From: AAAI-98 Proceedings. Copyright © 1998, AAAI (www.aaai.org). All rights reserved. Concepts From Time Series Michael T. Rosenstein and Paul R. Cohen Computer Science Department, LGRC University of Massachusetts Box 34610 Amherst, MA01003-4610 {mtr, cohen}@cs.umass.edu Abstract membership). A similar definition follows from the in- fluential work of Rosch involving categories and their This paper describes a way of extracting concepts from abstractions, called prototypes (Rosch & Lloyd 1978). streams of sensor readings. In particular, we demon- A protoype is best understood as a representative cat- strate the value of attractor reconstruction techniques egory member that may or may not correspond to any for transforming time series into clusters of points. observed instance. (In fact, the latter case is true for These clusters, in turn, represent perceptual categories the results in this paper.) Thus, whenever such abstrac- with predictive value to the agent/environment system. tions can take the place of a category, one can think of Wealso discuss the relationship between categories and concepts, with particular emphasis on class member- a concept as a category prototype plus its meaning or ship and predictive inference. predictive inferences. FUNCTIONAL EPISTEMIC Introduction Time Series Instance This research is part of an effort to explain how senso- I rimotor agents develop symbolic, conceptual thought, clustering as every human child does. As in (Cohen et al. 1996) Concept Category Discovery Cluster we are trying to "grow" an intelligent agent fi’om min- I imal beginnings by having it interact with a complex observation environment. One problem with such projects is the Outc~omes Entailments transformation of streams of sensor data into symbolic concepts, cf. the symbol grounding problem (Harnad Time Series Instance 1990). Hence, the focus of this paper is an unsuper- I vised learning mechanism for extracting concepts from recognition time series of sensor readings. Concept Cluster Prototype Category Prototype Concepts are abstractions of experience that confer Use I a predictive ability for new situations. Moreover, for prediction this project we assume a predictive semantics where Outc~ome Meaning the meaning of a concept is the predictions it makes. This working definition applies equally well to both Figure 1: Functional view of concepts and the corresponding objects and activities and depends upon a notion of epistemic terms. category. For instance, just as one can form the cat- egory TOY for objects BALL and TOP, one can also create the category PLAYfor activities BOUNCEand Figure 1 illustrates the way we realize these defini- tions of concepts. In particular, we begin with time SPIN. From the interactionist perspective (Lakoff 1984; Johnson 1987), this correspondence is a natural one. In- series data and form clusters of points that have some- deed, objects and activities seem to be duals -- linked thing in common. We then observe the subsequent out- by sensorimotor experience -- with a category of one comes, i.e., the future properties of the cluster mem- connected to a category of the other. bers. Since we equate clusters with categories and out- Epistemically, a category is simply a collection of in- comes with entailments, we also equate the correspond- stances (of objects or activities) and a concept is a cat- ing acts of clustering and observation with the discovery egory plus its entailments (or consequences of category of concepts. Concept use then follows a similar two-step procedure: (1) Recognition. Given a new time series, Copyright © 1998, American Association for Artificial find the cluster prototype most like the new instance. Intelligence (www.aaai.org). All rights reserved. (2) Prediction. Report the most likely outcome for the cluster referred to by the matching prototype. CRASHor KISS, and interaction mapsfor predicting out- comes such as CONTACTor NO-CONTACT.These repre- Experimental Environment sentations, in turn, were based on phase portraits and To demonstrate concept discovery and use, a simple ex- basins of attraction -- commontools used by dynami- perimental environment was created where two agents cists for understanding system behavior. interact based on their predetermined movementstrate- During a learning phase, a library of activity maps gies. These agents are entirely reactive and pursue or was constructed by running thousands of trials in the avoid one another using one of four basic behaviors: simulator while recording the movementpatterns of each agent as well as the outcomeof every trial. In a su- 1. NONE.The agent follows a line determined by its pervised manner, one behavior mapwas built for each of initial Velocity. Noattention is paid to the opponent’s eight agent types (four basic movementstrategies with position. three levels of intensity for AVOIDand CRASH), and one 2. AVOID.The agent attempts to move away from its interaction mapwas built for each of the 36 distinct opponent and avoid contact. pairs of behaviors (64 possible pairs minus 28 symmet- 3. CRASH.The agent attempts to move toward its op- rically equivalent pairs). With the pursuit/avoidance ponent and initiate contact. simulator, this library of activity mapsproved sufficient for recognizing the participants of a new trial and for 4. KISS. The agent slows down before making contact predicting a CONTACTor NO-CONTACToutcome. (implemented as a combination of AVOIDand CRASH). Table 1 illustrates the performance of the recogni- Figure 2 shows two examples from the pur- tion algorithm in (Rosenstein et al. 1997). Interest- suit/avoidance simulator. During each trial, the agents ingly, this confusion table demonstrates the misinter- interact only whenthey are close enough to detect one pretation of the various behaviors. For example, 66% another, as represented by the inner circle in Figure of the time the algorithm confused KISSwith one of the 2. A trial ends when the agents make contact, when CRASHmovement strategies, yet rarely mistook KISS for they get too far apart, or whenthe trial exceeds a time one of the AVOIDtypes. The reason for this sort of limit whichis large comparedto the duration of a typi- confusion is that apparent behavior is dependent upon cal interaction. The simulator implements a movement not only the agent’s predetermined movementstrategy, strategy by varying the agent’s acceleration in response but also the circumstances, i.e., initial velocity and op- to the relative position of its opponent. (See (Rosen- ponent behavior. Actually, in some situations, a KISS stein et al. 1997) for details.) In particular, movement agent reacts just like a CRASHtype, and vice versa. strategies are equations of the form These results suggest that another way to categorize a = i. f(d, [l), (1) interactions is by the nature of the interaction itself. In fact, the explicit step of behavior recognition is no where a is acceleration, d is distance between agents, longer necessary with the clustering approach described f is a function that gives one of the basic movement shortly. strategies, and i is a scale factor that represents the agent’s strength or intensity. Recognizer Response Actual N A- A AT C- C G-t- K N 40 21 7 3 26 3 0 A- 18 53 19 5 5 0 0 A 2 16 7*4 8 0 0 0 AT 0 2 8 90 0 0 0 C- 26 6 3 1 35 23 6 Figure 2: Simulator screen dump showing a representative C 2 2 0 1 21 25 21 28 trial of: (a) AVOIDVS. CRASH; (b) KISS VS. KISS. C÷ 0 0 0 0 1 9 7.’/" 13 Activity Maps K 8 1 0 0 21 20 25 25 Our previous approach to concept discovery was based on representations of dynamics called activity Table 1: Confusion table illustrating recognition perfor- maps (Rosenstein et al. 1997). In keeping with the mance with agent behaviors chosen randomly. Shown axe functional view of concepts in Figure 1, we also made response percentages, where behavior names are shortened the distinction betweentwo types of activity maps: be- to first letters only, and - and + indicate weak and strong havior maps for recognizing agent behaviors such as forms, respectively. Despite the success of activity mapsat recognizing behaviors and predicting outcomes, the approach has ESCAPE/ two drawbacks worth mentioning. First, even for a . .,-’"’""" simple simulator with just two agents, the size of the 0.8 map library scales as O(n2), where n is the number of movementstrategies. Preferably, the concept library should have size proportional to the number of needed concepts. Second, thousands of trials are necessary to build just one mapand the associated learning algo- rithm must operate in a supervised fashion. Instead, an agent should find the relevant categories for itself, without the imposed biases of an external teacher. For these reasons, we developed the current approach to CHASE concept discovery based on clustering techniques. Be- 0.2 low, we showthat few clusters and few experiences are CONTACT needed to recognize situations and predict outcomes -- ....’" / without a supervisor -- and to do so quite accurately. / I I I I 0.2 0.4 0.6 0.8 The Method of Delays obj ect-distance [ t-5 ] The formation of clusters requires a metric and so one must first devise a suitable metric space for the data. Figure 3: Two-dimensionaldelay portrait with J = 5 and In this work we makeuse of an attractor reconstruction trajectories illustrating CONTACT, ESCAPE, and CHASE. technique called the methodof delays or delay-space em- bedding. Takens proved that the method of delays pro- vides a means for mappinga time series to a topolog- this: if the state of the environmentat time t is uncer- ically equivalent spatial representation (an embedding) tain because the agent has, say, one sensor, then exam- of aa underlying dynamical system (Takens 1981). This ination of sensor readings prior to time t will reduce the mapping is accomplished by forming an m-dimensional ambiguity.