Distributed Indexing and Querying in Sensor Networks

using Statistical Models

Arnab Bhattacharyaarnabb@iitk.ac.in

Indian Institute of Technology (IIT), Kanpur

Jul 17, 2008 CS, ULB 2

Wireless sensor networks

• “Sensor” is a tiny, cheap communicating device with limited memory, communication bandwidth and battery life– Communication is precious

• Provides monitoring of physical phenomena• Wireless sensor network (WSN): a collection

of such sensors– Enables spatio-temporal monitoring of events– Inter-communication among neighboring sensors– Base station as a centralized point of entry

Jul 17, 2008 CS, ULB 3

Semantic modeling

• Uses of WSNs– How many rooms are occupied?– Is there a fire in any room?– What is the pattern of birds’ movements?

• Low-level individual sensor readings do not provide semantics

• Content summarization by modeling• Which models to use?• Where and when to model?

Jul 17, 2008 CS, ULB 4

Outline

• Semantic modeling– Which models to use?– Where and when to build the models?

• MIST: An index structure

• Query algorithms

• Experiments

• Conclusions

Jul 17, 2008 CS, ULB 5

How to model?

• Zebranet

• Track movement of zebras by velocity sensors

• Three discrete states:– Grazing (G)– Walking (W)– Fast-moving (F)

• Zebras’ behavior by state sequence– G W W W W F F G G, G G F F F W W W

Jul 17, 2008 CS, ULB 6

Statistical models• Markov Chain (MC)

– Provides inference about behavior in general

– τ: transition probabilities– π: start state probabilities

• Hidden Markov Model (HMM)– Try to infer the causes of

such behavior– ξ: emission probabilities

• Use of either model depends on the context

Zebra Mobility: HMM

Zebra Mobility: MC

Jul 17, 2008 CS, ULB 7

When and where: Queries• Identify interesting behaviors in the network

– Example: Identify all zebras (sensors) that observed the behavior pattern FFFF with likelihood > 0.8

• May denote possible predator attack

• Sequence queries– Range query: Return sensors that observed a particular

behavior with likelihood > threshold– Top-1 query: Which sensor is most likely to observe a

given behavior?

• Model queries– 1-NN query: Which sensor is most similar to a given

pattern (model)?

Jul 17, 2008 CS, ULB 8

Centralized solution

• Each sensor– Builds a model– Transmits the model to the base station (BS)

• Queries come to BS

• BS answers them– No query communication

• Each update in a sensor is transmitted– Huge update costs

Jul 17, 2008 CS, ULB 9

Slack-based centralized solution

• To save update costs• Introduce slack locally at each sensor• No update if new parameter is within slack of old

parameter– Update costs reduced

• BS knows slack– Finds range for likelihood from each sensor– If cannot be answered by cached models, then query

transmitted to the sensor– Query communication costs are introduced

Jul 17, 2008 CS, ULB 10

Outline

• Semantic modeling• MIST: An index structure

– Correlation among models– Composition of models– Hierarchical aggregation of index– Dynamic maintenance

• Query algorithms• Experiments• Conclusions

Jul 17, 2008 CS, ULB 11

MIST (Model-based Index Structure)

• Overlay a tree on the network

• Each sensor trains a model (MC/HMM) using observed sequences

• Aggregation of child models into parent using correlation among models

• Two types of composite models

• Bottom-up aggregation of index models

• Update in models handled by slack

Jul 17, 2008 CS, ULB 12

Correlation among models

• Models λ1,..., λm are (1-ε)-correlated if for all corresponding parameters σ1,...,σm:

• ε →0: High correlation– Models are similar

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Jul 17, 2008 CS, ULB 13

Outline

• Semantic modeling• MIST: An index structure

– Correlation among models– Composition of models– Hierarchical aggregation of index– Dynamic maintenance

• Query algorithms• Experiments• Conclusions

Jul 17, 2008 CS, ULB 14

Average index model• λavg maintains

– Average of all corresponding parameters:

– ε’: Correlation parameter between λavg and any λi

– βmax, βmin: maximum and minimum of all parameters from constituent models

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Jul 17, 2008 CS, ULB 15

Min-max index models

• λmin and λmax maintains

– Minimum and maximum of all corresponding parameters:

– No extra parameter

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Jul 17, 2008 CS, ULB 16

Comparison• Statistical properties

– Average: Valid statistical models• Transition and start state probabilities add up to 1

– Min-max: Pseudo-models• Probabilities, in general, do not add up to 1

• Parameters– Average: 3 extra parameters

• Total n+3 parameters

– Min-max: no extra parameter• Total 2n parameters

Jul 17, 2008 CS, ULB 17

Outline

• Semantic modeling• MIST: An index structure

– Correlation among models– Composition of models– Hierarchical aggregation of index– Dynamic maintenance

• Query algorithms• Experiments• Conclusions

Jul 17, 2008 CS, ULB 18

Hierarchical index• Average model

– Correlation parameter ε’

• Correlation gets reduced

– βmax (βmin)

• Maximum (minimum) of βmax (βmin) ’s of children

– Bounds become larger

• Min- (max-) model– Aggregation of min- (max-)

model parameters– Min (max) becomes smaller

(larger)

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Jul 17, 2008 CS, ULB 19

Dynamic maintenance

• Observations and therefore models change• Slack parameter δ• Models re-built with period d• Last model update time u• No update if λ(t+d) is within (1- δ) correlation with λ(u)

• Correlation parameter εslack maintained in the parent as

• Hierarchical index construction assumes εslack

)1)(1()1( noslackslack

Jul 17, 2008 CS, ULB 20

Outline

• Semantic modeling

• MIST: An index structure

• Query algorithms– Sequence queries– Model queries

• Experiments

• Conclusions

Jul 17, 2008 CS, ULB 21

Queries

• Sequence queries– Query sequence of symbols: q = q1q2...qk

– Range query: Return sensors that have observed q with a probability > χ

– Top-1 query: Given q, return the sensor that has the highest probability of observing q

• Model queries– Query model: Q = {π,τ}– 1-NN query: Return the sensor model that is

most similar to Q

Jul 17, 2008 CS, ULB 22

Range query• Probability of observing q from λ is

– q is of length k– σi is the ith parameter in P(q| λ)– For MC λ = {π,τ},

– For HMM, P(q| λ) is calculated as a sum along all possible state paths, each having 2k terms

• Idea is to bound every parameter σi separately

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Jul 17, 2008 CS, ULB 23

Bounds

• Average model

– Use of δ and εslack to correct for changes after the last update

– Therefore, bounds for P(q| λ) are

• Min-max model

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Jul 17, 2008 CS, ULB 24

Top-1 query

• For any internal node– Each subtree has a lower bound and an upper

bound of observing q– Prune a subtree if its lower bound is higher than

upper bound of some other subtree• Guarantees that best answer is not in this subtree

• Requires comparison of bounds across subtrees

• Pruning depends on dissimilarity of subtree models

Jul 17, 2008 CS, ULB 25

Model (1-NN) query

• Requires notion of distance between models• Euclidean distance or L2 norm

– Corresponding parameters are considered as dimensions

• Straightforward for MCs• For HMMs, state correspondence needs to be

established– Domain knowledge– Matching

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Jul 17, 2008 CS, ULB 26

Average models

• M-tree like mechanism– 1-nearest-neighbor (1-NN) query

• “Model distance” space is a metric space

• Topology is the overlaid communication tree

• Average model maintains radius as largest possible distance to any model in the subtree

• For each parameter

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Jul 17, 2008 CS, ULB 27

Min-max models

• R-tree like mechanism– 1-nearest-neighbor (1-NN) query

• “Model parameter” space is a vector space• Topology is the overlaid communication tree

• For each parameter σi, there is a lower (σimin.

(1-δ)) and an upper bound (σimax/(1-δ))

• The min-max models thus form a bounding rectangle– Similar to MBRs

Jul 17, 2008 CS, ULB 28

“Curse of dimensionality”

• Dimensionality = number of model parameters

• No “curse” for sequence queries– Each index model computes two bounds of P(q|λ)– Pruning depends on whether χ (threshold) falls

within these bounds– Bounds are real numbers between 0 and 1– Single dimensional space – probability line

• “Curse” exists for model queries– R-tree, M-tree like pruning on parameter space

Jul 17, 2008 CS, ULB 29

Outline

• Semantic modeling

• MIST: An index structure

• Query algorithms

• Experiments– Experimental setup– Effects of different parameters– Fault-tolerance

• Conclusions

Jul 17, 2008 CS, ULB 30

Optimal slack• Large slack minimizes updates but querying cost

goes up• Reverse for small slack• Optimal can be chosen by analyzing expected total

costs• Non-linear optimization

– Difficult for local nodes– Almost impossible over the entire network– Changes in the models require re-computation

• Experimental method

Jul 17, 2008 CS, ULB 31

Fault-tolerance

• Periodic heartbeat messages from child to parent– Extra messages

• When parent fails or child-parent link fails– Child finds another parent– Sends model parameters– Model, correlation, etc. is calculated afresh in parent

• When node or link comes up– Child switches to original parent– Old parent notified– Parents update their models, correlation, etc.

Jul 17, 2008 CS, ULB 32

Outline

• Semantic modeling

• MIST: An index structure

• Query algorithms

• Experiments– Experimental setup– Effects of different parameters– Fault-tolerance

• Conclusions

Jul 17, 2008 CS, ULB 33

Experimental setup• Two datasets

– Real dataset• Laboratory sensors• Temperature readings• Readings for every 30s for 10 days• 4 rooms, each having 4 sensors• States: C (cold, <25°C), P (pleasant), H (hot, >27°C)

– Synthetic dataset• Network size varied from 16 to 512• State size varied from 3 to 11• Correlation parameter ε varied from 0.001 to 0.5

• Both MCs and HMMs• Metric to measure

– Communication cost in bytes

Jul 17, 2008 CS, ULB 34

Compared techniques

• Centralized with no slack– Node transmits all updates to BS– Zero querying cost

• Centralized with slack– Node maintains slack– Query sent to sensor nodes if cached models at BS

cannot answer

• MIST schemes– Average/min-max models– With/without slack

Jul 17, 2008 CS, ULB 35

Effect of query rate

• Slack-based schemes win at small query rates• Centralized scheme with no slack is the best at

very high query rates

Jul 17, 2008 CS, ULB 36

Update costs

• No-slack schemes have almost double costs• MIST’s slack schemes are better since updates are

pruned at every level in the hierarchy

Jul 17, 2008 CS, ULB 37

Query costs

• Costs increase with decreasing correlation (1-ε)• At high correlation (low ε), no-slack schemes

(including centralized) perform better

Jul 17, 2008 CS, ULB 38

Optimal slack

• Minimum exists for MIST’s schemes• Centralized: Due to low query rate, update costs

dominated over querying costs

Jul 17, 2008 CS, ULB 39

Network size

• No-slack schemes are better• Querying cost increases due to higher bounds and

longer path lengths to leaf nodes

Jul 17, 2008 CS, ULB 40

Number of states: update costs

• Update costs increase with number of states• MIST schemes are scalable due to

hierarchical pruning

Jul 17, 2008 CS, ULB 41

Number of states: query costs

• Querying cost decreases– Each model parameter σ decreases– Probability of observing q, i.e., P(q|λ) decreases – Therefore, bounds decrease

Jul 17, 2008 CS, ULB 42

Number of states: total costs

• For sequence queries, no “curse of dimensionality”

Jul 17, 2008 CS, ULB 43

Number of states: model query

• For model queries, “curse of dimensionality” sets in– Scalable up to reasonable state sizes

Jul 17, 2008 CS, ULB 44

Fault-tolerance experiments

• Costs increase moderately due to parent switching

• Scalable with probability of failure

Jul 17, 2008 CS, ULB 45

Outline

• Semantic modeling

• MIST: An index structure

• Query algorithms

• Experiments

• Conclusions– Future work

Jul 17, 2008 CS, ULB 46

Conclusions• A hierarchical in-network index structure for sensor

networks using statistical models• Hierarchical model aggregation schemes

– Average model– Min-max models

• Queries– Sequence queries– Model query

• Experiments– Better than centralized schemes in terms of update,

querying and total communication costs– Scales well with network size and number of states

Jul 17, 2008 CS, ULB 47

Future work

• How to overlay the tree?– Similar models should be in the same subtree

• “Quality” of tree

– Distributed solutions– What happens when models are updated?

• Fault-tolerance– How to find the best parent during faults?– Whether to switch back or stay after recovery– How to replicate information in siblings?

• Deployment