Wavelet Models for Video Time-Series 
Sherig Ma and Chuanyi Ji 
Department of Electrical, Computer, and Systems Engineering 
Rensselaer Polytechnic Institute, Troy, NY 12180 
e-mail: shengm@ecse.rpi.edu, chuanyi@ecse.rpi.edu 
Abstract 
In this work, we tackle the problem of time-series modeling of video 
traffic. Different from the existing methods which model the time- 
series in the time domain, we model the wavelet coefficients in the 
wavelet domain. The strength of the wavelet model includes (1) a 
unified approach to model both the long-range and the short-range 
dependence in the video traffic simultaneously, (2) a computation- 
ally efficient method on developing the model and generating high 
quali_ty video traffic, and (3) feasibility of performance analysis us- 
ing the model. 
I Introduction 
As multi-media (compressed Variable Bit Rate (VBR) video, data and voice) traffic 
is expected to be the main loading component in future communication networks, 
accurate modeling of the multi-media traffic is crucial to many important appli- 
cations such as video-conferencing and video-on-demand. From modeling stand- 
point, multi-media traffic can be regarded as a time-series, which can in principle 
be modeled by techniques in time-seres modeling. Modeling such a time-series, how- 
ever, turns out to be difficult, since it has been found recently that real-time video 
and Ethernet traffic possesses the complicated temporal behavior which fails to be 
modeled by conventional methods[3][4]. One of the significant statistical properties 
found recently on VBR video traffic is the co-existence of the long-range (LRD) and 
the short-range (SRD) dependence (see for example [4][6] and references therein). 
Intuitively, this property results from scene changes, and suggests a complex behav- 
ior of video traffic in the time domain[7]. This complex temporal behavior makes 
accurate modeling of video traffic a challenging task. The goal of this work is to de- 
velop a unified and computationally efficient method to model both the long-range 
and the short-range dependence in real video sources. 
Ideally, a good traffic model needs to be (a) accurate enough to characterize perti- 
nent statistical properties in the traffic, (b) computationally efficient, and (c) fea- 
916 S. Ma and C. Ji 
sible for the analysis needed for network design. The existing models developed 
to capture both the long-range and the short-range dependence include Fractional 
Auto-regressive Integrated Moving Average (FARIMA) models[4], a model based 
on Hosking's procedure[6], Transform-Expand-Sample (TES) model[9] and scene- 
based models[7]. All these methods model both LRD and SRD in the time domain. 
The scene-based modeling[7] provides a physically interpretable model feasible for 
analysis but difficult to be made very accurate. TES method is reasonably fast but 
too complex for the analysis. The rest of the methods suffer from computational 
complexity too high to be used to generate a large volume of synthesized video 
traffic. 
To circumvent these problems, we will model the video traffic in the wavelet domain 
rather than in the time domain. Motivated by the previous work on wavelet rep- 
resentations of (the LRD alone) Fractional Gaussian Noise (FGN) process (see [2] 
and references therein), we will show in this paper simple wavelet models can simul- 
taneously capture the short-range and the long-rage dependence through modeling 
two video traces. Intuitively, this is due to the fact that the (deterministic) similar 
structure of wavelets provides a natural match to the (statistical) self-similarity of 
the long-range dependence. Then wavelet coefficients at each time scale is modeled 
based on simple statistics. Since wavelet transforms and inverse transforms is in 
the order of O(N), our approach will be able to attain the lowest computational 
complexity to generate wavelet models. Furthermore, through our theoretical anal- 
ysis on the buffer loss rate, we will also demonstrate the feasibility of using wavelet 
models for theoretical analysis. 
1.1 Wavelet Transforms 
In L2(R) space, discrete wavelets qbjn(t)'s are ortho-normal basis which can be rep- 
resented as qb?(t) = 2-J/2qJ(2-Jt - m), for t e [0,2 K- 1] with K _> 1 being an 
integer. qb(t) is the so-called mother wavelet. i _< j _< K and 0 _< rn _< 2 K-j - 1 
represent the time-scale and the time-shift, respectively. Since wavelets are the di- 
lation and shift of a mother wavelet, they possess a deterministic similar structure 
at different time scales. For simplicity, the mother wavelet in this work is chosen 
to be the Haar wavelet, where qb(t) is 1 for 0 _< t < 1/2,-1 for 1/2 <_ t < 1 and 0 
otherwise. 
Let djn's be wavelet coefficients of a discrete-time process x(t) (t e [0,2 K- 
1]). Then djn can be obtained through the wavelet transform djn - 
Z2 '" -1 
t=0 x(t)qJjn(t)  x(t) can be represented through the inverse wavelet transform 
L,,=0 dO?(t) + 0o, where 00 is equal to the average of x(t). 
2 Wavelet Modeling of Video Traffic 
2.1 The Video Sources 
Two video sources are used to test our wavelet models: (1) "Star Wars"[4], where 
each frame is encoded by JPEG-like encoder, and (2) MPEG coded videos at Group 
of Pictures (GOP) level[7][11] called "MPEG GOP" in the rest of the paper. The 
modeling is done at either the frame level or the GOP level. 
Wavelet Models f o r Vdeo Time-Series 917 
I1 
+:AR(1} 
o 
o 
o 
$ 8 10 12 14 16 0 2 4 6 6 10 12 
Figure 1: Log 2 of Variance of d 3 versus 
the time scale j 
Figure 2: Log 2 of Variance of d3 versus 
the time scale j 
0.4 
0.2 
'20 2 4 6 6 10 12 14 16 18 20 
g 
Figure 3: The sample autocorrelations of d. 
2.2 The Variances ad Auto-correlation of Wavelet Coefficients 
As the first step to understand how wavelets capture the LRD and SRD, we plot in 
Figure (1) the variance of the wavelet coefficients dn's at different time scales for 
both sources. To understand what the curves mean, we also plot in Figure (2) the 
variances of wavelet coefficients for three well-known processes: FARIMA(0, 0.4, 0), 
FARIMA(1, 0.4, 0), and AR(1). FARIMA(0, 0.4,0) is a long-range dependent po- 
cess with Hurst parameter H -- 0.9. AR(1) is a short-range dependent process, and 
FARIMA(1, 0.4, 0) is a mixture of the long-range and the short-range dependent 
process. 
As observed, for FARIMA(0,0.4,0) process (LRD alone), the variance increases 
with j exponentially for all j. For AR(1) (SRD alone), the variance increases at 
an even faster rate than that of FARIMA(0, 0.4, 0) when j is small but saturates 
when j is large. For FARIMA(1, 0.4, 0), the variance shows the mixed properties 
from both AR(1) and FARIMA(0, 0.4, 0). The variance of the video sources behaves 
similarly to that of FARIMA(1, 0.4, 0), and thus demonstrate the co-existence of the 
SRD and LRD in the video sources in the wavelet domain. 
Figure 3 gives the sample auto-correlation of d in terms of m's. The auto- 
correlation function of the wavelet coefficients approaches zero very rapidly, and 
918 S. Ma and C. Ji 
Figure 4: Quantile-Quantile of d for j = 3. Left: Star Wars. Right: GOP. 
thus indicates the short-range dependence in the wavelet domain. This suggests 
that although the autocorrelation of the video traffic is complex in the time-domain, 
modeling wavelet coefficients may be done using simple statistics within each time 
scale. Similar auto-correlations have been observed for the other j's. 
2.3 Marginal Probability Density Functions 
Is variance sufficient for modeling wavelet coefficients? Figure (4) plots the Q - Q 
plots for the wavelet coefficients of the two sources at j = 31. The figure shows that 
the sample marginal density functions of wavelet coefficients for both the "Star 
Wars" and the MPEG GOP source at the given time scale have a much heavier tail 
than that of the normal distribution. Therefore, the variance alone is only sufficient 
when the marginal density function is normal, a. nd in general a marginal density 
function should be considered as another pertinent statistical property. 
It should be noted that correlation among wavelet coefficients at different time 
scales is neglected in this work for simplicity. We will show both empirically and 
theoretically that good performance in terms of sample auto-correlation and sample 
buffer loss probability can be obtained by a corresponding simple algorithm. More 
careful treatment can be found in [8]. 
2.4 An Algorithm for Generating Wavelet Models 
The algorithm we derive include three main steps: (a) obtain sample variances 
of wavelet coefficients at each time scale, (b) generate wavelet coefficients inde- 
pendently from the normal marginal density function using the sample mean and 
variance 2, and (c) perform a transformation on the wavelet coefficients so that the 
a Similar behaviors have been observed at the other time scales. A Q - Q plot is a 
standard statistical tool to measure the deviation of a marginal density function from a 
normal density. The Q - Q plots of a process with a normal marginal is a straight line. 
The deviation from the line indicates the deviation from the normal density. See [4] and 
references therein for more details. 
2The mean of the wavelet coefficients can be shown to be zero for stationary processes. 
Wavelet Models for Vdeo Time-Series 919 
resulting wavelet coefficients have a marginal density function required by the traf- 
fic. The obtained wavelet coefficients form a wavelet model from which synthesized 
video traffic can be generated. The algorithm can be summarized as follows. 
Let &(t) be the video trace of length N. 
Algorithm 
1. Obtain wavelet coefficients from &(t) through the wavelet transform. 
2. Compute the sample variance dj of wavelet coefficients at each time scale 
j. 
3. Generate new wavelet coefficients djn's for all j and rn independently 
through Gaussian distributions with variances dj's obtained at the previous 
step. 
4. Perform a transformation on the wavelet coefficients so that the marginal 
density function of wavelet coefficients is consistent with that determined 
by the video traffic(see [6] for details on the transformation). 
5. Do inverse wavelet transform using the wavelet coefficients obtained at the 
previous step to get the synthesized video traffic in the time domain. 
The computational complexity of both the wavelet transform (Step 1) and the 
inverse transform (Step 5) is O(N). So is for Steps 2, 3 and q. Then O(N) is 
the computational cost of the algorithm, which is the lowest attainable for traffic 
models. 
2.5 Experimental Results 
Video traces of length 171,000 for "Star Wars" and 66369 for "MPEG GOP" are 
used to obtain wavelet models. FARIMA models with 45 parameters are also ob- 
tained using the same data for comparison. The synthesized video traffic from 
both models are generated and used to obtain sample auto-correlation functions in 
the time-domain, and to estimate the buffer loss rate. The results 3 are given in 
Figure (6). Wavelet models have shown to outperform the FARIMA model. 
For the computation time, it takes more than 5-hour CPU time 4 on a SunSPARC 
5 workstation to develop the FARIMA model and to generate synthesized video 
traffic of length 171,000 . It only takes 3 minutes on the same machine for our 
algorithm to complete the same tasks. 
3 Theory 
It has been demonstrated empirically in the previous section that the wavelet model, 
which ignores the correlation among wavelet coefficients of a video trace, can match 
well the sample auto-correlation function and the buffer loss probability. To further 
evaluate the feasibility of the wavelet model, the buffer overflow probability has 
been analyzed theoretically in [8]. Our result can be summarized in the following 
theorem. 
3Due to page limit, we only provide plots for JPEG. GOP has similar results and was 
reported in [8]. 
4Computation time includes both parameter estimation and synthesized traffic 
generation. 
5The computational complexity to generate synthesized video traffic of length N is 
O(N ) for an FARIMA model[5][4]. 
920 S. Ma and C. Ji 
o, 
o8 
0.7 
i O. 
O   
00 200 300 400 500 
Figure 5: "-": Autocorrelation of "Star 
Wars"; "--"' ARIMA(25,d,20); "..": 
Our Algorithm 
.-4.5' 
04 0.45 O o. 
0.35 08 OEE 0.7 0.8 
Figure 6: Loss rate attained via 
simulation. Vertical axis: log0(Loss 
Rate); horizontal axis: work load. 
"-"' the single video source; ".."' 
FARIMA(25,d,20); "-" Our algorithm. 
The normalized buffer size: 0.1, 1, 10,30 
and 100 from the top down. 
Theorem Let BN and tN be the buffer sizes at the N th time slot due to the syn- 
thesized traffic by the our wavelet model, and by the FGN process, respectively. Let 
C and B represent the capacity, and the maximum allowable buffer size respectively. 
Then 
lnPr(BN , B) 
lnPr(//v > B) 
(C - 1)2( CB--_ )2(1-I-I)(1- ) 2H 
2a2(1 - H) 2 
(1) 
where  < H < i is the Hurst parameter. tt and a 2 is the mean and the variance 
of the traffic, respectively. B is assume to be (C - tt)2 k, where ko is a positive 
integer. 
This result demonstrates that using our simple wavelet model which neglects the 
correlations among wavelet coefficients, buffer overflow probability obtained is sim- 
ilar to that of the original FGN process as given in[10]. In other words, it shows 
that the wavelet model for a FGN process can have good modeling performance in 
terms of the buffer overflow criterion. 
We would like to point out that the above theorem is held for a FGN process. 
Further work are needed to account for more general processes. 
4 Conclusions 
In this work, we have described an important application on time-series model- 
ing: modeling video traffic. We have developed a wavelet model for the time- 
series. Through analyzing statistical properties of the time-series and comparing 
the wavelet model with FARIMA models, we show that one of the key factors to suc- 
cessfully model a time-series is to choose an appropriate model which naturally fits 
the pertinant statistical properties of the time-series. We have shown wavelets are 
particularly feasible for modeling the self-similar time-series due to the video traffic. 
Wavelet Models for lhdeo Time-Series 921 
We have developed a simple algorithm for the wavelet models, and shown that the 
models are accurate, computationally efficient and simple enough for analysis. 
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