Uncertain Time-Series Similarity: Return to the Basics Dallachiesa - - PowerPoint PPT Presentation

Uncertain Time-Series Similarity: Return to the Basics

Dallachiesa et al., VLDB 2012 Li Xiong, CS730

Problem

• Problem: uncertain time-series similarity
• Applications:

– location tracking of moving objects; traffic monitoring; remote sensing

• Uncertain time-series is pervasive

– Imprecision of sensor observations – Privacy preserving transformations

• Similarity matching is basis for many analysis and

mining

– Clustering – Shapelet – Motif – …

Overview

• Review of 3 state-of-art techniques for similarity

matching in uncertain time series

– MUNICH, PROUD, DUST

• Experimental comparison of the techniques for

similarity matching on 17 real (perturbed) datasets

• Two additional (simple) similarity measures

which unexpectedly outperforms the state-of-art

• Discussion of research directions

Modeling/Representing uncertain time-series

• Repeated measurements (samples)
• Probability density function (pdf) over the

uncertain values

Modeling/Representing uncertain time-series

• Repeated measurements (samples)

Modeling/Representing uncertain time-series

• Probability density function (pdf) over the

uncertain values

Similarity metrics

• Euclidean Distance (ED)
• Dynamic Time Warping (DTW)

Similarity based range query

• Range query: given a collection of time-series

C, a query sequence Q, find similar series S in C

• Probabilistic range query

State-of-the-Art

• MUNICH

– Repeated observation model

• PROUD

– Random variable model

• DUST

– Random variable model

MUNICH

• Repeated observation model
• Euclidean distance (Lp-norm) and Dynamic

Time Warping (DTW)

10 21/2/2011

MUNICH

• Materialize uncertain sequences X and Y to all

possible certain sequences

• Define the set of distances between all

possible sequences

• Uncertain distance

MUNICH

• Naïve Computation: exponential computation

cost (note the typo)

12 CAO Chen, DB Group, CSE, HKUST 21/2/2011

MUNICH

• Lower bounding and upper bounding the

distance/probability

• Approximate the samples using minimum

bounding intervals

MUNICH

• Minimum bounding interval

MUNICH

• Compute upper bound and lower bound of

distances between all possible interval sequences

MUNICH

• Recall uncertain distance and probabilistic

range query

• Compute lower bound and upper bound for Pr

MUNICH

• Pruning based on lower and upper bound
• Stepwise refinement

True Hit True Drop

PROUD

• Pdf model and Euclidean distance
• Probabilistic distance model

PROUD

• Probabilistic distance model
• The distance approaches a normal distribution

when number of time points sufficiently large (central limit theorem)

PROUD

• Recall probabilistic range query
• CDF of normal distribution expressed as error

function and compute

• Compute normalized epsilon and test

DUST

• Probability model
• DUST similarity metric
• Bayesian probability computation

DUST: A Generalized Notion of Similarity between Uncertain Time Series

Smruti R. Sarangi and Karin Murthy IBM Research Labs, Bangalore, India

Resolving the Question

• T2 should be closer to T1 than T3

– This is because it is possible that T2 and T1 are the same time series. T2 just has some additional error. – T3 and T1 can never be the same time series because the last value has a very large divergence

23

T1 T2 T3 time

value

T2 or T3 ???

Euclidean distance (EUCL) and Dynamic Time Warping (DTW)

T3 DUST T2

Extending Prior Work

24

Two time series are considered similar if : P(DIST(T1,T2) ≤ ε) ≥ τ DIST(T1, T2) = sqrt(Σi dist(T1[i], T2[i])2) dist(x,y) = |x-y| Assumption P(DIST(T1,T2) ≤ ε) = p(DIST(T1,T2) = 0) ε (irrespective of the size of ε) Prior Work

25

• log (φ(|T1[i] – T2[i]|)

Some Algebra

P(DIST(T1,T2) ≤ ε) > P(DIST(T1,T3) ≤ ε) p(DIST(T1,T2) = 0) > p(DIST(T1,T3) = 0) Πi p(dist(T1[i], T2[i]) = 0) > Πi p(dist(T1[i], T3[i]) = 0) Σi –log(p(dist(T1[i], T2[i]) = 0)) ≤ Σi –log(p(dist(T1[i], T3[i]) = 0))

≈

φ(x) = p(dist(0,x) = 0)

dist(x,y) is only dependent on |x-y| proved in the paper

dust(x,y) = -log(φ(|x-y|)) + log(φ(0)

Definition

DUST

• Compute
• Bayes Theorem
• Require

– Data distribution (uniform) – Error distribution

Comparison

• Common assumption: value at each

timestamp independent

– Correlations neglected

Comparison

MUNICH PROUD DUST Uncertainty modeling Multiple

• bservations

Random variable Random variable A priori knowledge Mean and standard deviation Data distribution and error distribution Distance metric Euclidean, DTW Euclidean DUST, Euclidean, DTW Similarity queries Probabilistic range queries Probabilistic range queries kNN queries

Experimental Study

• Data

– 17 real datasets from UCR: time series with exact values as ground truth – (not real) Perturbation with uniform, normal and exponential error distributions

• Similarity matching: probabilistic range

queries

• Metric: F1 metric
• Baseline: Euclidean distance

Moving average filters

• Uncertain moving average (UMA) – weigh less

the observations with larger errror standard deviation

• Uncertain exponential moving average

(UEMA) – weigh more the nearest neighbors

Discussion

• Experiment and Analysis track paper
• Good analytical and experimental survey
• Unexpected results

Discussion

• What’s realistic prior knowledge to assume?
• How to model correlations between time

points?