{"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. 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IEEE  Journal  on  Selected  Area  of Communications,  15,  to appear  in \nMay  1997. \n\n[8]  S.  Ma and C. 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}]}