Quick Detection of Changes in Trac Statistics: Ivy Hsu and Jean Walrand 2. Department of EECS, University of California, Berkeley, CA 94720

Transcription

1 Quic Detection of Changes in Trac Statistics: Alication to Variable Rate Comression Ivy Hsu and Jean Walrand 2 Deartment of EECS, University of California, Bereley, CA resented at the 32nd Annual Allerton Conference on Communication, Control, and Comuting, Urbana-Chamion, IL, Set. 994 Abstract In many communication alications, sources generate data at a variable rate. This variation can be catured by a two-level model. At a larger time scale, the source alternates between a nite number of models at the higher level, while each model reresents the local uctuation at the smaller time scale. Tyically the data goes through a lossy comression at the encoder, and then is entered into a buer before being transmitted over a ed-rate or a regulated-rate (e.g., Leay Bucet) channel. Since the buer size is nite, the choice of comression arameters must balance the tradeo between the distortion due to lossy comression and the distortion caused by buer overow. As the otimal comression arameters may be dierent for each individual model, a method is needed to detect changes from one model to another. We show via eamles that for many common trac models, the roblem of detecting changes in trac statistics can be reduced to the following general change oint detection roblem: Let A ; A 2 ; : : : be a series of indeendent observations such that A ; A 2 ; : : :; A m? are i.i.d. distributed according to an unnown distribution F, and A m ; A m+ ; : : : are i.i.d. distributed according to another unnown distribution G. m is the unnown change oint. The objective is to determine that a change has occurred as soon as ossible, while maintaining a low rate of false alarms. In this aer we roose a sequential nonarametric change oint detection algorithm using Kolmogorov-Smirnov statistics, and demonstrate that it is asymtotically otimal, in that under the constraint E[time until a false alarm] T, our algorithm is such that E[detection delay] = O(log(T )), as T!. This roerty is nown to be the best that can be achieved. Furthermore, unlie other nonarametric schemes, the erformance of the algorithm is indeendent of the distributions F and G. Introduction Consider communication alications such as MPEG video, in which the outut stream generated by the source has a variable rate due to statistical variations in the data. The outut stream is released onto a channel at a constant rate or at a regulated rate that allows some burstiness. Fied bandwidth radio channels for wireless communication are This research is suorted in art by a grant of the Networing and Communications Research Program of the National Science Foundation. 2 addresses: ihsu@eecs.bereley.edu, wlr@eecs.bereley.edu

2 an eamle of the former. The Leay Bucet scheme roosed for the Asynchronous Transfer Mode (ATM) networs reresents the latter. In both cases, unless the transmission rate eceeds the ea rate of the source, it is necessary to lace a buer between the source encoder and the channel to store data when they are generated faster than what the channel can transmit. First suose that the source can be characterized by a time invariant model. Then for any given comression scheme, the quantities of interest are its mean rate and mean-square distortion measure for the given source model. Since the buer size is nite, one must consider the tradeo between this distortion measure and the distortion caused by lost data when the buer overows. A ner comression reduces distortion, but generates more data and thus leads to higher overow robability. Dene the distortion-rate curve as the minimum distortion achievable by any scheme under an average rate constraint. Tse et al. [8] suggest that if the distortion-rate curve is concave at the channel rate R c, then one can achieve a lower distortion by timesharing between two comression schemes. Dene D T (R c ) as the minimum distortion achievable by any time-sharing scheme under an average rate constraint R c. Then the objective is for the buer overow robability and the steady state average distortion when the buer is not full to aroach zero and D T (R c ), resectively, as the buer size becomes large. The control scheme roosed in [8] laces a threshold at the half oint of the buer, and uses a coarse comression with average rate R < R c when the buer occuancy eceeds the threshold and a ne comression with rate R 2 > R c otherwise. Under this scheme, both the overow robability and the average distortion aroach their asymtotic values eonentially fast as the buer size goes to innity. Under a more rened assumtion, the source can be viewed to be alternating between a nite number of models, at a time scale much larger than the time scale of uctuation catured by each individual model. For eamle, a video sequence can contain active and quiet scenes, with a mean scene duration of several seconds. The active scenes in general cannot be comressed as well as the quiet ones. Therefore the otimal arameters of the comression scheme would be dierent in each model. Thus the encoder must be able to resond quicly when a change in the source model occurs and identify the new model. One aroach is to start with a ed set of source models and arameters for which the control arameters are nown, and conduct a arametric detection, i.e., comare the incoming trac with all eisting models, and determine the model that is closest. Then aly the corresonding control scheme. This aroach, although simle in nature, has a limited accuracy and requires etensive analyses of all ossible trac statistics. A more general aroach is to mae no rior assumtion about source models, and adatively determine the control arameters to use for each model. Thus the algorithm for change oint detection must be nonarametric. The system maintains a library of ast source statistics and corresonding control arameters. Once a change is detected, the new statistics are comared with entries in the library in order to determine if it belongs to one of the eisting models. If so, the control arameters associated with the identied model are used. Otherwise, a new entry is created in the library, and the control arameters for the closest model are selected as its starting oint. In both cases, these arameters are then adjusted over time, as the measured distortion is comared with the target. As an eamle, Fig. illustrates the bloc diagram of a ossible control scheme

3 MPEG video source variable bit rate acetization ATM cells buffer leay bucet rate control ATM networ change detection library + control toen buffer toens Figure : Transorting a variable rate MPEG sequence over an ATM networ with rate constraint. for variable rate video alications. The acetization bloc segments data generated by the video source into ATM cells of two riorities, in a way that incororates the multiresolution rincile: decoding with the high riority cells alone results in a coarser quality of video, while decoding the combined riorities results in a ner quality. The roortion of the two riorities is controlled based on the source statistics as mentioned above. The buer oerates under a artial buer sharing mechanism [5], in which only high riority cells can enter the buer when the occuancy is above a threshold. The data is emtied from the buer through a leay bucet before entering an ATM networ. In this aer we address the issue of detecting changes in trac statistics. The general change oint (or disorder) detection roblem has undergone much investigation since 950s. For the alication at hand we are interested in a sequential nonarametric detection method. The main criterion of an eective sequential scheme is that it should resond to changes as soon as ossible, while maintaining a low frequency of false alarms. In addition, it is desirable for the decision rule to be distribution-free, i.e., its eectiveness should not vary for dierent airs of statistics before and after the change oint. Finally, from the imlementation oint of view, the scheme should require only a bounded number of oerations and memory at each decision oint. Several sequential nonarametric detection methods have been roosed reviously. For a comrehensive review, see [2]. Most of these results are concentrated on the case in which the observed sequence is a concatenation of two i.i.d. random sequences, with some limited etension to n-deendent random sequences. Their common drawbac, however, is the lac of the distribution-free roerty. For eamle, the detection scheme in [] comares the emirical means before and after each decision oint. Consequently the false alarm frequency deends on the variance of the distribution before the change. In 2, we resent a general detection algorithm based on comarisons of emirical distributions. We show via eamles how this algorithm can be alied in the contet of detecting changes in incoming trac for many common models. The statistic used in this case is the Kolmogorov-Smirnov distance, which has the desirable roerty of being distribution-free [4]. This statistic has been used in [3] for the a osteriori change oint roblem, in which the emhasis is on accurate identication of the most liely oint of change within a given set of observations. 3 resents the main theorem of the aer. We show that the ratio of the eected delay of a correct detection to the logarithm of the eected time until a false alarm is uer bounded by a constant, as the memory size of the algorithm increases. This logarithmic relation imlies that the roosed scheme belongs to the class of asymtotically otimal algorithms, as indicated in [, 6]. We resent some reliminary numerical results in 4 and conclude in 5.

4 2 Change Point Detection Algorithm 2. Assumtions Let fa n g be the observed sequence and m the unnown change oint. Denote () the indicator function. Assume that A n = n (n<m) + n (nm) ; n, where n 's (res. n 's) are i.i.d. random variables with a continuous distribution F (res. G). Furthermore, assume that there eists a constant " > 0 such that the set A = f 2 R : jf ()?G()j "g satises either R A df () > 0 or R A dg() > 0. At the rst glance the i.i.d. assumtion may aear restrictive. But as the following eamles demonstrate, A n can be derived in many comle trac models: Eamle Marov Modulated Sources One common model for bursty sources is the Marov modulated uid, in which the instantaneous arrival rate from a source deends on the state of an underlying Marov chain. Consider, for eamle, an ON-OFF rocess, in which the Marov chain alternates between two states, and the source generates data at a constant rate when the chain is ON and no data otherwise. The source statistics can dier when there is a change in the transition rates of the Marov chain, or in the outut rate, or both. Choose the regenerative oints t n as the successive times at which the rocess returns to the OFF state. Then dene A n (total outut in [t n ; t n+ ))=(t n+?t n ). Eamle 2 Periodic Sources Periodic sources arise when inut data is encoded at ed time intervals (e.g., PCM of voice, frame-by-frame encoding of video). Thus A n can be taen as the data generated over any constant multile of a eriod. 4 gives a more comle eamle in the case of MPEG video, where the source can be treated as a suerosition of several eriodic streams. as 2.2 Algorithm The algorithm is characterized by three arameters: the memory size N, the windowing arameter a, and the threshold c. At each time n, only the last N observations need to be stored in memory. The arameter a is a constant in (0; ]. For simlicity in notation, 2 let a be chosen such that an 2 N. Given the memory of N oints, a denes a window such that each in fan; an + ; : : : ; (? a)ng is a candidate change oint. Dene, for n N and 2 [an; (? a)n], D N (; n) = ma n?n +in E N (n) = n?n X+ j=n?n + (Aj A i )? N? nx j=n?n ++ (Aj A i ) ; () ma D N(; n): (2) an(?a)n Let the decision rule be d N (n) = (EN (n)>c), where 0 < c <. The detection time is then (N) = minfn N : d N (n) = g. The algorithm can be interreted as follows: D N (; n) is the Kolmogorov-Smirnov distance, namely the maimum absolute dierence, between the emirical distributions

5 of two sequences, (A n?n + ; : : : ; A n?n + ) and (A n?n +? ; : : : ; A n ). For each an (? a)n, we calculate the Kolmogorov-Smirnov distance and declare that a change has occurred if at any oint the distance eceeds the threshold c. 3 Theorem Let P () (res. E ()) reresent the conditional robability (res. eectation) given that no change oint ever occurs. Similarly, let P m () (res. E m ()) be the conditional robability (res. eectation) given that change occurs at time m. As mentioned earlier, we are interested in two quantities: () E [(N)? N] eected time until a false alarm (2) E m [((N)? m)j(n) m] eected detection delay The objective is to detect the change as fast as ossible, while maintaining a large eected time between false alarms. The following theorem relates the two quantities under the above algorithm. Theorem Given that " c +! =2? 2a(? a) log(? c=2) ; (3) lim su N! su mn E m [((N)? m)j(n) m] log E [(N)? N] Constant: (4) It is ointed out in [, 6] that for the roblem of minimizing su m E m [((N)? m)j(n) m] under the constraint E [(N)?N] T, the otimal sequential detection rule admits an asymtotic behavior su m E m [((N)? m)j(n) m] = O(log(T )), as T!. Thus by virtue of theorem, the roosed algorithm is asymtotically otimal. Theorem follows from Lemmas 2 and 3 below. We rst derive an uer bound on the robability of false alarm at time N +, for all 0. Lemma P ((N)? N = ) L (N)Ne?L 2(N )N ; 8 0; (5) where L (N) and L 2 (N) are dened below. Proof Dene N (n) = P (d N (n) = ) for all n N, then P ((N)? N = ) N (N + ). Given that no change taes lace, the samles are i.i.d. F. Thus N (n) N (N), for all n N. Then N (N) = P ma an(?a)n D N(; N) > c! X an(?a)n P (D N (; N) > c): (6)

6 P (D N (; N) > c) 0 (a) ma j in 0 (b) ma j in X j= X (Aj A i )? F (A i )j + j N? NX j=+ (Aj A i )? F (A i )j > c (Aj A i )? F (A i )j + ma j (Aj A in N? i )? F (A i )j > c j= j=+ 0 (c) su j X (Aj )? F ()j > c=2a + j= 0 + NX su j (Aj )? F ()j > c=2a ; (7) N? j=+ where (a) is due to the triangular inequality, and (b) follows from the fact that the maimum of the sum is less than or equal to the sum of the maima. (c) results from P (X + Y > c) P (X > c=2 or Y > c=2) P (X > c=2) + P (Y > c=2), and that the maimum distance between an emirical distribution and the actual distribution at the samle oints is uer bounded by the suremum distance. P D su j j= (Aj )?F ()j is the Kolmogorov-Smirnov one-samle statistic based on samle oints. It can be eressed as the maimum of two non-negative onesided Kolmogorov-Smirnov statistics D + and D?, dened as 2 3 D + su 4 X (Aj )? F () 5 i = ma i? F (A (i)) ; (8) and D? su 2 j= 4 F ()? X j= (Aj ) 3 5 = ma i NX F (A (i))? i? A A ; (9) where A < () A < < (2) A are the order statistics of A () ; : : : ; A. The suerscrit reresents the total number of samle oints. Since F is continuous, by the robability-integral transformation theorem, given a random variable A F, the random variable U generated by the transformation U = F (A) has a Uniform(0, ) distribution. Therefore, U = F (i) (A ) is the (i) ith order statistic of i.i.d. Uniform(0, ) random variables. Thus for any ed > 0, P ( ma i i? F (A (i)) > ) = P ( ma i i? U (i) > ) X i= P (U (i) < i? ): (0) Note that for i= <, P (U < i? ) = 0. For (i) i=, P (U < i (i)? ) is the robability that out of i.i.d. Uniform(0, ) random variables, at least i of them are less than i=?. Let W j (U j < i=? ). Then P (U < i? ) = (i) P (P j= W j i), which is the robability that the samle average is greater than i=, while E[W ] = i=?. Dene '() log Ee W. Then by Marov inequality, P ( P j= W j i ) e(? i ) e( '()). In articular, P (P j= W j i ) e(? su [ i?'()]) e(?i + (i; ; )), where I + (i; ; ) = ( i log i= i=? + (? i ) log?i=?i=+ ; for i > ; for i : ()

7 Similarly, we can show that P (U (i)? i? > ) e(?i? (i; ; )), and I? (i; ; ) = ( i? + (? i? ; for i? log (i?)= ) log?(i?)= (i?)=+?(i?)=? ; for i? <?? : (2) Dene I N () = min an(?a)n min i minfi + (i; ; ); I? (i; ; )g. Then P (D > c=2) P (D + > c=2) + P (D? > c=2) 2 e(?i N (c=2)). Substituting into Eqn. 6, N (N) X an(?a)n X an(?a)n P (D > c=2) + P (D N? > c=2) 2 [ e(?i N (c=2)) + (N? ) e(?(n? )I N (c=2))] 4(? a)? e(?i N (c=2)) Ne?aI N (c=2)n L (N)Ne?L 2(N )N : (3) if Note that if P ((N)?N = j) for all j 0, then E [(N)?N] is minimized P ((N)? N = j) = 8 >< >: ; j j = b=c;? j ; j = j + ; 0; otherwise: Thus E [(N)? N] 2 (? o()), where o()! 0 as! 0. Since N(N)! 0 as N!, alying lemma, we obtain a lower bound on the eected time until a false alarm: Lemma 2 E [(N)? N]? e(?i N(c=2)) e ai N (c=2)n (? o()); (4) 8(? a)n where o()! 0 as N!. Before giving the uer bound on the eected detection delay, we rst state the Hoeding's inequality: Theorem 2 (Hoeding's Inequality) [7] Let Y ; Y 2 ; : : : ; Y N be indeendent random variables with zero means and bounded ranges: a i Y i b i. For each > 0, P (jy + + Y N j ) 2 e " #?2 P 2 Ni= (b i? a i ) 2 : (5) Lemma 3 Given condition 3, E m [(N)? mj(n) m] N + 2? an N! : (6)

8 Proof Dene N P m (d N (m + (? a)n? ) = 0), and q N P m (d N (m + N) = ). Then the following inequality is given in []:! E m [(N)? mj(n) m] N + N=q N : (7)? an N Denote F i () (res. G i ()) as the emirical distribution function generated by i i.i.d. samles with distribution F (res. G). Then D N (; m + (? a)n? ) = su Therefore, for any ed 0 2 A, j an F an() +? an G?aN ()? G N? ()j: (8) N P (su jf an ()? G (?a)n ()j c) P (jf an ( 0 )? G (?a)n ( 0 )j c): (9) By the triangular inequality, jf an ( 0 )?G (?a)n ( 0 )?F ( 0 )+G( 0 )j+jf an ( 0 )?G (?a)n ( 0 )j jf ( 0 )?G( 0 )j > ": (20) Since the condition imlies that " c, N P (jf an ( 0 )? F ( 0 ) + G( 0 )? G (?a)n ( 0 )j > "? c) = P (j a anx j= [ (j 0 )? F ( 0 )] +? a NX j=an + Alying the Hoeding's Inequality to Eqn. 2, we get ( b j = [? a F ( 0)]; for j an; [G(?a 0)]; for an + j N; ( a j = a [?F ( 0)]; for j an;?a [G( 0)? ]; for an + j N: [G( 0 )? (j 0 )]j > N("? c)): (2) Thus P N j= (b j?a j ) 2 = an=a 2 +(?a)n=(?a) 2 = N=a(?a), and N 2 e[?2na(? a)("? c) 2 ]: On the other hand, we obtain the following lower bound for q N : q N P (su jg an ()? G() + G()? G (?a)n ()j > c) (G an ()? G())? su(g (?a)n ()? G()) > c) P (su P (su(g an ()? G()) > c=2) P (su = P (D + an > c=2) P (D? (?a)n > c=2) = P ( ma ian (G (?a)n ()? G()) <?c=2) i an? U an (i) > c=2) P ( ma U (?a)n? i? (i) i(?a)n (? a)n > c=2) P (U an (an ) <? c=2) P (U (?a)n () > c=2) = (? c=2) N : (22) N 2q N for all N if e?2a(?a)("?c)2? c=2, or equivalently, if condition 3 holds. Note that I N (c=2) inf >c=2 [ log + (? ) log? ]; 8N, and thus Theorem?c=2?+c=2 follows.

9 4 Numerical Eamle In the variable-rate video coding schemes such as MPEG, the number of bits generated by each frame deends on both its information content and the encoding technique used. In MPEG, the choices of encoding modes are intraframe (I), redictive (P), and interolative (B), in the order of increasing coding eciency. A tyical sequence consists of eriodic interleaving of frames from each mode. For eamle, the sequence in our eamle has a eriod of =5 frames in the following attern: fi, B, B, P, B, B, P, B, B, P, B, B, P, B, Bg. One aroach is to dene A n as the total amount of data generated over a 5- frame eriod. However, using the nowledge of the framing attern, one can eect to achieve a faster detection by modifying the algorithm as follows. First the windowing arameter a = 0:5 and the memory size N = even multile of. We can treat frames from the three modes as three searate streams. Dene DN(aN; i n) as the signed Kolmogorov- Smirnov distance of the i th stream, i 2 fi; P; Bg. Let E N (n) be the absolute value of the weighted average of the three distances. That is, E N (n) = j 0 5 DB N(aN; n)j. 5 DI N (an; n)+ 4 5 DP N (an; n)+ Fig. 2 illustrates the outut of a broadcast-quality MPEG sequence, in Kb/frame. Note that the data rates of I, P, and B frames form three distinct grous. We aly the algorithm with memory size N = 20 and threshold c = 0:75. The circles at the to of the gure indicate the resulting detection oints. They corresond well with visual artitions of the sequence. Our reliminary eeriment using the scheme in Fig. shows that changing the roortion of the two riorities at these oints result in a smaller distortion than a ed-riority olicy. 5 Conclusion In this aer we roosed a sequential change oint detection algorithm that requires no assumtion of the distributions before and after the change. We roved that it is asymtotically otimal as the memory size becomes large. The algorithm is general enough to be alicable in many contets. It is resented here for the alication of detecting changes in trac models for variable-rate comression. By catering the comression arameters to each trac model, the system can achieve better utilization of the resources. Acnowledgements The authors are grateful to Profs. Peter Glynn and Vive Borar for their suggestions and Dr. Daniel Reininger for sulying the MPEG sequence. References [] B. S. Darovsy and B. E. Brodsy (987) \A Nonarametric Method for Fastest Detection of a Change in the Mean of a Random Sequence", Theory of Probability and its Alications, 987,

10 Kbits a a a a frame no. Figure 2: Outut er frame of an MPEG sequence. detection oints. The oen circles reresent the [2] B. E. Brodsy and B. S. Darhovsy (993) Nonarametric Methods in Change- Point Problems Kluwer Academic Publishers. [3] E. Carlstein (988) \Nonarametric Change-Point Estimation", Annals of Statistics, 988, vol. 6, no., [4] J. D. Gibbons and S. Charaborti (992) Nonarametric Statistical Inference, Marcel Deer. [5] H. Kroner, G. Hebuterne, P. Boyer, and A. Gravey (99) \Priority Management in ATM Switching Nodes", IEEE Journal on Selected Areas in Communications, Aril 99, vol. 9, no. 3, [6] M. Polla (985) \Otimal Detection of a Change in Distribution", Annals of Statistics, 985, [7] D. Pollard (984) Convergence of Stochastic Processes, Sringer-Verlag. [8] D. Tse, R. Gallager and J. Tsitsilis (993) \Otimal Buer Control for Variable- Rate Lossy Comression", 3st Allerton Conference.