Probabilistic Similarity Query on Dimension Incomplete Data

1. Probabilistic Similarity Query on Dimension Incomplete Data Wei Cheng1, Xiaoming Jin1, and Jian-Tao Sun2 Intelligent Data Engineering Group, School of Software, Tsinghua University1 Microsoft Research Asia2 ICDM 2009, Miami

2. Outline • Motivation & Problem • Our Solution • Experiments • Related Work • Summary and Future Work

3. Motivation • Multidimensional data are everywhere • Time series • stock data • data collected from sensor monitor • Feature vectors extracted from images or texts • …… • Similarity query on multidimensional data is important • data mining • database • information retrieval

4. Similarity query is challenging when the data is incomplete • Data incompleteness happens when: • Sensors do not work properly • Certain features are missing from particular feature vectors • ……. In order to process similarity query, imputation is necessary. (i.e. by “completing” the missing data by filling in specific values) Sensor data 2 X 3 12 … … Query Text vector 4 1 C1 Y 9 … … Image vector Z 5 2 11 … … … … … …

5. Dimension incomplete data • Dimension incomplete data satisfies: • (a) At least one of its data elements is missing; • (b) The dimension of the missing data element can not be determined. • E.g. • Observed data: • But we know the complete data should be of three dimensions • Data missing might happen on the first, second or third dimension. 3 6

6. Causes of dimension incomplete • Dimension incompleteness happens when: • Data missing happens while using the order as the implicit dimension indicator • The dimension indicator itself may also be lost • ……

7. Similarity query is more challenging when the dimension is incomplete • To measure the similarity between query and the dimension incomplete data object, we should first recover the incomplete data. • Enumerating all combination cases? – Time costing • E.g. Xobs : For an m-dimensional data object which has n elements missing, there will be Cmn cases to recover it. lost one dimension 3 6 3 possible results after data recovery X 3 6 Imputed element 3 6 X 3 6 X

8. Problem statement:

9. Two assumptions: • The probability of using each recovery result is equal. • The missing values obey normal distribution.

10. Efficient approach for PSQ-DID • A gradual refinement search strategy including two pruning methods: • Lower/upper bounds of confidence • Probability triangle inequality • Our Overall Query Process

11. Lower and upper bounds of confidence • The missing part and the observed part of the dimension incomplete data are treated separately. Since we use Euclidean distance, we have: Lower/upper bounds of the observed part, denoted by δLBobs and δUBobs. Lower/upper bounds of the missing part, denoted by δLBmis and δUBmis.

12. E.g. • Xobs=(2,8,7), Q=(1,4,5,6,7) • δ2LBobs(Q, Xobs)=(2-1)2+(8-6)2+(7-7)2 = 5 corresponding recovery version: (2,8,7,x1,x2) • For the imputed random variables Xmis={x1,x2}, If the imputation policy is using the mean value of the two adjacent observed elements as the expectation of the imputed random variables, then δ2LBmis(Q , Xmis )=(4-x1)2+(5-x2)2,(E(x1)=E(x2)=5), corresponding to Xrv =(2, , , 8, 7). 5 5

13. Lower and upper bounds of confidence • We prove that Denoted by: ,

14. Probability triangle inequality • Given a query Q and a multidimensional data object R (|Q| = |R|). For a dimension incomplete data object Xobs whose underlying complete version is X, we have: • (1) • (2) Calculated in advance and stored in the database O(|Xobs|(|Q|-|Xobs|)2) Calculated during query processing O(|Q|)

15. Experiments • Data sets: • Standard and Poor 500 index historical stock data(S&P500) (251 dimensions) • A new data set with 30 dimensions • by segmenting the S&P500 data set, resulting in 4328 data objects. • Corel Color Histogram data (IMAGE) • 68040 images • 32 dimensions • Dimension incomplete data set: • randomly removing some dimensions of each data object.

16. Experiment Setup • Ground truth: • Similarity query results on the complete data • Performance measures • Precision, recall, pruning power • Pruning power=Ndefinite/Nprocessed • Nprocessed : number of all data objects • Ndefinite: number of data objects judged as dismissals or search results by the pruner. • Query: 100 data objects randomly sampled from the data set

17. Effectiveness of probabilistic similarity query on dimension incomplete data Query precision on S&P500 data set Query recall on S&P500 data set

18. Effectiveness of probabilistic similarity query on dimension incomplete data Query precision on IMAGE data set Query recall on IMAGE data set

19. Effect of the confidence threshold • Missing ratio=0.1; r=60 for S&P500, r=0.7 for IMAGE data Confidence threshold vs precision-recall

20. Effectiveness of different pruners Pruning power of probability triangle inequality

21. Pruning Power of Four Pruners • Pruner1: probability triangle inequality using confidence lower bound confidence; Pruner2: probability triangle inequality using confidence upper bound confidence; Pruner3: confidence lower bound; Pruner4: confidence upper bound • missing ratio=10%, c= 0.1, number of assistant objects=20 Pruning power of four pruners

22. Comparison of query quality when neglecting naïve verification • For data objects that the four pruners can not judge, Pos simply outputs as query results, Neg, by contrast, judges them as dismissals. • c=0.1 Comparison of query quality

23. Performance analysis Time cost

24. Related Work • Few research papers discuss similarity search on dimension incomplete data • Incomplete data • Recovery • D. Williams et al. [ICML’05], K. Lakshminarayan et al. [Applied Intelligence’99],… • Indexing • G. Canahuate et al. [EDBT’06], B. C. Ooi et al. [VLDB’98],… • Uncertain data • J. Pei et al.[Sigmod’08], D. Burdick et al. [VLDB’05],… • Dimension incomplete data • Symbolic sequences • J. Gu et al. [DEXA’07]

25. Summary and Future Work • Problem: • Tackle the similarity query on a new uncertain form (dimension incomplete) • Solution: • Lower and upper bounds of confidence • So that we can avoid enumerate all C|Q||Xmis| recovery cases • Probability triangle inequality • Further boost the performance in query processing procedure • Future work • Other similarity measurements • Index dimension incomplete data

26. Many thanks!