained at the \n\nprevious step to get the synthesized video traffic in the time domain . \n\nThe computational complexity of both the wavelet transform (Step 1) and the \ninverse transform (Step 5) is O(N). So is for Steps 2, 3 and 4. Then O(N) is \nthe computational cost of the algorithm, which is the lowest attainable for traffic \nmodels. \n\n2.5 Experimental Results \n\nVideo traces of length 171, 000 for \"Star Wars\" and 66369 for \"MPEG GOP\" are \nused to obtain wavelet models. FARIMA models with 45 parameters are also ob(cid:173)\ntained using the same data for comparison. The synthesized video traffic from \nboth models are generated and used to obtain sample auto-correlation functions in \nthe time-domain, and to estimate the buffer loss rate. The results3 are given in \nFigure (6). Wavelet models have shown to outperform the FARIMA model. \n\nFor the computation time, it takes more than 5-hour CPU time4 on a SunSPARC \n5 workstation to develop the FARIMA model and to generate synthesized video \ntraffic of length 171, 0005 . It only takes 3 minutes on the same machine for our \nalgorithm to complete the same tasks. \n\n3 Theory \n\nIt has been demonstrated empirically in the previous section that the wavelet model, \nwhich ignores the correlation among wavelet coefficients of a video trace, can match \nwell the sample auto-correlation function and the buffer loss probability. To further \nevaluate the feasibility of the wavelet model, the buffer overflow probability has \nbeen analyzed theoretically in [8]. Our result can be summarized in the following \ntheorem. \n\n3Due to page limit, we only provide plots for JPEG. GOP has similar results and was \n\nreported in [8]. \n\n4Computation time includes both parameter estimation and synthesized \n\ntraffic \n\ngeneration. \n\n5The computational complexity to generate synthesized video traffic of length N is \n\nO(N2) for an FARIMA model[5][4]. \n\n\f.. -\n\n920 \n\nOJ \n\n01 \n\n0.2 \n\n0.1 \n\nFigure 5: \"-\": Autocorrelation of \"Star \nWars\"; \"- -\": ARIMA(25,d,20); \" \". \nOur Algorithm \n\nS. Ma and C. Ii \n\n-2 \n\n-2.5 \n\n-4.5 \n\n-5 \n\n~.5 \n\nI \nI \nI \n\n4~1 O.li \n\n0.4 \n\n0.45 \n\no.s \n\n0.&6 \n\n0.8 \n\n085 \n\n07 \n\n0.71 OJ \n\nLoss rate attained via \nFigure 6: \nsimulation. Vertical axis: \nloglO (Loss \nRate); horizontal axis: work load. \n\"-\": \nthe single video source; \"\". \nFARIMA(25,d,20); \"-\" Our algorithm. \nThe normalized buffer size: 0.1, 1, 10,30 \nand 100 from the top down. \n\nTheorem Let BN and EN be the buffer sizes at the Nth time slot due to the syn(cid:173)\nthesized traffic by the our wavelet model, and by the FGN process, respectively. Let \nC and B represent the capacity, and the maximum allowable buffer size respectively. \nThen \n\nInPr(BN > B) \n\nInPr(EN > B) \n\n(C - JL)2(i!:;?(1-H)e~7f-)2H \n\n20-2(1- H)2 \n\n(1) \n\nwhere ~ < H < 1 is the Hurst parameter. JL and 0-2 is the mean and the variance \nIt )2ko, where ko is a positive \nof the traffic, respectively. B is assume to be (C -\ninteger. \n\nThis result demonstrates that using our simple wavelet model which neglects the \ncorrelations among wavelet coefficients, buffer overflow probability obtained is sim(cid:173)\nilar to that of the original FGN process as given in[10]. In other words, it shows \nthat the wavelet model for a FG N process can have good modeling performance in \nterms of the buffer overflow criterion. \n\nWe would like to point out that the above theorem is held for a FGN process. \nFurther work are needed to account for more general processes. \n\n4 Concl usions \n\nIn this work, we have described an important application on time-series model(cid:173)\ning: modeling video traffic. We have developed a wavelet model for the time(cid:173)\nseries. Through analyzing statistical properties of the time-series and comparing \nthe wavelet model with FARIMA models, we show that one of the key factors to suc(cid:173)\ncessfully model a time-series is to choose an appropriate model which naturally fits \nthe pertinant statistical properties of the time-series. We have shown wavelets are \nparticularly feasible for modeling the self-similar time-series due to the video traffic. \n\n\fWavelet Models for Video Time-Series \n\n921 \n\nWe have developed a simple algorithm for the wavelet models, and shown that the \nmodels are accurate, computationally efficient and simple enough for analysis. \n\nReferences \n\n[1] I. Daubechies, Ten Lectures on Wavelets. Philadelphia: SIAM, 1992. \n[2] Patrick Flandrin, \"Wavelet Analysis and Synthesis of Fractional Brownian Mo(cid:173)\n\ntion\", IEEE transactions on Information Theory, vol. 38, No.2, pp.910-917, \n1992. \n\n[3] W.E Leland, M.S . Taqqu, W. Willinger and D.V. 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Ji, \"Modeling Video Traffic in Wavelet Domain\" , to appear IEEE \n\nINFO COM, 1998. \n\n[9] B. Melamed, D. Raychaudhuri, B. Sengupta, and J. Zdepski. Tes-based video \nIEEE \n\nsource modeling for performance evaluation of integrated networks. \nTransactions on Communications, 10, 1994. \n\n[10] Ilkka Norros, \"A storage model with self-similar input,\" Queuing Systems, \n\nvol.16, 387-396, 1994. \n\n[11] O. Rose. \"Statistical properties of mpeg video traffic and their impact on \ntraffic modeling in atm traffic engineering\", Technical Report 101, University \nof Wurzburg, 1995. \n\n, \n) \n\n\f", "award": [], "sourceid": 1437, "authors": [{"given_name": "Sheng", "family_name": "Ma", "institution": null}, {"given_name": "Chuanyi", "family_name": "Ji", "institution": null}]}