A Martingale Framework for Concept Change Detection in Time-Varying Data Streams

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1 A Martigale Framework for Cocept Chage Detectio i Time-Varyig Data Streams She-Shyag Ho sho@gmu.edu Departmet of Computer Sciece, George Maso Uiversity, 4400 Uiversity Drive, Fairfax, VA USA Abstract I a data streamig settig, data poits are observed oe by oe. The cocepts to be leared from the data poits may chage ifiitely ofte as the data is streamig. I this paper, we exted the idea of testig exchageability olie (Vovk et al., 2003) to a martigale framework to detect cocept chages i time-varyig data streams. Two martigale tests are developed to detect cocept chages usig: (i) martigale values, a direct cosequece of the Doob s Maximal Iequality, ad (ii) the martigale differece, justified usig the Hoeffdig-Azuma Iequality. Uder some assumptios, the secod test theoretically has a lower probability tha the first test of rejectig the ull hypothesis, o cocept chage i the data stream, whe it is i fact correct. Experimets show that both martigale tests are effective i detectig cocept chages i time-varyig data streams simulated usig two sythetic data sets ad three bechmark data sets. 1. Itroductio A challege i miig data streams is the detectio of chages i the data-geeratig process. Recet research icludes profilig ad visualizig chages i data streams usig velocity desity estimatio (Aggarwal, 2003), but reaches o coclusio o whether a chage takes place. Fa et al. (2004) proposed chage detectio (active miig) based o error estimatio of a model of the ew data stream without kowig the true class labels. Kifer et al. (2004) proposed a chage-detectio method with statistical guaratees of the reliability of detected chages, however Appearig i Proceedigs of the 22 d Iteratioal Coferece o Machie Learig, Bo, Germay, Copyright 2005 by the author(s)/ower(s). the method is impractical for high dimesioal data streams. Besides detectig chages, chage-adaptive methods for the so-called cocept drift problem, based o a slidig widow (istace selectio) (Klikeberg & Joachims, 2000; Widmer & Kubat, 1996), istace weightig (Klikeberg, 2004), ad esemble learig (Chu et al., 2004; Kolter & Maloof, 2003; Wag et al., 2003), are also suggested. The problem of detectig chages i sequetial data was first studied by statisticias ad mathematicias. I the olie settig, data are observed oe by oe from a source. The disruptio of stochastic homogeeity of the data might sigal a chage i the data-geeratig process, which would require decisiomakig to avoid possible losses. This problem is geerally kow as a chage-poit detectio. Methods of chage detectio first appeared i the Forties based o Wald s sequetial aalysis (Wald, 1947), ad later, Page itroduced the cumulative sum method (Page, 1957). These methods are parametric ad work oly for low-dimesioal data streams. A effective chage-detectig algorithm requires that (i) the mea (or media) delay time betwee a true chage poit ad its detectio be miimal, (ii) the umber of miss detectios be miimal, ad (iii) data streams be hadled efficietly. I this paper, we propose a martigale framework that effectively ad efficietly detects cocept chages i time-varyig data streams. I this framework, whe a ew data poit is observed, hypothesis testig usig a martigale takes place to decide whether chage occurs. Two tests are show to be effective usig this framework: testig exchageability usig (i) a martigale value (Vovk et al., 2003) ad (ii) the martigale differece. The first test is a direct cosequece of the Doob s Maximal Iequality. We provide detailed justificatio for the secod test usig the Hoeffdig-Azuma Iequality. Uder some assumptios, this secod test has a much lower probability tha the first test of rejectig the ull hypothesis, o cocept chage i the

2 data stream, whe it is i fact correct. The efficiecy of the martigale tests depeds o the speed of the classifier used for the costructio of the martigale. Our martigale approach is a efficiet, oe-pass icremetal algorithm that (i) does ot require a slidig widow o the data stream, (ii) does ot require moitorig the performace of the base classifier as data poits are streamig, ad (iii) works well for high dimesioal, multi-class data stream. I Sectio 2, we review the cocept of martigale ad exchageability. I Sectio 3, we describe ad justify two tests usig martigales. I Sectio 4, we examie both tests i time-varyig data streams simulated usig two sythetic data sets ad three bechmark data sets. 2. Martigale ad Exchageability Let {Z i : 1 i < } be a sequece of radom variables. A fiite sequece of radom variables Z 1,, Z is exchageable if the joit distributio p(z 1,, Z ) is ivariat uder ay permutatio of the idices of the radom variables. A martigale is a sequece of radom variables {M i : 0 i < } such that M is a measurable fuctio of Z 1,, Z for all = 0, 1, (i particular, M 0 is a costat value) ad the coditioal expectatio of M +1 give M 0,, M is equal to M, i.e. E(M +1 M 1,, M ) = M (1) Vovk et al. (2003) itroduced the idea of testig exchageability olie usig the martigale. After observig a ew data poit, a learer outputs a positive martigale value reflectig the stregth of evidece foud agaist the ull hypothesis of data exchageability. Cosider a set of labeled examples Z = {z 1,, z 1 } = {(x 1, y 1 ),, (x 1, y 1 )} where x i is a object ad y i { 1, 1}, its correspodig label, for i = 1, 2,, 1. Assumig that a ew labeled example, z, is observed, testig exchageability for the sequece of examples z 1, z 2,, z cosists of two mai steps (Vovk et al., 2003): A. Extract a p-value p for the set Z {z } from the strageess measure deduced from a classifier The radomized p-value of the set Z {z } is defie as V (Z {z }, θ ) = #{i : α i > α } + θ #{i : α i = α } (2) where α i is the strageess measure for z i, i = 1, 2,, ad θ is radomly chose from [0, 1]. The strageess measure is a way of scorig how a data poit is differet from the rest. Each data poit z i is assiged a strageess value α i based o the classifier used (e.g. support vector machie (SVM), earest eighbor rule, ad decisio tree). I our work, the SVM is used to compute the strageess measure, which ca be either the Lagrage multipliers or the distaces from the hyperplae for the examples i Z {z }. The p-values p 1, p 2, output by the radomized p- value fuctio V are distributed uiformly i [0, 1], provided that the iput examples z 1, z 2, are geerated by a exchageable probability distributio i the iput space (Vovk et al., 2003). This property of output p-values o loger holds whe the exchageability coditio is ot satisfied (see Sectio 3). B. Costruct the radomized power martigale A family of martigales, idexed by ɛ [0, 1], ad referred to as the radomized power martigale, is defied as M (ɛ) = i=1 ( ) ɛp ɛ 1 i (3) where the p i s are the p-values output by the radomized p-value fuctio V, with the iitial martigale M (ɛ) 0 = 1. We ote that M (ɛ) = ɛp ɛ 1 M (ɛ) 1. Hece, it is ot ecessary to stored the previous p-values. I our experimets, we use ɛ = 0.92, which is withi the desirable rage where the martigale value is more sesitive to a violatio of the exchageability coditio (Vovk et al., 2003). Whe θ = 1, the p-value fuctio V is determiistic, the martigale costructed is also determiistic. We use this determiistic martigale i our justificatio for the secod test i Sectio Testig for Chage Detectio Ituitively, we assume that a sequece of data poits with a cocept chage cosists of cocateatig two data segmets, S 1 ad S 2, such that the cocepts of S 1 ad S 2 are C 1 ad C 2 respectively ad C 1 C 2. Switchig a data poit z i from S 2 to a positio i S 1 will make the data poit stads out i S 1. The exchageability coditio is, therefore, violated. Exchageability is a ecessary coditio for a coceptually stable data stream. The absece of exchageability would suggest cocept chages.

3 Whe a cocept chage occurs, the p-values output from the radomized p-value fuctio (2) become skewed ad the p-value distributio is o loger uiform. By the Kolmogorov-Smirov Test (KS-Test) 1, the p-values are show ot to be distributed uiformly after the cocept chages. The ull hypothesis the p-values output by (2) are uiformly distributed is rejected at sigificace level α = 0.05, after sufficiet umber of data poits are observed (see the example i Figure 1). The skewed p-value distributio plays a importat role i our martigale test for chage detectio as small p-values iflate the martigale values. We ote that a immediate detectio of a true chage is practically impossible. Hece, a short delay time betwee a chage ad its detectio is highly desirable. the Hoeffdig-Azuma Iequality respectively. Cosider the simple ull hypothesis H 0 : o cocept chage i the data stream agaist the alterative H 1 : cocept chage occurs i the data stream. The test cotiues to operate as log as Martigale Test 1 (MT1): OR 0 < M (ɛ) < λ (4) where λ is a positive umber. Oe rejects the ull hypothesis whe M (ɛ) λ. Martigale Test 2 (MT2): 0 < M (ɛ) M (ɛ) 1 < t (5) where t is a positive umber. Oe rejects the ull hypothesis whe M (ɛ) M (ɛ) 1 t Justificatio for Martigale Test 1 (MT1) Assumig that {M k : 0 k < } is a oegative martigale, the Doob s Maximal Iequality (Steele, 2001) states that for ay λ > 0 ad 0 <, ( ) λp max M k λ E(M ) (6) k Figure 1. The 10-dimesioal data poits simulated usig the ormally distributed clusters data geerator (see Sectio 4.1.2) are observed oe by oe from the 1st to the 2000th data poit with cocept chage startig at the 1001th data poit. The reader should ot cofuse the p- values from the KS-Test ad the p-values computed from (2) Martigale Framework for Detectig Chages I the martigale framework, whe a ew data poit is observed, hypothesis testig takes place to decide whether a cocept chage occurs i the data stream. The decisio is based o whether the exchageability coditio is violated, which, i tur, is based o the martigale value. Two hypothesis tests based o the martigale (3) are proposed based o the Doob s Maximal Iequality ad 1 Kifer et al. (2004) proposed usig Kolmogorov- Smirov Test (KS-Test) for detectig chages usig two slidig widows ad a discrepacy measure which was tested oly o 1D data stream. Hece, if E(M ) = E(M 1 ) = 1, the ( ) P max M k λ 1 k λ (7) This iequality meas that it is ulikely for ay M k to have a high value. Oe rejects the ull hypothesis whe the martigale value is greater tha λ. But there is a risk of aoucig a chage detectio whe there is o chage. The amout of risk oe is willig to take will determie what λ value to use Justificatio for Martigale Test 2 (MT2) Theorem 1 (Hoeffdig-Azuma Iequality) Let c 1,, c m be costats ad let Y 1,, Y m be a martigale differece sequece with Y k c k, for each k. The for ay t 0, ( ) m P Y k t k=1 ) 2 exp ( t2 2 c 2 k (8) To use this probability boud to justify our hypothesis test, we eed the martigale differece to be bouded, i.e. Y i = M i M i 1 K such that M i ad M i 1

4 are two arbitrary cosecutive martigale values ad K R +. This bouded differece coditio states that the process does ot make big jumps. Moreover, it is ulikely that the process waders far from its iitial poit. Hece, before usig (8) to costruct the probability upper boud to justify MT2, we eed to show that the differece betwee two cosecutive power martigale values is bouded for some fixed ɛ. As metioed earlier i Sectio 2, we use the determiistic power martigale i our proof. We set θ = 1, for Z + i the radomized p-value fuctio (2). A output p-values p is a multiple of 1 betwee 1 ad 1. The martigale differece is However, the probability upper boud (13) for MT2 also depeds o, the umber of data poits used. As icreases, the upper boud also icreases. The probability of rejectig the ull hypothesis whe it is correct icreases. To maitai a much better probability boud for larger, t ca be icreased (see Figure 2) at the expese of a higher delay time (see Sectio 4.2). d = 1 i=1 ( ) ɛp ɛ 1 i (ɛp ɛ 1 1) (9) For p = u, 1 u, if ( ) log ɛ p < exp 1 ɛ (10) we have d > 0; otherwise d < 0. The most egative d occurs whe p = 1 ad the most positive d occurs whe p = 1. This most positive value is higher tha the most egative value ad, therefore, p = 1 will be used i the bouded differece coditio. Whe m = 1, the Hoeffdig-Azuma Iequality (8) becomes ) P ( Y 1 t) 2 exp ( t2 (11) ad hece, for ay, P ( M (ɛ) M (ɛ) 1 t) 2 exp ( 2 ɛ ( 1 t 2 2c 2 1 ) ɛ 1 1 ) 2 (M (ɛ) 1 ) 2 (12) Assumig that every testig step is a ew testig step based o a ew martigale sequece, we set the previous martigale value M (ɛ) 1 = M (ɛ) 0 = 1 o the righthad side of the iequality (12). Hece, we have P ( M (ɛ) M (ɛ) 1 t) 2 exp If we oly cosider M (ɛ) ( 2 ɛ ( 1 t 2 ) ) ɛ 1 2 (13) 1 > M (ɛ) 1, the upper boud is less tha the right-had side of (13). Like MT1, oe selects t accordig to the risk oe is willig to take. Figure 2. Compariso of the upper boud of the probability of the martigale differece for some t values ad ɛ = 0.92, ad the fixed probability upper boud for the martigale value whe λ = 20 o a data stream cosistig of data poits. To have a upper boud for MT1 that matches the upper boud for a particular t value (say, 3 4) for a small (< 5000), λ has to be very large. From Figure 2, oe observes that if a slidig widow is ot used for MT2, the classifier used to extract the p- value should dyamically remove old data poits from its memory whe the upper boud exceeds a predefied value. I our experimets, we use a pseudoadaptive approach for the widow size. Our widow starts from the previous detected poit ad icreases i size util the ext chage poit is detected, as log as the probability upper boud does ot exceed a fixed value we specified for a particular chose t. Otherwise, we remove the earliest data poit from the memory. We ote that i our experimets the iterval betwee two true chage poits is small (< 2, 000) ad the performace of MT2 is ot affected by the upper boud (13) as icreases. 4. Experimets Experimets are performed to show that the two tests are effective i detectig cocept chages i timevaryig data streams simulated usig two sythetic data sets ad three bechmark data sets. The five differet simulated data streams are described i Sectio 4.1.

5 We examie the performace of both tests based o the retrieval performace idicators, recall ad precisio, ad the delay time for chage detectios for various λ ad t values o two time-varyig data streams simulated usig the two sythetic data sets. The retrieval performace idicators are defied i our cotext as: Precisio = Recall = Number of Correct Detectios Number of Detectios Number of Correct Detectios Number of True Chages Precisio is the probability that a detectio is actually correct, i.e. detectig a true chage. Recall is the probability that a chage detectio system recogizes a true chage. The delay time for a detected chage is the umber of time uits from the true chage poit to the detected chage poit, if ay. We also show that both martigale tests are feasible o high dimesioal (i) umerical, (ii) categorical, ad (iii) multi-class data streams. I the experimets, a fast adiabatic icremetal SVM (Cauweberghs & Poggio, 2000), usig the Gaussia kerel ad C = 10, is used to deduce the strageess measure for the data poits. A ecessary coditio for both tests to work well is that the classifier must have a reasoable classificatio accuracy. At a fixed ɛ, the performace of the two tests deped o the λ or t. Experimetal results are reported i Sectio Simulated Data Stream Descriptios I this subsectio, we describe how the five data streams with cocept chages are simulated by (i) usig rotatig hyperplae (Hulte et al., 2001) (Sectio 4.1.1), (ii) usig the ormally distributed clusters data geerator (NDC) (Musicat, 1998) (Sectio 4.1.2), (iii) combiig rigorm ad twoorm data sets (Breima, 1996) (Sectio 4.1.3), (iv) modifyig UCI ursery data set (Blake & Merz, 1998) (Sectio 4.1.4), ad (v) modifyig the USPS hadwritte digits data set (LeCu et al., 1989) (Sectio 4.1.5) Simulated Data Stream usig Rotatig Hyperplae A data stream is simulated by usig a rotatig hyperplae to geerate a sequece of 100,000 data poits cosistig of chages occurrig at poits (1, 000 i)+1, for i = 1, 2,, 99. First we radomly geerate 1,000 data poits with each compoet s values i the closed iterval [ 1, 1]. These data poits are labeled positive ad egative based o the followig equatio: m { < c : egative w i x i = c : positive i=1 (14) where c is a arbitrary fixed costat, x i is the compoet of a data poit, x, ad the fixed compoets, w i, of a weight vector are radomly geerated betwee -1 ad 1. Similarly, the ext 1,000 radom data poits are labeled usig (14) with a ew radomly geerated fixed weight vector. This process cotiues util we get a data stream cosistig of 100 segmets of 1,000 data poits each. Noise is added by radomly switchig the class labels of p% of the data poits. I our experimet, p = 5 ad m = Simulated Data Stream usig the ormally distributed clusters data geerator (NDC) Liearly o-separable biary-class data streams of 100,000 data poits cosistig of chages occurrig at poits (1, 000 i) + 1, for i = 1, 2,, 99 is simulated usig the NDC i R 10 with radomly geerated cluster meas ad variaces. The values for each dimesio are scaled to rage i [ 1, 1]. The geeratig process for the data stream is similar to that used for the rotatig hyperplae data stream described i Sectio Numerical High Dimesioal Datasets: Rigorm ad Twoorm We combied the rigorm (RN) (two ormal distributio, oe withi the other) ad twoorm (TN) (two overlappig ormal distributio) data sets to form a ew biary-class data stream of 20 umerical attributes cosistig of 14, 800 data poits. The 7,400 data poits from the RN are partitioed ito 8 subsets with the first 7 subsets (RN i, i = 1,, 7) cosistig of 1,000 data poits each ad RN 8 cosistig of 400 data poits. Similarly, the 7,400 data poits from TN are also partitioed ito 8 subsets with the first 7 subsets (T N i, i = 1,, 7) cosistig of 1,000 data poits each ad the T N 8 cosistig of 400 data poits. The ew data stream is a sequece of data poits arraged as follows: {RN 1 ; T N 1 ; RN 7 ; T N 7 ; RN 8 ; T N 8 } with 15 chages at data poits 1000i + 1 for i = 1,, 14, ad 14, Categorical High Dimesioal Dataset: Nursery bechmark We modified the ursery data set, which cosists of 12,960 data poits i 5 classes with 8 omial attributes, to form a ew biary-class data stream.

6 Segmet Digit 1 Digit 2 Digit 3 Total Chage Poit (0) 502 (1) 731 (2) (0) 658 (3) 652 (4) (1) 556 (5) 664 (6) (7) 542 (8) 644 (9) Table 1. Three-Digit Data Stream: TR (D): TR is the umber of data poits ad D is the true digit class of the data poits. First, we combied three classes (ot recommeded, recommeded, ad highly recommeded) ito a sigle class cosistig of 4,650 data poits labeled as egative examples. The set RN is formed by radomly selectig 4,000 out of the 4,650 data poits. The priority class cotais 4,266 data poits that are labeled as positive examples. We radomly selected 4,000 out of the 4,266 data poits to form the set P P. The special priority class, which cotais 4,044 data poits, is split ito two subsets cosistig of 2,000 data poits each, a set (SP P ) of positive examples, ad a set (SP N) of egative examples. The other 44 data poits are removed. New subsets of data poits are costructed as follows: Set A i : 500 egative examples from RN ad 500 positive examples from P P. Set B i : 500 egative examples from SP N ad 500 positive examples from P P. Set C i : 500 egative examples from RN ad 500 positive examples from SP P. The data stream S is costructed as follows: {A 1 ; B 1 ; C 1 ; A 2 ; B 2 ; C 2 ; A 3 ; B 3 ; C 3 ; A 4 ; B 4 ; C 4 } cosistig of 12,000 examples with 11 chage poits Multi-class High Dimesioal Data: three-digit data stream from USPS hadwritte digits data set. The USPS hadwritte digits data set, which cosists of 10 classes of dimesio 256 ad icludes 7,291 data poits, is modified to form a data stream as follows. There are four differet data segmets. Each segmet draws from a fixed set of three differet digits i a radom fashio. The three-digit sets chage from oe segmet to the ext. The compositio of the data stream ad groud truth for the chage poits are summarized i Table 1. We ote that the chage poits do ot appear at fixed itervals. The oe-agaist-the-rest multi-class SVM is used to extract p-values. For the three-digit data stream, three oe-agaist-therest SVM are used. Hece, three martigale values are computed at each poit to detect chage (see Figure 7). Whe oe of the martigale values is greater tha λ (or t), chage is detected Results Figure 3 ad 4 show the recall, precisio, ad delay time of the two martigale tests o the data streams simulated usig the rotatig hyperplae ad NDC respectively. As ca be see from Figure 3 ad 4 (first row), the recall is cosistetly greater tha 0.95 o both simulated data streams for various λ ad t values. Both tests recogize cocept chages with high probability. As λ or t icreases, oe observes that the precisio icreases. As λ icreases from 4 to 100, the upper boud (7) becomes tighter, decreasig from 0.25 to 0.01, for MT1. This correspods to the precisio icreasig from 0.82 to 1 (see Figure 3), decreasig the false alarm rate. O the other had, as t icreases from 1.5 to 5, precisio icreases from 0.88 to 1. The upper boud (13) for MT2 is cosistetly small as log as the data stream used for computig the martigale is short (e.g. at = 1, 000, whe t = 1.5, the upper boud is ad whe t = 5, the upper boud is ). This is a plausible explaatio for MT2 havig a higher precisio tha MT1. A similar tred also appears i simulated data streams usig the NDC (see Figure 4). To this ed, it seems that for high recall ad precisio, a large λ or t should be used. Figure 3 ad 4 (secod row) reveal, usurprisigly, that a higher precisio (usig higher λ or t) comes at the expese of a higher mea (or media) delay time for both tests. The mea (or media) delay time for the two tests do ot differ sigificatly. With a box-plot o the delay time, oe ca observe that the delay time distributio skews toward large values (i.e. small values are packed tightly together ad large values stretch out ad cover a wider rage), idepedet of the λ or t value. The delay time is very likely to be less tha the mea delay time. I real applicatios, λ or t must be chose to miimize losses (or cost) due to delay time, missed detectios, ad false alarms. Figure 5, 6, ad 7 show the feasibility of MT1 ad MT2 o high dimesioal (i) umerical (combiig rigorm ad twoorm data sets), (ii) categorical (modified UCI ursery data set), ad (iii) multi-class (modified USPS hadwritte digit data set) data streams, respectively. From the figures, oe observes that for MT2, whe chages are detected, there are more variatios i the martigale values. To this ed, oe sees

7 Figure 3. Simulated data streams usig the rotatig hyperplae. Left Colum: MT1 (with λ 1 scaled by a factor of 100 for easier visualizatio of the probability); Right Colum: MT2. First Row: Precisio ad Recall; Middle Row: Mea ad Media Delay time for various λ ad t values. that a chage is detected by either of the tests whe the martigale value deviates from its iitial value, M 0 = Coclusio I this paper, we describe a martigale framework for detectig cocept chages i time-varyig data streams based o the violatio of exchageability coditio. Two tests usig martigales to detect chages are used to demostrate this framework. Oe test usig the martigale value (MT1) for chage detectio is easily justified usig the Doob s Maximal Iequality. The other test, based o the martigale differece (MT2), is justified usig the Hoeffdig-Azuma Iequality. Uder some assumptios, MT2 theoretically has a much lower probability tha MT1 of rejectig the ull hypothesis o cocept chage i the data stream whe it is i fact correct. Our experimets show that both martigale tests detect cocept chages with high probability. Precisio icreases with the icrease of λ or t values, but at the expese of a higher mea (or media) delay time. Experimets also show the effectiveess of the two tests for cocept chage detectio o high-dimesioal (i) umerical, (ii) categorical, ad (iii) multi-class data streams. Figure 4. Simulated data streams usig the NDC data geerator. Left Colum: MT1; Right Colum: MT2. (Explaatio: See Captio for Figure 3.) Ackowledgmets The author thaks the reviewers for useful commets, Alex Gammerma ad Vladimir Vovk for the mauscript of (Vovk et al., 2005) ad useful discussios, ad Harry Wechsler for guidace ad discussios. Refereces Aggarwal, C. C. (2003). A framework for chage diagosis of data streams. Proc. ACM SIGMOD It. Cof. o Maagemet of Data (pp ). ACM. Blake, C., & Merz, C. (1998). UCI repository of machie learig databases. Breima, L. (1996). Bias, variace, ad arcig classifiers (Techical Report 460). Statistics Departmet, Uiversity of Califoria. Cauweberghs, G., & Poggio, T. (2000). Icremetal support vector machie learig. Advaces i Neural Iformatio Processig Systems 13 (pp ). MIT Press. Chu, F., Wag, Y., & Zaiolo, C. (2004). A adaptive learig approach for oisy data streams. Proc. 4th IEEE It. Cof. o Data Miig (pp ). IEEE Computer Society. Fa, W., Huag, Y.-A., Wag, H., & Yu, P. S. (2004). Active miig of data streams. Proc. 4th SIAM It. Cof. o Data Miig. SIAM.

8 Figure 5. Simulated data streams usig the Rigorm ad Twoorm data sets: The martigale values of the data stream. represet detected chage poits. Top Graph: MT1 (λ = 20), mea (media) delay time is (26) with 2 false alarms. Bottom Graph: MT2 (t = 3.5), a miss detectio at The mea (Media) delay time is (24.5). Figure 6. Simulated data streams usig the UCI ursery dataset: The martigale values of the data stream. represet detected chage poits. Top Graph: MT1 (λ = 6), the mea (media) delay time is (81). Bottom Graph: MT2 (t = 3.5), the mea (media) delay time is (92). Hulte, G., Specer, L., & Domigos, P. (2001). Miig time-chagig data streams. Proc. 7th ACM SIGKDD It. Cof. o Kowledge Discovery ad Data Miig (pp ). ACM. Kifer, D., Be-David, S., & Gehrke, J. (2004). Detectig chage i data streams. Proc. 13th It. Cof. o Very Large Data Bases (pp ). Morga Kaufma. Klikeberg, R. (2004). Learig driftig cocepts: examples selectio vs example weightig. Itelliget Data Aalysis, Special Issue o Icremetal Learig Systems capable of dealig with cocept drift, 8, Klikeberg, R., & Joachims, T. (2000). Detectig cocept drift with support vector machies. Proc. 17th It. Cof. o Machie Learig (pp ). Morga Kaufma. Kolter, J. Z., & Maloof, M. A. (2003). Dyamic weighted majority: A ew esemble method for trackig cocept drift. ICDM (pp ). IEEE Computer Society. LeCu, Y., Boser, B., Deker, J. S., Hederso, D., Howard, R. E., Hubbard, W., & Jackel, L. J. (1989). Backpropagatio applied to hadwritte zip code recogitio. Neural Computatio, 1, Musicat, D. R. (1998). Normally distributed clustered datasets. Computer Scieces Departmet, Uiversity of Wiscosi, Madiso, Page, E. S. (1957). O problem i which a chage i a parameter occurs at a ukow poit. Biometrika, 44, Steele, M. (2001). Stochastic calculus ad fiacial applicatios. Spriger Verlag. Figure 7. Simulated three-digit data stream usig the USPS hadwritte digit data set: The martigale values of the data stream. represet detected chage poits. Left Graph: MT1 (λ = 10), the delay time are 45, 99 ad 62. There is oe false alarm. Right Graph: MT2 (t = 2.5), the delay time are 88, 81 ad 73. There is oe false alarm. Vovk, V., Gammerma, A., & Shafer, G. (2005). Algorithmic learig i a radom world. Spriger. Vovk, V., Nouretdiov, I., & Gammerma, A. (2003). Testig exchageability o-lie. Proc. 20th It. Cof. o Machie Learig (pp ). AAAI Press. Wald, A. (1947). Sequetial aalysis. Wiley, N. Y. Wag, H., Fa, W., Yu, P. S., & Ha, J. (2003). Miig cocept-driftig data streams usig esemble classifiers. Proc. 9th ACM SIGKDD It. Cof. o Kowledge Discovery ad Data Miig (pp ). ACM. Widmer, G., & Kubat, M. (1996). Learig i the presece of cocept drift ad hidde cotexts. Machie Learig, 23,