Periodic Episode Discovery Over Event Streams

Transcription

1 Periodic Episode Discovery Over Event Streams Julie Soulas 1,2(B) and Philippe Lenca 1,2 1 UMR 6285 Lab-STICC, Institut Mines-Telecom, Telecom Bretagne, Technopôle Brest Iroise CS 83818, Brest Cedex 3, France {julie.soulas,philippe.lenca}@telecom-bretagne.eu 2 Université Européenne de Bretagne, Rennes, France Abstract. Periodic behaviors are an important component of the life of most living species. Daily, weekly, or even yearly patterns are observed in both human and animal behaviors. These behaviors are searched as frequent periodic episodes in event streams. We propose an efficient algorithm for the discovery of frequent and periodic episodes. Update procedures allow us to take into account that behaviors also change with time, or because of external factors. The interest of our approach is illustrated on two real datasets. 1 Introduction The discovery of patterns based on the temporality in their occurrences is of great interest in a wide range of applications, such as social interactions analysis [9], biological sustainability studies [1], elderly people monitoring [15], mobility data analysis [2], etc. The rhythm of the patterns appearances is studied in order to determine whether the patterns occur regularly (the time gaps between the occurrences are bounded [1, 16]), periodically (some occurrences form repeating cycles of time intervals), or mostly in a specific time interval. Periodicity highlights habits. For example, Li et al. [1] studied the travel behaviors of animals, building rules such as: From 6 pm to 6 am, it has 9% probability staying at location A. Soulas et al. [15] studied the living habits of elderly people, and discovered periodic episodes as such The user has 7% probability having breakfast around 9:22 ± 4 min. The discovery of such behaviors enhances understanding of the needs of the living beings, and the factors governing their behavior. It is also useful to detect anomalies in routines due to major events such as environmental change [1] or the onset of disorders [15]. The discovery of periodic behaviors and their evolution involves three subproblems: (i) the detection of the periods (daily, weekly, etc), (ii) the discovery of periodic behaviors and (iii) the update of the periods and patterns when new data arrives. We here focus on sub-problems (ii) and (iii). Period determination has already been extensively studied [2, 1]. Moreover, the experts in the target applications domains usually have background knowledge on the interesting periods, or require particular periods to be investigated: e.g., the physicians monitoring elderly people express the need to study daily and weekly behaviors. c Springer International Publishing Switzerland 215 F. Pereira et al. (Eds.) EPIA 215, LNAI 9273, pp , 215. DOI: 1.17/

2 548 J. Soulas and P. Lenca The main contributions of the paper are: a new frequent parallel episode mining algorithm on data streams; and a heuristic for the online estimation of the periodicity of the episodes. The rest of the paper is organized as follows: section 2 presents some prominent related work. Section 3 details the proposition for frequent periodic pattern mining and updating. Experiments (section 4) on two real datasets illustrate the interest of this approach. Finally, some conclusions are drawn, and ideas for future work are presented. 2 Related Work Frequent episode mining has attracted a lot of attention since its introduction by Mannila et al. [12]. The algorithms (e.g. [11 13, 18]) differ from one another by their target episodes (sequential or parallel), their search strategies (breadth or depth first search), the considered occurrences (contiguous, minimal, overlapping, etc.), and the way they count support. However, most algorithms consider only static data. The formalism used in this paper (see section 3.1) is loosely inspired from the formalism used in [11] and [18]. With the rapidly increasing amount of data recording devices (network traffic monitoring, smart houses, sensor networks,...), stream data mining has gained major attention. This evolution led to paradigm shifts. For an extensive problem statement and review of the current trends, see [6]. In particular, item set [3,17] and episode [11,13,14] mining in streams have been investigated. The application context of [14] is close to the behavior we are searching for: the focus is set on the extraction of human activities from home automation sensor streams. However, periodicity is not taken into account. Due to their powerful descriptive and predictive capabilities, periodic patterns are studied in several domains. For instance, Kiran and Reddy [8] discover frequent and periodic patterns in transactional databases. Periodicity is also defined and used with event sequences, for example with the study of parallel episodes in home automation sensor data for the monitoring of elderly people [7, 15]. These three periodic pattern mining algorithms process only static data. To the best of our knowledge, few studies have focused on mining both frequent and periodic episodes over data streams. One can however point out some rather close studies: Li et al. [1] and Baratchi et al. [2] both use geo-spatial data in order to detect areas of interest for an individual (respectively eagles and people) and periodic movement patterns. They both also determine the period of the discovered patterns. However, their periodicity descriptions are based on single events, not episodes. 3 Frequent Periodic Pattern Discovery and Update Over Data Streams 3.1 Problem Statement Behavioral patterns are searched in the form of episodes (definition 2)inanevent (definition 1) sequence, which is processed using the classical sliding window

3 Periodic Episode Discovery Over Event Streams 549 framework (length of the window: T W ). Indeed, recent behaviors are observable in recent events. Definition 1 (Event). An event is a (e, t) pair, where e is the event label, taking values in a finite alphabet A; andt is the timestamp. Definition 2 (Episode, episode length). An episode E is a set of n event labels {e 1,..., e n } taken from the alphabet A. Thelength of episode E is n. Definition 3 (Episode occurrence, occurrence duration). An episode E = {e 1,..., e n } occurs if there are n events whose labels match the n items in E. Formally, there is an occurrence o of E at time t 1 if there exists a permutation σ on (1,...n) and n timestamps t 1... t n, such that o = (e σ(1),t 1 ),..., (e σ(n),t n ) is a subsequence of the event stream. The duration of o is δt o = t n t 1. The label order in the occurrence is not taken into account: it corresponds to the episodes referred to as parallel in the problem definition of episode mining [12]. The events making the occurrence may be interleaved with other events. The events occurring in the vicinity of each other are more likely linked to a same behavior than distant events. A stricter constraint T ep on the maximal episode duration can thus be set (T W is used otherwise). T ep exploits expert or statistical knowledge regarding the expected behavior durations. It also serves as a heuristic for the reduction of the search space (see section 3.2). Definition 4 (Minimal occurrence - MO). Let E = {e 1,...e n } be an episode, occurring on o = (e σ(1),t 1 ),...(e σ(n),t n ). o is a minimal occurrence if there is no other, shorter occurrence that occurs within the time span of o. That is to say, o = (e σ (1),t 1),...(e σ (n),t n) such that t 1 t 1, t n t n and t n t 1 <t n t 1. Definition 5 (Time queue - TQ). The time queue of an episode E (noted TQ E ) is the list containing the distinct pairs of beginning and end timestamps of its minimal occurrences. We consider here only the minimal occurrences. The support of an episode is the length of its time queue. An episode is frequent if its support is greater than a minimal support threshold S min. Minimal occurrences have convenient properties for the mining of frequent episodes, namely: An episode E has at most one time queue entry that starts (respectively finishes) at a given timestamp t (observation 1); LetEbean episode, and E a subepisode (subset) of E. For every entry in thetqofe, there is at least one entry in the TQ of E (observation 2); As a consequence, the support of E is greater or equal to this of E: the support verifies the downward closure property (observation 3); A new event (e, t) can be part of an occurrence of episode E = {e} E (where e E) if the latest entry in the TQ of E started less than T ep (or T W ) ago (observation 4a). This occurrence is minimal if the beginning of the latest entry in TQ E starts strictly after the latest entry in TQ E (observation 4b). This gives a particular importance to the recently observed episodes.

4 55 J. Soulas and P. Lenca a b c a c c b a c d b t Fig. 1. Example: a segment of an event stream Fig. 2. Example of an histogram representing the observed occurrence times for some daily habits of an elderly person living in a smart home Example. Figure 1 presents an example of a event stream segment. The current window contains 11 events (a, 5), (b, 51), etc. The last seen event is (b, 6), and the labels take values in the alphabet A = {a, b, c, d}. With T ep = 3, episode {a, c} occurs on (a, 5), (c, 52), (c, 52), (a, 53), (a, 53), (c, 54), (a, 53), (c, 55) etc: (a, 53), (c, 54) is minimal, but (a, 53), (c, 55) is not. The time queue for episode {a, b, c} is [(5, 52), (51, 53), (53, 56), (55, 57), (56, 58), (57, 6)], and its support is 6. (54, 57) is not in the TQ because it does not correspond to a MO. A one-to-one mapping between a time queue entry and a MO is not guarantied: the time queue entry (53, 56) of episode {a, b, c}, corresponds to two MO: (a, 53), (c, 54), (b, 56) and (a, 53), (c, 55), (b, 56). Periodicity. Humans and animals tend to follow routines [1, 15]. A typical example of periodic behavior is the daily occurrences of some human activities of daily living. Figure 2 presents the occurrence times of three such activities, recorded in a smart home over a six-month period (CASAS dataset, presented in section 4.1). It highlights some of the characteristics these activities may have: The occurrence times vary from one day to the next, and this variability is user- and activity-dependent: here go to bed is less variable than wake up, Activities may have several components. Here, there seems to be two meals at home a day: a breakfast around 8: and a dinner around 18:, Each component has its own preferential occurrence time (mean μ) and variability (standard deviation σ), Some occurrences do not follow the periodic patterns. This leads us to describe the periodicity of an episode as a distribution of its relative occurrence times within the period of interest (e.g 1 day, 1 week). This is done thanks to Gaussian Mixture Models (GMM), since they take into account the aforementioned characteristics.

5 Periodic Episode Discovery Over Event Streams 551 Root a d b c (52, 52), (5, 5), (51, 51), (54, 54), (53, 53), (56, 56), {a} {b} (55, 55), {c} {d} (57, 57) (6, 6) (58, 58) s = 3 s = 3 (59, 59) b c s = 4 a c a b s =1 (5, 51), (5, 52), (51, 53), (52, 53), (53, 56), {a, b} (53, 54), {a, c} {b, c} (56, 57), (57, 6) s = 5 (55, 57), (57, 58) c s = 5 b a {a, b, c} (5, 52), (51, 53), (53, 56), (55, 57), (56, 58), (57, 6) s = 6 Legend: ls { } episode support frequent episode that was recently observed Update with (b, 6): (6, 6) the TQ elements (51, 52), in bold face correspond to newly (55, 56), (56, 58), (58, 6) added elements s = 4 { } node created with the arrival of (b, 6) Fig. 3. Lattice corresponding to the example stream figure 1, when(b, 6) is the last seen event. The update when event (b, 6) arrives is highlighted in boldface. For a periodicity model {period T, n components (μ 1,σ 1 ),...(μ n,σ n )}, an occurrence is expected to occur in every time interval k T + μ i (±σ i ), such that 1 i n and for every integer k such that k T +μ i is in the current window. The quality of a periodicity description is evaluated on its accuracy, that is to say the proportion of the expected occurrences that were actually observed. 3.2 Frequent Episode Discovery and Updating In order to be habits, episodes need to be frequent. However, the periodic episodes are not necessarily the most frequent ones, which is why the support threshold should remain rather low. We propose to handle the task of frequent episode mining over an event stream thanks to a frequent episode lattice (FEL). Episode Lattice. The frequent episodes and their time queues are stored in a frequent episode lattice (FEL). The nodes in the FEL correspond either to length-1 episodes, or to frequent episodes. Length-1 episodes are kept even if they are not frequent (yet) in order to build longer episodes when they do. The parents of a node (located at depth d) correspond to its sub-episodes of length d 1, and its children to its super-episodes of length d + 1. The edges linking two episodes are indexed on the only event label that is present in the child episode but not in the parent. Each node retains the TQ of the corresponding episode and the GMM description that best fits the episode (see section 3.3). The episode lattice corresponding to the example in figure 1 is given in figure 3. In spite of its possibly big edge count, the lattice structure was chosen over the standard prefix tree because it allows faster episode retrieval and update.

6 552 J. Soulas and P. Lenca Algorithm 1. Computation of E = E 1 E 2 s TQ from the TQ of E 1 and E 2 Input: TQ 1 (resp. TQ 2 the time queue of E 1 (resp. E 2), indexed on i (resp. j) 1: i ; j ; TQ [ ]; support s 2: while i< TQ 1 and j< TQ 2 do 3: if TQ 1[i] finishes after TQ 2[j] then 4: Increment j as long as TQ 2[j] ends before TQ 1[i] 5: else 6: Increment i as long as TQ 1[i] ends before TQ 2[j] 7: start min(tq 1[i][],TQ 2[j][] 8: end min(tq 1[i][1],TQ 2[j][1] 9: if end start < T ep then 1: Add (start, end) totq; s s +1 /* New minimal occurrence */ 11: Increment the index of the TQ whose current element started earlier (both if TQ 1[i][] == TQ 2[j][]) 12: return TQ,s Update with a New Event. We keep track of the recently modified nodes (RMN, the nodes describing an episode that occurred recently, i.e. less than T ep, or T W, ago). Indeed (see observations 4a and 4b), the recent occurrences of these episodes can be extended with new, incoming events to form longer episodes. The RMN are stored in a collection of lists (nodes at depth 1, depth 2, etc). The TQ of a newly frequent length-n episode is computed thanks to the time queues of a length-(n-1) sub-episode and the length-1 episode containing the missing item, using algorithm 1. When a new event (e, t) arrives, it can be a new occurrence (and also a MO) of the length-1 episode {e}. It can also form a new MO of an episode E = E {e}, where E is a recently observed episode. The lattice update follows these steps: 1. If label e is new: create a node for episode {e} and link it to the FEL root; 2. Update the time queue of episode {e}; 3. If {e} is frequent: (a) Add it to the RMN list; (b) For each node N E in the RMN list, try to build a new occurrence of E {e}, following algorithm 2, which takes advantage of observations 1 4. If an episode E = E {e} becomes frequent, a new node N E is created, and is linked to its parents in the lattice. The parents are the nodes describing the episodes E \{e } for each e E, and are accessible via N E.parent(e ).child(e), where N E is the node for the known subset E. Since the RMN list is layered, and explored by increasing node depth, N E.parent(e ).child(e) is always created before N E tries to access it. The update process of the FEL is illustrated with the arrival of a new event (b, 6) in figure 3. (b, 6) makes {b} a frequent episode. The nodes the RMN list ({a}, {c} and {a, c}) are candidate for the extension with the (frequent) new event. This allows the investigation of episodes {a, b} (extension of {a}), {b, c} (extension of {c}), and {a, b, c} (extension of {a, c}), which indeed become frequent.

7 Periodic Episode Discovery Over Event Streams 553 Algorithm 2. RMN-based update when a new event (e, t) arrives Input: new event (e, t); recently modified node N E, characterizing episode E 1: if e E then 2: pass /* E cannot be extended with label e: E already contains it */ 3: else 4: if E.lastMO starts before t T ep then 5: Remove N E from RMN list: the last MO of E is too old to be extended 6: else 7: if N E has a child N E on label e then /* E is already frequent */ 8: if E.lastMO starts strictly after E.lastMO then /* New MO */ 9: Add new entry to N E.T Q; AddN E.T Q to the RMN list 1: else /* There is already a MO for E starting in E.lastMO.start: there cannot be another one */ 11: pass 12: else /* E may become frequent */ 13: TQ,S algorithm1(tq E,TQ E ) 14: if S S min then 15: Create node N E for E. Link it to its parents. 16: Add N E to the RMN list 17: return /* The FEL is updated with the information from event (e, t) */ Removal of Outdated Information. Events older than T W are outdated, and their influence in the FEL needs to be removed. The TQ construction makes it so that its entries are ordered by start timestamp: the entries that need to be removed are thus at the beginning of the nodes TQ. Moreover, according to observation 2, for every TQ entry (and thus every outdated entry) there is at least one (outdated) entry in the TQ of one of the parent nodes. The FEL can thus be traversed from the root using a breadth-first search algorithm, where nodes are investigated and updated only if at least one of their parents presents outdated occurrences. Episodes becoming rare are removed from the FEL. 3.3 Periodicity Discovery The periodicity of an episode is described thanks to a GMM. Each node in the FEL is associated with a GMM describing the periodicity of the episode, which is updated when new MO are observed or occurrences removed. Usually, a GMM is trained with the Expectation-Maximization algorithm [5] (EM): for each component of the GMM (the number of components being a user-given parameter), and each data point x, the probability that x was generated by the component is computed. The components characteristics (mean, standard deviation) are then tweaked to maximize the likelihood of the data point/component attribution. But streaming data may be non-stationary, the number of components may evolve, as well as their characteristics. It is not acceptable to ask the user for the number of components, especially since the suitable number depends on the considered episode. We here extend EM with heuristics for the addition, removal and merging of components.

8 554 J. Soulas and P. Lenca Algorithm 3. Overview of the periodicity update (comp=gmm component) New occ. Comp. match? Iteration of EM yes Distribution update Empty comp.? no Close comps.? no New occ.? no no yes yes Create a component Remove component Merge components yes Algorithm 3 presents the general workflow for the periodicity update. When a new MO is detected for the episode, the position of the timestamp in the period t r = timestamp modulo period is computed. If t r does not match any of the existing components, i.e. for each component (μ, σ), t r μ >σ, a new component is added. When outdated data is removed, some components lose their importance. When too rare, they are removed from the GMM. Finally, when two components (μ 1,σ 1 ), (μ 2,σ 2 ) become close to one another, i.e if μ 1 μ 2 < a (σ 1 + σ 2 ) (with a =1.5 in the experiments), the two components are merged. In the general case, GMM updates do not change much the model. Thus, when the number of components does not change, a single EM iteration is necessary to update the characteristics of the components. The interest of this approach was evaluated on synthetic data, following know mixture of gaussian models evolving with time. The heuristics allow the detection of the main trends in the data: emergence of new components, disparition of old and rare components, shiftings in characteristics of the components. 4 Experimentation A prototype was implemented in Python. It was also instrumented to record the episodes and lattice updates. The instrumentation slows down the experimentations: the execution times given in the next subsections are over-estimated. 4.1 Ambient Assisted Living Dataset The CASAS project [4] uses home automation devices to improve ageing at home. Over the years, they collected and published several datasets. We present here our experimentations on the Aruba dataset 1. The house of an elderly woman was equipped with motion detectors and temperature sensors. The obtained information was annotated with activities (11 labels, such as Sleeping, Housekeeping, etc.; 22 when dissociating the begin and end timestamp of each activity). These annotations are our events ( events, from Nov. 21 to Jun. 211). 1 number 17, consulted on Dec 4 th, 214.

9 Periodic Episode Discovery Over Event Streams Dec Feb Apr Jun Time (a) Event count in the window frequent periodic Dec Feb Apr Jun Time (b) Interesting episode count 8 s 6 s 4 s 2 s s Dec Feb Apr Jun Time (c) Cumulative execution time Fig. 4. Execution log for the CASAS Aruba annotation dataset The dataset was processed using a period of one day, a window T W of 3 weeks, a minimal support S min of 15, a maximal episode duration T ep of 3 minutes, and an accuracy threshold of 7%. The parameter setting was reinforced by a descriptive analysis of the data (e.g., it showed that most activities last less than 3 minutes). The results obtained throughout the course of the execution are given in figure 4. During the first 3 weeks, the sliding window fills with the incoming events, and the first frequent and periodic episodes appear. Then, the number of events in the window remains quite stable, but the behaviors keep evolving. The execution time (figure 4c) shows the scalability of the approach for this kind of application. The contents of the FEL in the last window is investigated, some of periodic episodes with the highest accuracy A are: {Sleeping end}: 5MO,1component,μ = 6:, σ = 2 hours, A = 1% {Sleeping end, Meal Preparation begin, Meal Preparation end, Relax begin}: 26 MO, 1 component, μ =6:, σ =1.45 hours, A = 82% {Enter Home begin, Enter Home end}: 61 MO, 1 component, μ =14:, σ =3 hours, A = 88% These patterns can be interpreted as habits: the person woke up every morning around 6:, and also had breakfast in 82% of the mornings. The third episode describes a movement pattern: the inhabitant usually goes out of home at some time (it is another episode), and comes back in the early afternoon. Figure 5 presents the influence of the minimal support S min, maximal episode duration T ep, and window length T W on the maximal size of the FEL and the execution time. In particular, it shows that the execution time is reasonable and scalable. The duration of the episodes also has a large impact on the size of the FEL. 4.2 Travian Game Dataset Travian is a web-browser game, where players, organized into alliances, fight for the fulfilling of objectives and the control of territories. The game company releases each day a snapshot of the server status: it contains information on the players (villages, alliance membership). These daily updates were collected for the 214 fr5 game round, from July 8 th to November 23 rd. We focus here on the

10 556 J. Soulas and P. Lenca Max. episode count frequent episodes periodic episodes Minimal support (a) Minimal support Time (in minutes) Total execution time Minimal support (b) Minimal support Max. episode count frequent episodes periodic episodes Maximal episode duration (hours) (c) Maximal episode duration Time (in minutes) 3 25 Total execution time Maximal episode duration (hours) (d) Maximal episode duration Max. episode count frequent episodes periodic episodes Window duration (weeks) (e) Window duration Time (in minutes) Total execution time Window duration (weeks) (f) Window duration Fig. 5. Influence of the algorithm configuration on the frequent and periodic episode counts, and on the execution time for the CASAS Aruba dataset Aug Sep Oct Nov Time (a) Event count in the window frequent periodic Aug Sep Oct Nov Time (b) Interesting episode count 15 s 1 s 5 s s Aug Sep Oct Nov Time (c) Cumulative execution time Fig. 6. Execution log for the Travian fr5 alliance membership dataset

11 Periodic Episode Discovery Over Event Streams 557 players alliance shifts: the event labels look like Player P [joined left] alliance A such events are recorded, but most labels are rare (25985 labels). The dataset was processed with a period of one week, a window T W =six weeks, a minimal support S min = 5 and a maximal episode duration T ep =1day. Figure 6 presents the evolution of the window size, episode counts, and execution time during the mining. The results are fairly different from those of the home automation dataset, but were explained by a player (picturing a domain expert). During the first 6 weeks, the window fills rapidly with events: new players register onto the game, and the diplomacy begins. The players join or switch alliances. After the 6 weeks, the event count in the window decreases with time. Several explanations: (i) the opening of a new game round (on August 22 nd ) slowed down the number of new player registrations (players tend to join the most recent game round); (ii) most players have found an alliance they like: they stop changing alliances. Until October, little frequent and periodic patterns are detected, but their number increases rapidly after that. The periodic episodes discovered in the Sep, 18 th Oct, 3 th (maximal count of periodic episodes) contain notably: {1SixCentDix8 left Vtrans, 1SixCentDix8 joined ichiefs}: 8 MO, 2 components, μ 1 = Fri. :, μ 2 = Mon. :, σ 1 = σ 2 =,A = 8% {1SixCentDix8 left ichiefs, 1SixCentDix8 joined Vtrans}: 8 MO, 2 components, μ 1 = Sat. :, μ 2 =Tue. :, σ 1 = σ 2 =,A = 8% {Jill left Bakka, Jill joined LI}: 1 MO, 2 components, μ 1 = Mon. 18:, σ 1 =1day,μ 2 = Fri. :, σ 2 =,A = 75% Some players periodically change of alliance: 1SixCentDix8 leaves Vtrans for ichiefs on Mondays and Fridays, and goes back to Vtrans one day later. Jill goes from Bakka to LI either on Mondays or Tuesdays, as well as on Fridays. This actually highlights a strategy allied alliances (ichiefs and Vtrans on one side, and Bakka and LI on the other side) have developed to share with one another the effects of artifacts owned by players 1SixCentDix8 and Jill, respectively. 5 Conclusion Behavior pattern (episode) mining over event sequences is an important data mining problem, with many applications, in particular for ambient assisted living, or wildlife behavior monitoring. Several frequent episode mining algorithms have been proposed for both static data and data streams. But while periodicity can also be an interesting characteristic for the study of behaviors, very few algorithm have addressed frequent and periodic patterns. We propose an efficient algorithm to mine frequent periodic episodes in data streams. We briefly illustrate the interest of this algorithm with two case studies. As a perspective of this work, the experiments can be extensively increased, and applied to other application domains. It also be interesting to include a period-determination algorithm in order to automatically adapt the period to each pattern. Closed episodes and non-overlaping occurrences could also be investigated.

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