Evaluating Probabilistic Queries over Imprecise Data

Transcription

1 Evaluating Probabilistic Queries over Imprecise Data Reynold Cheng Dmitri V. Kalashnikov Sunil Prabhakar Department o Computer Science, Purdue University {ckcheng,dvk,sunil}@cs.purdue.edu ABSTRACT Many applications employ sensors or monitoring entities such as temperature and wind speed. A centralized database tracks these entities to enable query processing. Due to continuous changes in these values and limited resources (e.g., network bandwidth and battery power), it is oten ineasible to store the exact values at all times. A similar situation exists or moving object environments that track the constantly changing locations o objects. In this environment, it is possible or database queries to produce incorrect or invalid results based upon old data. However, i the degree o error (or uncertainty) between the actual value and the database value is controlled, one can place more conidence in the answers to queries. More generally, query answers can be augmented with probabilistic estimates o the validity o the answers. In this paper we study probabilistic query evaluation based upon uncertain data. A classiication o queries is made based upon the nature o the result set. For each class, we develop algorithms or computing probabilistic answers. We address the important issue o measuring the quality o the answers to these queries, and provide algorithms or eiciently pulling data rom relevant sensors or moving objects in order to improve the quality o the executing queries. Extensive experiments are perormed to examine the eectiveness o several data update policies. 1. INTRODUCTION In many applications, sensors are used to continuously track or monitor the status o an environment. Readings rom the sensors are sent back to the application, and decisions are made based on these readings. For example, temperature sensors installed in a building are used by a central air-conditioning system to decide whether the temperature o any room needs to be adjusted or to detect other problems. Sensors distributed in the environment can be used to Portions o this work were supported by NSF CAREER grant IIS , NSF grant 144-CCR, NSF grant CCR and Intel PhD Fellowship Permission to make digital or hard copies o all or part o this work or personal or classroom use is granted without ee provided that copies are not made or distributed or proit or commercial advantage and that copies bear this notice and the ull citation on the irst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speciic permission and/or a ee. SIGMOD 23, June 9-12, San Diego, CA. Copyright 23 ACM X/3/6...$5.. detect i hazardous materials are present and how they are spreading. In a moving object database, objects are constantly monitored and a central database may collect their updated locations. The ramework or many o these applications includes adatabaseorservertowhichthereadingsobtainedbythe sensors or the locations o the moving objects are sent. Users query this database in order to ind inormation o interest. Due to several actors such as limited network bandwidth to the server and limited battery power o the mobile device, it is oten ineasible or the database to contain the exact status o an entity being monitored at every moment in time. An inherent property o these applications is that readings rom sensors are sent to the central server periodically. In particular, at any given point in time, the recorded sensor reading is likely to be dierent rom the actual value. The correct value o a sensor s reading is known only when an update is received. Under these conditions, the data in the database is only an estimate o the actual state o the environment at most points in time. x1 x Recorded Temperature Possible Current Temperature (a) y y1 x x y x y (b) y Bound or Current Temperature Figure 1: Example o sensor data and uncertainty. This inherent uncertainty o data aects the accuracy o answers to queries. Figure 1(a) illustrates a query that determines the sensor (either x or y) thatreportsthelower temperature reading. Based upon the data available in the database (x and y ), the query returns x astheresult. In reality, the temperature readings could have changed to values x 1 and y 1,inwhichcase y isthecorrectanswer. This example demonstrates that the database does not always truly capture the state o the external world, and the value o the sensor readings can change without being rec- x x (c) y y

2 ognized by the database. Sistla et. al [12] identiy this type o data as a dynamic attribute, whose value changes over time even i it is not explicitly updated in the database. In this example, because the exact values o the data items are not known to the database between successive updates, the database incorrectly assumes that the recorded value is the actual value and produces incorrect results. Given the uncertainty o the data, providing meaningul answers seems to be a utile exercise. However, one can argue that in many applications, the values o objects cannot change drastically in a short period o time; instead, the degree and/or rate o change o an object may be constrained. For example, the temperature recorded by a sensor may not change by more than a degree in 5 minutes. Such inormation can help solve the problem. Consider the above example again. Suppose we can provide a guarantee that at the time the query is evaluated, the actual values monitored by x and y could be no more than some deviations, d x and d y,rom x and y,respectively,asshowninfigure1(b). Withthis inormation, we can state with conidence that x yields the minimum value. In general, the uncertainty o the objects may not allow us to identiy a single object that has the minimum value. For example, in Figure 1(c), both x and y have the possibility o recording the minimum value since the reading o x may not be lower than that o y. Asimilarsituationexistsorother types o queries such as those that request a numerical value (e.g., What is the lowest temperature reading? ). For these queries too, providing a single value may be ineasible due to the uncertainty in each object s value. Instead o providing adeiniteanswer,thedatabasecanassociatedierentlevels o conidence with each answer (e.g., as a probability) based upon the uncertainty o the queried objects. The notion o probabilistic answers to queries over uncertain data has not been well studied. Wolson et. al briely touched upon this idea [14] or the case o range queries in the context o a moving object database. The objects are assumed to move in straight lines with a known average speed. The answers to the queries consist o objects identities and the probability that each object is located in the speciied range. Cheng et. al [3] considered the problem o answering probabilistic nearest neighbor queries under a moving-object database model. In this paper we extend the notion o probabilistic queries to cover a much broader class o queries. The class o queries considered includes aggregate queries that compute answers over a number o objects. We also discuss the importance o the nature o answer requested by a query (identity o object versus the value). For example, we show that there is a signiicant dierence between the ollowing two queries: Which object has the minimum temperature? versus What is the minimum temperature?. Furthermore, we relax the model o uncertainty so that any reasonable model can be used by the application. Our techniques are applicable to the common models o uncertainty that have been proposed elsewhere. The probabilities in the answer allow the user to place appropriate conidence in the answer as opposed to having an incorrect answer or no answer at all. Depending upon the application, one may choose to report only the object with the highest probability as having the minimum value, or only those objects whose probability exceeds a minimum probability threshold. Our proposed work will be able to work with any o these models. Answering aggregate queries (such as minimum or average) ismuchmorechallengingthanrangequeries,especially in the presence o uncertainty. The answer to a probabilistic range query consists o a set o objects along with a non-zero probability that the object lies in the query range. Each object s probability is determined by the uncertainty o the object s value and the query range. However, or aggregate queries, the interplay between multiple objects is critical. The resulting probabilities are greatly inluenced by the uncertainty o attribute values o other objects. For example, in Figure 1(c) the probability that x has the minimum value is aected by the relative value and bounds or y. Aprobabilisticansweralsorelectsacertainlevelouncertainty that results rom the uncertainty o the queries object values. I the uncertainty o all (or some) o the objects was reduced (or eliminated completely), the uncertainty o the result improves. For example, without any knowledge about the value o an object, one could arguably state that it alls within a query range with 5% probability. On the other hand, i the value is known perectly, one can state with 1% conidencethattheobjecteitherallswithinor outside the query range. Thus the quality o the result is measured by degree o ambiguity in the answer. We thereore need metrics to evaluate the quality o a probabilistic answer. We propose metrics or evaluating the quality o the probabilistic answers. As we shall see, it turns out that dierent metrics are suitable or dierent classes o queries. It is possible that the quality o a query result may not be acceptable or certain applications a more deinite result may be desirable. Since the poor quality is directly related to the uncertainty in the object values, one possibility or improving the results is to delay the query until the quality improves. However this is an unreasonable solution due to the increased query response time. Instead, the database could request updates rom all objects (e.g., sensors) this solution incurs a heavy load on the resources. In this paper, we propose to request updates only rom objects that are being queried, and urthermore those that are likely to improve the quality o the query result. We present several heuristics and an experimental evaluation. These policies attempt to optimize the use o the constrained resource (e.g., network bandwidth to the server) to improve average query quality. It should be noted that the imprecision in the query answers is inherent in this problem (due to uncertainty in the actual values o the dynamic attribute), in contrast to the problem o providing approximate answers or improved perormance wherein accuracy is traded or eiciency. To sum up, the contributions o this paper are: Abroadclassiicationoprobabilisticqueriesoveruncertain data, based upon a lexible model o uncertainty; Techniques or evaluating probabilistic queries, including optimizations; Metrics or quantiying the quality o answers to probabilistic queries; Policies or improving the quality o answers to probabilistic queries under resource constraints. The rest o this paper is organized as ollows. In Section 2 we describe a general model o uncertainty, and the concept o probabilistic queries. Sections 3 and 4 discuss

3 the algorithms or evaluating dierent kinds o probabilistic queries. Section 5 discusses quality metrics that are appropriate to these queries. Section 6 proposes update policies that improve the query answer quality. We present an experimental evaluation o the eectiveness o these update policies in Section 7. Section 8 discusses related work and Section 9 concludes the paper. 2. PROBABILISTIC QUERIES In this section, we describe the model o uncertainty considered in this paper. This is a generic model, as it can accommodate a large number o application paradigms. Based on this model, we introduce a number o probabilistic queries. 2.1 Uncertainty Model One popular model or uncertainty or a dynamic attribute is that at any point in time, the actual value is within a certain bound, d o its last reported value. I the actual value changes urther than d, then the sensor reports its new value to the database and possibly changes d. For example, [14] describes a moving-object model where the location o an object is a dynamic attribute, and an object reports its location to the server i its deviation rom the last reported location exceeds a threshold. Another model assumes that the attribute value changes with known speed, but the speed may change each time the value is reported. Other models include those that have no uncertainty. For example, in [4], the exact speed and direction o movement o each object are known. This model requires updates at the server whenever an object s speed or direction changes. For the purpose o our discussion, the exact model o uncertainty is unimportant. All that is required is that at the time o query execution the range o possible values o the attribute o interest are known. We are interested in queries over some dynamic attribute, a, oasetodatabaseobjects, T. Also, weassumethata is a real-valued attribute, although our models and algorithms can be extended to other domains e.g., integer and coordinates easily. We denote the ith object o T by T i and the value o attribute a o T i by T i.a (where i =1,..., T ). Throughout this paper, we treat T i.a as a continuous random variable. The uncertainty o T i.a can be characterized by the ollowing two deinitions (we use pd to abbreviate the term probability density unction ): Deinition 1: An uncertainty interval o T i.a at time instant t, denotedbyu i(t), is an interval [l i(t),u i(t)] such that l i(t) andu i(t) arereal-valuedunctionsot, andthat the conditions u i(t) l i(t) andt i.a [l i(t),u i(t)] hold. Deinition 2: An uncertainty pd o T i.a at time t, denoted by i(x, t), is a pd o random variable T i.a, suchthat i(x, t)= i x/ U i(t). Notice that since i(x, t) isapd,ithasthepropertythat u i (t) l i (t) i(x, t)dx =1. Theabovedeinitionspeciiestheuncertainty o T i.a at time instant t in terms o a closed interval and the probability distribution o T i.a. Notice that this deinition does not speciy how the uncertainty interval evolves over time, and what the nature o the pd i(x, t) is inside the uncertainty interval. The only requirement or i(x, t) isthatitsvalueisoutsidetheuncertaintyinterval. Usually, the scope o uncertainty is determined by the last recorded value, the time elapsed since its last update, and some application-speciic assumptions. For example, one may decide that U A(t) containsallthevalueswithina distance o (t t update ) v rom its last reported value, where t update is the time that the last update was obtained, and v is the maximum rate o change o the value. One may also speciy that T i.a is uniormly distributed inside the interval, i.e., i(x, t) =1/[u i(t) l i(t)] or x U i(t), assuming that u i(t) >l i(t). It should be noted that a uniorm distribution represents the worst-case uncertainty (highest entropy) over agiveninterval. 2.2 Classiication o Probabilistic Queries We now present a classiication o probabilistic queries and examples o common representative queries or each class. We identiy two important dimensions or classiying database queries. First, queries can be classiied according to the nature o the answers. An entity-based query returns asetoobjectsthatsatisytheconditionothequery. A value-based query returns a single value, examples o which include querying the value o a particular sensor, and computing the average value o a subset o sensor readings. The second property or classiying queries is whether aggregation is involved. We use the term aggregation loosely to reer to queries where interplay between objects determines the result. In the ollowing deinitions, we use the ollowing naming convention: the irst letter is either E (or entitybased queries) or V (or value-based queries). 1. Value-based Non-Aggregate Class This is the simplest type o query in our discussions. It returns an attribute value o an object as the only answer, and involves no aggregate operators. One example o a probabilistic query or this class is the VSingleQ: Deinition 3: Probabilistic Single Value Query (VSingleQ) Given an object T k,avsingleqreturnsl, u, {p(x) x [l, u]} where l, u Rwith l u, andp(x) isthe pd o T k.a such that p(x) =whenx<lor x>u. An example o VSingleQ is What is the wind speed recorded by sensor s 22? Observe how this deinition expresses the answer in terms o a bounded probabilistic value, instead o a u single value. Also notice that p(x)dx =1. l 2. Entity-based Non-Aggregate Class This type o query returns a set o objects, each o which satisies the condition(s) o the query, independent o other objects. A typical example o this class is the ERQ: Deinition 4: Probabilistic Range Query (ERQ) Given aclosedinterval[l, u], where l, u Rand l u, anerq returns a set o tuples (T i,p i), where p i is the non-zero probability that T i.a [l, u]. An ERQ returns a set o objects, augmented with probabilities, that satisy the query interval. 3. Entity-based Aggregate Class The third class o query returns a set o objects which satisy an aggregate condition. We present the deinitions o three typical queries or this class. The irst two return objects with the minimum or maximum value o T i.a: Deinition 5: Probabilistic Minimum (Maximum) Query (EMinQ (EMaxQ)) An EMinQ (EMaxQ) returns a set R o tuples (T i,p i), where p i is the non-zero probability that T i.a is the minimum (maximum) value o a among all objects in T. Aone-dimensionalnearestneighborquery,whichreturns object(s) having a minimum absolute dierence o T i.a and

4 Query Class Entity-based Value-based Aggregate ENNQ, EMinQ, EMaxQ VAvgQ, VSumQ, VMinQ, VMaxQ Non-Aggregate ERQ VSingleQ Answer (Probabilistic) {(T i,p i ) 1 i T p i > } l, u, {p(x) x [l, u]} Answer (Non-probabilistic) {T i 1 i T } x x R Table 1: Classiication o Probabilistic Queries. agivenvalueq, isalsodeined: Deinition 6: Probabilistic Nearest Neighbor Query (ENNQ) Given a value q R,anENNQreturnsasetR o tuples (T i,p i), where p i is the non-zero probability that T i.a q is the minimum among all objects in T. Note that or all the queries we deined in this class the condition pi =1holds. T i R 4. Value-based Aggregate Class The inal class involves aggregate operators that return a single value. Example queries or this class include: Deinition 7: Probabilistic Average (Sum) Query (VAvgQ (VSumQ)) AVAvgQ(VSumQ)returnsl, u, {p(x) x [l, u]}, wherel, u Rwith l u, X is a random variable or the average (sum) o the values o a or all objects in T,andp(x) isapdox satisying p(x) =ix<l or x>u. Deinition 8: Probabilistic Minimum (Maximum) Value Query (VMinQ (VMaxQ)) AVMinQ(VMaxQ)returns l, u, {p(x) x [l, u]} where l, u Rwith l u, X is a random variable or minimum (maximum) value o a among all objects in T,andp(x) isapdox satisying p(x) = i x<uor x>l. All these aggregate queries return answers in the orm o a probabilistic distribution p(x) inaclosedinterval[l, u], such u that p(x)dx =1. l Table 1 summarizes the basic properties o the probabilistic queries discussed above. For illustrating the dierence between probabilistic and non-probabilistic queries, the last row o the table lists the orms o answers expected i probability inormation is not augmented to the result o the queries e.g., the non-probabilistic version o EMaxQ is a query that returns object(s) with maximum values based only on the recorded values o T i.a. It can be seen that the probabilistic queries provide more inormation on the answers than their non-probabilistic counterparts. Recorded Value Bound or Current Value u q l Figure 2: Illustrating the probabilistic queries. s1 s2 s3 s4 Example. We illustrate the properties o the probabilistic queries with a simple example. In Figure 2, readings o our sensors s 1,s 2,s 3 and s 4,eachwithadierentuncertainty interval, are being queried at time t. Suppose we have a set T o our database objects to record the inormation o these sensors, and each object has an attribute a to store the sensor reading. Further, assume that or each reading, its actual value has an even chance o being located at every point in its uncertainty interval, and the actual value has no chance o lying outside the interval i.e., s1 (x, t ), s2 (x, t ), s3 (x, t ), s4 (x, t )arebounded uniorm distribution unctions. Also let the uncertainty intervals rom the readings o s 1, s 2, s 3 and s 4 at time t be [l s1,u s1 ], [l s2,u s2 ], [l s3,u s3 ]and[l s4,u s4 ]respectively.a VSingleQ applied on s 4 at time t will give us the result: l s4,u s4, 1/(u s4 l s4 ). When an ERQ (represented by the interval [l, u]) is invoked at time t to ind out how likely each reading is inside [l, u], we see that the reading o s 1 is always inside the interval. It thereore has a probability o 1 or satisying the ERQ. The reading o s 4 is always outside the rectangle, thus it has a probability o o being located inside [l, u]. Since U s2 (t )andu s3 (t )partially overlap [l, u], s 2 and s 3 have some chance o satisying the query. In this example, the result o the ERQ is: {(s 1, 1), (s 2,.7), (s 3,.4)}. In the same igure, an EMinQ is issued at time t.weobserve that s 1 has a high probability o having the minimum value, because a large portion o the U s1 (t )hasasmaller value than others. The reading o s 1 has a high chance o being located in this portion because s1 (x, t) isauniorm distribution. The reading o s 4 does not have any chance o yielding the minimum value, since none o the values inside U s4 (t )issmallerthanothers. TheresultotheEMinQ or this example is: {(s 1,.7), (s 2,.2), (s 3,.1)}. On the other hand, an EMaxQ will return {(s 4, 1)} as the only result, since every value in U s4 (t )ishigherthananyreadings rom any other sensors, and we are assured that s 4 yields the maximum value. An ENNQ with a query value q is also shown, where the results are: {(s 1,.2), (s 2,.5), (s 3,.3)}. When a value-based aggregate query is applied to the scenario in Figure 2, a bounded pd p(x) isreturned. Ia VSumQ is issued, the result is a distribution in [l s1 + l s2 + l s3 + l s4,u s1 + u s2 + u s3 + u s4 ]; each x in this interval is the sum o the readings rom the our sensors. The result o a VAvgQ is a pd in [(l s1 + l s2 + l s3 + l s4 )/4, (u s1 + u s2 + u s3 + u s4 )/4]. The results o VMinQ and VMaxQ are probability distributions in [l s1,u s1 ]and[l s4,u s4 ]respectively,since only the values in these ranges have a non-zero probability value o satisying the queries. 3. EVALUATING ENTITY- QUERIES In this section we examine how the probabilistic entitybased queries introduced in the last section can be answered. We start with the discussion o an ERQ, ollowed by a more

5 complex algorithm or answering an ENNQ. We also show how the algorithm or answering an ENNQ can be easily changed or EMinQ and EMaxQ. 3.1 Evaluation o ERQ Recall that ERQ returns a set o tuples (T i,p i)wherep i is the non-zero probability that T i.a is within a given interval [l, u]. Let R be the set o tuples returned by the ERQ. The algorithm or evaluating the ERQ at time instant t is described in Figure R 2. or i 1 to T do (a) OI U i(t) [l, u] (b) i (width o OI )then i. p i OI i(x, t)dx ii. i p i then R R (T i,p i) 3. return R Figure 3: ERQ Algorithm. In this algorithm, each object in T is checked once. To evaluate p i or T i,weirstcomputetheamountoverlapping interval OI o the two intervals: U i(t) and[l, u] (Step2a). I OI has zero width, we are assured that T i.a does not lie in [l, u], and by the deinition o ERQ, T i will not be included in the result. Otherwise, we calculate the probability that T i.a is inside [l, u]byintegrating i(x, t) overoi, andputtingthe result into R i p i (Step2b). Thesetotuples(T i,p i), stored in R, arereturnedinstep Evaluation o ENNQ be applied to multiple attributes, such as coordinates. In particular, it could be a nearest-neighbor query or moving objects. Unlike the case o ERQ, we can no longer determine the probability or an object independent o the other objects. Recall that an ENNQ returns a set o tuples (T i,p i) where p i denotes the non-zero probability that T i has the minimum value o T i.a q. Let S be the set o objects to be considered by the ENNQ, and let R be the set o tuples returned by the query. The algorithm presented here consists o 4 phases: projection, pruning, bounding and evaluation. The irst three phases ilter out objects in T whose values o a have no chance o being the closest to q. The inal phase, evaluation, isthecoreooursolution:orevery object i that remains ater the irst three phases, the probability that T i.a is the nearest to q is computed. 1. Projection Phase. In this phase, the uncertainty interval o each T i.a is computed based on the application s uncertainty model. Figure 4(a) shows the last recorded values o T i.a in S at time t,andtheuncertaintyintervalsareshowninfigure4(b). 1. or i 1 to S do (a) i (q U i(t)) then N i q (b) else i. i ( q l i(t) < q u i(t) ) then N i l i(t) ii. else N i u i(t) (c) i ( q l i(t) < q u i(t) ) then F i u i(t) (d) else F i l i(t) 2. Let = min i=1,..., S F i q and m = S 3. or i 1 to m do i ( N i q >) then S S T i 4. return S T5.a 6 U5 T5.a Figure 5: Algorithm or the Pruning Phase. q T2.a 25 T1.a 2 T3.a T4.a U1 n1 q 1 (a) U2 n2 (c) n3 U n4 U4 q q T1.a U1 n1 U1 U2 T2.a n5 n1 n2 n3 n4 T3.a T4.a (b) U3 U4 U2 n2 n3 n4 U3 U4 (d) 2. Pruning Phase. Consider two uncertainty intervals U 1(t) andu 2(t). I the smallest dierence between U 1(t) andq is longer than the largest dierence between U 2(t) andq, wecanimmediately conclude that T 1 is not a part o the answer to the ENNQ: even i the actual value o T 2.a is as ar as possible rom q, T 1.a still has no chance to be closer to q than T 2.a. Based on this observation, we can eliminate objects rom T by the algorithm shown in Figure 5. In this algorithm, N i and F i record the closest and arthest possible values o T i.a to q, respectively. Steps 1(a) to 1(d) assign proper values to N i and F i.iq is inside interval U i(t), then N i is taken as point q itsel. Otherwise, N i is either l i(t) oru i(t), depending on which value is closer to q. F i is assigned in a similar manner. Ater this phase, S contains the minimal set o objects which must be considered by the query; any o them can have a value o T i.a closest to q. Figure 4(b) illustrates how this phase removes T 5,whichisirrelevanttotheENNQ,romS. Figure 4: Phases o the ENNQ algorithm. Processing an ENNQ involves evaluating the probability o the attribute a o each object T i being the closest to (nearest-neighbor o) a value q. Ingeneral, thisquerycan 3. Bounding Phase. For each object in S, weonlyneedtolookatitsportiono uncertainty interval located no arther than rom q. Wedo this conceptually by drawing a bounding interval B o length 2, centeredatq. Any portion o the uncertainty interval outside B can be ignored. Figure 4(c) shows a bounding

6 interval with length 2, andfigure4(d)illustratestheresult o this phase. The phases we have just described attempt to reduce the number o objects to be evaluated, and derive tight bounds on the range o values to be considered. 4. Evaluation Phase. Based on S and the bounding interval B, ouraimisto calculate, or each object in S, the probability that it is the nearest neighbor o q. In the pruning phase, wehave already ound N i,thepointinu i(t) nearesttoq. Letuscall N i q the near distance o T i,orn i. Denote the interval with length 2r, centeredatq by I q(r). Also, let P i(r) be the probability that T i is located inside I q(r), and pr i(r) be the pd o R i,wherer i = T i.a q. Figure 6 presents the algorithm or this phase. Note that i T i.a has no uncertainty i.e., U i(t) isexactly equal to T i.a, theevaluationphasealgorithmneedstobe modiied. Our technical report [2] discusses how this algorithm can be changed to adapt to such situations. In the rest o this section, we will explain how the evaluation phase works, assuming non-zero uncertainty. 1. R 2. Sort the elements in S in ascending order o n i,and rename the sorted elements in S as T 1,T 2,...,T S 3. n S or i 1 to S do (a) p i (b) or j i to S do i. p n j+1 pr n j i(r) ii. p i p i + p (c) R R (T i,p i) 5. return R j k=1 k i (1 P k(r)) dr Figure 6: Algorithm or the Evaluation Phase. Evaluation o P i(r) and pr i(r) To understand how the evaluation phase works, it is crucial to know how to obtain P i(r). As introduced beore, P i(r) istheprobabilitythat T i.a is located inside I q(r). We illustrate the evaluation o P i(r) infigure7. 1. i r n i return 2. i r q F i, return 1 3. O i U i(t) I q(r) 4. return Oi i(x, t)dx Figure 7: Computation o P i (r). We also need to compute pr i(r) apddenotingt i.a equals either the upper or lower bound o I q(r). I P i(r) isa dierentiable unction o r, pr i(r) isthederivativeop i(r). Evaluation o p i. We can now explain how p i,theprobability that T i.a is closest to q, iscomputed. LetProb(r) be the probability that (1) T i.a q = r and (2) T i.a q = min k=1,..., S T k.a q. ThenEquation1outlinesthestructure o our solution: p i = n i Prob(r) dr (1) The correctness o Equation 1 depends on whether it can correctly consider the probability that T i.a is the nearest neighbor or every possible value in the interval U i(t), and then sum up all those probability values. Recall that n i represents the shortest distance rom U i(t) toq, while[q,q + ] istheboundingintervalb, beyondwhichwedo not need to consider. Equation 1 expands the width o the interval I q(r) rom2n i to 2. Each value in U i(t) must thereore lie on either the upper bound or the lower bound o some I q(r), where r [n i,]. In other words, by gradually increasing the width o I q(r), we visit every value x in U i(t) and evaluates the probability that i T i.a = x, thent i.a is the nearest-neighbor o q. Wecanrewritetheaboveormula by using pr i(r) andp k (r) (wherek i), as ollows: p i = = n i Prob( T i.a q = r) Prob( T k.a q >r)dr (2) n i p i(r) S k=1 k i (1 P k (r)) dr (3) Observe that each 1 P k (r) termregisterstheprobability that T k.a is arther rom q than T i.a. Eicient Computation o p i The computation time or p i can be improved. Note that P k (r) hasavalueoi r n k.thismeanswhenr n k,1 P k (r) isalways1,and T k has no eect on the computation o p i.insteadoalways considering S 1 objects in the term o Equation 3 throughout [n i,], we may actually consider ewer objects in some ranges, resulting in a better computation speed. This can be achieved by irst sorting the objects according to their near distance rom q. Next,theintegrationinterval [n i,]isbrokendownintoanumberointervals,withend points deined by the near distance o the objects. The probability o an object having a value o a closest to q is then evaluated or each interval in a way similar to Equation 3, except that we only consider T k.a with non-zero P k (r). Then p i is equal to the sum o the probability values or all these intervals. The inal ormula or p i becomes: p i = S j=i n j+1 n j pr i(r) j k=1 k i (1 P k (r)) dr (4) Here we let n S +1 be or notational convenience. Instead o considering S 1objectsinthe term, Equation 4 only handles j 1 objectsininterval[n j,n j +1]. This optimization is shown in Figure 6. Example Let us use our previous example to illustrate how the evaluation phase works. Ater 4 objects T 1,...,T 4 were captured (Figure 4(d)), Figure 8 shows the result ater these objects have been sorted in ascending order o their near distance, withther-axis being the absolute dierence o T i.a rom q, andn 5 equals. The probability p i o each T i.a being the nearest neighbor o q is equal to the integral o the probability that T i.a is the nearest neighbor over the interval [n i,n 5]. Let us see how we evaluate uncertainty intervals when computing p 2. Equation 4 tells us that p 2 is evaluated by integrating over [n 2,n 5]. Since objects are sorted according

7 U1 U2 U3 U4 r n 1 n 2 n3 n4 n 5 = Figure 8: Illustrating the evaluation phase. to n i,wedonotneedtoconsiderall5othemthroughout [n 2,n 5]. Instead, we split [n 2,n 5]into3sub-intervals, namely [n 2,n 3], [n 3,n 4]and[n 4,n 5], and consider possibly ewer uncertainty intervals in each sub-interval. For example, in [n 2,n 3], only U 1 and U 2 need to be considered. 3.3 Evaluation o EMinQ and EMaxQ We can treat EMinQ and EMaxQ as special cases o ENNQ. In act, answering an EMinQ is equivalent to answering an ENNQ with q equal to the minimum lower bound o all U i(t) int. We can thereore modiy the ENNQ algorithm to solve an EMinQ as ollows: ater the projection phase, we evaluate the minimum value o l i(t) amongalluncertainty intervals. Then we set q to that value. We then obtain the results to the EMinQ ater we execute the rest o the ENNQ algorithm. Solving an EMaxQ is symmetric to solving an EMinQ in which we set q to the maximum o u i(t) aterthe projection phase o ENNQ. 4. EVALUATING VALUE- QUERIES In this section, we discuss how to answer the probabilistic value-based queries deined in Section Evaluation o VSingleQ Evaluating a VSingleQ is simple, since by the deinition o VSingleQ, only one object, T k,needstobeconsidered. Suppose VSingleQ is executed at time t. Then the answer returned is the uncertainty inormation o T k.a at time t, i.e., l k (t), u k (t) and{ k (x, t) x [l k (t),u k (t)]}. 4.2 Evaluation o VSumQ and VAvgQ Let us irst consider the case where we want to ind the sum o two uncertainty intervals [l 1(t),u 1(t)] and [l 2(t),u 2(t)] or objects T 1 and T 2. Notice that the values in the answer that have non-zero probability values lie in the range [l 1(t) +l 2(t),u 1(t) +u 2(t)]. For any x inside this interval, p(x) (thepdorandomvariablex = T 1.a + T 2.a) is: min{u 1 (t),x l 2 (t)} max{l 1 (t),x u 2 (t)} 1(y, t) 2(x y, t)dy (5) The lower (upper) bound o the integration interval are evaluated according to the possible minimum (maximum) value o T 1.a. We can generalize this result or summing the uncertainty intervals o T objects by picking two intervals, summing them up using the above ormula, and using the resulting interval to add to another interval. The process is repeated until we inish adding all the intervals. The resulting interval should have the ollowing orm: T [ i=1 l i(t), T i=1 u i(t)] VAvgQ is essentially the same as VSumQ except or a division by the number o objects over which the aggregation is applied. 4.3 Evaluation o VMinQ and VMaxQ To answer a VMinQ, we need to ind a lower bound l, an upper bound u, andapdp(x) wherep(x) isthepdothe minimum value o a. Recallthat(Section3.3)toansweran EMinQ, we set q to be the minimum o the lower bound o the uncertainty intervals, and then obtain a bounding interval B in the bounding phase, within which we explore the uncertainty intervals to ind the item with the minimum value. Interval B is exactly the interval [l, u], since B determines the region where the possible minimum values are. In order to answer a VMinQ, we can use the irst three phases o EMinQ (projection, pruning, bounding) to ind B as the interval [l, u]. The evaluation phase is replaced by the algorithm shown in Figure R 2. Sort the elements in S in ascending order o n i,and rename the sorted elements in S as T 1,T 2,...,T S 3. n S or j 1to S do (a) or r [n j,n j+1] do i. p ii. or i 1toj do p p + pr i(r) iii. R R (r, p) 5. return {n 1,,R} Figure 9: Evaluation Phase o VMinQ. j k=1 k i (1 P k(r)) Again, Steps 2 and 3 sort the objects in ascending order o n i. Ater these steps, B = [n 1,] represents the range o possible values. Step 4 evaluates p(r) oreachvalue r [n 1,]. Since we have sorted the uncertainty intervals in ascending order o n i,weneednotconsiderall T objects throughout [n 1,]. Speciically, or any value r in the interval [n j,n j+1], we only need to consider objects T 1,...,T j in evaluating p(r). Hence or r [n j,n j+1], p(r) isgivenby the ollowing ormula: p(r) = j i=1 (pr i(r) j k=1 k i (1 P k (r))) (6) The pair (r, p) representsr, p(r). It is inserted into the set R in Step 4(a)iii. Finally, Step 5 returns the lower bound (n 1), upper bound () otheminimumvalue,andthedistribution o the minimum values in [n 1,](R). VMaxQ is handled in an analogous ashion.

8 5. QUALITY OF PROBABILISTIC RESULTS In this section, we discuss several metrics or measuring the quality o the results returned by probabilistic queries. While other metrics (e.g., standard deviation) exist, the metrics described below measures the quality reasonably well. It is interesting to see that dierent metrics are suitable or dierent query classes. 5.1 Entity-Based Non-Aggregate Queries For queries that belong to the entity-based non-aggregate query class, it suices to deine the quality metric or each (T i,p i)individually,independentoothertuplesintheresult. This is because whether an object satisies the query or not is independent o the presence o other objects. We illustrate this point by explaining how the metric o ERQ is deined. For an ERQ with query range [l, u], the result is the best i we are sure either T i.a is completely inside or outside [l, u]. Uncertainty arises when we are less than 1% sure whether the value o T i.a is inside [l, u]. We are conident that T i.a is inside [l, u] ialargepartou i(t) overlaps[l, u] i.e.,p i is large. Likewise, we are also conident that T i.a is outside [l, u] ionlyaverysmallportionou i(t) overlaps[l, u] i.e., p i is small. The worst case happens when p i is.5, where we cannot tell i T i.a satisies the range query or not. Hence areasonablemetricorthequalityop i is: p i.5 (7).5 In Equation 7, we measure the dierence between p i and.5. Its highest value, which equals 1, is obtained when p i equals or 1, and its lowest value, which equals, occurs when p i equals.5. Hence the value o Equation 7 varies between to 1, and a large value represents good quality. Let us now deine the score o an ERQ: Score o an ERQ = 1 p i.5 R i R.5 (8) n i=1 The entropy, H(X), measures the average number o bits required to encode X, ortheamountoinormationcarried in X [11]. I H(X) equals,thereexistssomei such that p(x i)=1,andwearecertainthatx i is the message, and there is no uncertainty associated with X. Ontheother hand, H(X) attainsthemaximumvaluewhenallthemessages are equally likely, in which case H(X) equalslog 2 n. Recall that the result to the queries we deined in this class is returned in a set R consisting o tuples (T i,p i). We can view R as a set o messages, each o which has a probability p i. Moreover, the property that pi =1holds. Then H(R) measurestheuncertaintyotheanswertothese queries; the lower the value o H(R), the more certain is the answer. Bounding Interval. Uncertainty o an answer also depends on another important actor: the bounding interval B. Recall that beore evaluating one o these aggregate queries, we need to ind B that indicates all possible values we have to consider. Then we consider all the portions o uncertainty intervals that lie within B. Notethatthedecisionowhich object satisies the query is only made within this interval. Also notice that the width o B is determined by the width o the uncertainty intervals associated with objects; a large width o B is the result o large uncertainty intervals. Thereore, i B is small, it indicates that the uncertainty intervals o objects that participate in the inal result o the query are also small. In the extreme case, when the uncertainty intervals o participant objects have zero width, then the width o B is zero too. The width o B thereore gives us a good indicator o how uncertain a query answer is. U1 U2 U1 U2 U1 U2 U1 U2 where R is the set o tuples (T i.a, p i)returnedbyanerq. Essentially, Equation 8 evaluates the average over all tuples in R. Notice that in deining the metric o ERQ, Equation 7 is deined or each T i,disregardingotherobjects. Ingeneral, to deine quality metrics or the entity-based non-aggregate query class, we can deine the quality o each object individually. The overall score can then be obtained by averaging the quality value or each object. 5.2 Entity-Based Aggregate Queries Contrary to an entity-based non-aggregate query, we observe that or an entity-based aggregate query, whether an object appears in the result depends on the existence o other objects. For example, consider the ollowing two sets o answers to an EMinQ: {(T 1.a,.6), (T 2.a,.4)} and {(T 1.a,.6), (T 2.a,.3), (T 3.a,.1)}. Howcanwetellwhichanswerisbetter? We identiy two important components o quality or this class: entropy and interval width. Entropy. Let X 1,...,X n be all possible messages, with respective probabilities p(x 1),...,p(X n)suchthat 1. The entropy o a message X {X 1,...,X n} is: H(X) = n i=1 p(x i)log 2 1 p(x i) n i=1 p(xi) = (9) (a) (b) Figure 1: Illustrating how the entropy and the width o B aect the quality o answers or entity-based aggregate queries. The our igures show the uncertainty intervals (U 1 (t ) and U 2 (t ))insideb ater the bounding phase. Within the same bounding interval, (b) has a lower entropy than (a), and (d) has a lower entropy than (c). However, both (c) and (d) have less uncertainty than (a) and (b) because o smaller bounding intervals. An example at this point will make our discussions clear. Figure 1 shows our dierent scenarios o two uncertainty intervals, U 1(t )andu 2(t ), ater the bounding phase or an EMinQ. We can see that in (a), U 1(t )isthesameasu 2(t ). I we assume uniorm distribution or both uncertainty intervals, both T 1 and T 2 will have equal probability o having the minimum value o a. In(b),itisobviousthatT 2 has a much greater chance than T 1 to have the minimum value o a. UsingEquation9,wecanobservethattheanswerin(b) enjoys a lower degree o uncertainty than (a). In (c) and (d), all the uncertainty intervals are halved o those in (a) and (c) (d)

9 (b) respectively. Hence (d) still has a lower entropy value than (c). However, since the uncertainty intervals in (c) and (d) are reduced, their answers should be more certain than those o (a) and (b). Notice that the widths o B or (c) and (d) are all less than (a) and (b). The quality o entity-based aggregate queries is thus decided by two actors: (1) entropy H(R) o the result set, and (2) width o B. Their score is deined as ollows: Score o Entity, Aggr Query = H(R) width o B (1) Notice that the query answer gets a high score i either H(R) is low, or the width o B is low. In particular, i either H(R) or the width o B is zero, then H(R) =isthemaximum score. 5.3 Value-Based Queries Recall that the results returned by value-based queries are all in the orm l, u, {p(x) x [l, u]}, i.e.,aprobability distribution o values in interval [l, u]. To measure the quality o such queries, we can use the concept o dierential entropy, deinedasollows: Ĥ(X) = l u p(x)log 2 p(x)dx (11) where Ĥ(X) isthedierentialentropyoacontinuousrandom variable X with probability density unction p(x) deined in the interval [l, u] [11]. Similar to the notion o entropy, Ĥ(X) measures the uncertainty associated with the value o X. Moreover, X attains the maximum value, log 2(u l) whenx is uniormly distributed in [l, u]. When u l =1,Ĥ(X) =. Thereore,iarandomvariablehas more uncertainty than the uniorm distribution in [, 1], it will have a positive entropy value; otherwise, it will have a negative entropy value. We use the notion o dierential entropy to measure the quality o value-based queries. Speciically, we apply Equation 11 to p(x) asameasureohowmuchuncertaintyisinherent to the answer o a value-based query. The lower the dierential entropy value, the more certain is the answer, and hence the better quality is the answer. In particular, i there is a value y in [l, u] suchthatthevalueop(y) ishigh, then the entropy will be low. We now deine the score o a probabilistic value-based query: Score o a Value-Based Query = Ĥ(X) (12) The quality o a value-based query can thus be measured by the uncertainty associated with its result: the lower the uncertainty, the higher score can be obtained as indicated by Equation IMPROVING ANSWER QUALITY In this section, we discuss several update policies that can be used to improve the quality o probabilistic queries, deined in the last section. We assume that the sensors cooperate with the central server i.e., a sensor can respond to update requests rom the sensor by sending the newest value to the server, as in the system model described in [9]. Suppose ater the execution o a probabilistic query, some slack time is available or the query. The server can improve the quality o the answers to that query by requesting updates rom sensors, so that the uncertainty intervals o some sensor data are reduced, potentially resulting in an improvement o the answer quality. Ideally, a system can demand updates rom all sensors involved in the query; however, this is not practical in a limited-bandwidth environment. The issue is, thereore, to improve the quality with as ew updates as possible. Depending on the types o queries, we propose anumberoupdatepolicies. Improving the Quality o ERQ The policy or choosing objects to update or an ERQ is very simple: choose the object with the minimum value computed in Formula 7, with an attempt to improve the score o ERQ. Improving the Quality o Other Queries Several update policies are proposed or queries other than ERQ: 1. Glb RR. This policy updates the database in a roundrobin ashion using the available bandwidth i.e., it updates the data items one by one, making sure that each item gets a air chance o being rereshed. 2. Loc RR. This policy is similar to Glb RR, except that the round-robin policy is applied only to the data items that are related to the query, e.g., the set o objects with uncertainty intervals overlapping the bounding interval o an EMinQ. 3.. An object with its lower bound o the uncertainty interval equal to the lower bound o B is chosen or update. This attempts to reduce the width o B and improve the score. 4.. This heuristic simply chooses the uncertainty interval with the maximum width to update, with an attempt to reduce the overlapping o the uncertainty intervals. 5. MinExpEntropy. Another heuristic is to check, or each T i.a that overlaps B, theeecttotheentropyi we choose to update the value o T i.a. Suppose once T i.a is updated, its uncertainty interval will shrink to asinglevalue.thenewuncertaintyisthenapointin the uncertainty interval beore the update. For each value in the uncertainty interval beore the update, we evaluate the entropy, assuming that U i(t) shrinksto that value ater the update. The mean o these entropy values is then computed. The object that yields the minimum expected entropy is updated. 7. EXPERIMENTAL RESULTS In this section, we experimentally study the relative behaviors o the various update policies described above, with respect to improving the quality o the query results. We will discuss the simulation model ollowed by the results. 7.1 Simulation Model The evaluation is conducted using a discrete event simulation representing a server with a ixed network bandwidth (B messages per second) and 1 sensors. Each update rom a sensor updates the value and the uncertainty interval or the sensor stored at the server. Our uncertainty model is as ollows: An update rom sensor T i at time t update speciies the current value o the sensor, T i.a srv, and the rate, T i.r srv at which the uncertainty region (centered at T i.a srv) grows. Thusatanytimeinstant,t, ollowingthe update, the uncertainty interval (U i(t)) o sensor T i is given

10 by T i.a srv ± T i.r srv (t T i.t update ). The distribution o values within this interval is assumed to be uniorm. The actual values o the sensors are modeled as random walks within the normalized domain as in [9]. The maximum rate o change o individual sensors are uniormly distributed between and R max. At any time instant, the value o a sensor lies within its current uncertainty interval speciied by the last update sent to the server. An update rom the sensor is necessitated when a sensor is close to the edge o its current uncertainty region. Additionally, in order to avoid excessively large levels o uncertainty, an update is sent i either the total size o the uncertainty region or the time since the last update exceed threshold values. The representative experiments presented considered either EMinQ or VMinQ queries only. In each experiment the queries arrive at the server ollowing a Poisson distribution with arrival rate λ q. Each query is executed over asubsetothesensors. Thesubsetsareselectedrandomly ollowing the 8-2 hot-cold distribution (2% o the sensors are selected 8% o the time). The cardinality o each set was ixed at N sub =1. The maximum number o concurrent queries was limited to N q =1. Each query is allowed to request at most N msg updates rom sensors in order to improve the quality o its result. In order to study dierent aspects o the policies, query termination can be speciied either as (i) a ixed time interval (T active) aterwhichthequeryiscompletedeveniits requested updates have not arrived (due to network congestion) or (ii) when a target quality (G) isachieved.depending upon the policy, we study either the average achieved quality (score), the average size o the uncertainty region, or the average response time needed to achieve the desired quality. All measurements were made ater a suitable warmup period had elapsed. For airness o comparison, in each experiment, the arrival o queries as well as the changes to the sensor values were identical. Table 2 summarizes the major parameters and their deault values. The simulation parameters were chosen such that average cardinality o the result sets achieved by the best update policies was between 3 and 1. Param Deault Meaning D [, 1] Domain o attribute a R max.1 Maximum rate o change o a (sec 1 ) N q 1 Maximum # o concurrent queries λ q 2 Query arrival rate (query/sec) N sub 1 Cardinality o query subset T active 5 Query active time (sec) B 35 Network bandwidth (msg/sec) N msg 5 Maximum # o updates per query N conc 1 The # o concurrent updates per query Table 2: Simulation parameters and deault values. 7.2 Results Due to limited space, we only show the most important experimental results. Interested readers are reerred to our technical report [2] or more detailed discussions o our experiments. All igures in this section show averages. Bandwidth. Figure 11 shows scores or EMinQ achieved by various update policies or dierent values o bandwidth. Score MinExpEntr Bandwidth Figure 11: EMinQ score as unction o B The quality metric in this case is negated entropy times the size o the uncertainty region o the result set. Figure 12 is analogous to Figure 11 but shows scores or VMinQ instead o EMinQ. The score or VMinQ queries is negated continuous entropy. In Figures 11 and 12, the scores increase as bandwidth increases or all policies, approaching the perect score o zero or EMinQ. This is explained by the act that with higher bandwidth the updates requested by the queries are received aster. Thus or higher bandwidth the uncertainty regions or reshly updated sensors tend to be smaller than those using lower bandwidth. Smaller uncertainty regions translate into smaller uncertainty o the result set, and consequently higher score. The reduction in uncertainty regions with increasing bandwidth can be observed rom Figure 13. All schemes that avor updates or sensors being queried signiicantly outperorm the only scheme that ignores this inormation: Glb RR. The best perormance is achieved by the policy, which updates a sensor with the lower bound o the uncertainty region l i(t) equaltotheminimum lower bound among all sensors considered by the query. The MinExpEntropy policy showed worse results than the Min- Min and policies in Figures 11 and 13 and worse results than those o the policy or VMinQ queries, Figure 12. When comparing the and policies, the better score o the policy is explained by the act that the sensor picked or an update by the Min- Min policy tends to have large uncertainty too in act, the uncertainty interval is at least as large as the width o the bounding interval. In addition the value o its attribute a tends to have higher probability o being minimum. Response Time. Figure 14 shows response time as a unction o available bandwidth or EMinQ. Unlike the other experiments, in this experiment a query execution is stopped as soon as the goal score G (-.6) is reached. Once again the strategy showed the best results, reaching the goal score aster than the other policies. The dierence in response time is especially noticeable or smaller values o bandwidth, where it is almost twice as good as the other strategies. Predictably, the response time decreases when more bandwidth becomes available. Arrival Rate. Figures 15 and 16 show the scores achieved by EMinQ and VMinQ queries or various update policies as aunctionoqueryarrivalrateλ q.asλ q increases rom 5 to 25, more queries request updates and reduce the uncertainty