1. RACE: Time Series Compression with Rate Adaptivity and Error Bound for Sensor Networks Huamin Chen, Jian Li, and Prasant Mohapatra Presenter: Jian Li

2. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Agenda Motivation Background RACE Algorithm Numerical Evaluation Conclusion

3. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Motivation Sensor Networks  Limited energy source  Limited link bandwidth, may be time-varying Monitoring process  Continuous data generation and dissemination  Data rate may be large, and time-varying How to disseminate efficiently?  Compression and aggregation

4. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Data Quality: Impact factors Sampling frequency Number of sampling nodes Data dissemination  Compression  Aggregation

5. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Why Compress? How to get “properly small” data rate?  Lower sampling frequency  Reduce the number of sensors  Lossy/lossless compression Low sampling frequency is not equivalent to (lossy) compression of higher-precision raw data.  E.g.: whether detailed features along timeline can be retained? Lossy compression is able to adapt to various link constraints.

6. Networks Lab @ UC Davis lijian@cs.ucdavis.edu But, how about Error Bound? Volatile physical process  Data rate of time series could vary in a large range  Different compressibility at different time instances Lossy compression cannot guarantee error bound, given a target output data rate Consistency of data quality?  Multihop network transmission  Multiple time series compression

7. Networks Lab @ UC Davis lijian@cs.ucdavis.edu So, Our goal is … Adaptive compression  Compress time series into CBR/LBR flow  Trade-off: network capacity v.s. data quality Improve data quality  Exploit different compressibility along timeline to achieve certain error bound  Consistency of data quality among multiple time series compression

8. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Error norm of time series Data Quality: Error Norm Normalized data element Normalized data error e i =

9. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Haar Wavelet Transformation Compute neighboring elements’ average and difference  Average: trend of time series  Difference: details of time series An example: original time series is [2, 6, 5, 11], we get transformation output [6, -2, -2, -3].

10. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Wavelet coefficient tree Time series: [3, 4, 3, 2, 6, 8, 9, 7, 2, 3, 1, 2, 10, 8, 7, 9] Output coefficients: [5.25, 0, -2.25, -3.25, 0.5, -0.5, 0.5, 0.5, -0.5, 0.5, -1, 1, -0.5, -0.5, 1, -1]

11. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Data Element Reconstruction and, C j is individual coefficient.

12. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Reconstruction: example Calculation: +(5.25) +(0) -(-2.25) +(-0.5) +(-1)  6

13. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Magnitude-based zeroing Given a threshold a  if coefficient Cj < a, then this coefficient leaf is cut off and does not participate in reconstruction process.

14. Networks Lab @ UC Davis lijian@cs.ucdavis.edu RACE Algorithm Generating gradient error tree Error-based zeroing (i.e., compression process) Smoothing error bound via patching process

15. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Gradient Error Tree Gradient Error G(V)  V is a coefficient in wavelet coefficient tree  G(V) is defined as the max error that is incurred when the sub- tree rooted from node V is cut off: Gradient Error Tree  Computed from corresponding wavelet coefficient tree

16. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Gradient Error Tree: an example Time series: [3, 4, 3, 2, 6, 8, 9, 7, 2, 3, 1, 2, 10, 8, 7, 9] Coefficients: [5.25, 0, -2.25, -3.25, 0.5, -0.5, 0.5, 0.5, -0.5, 0.5, -1, 1, -0.5, -0.5, 1, -1]

17. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Error based zeroing Using error bound as threshold value, according to gradient error tree, apply magnitude-based zeroing to wavelet coefficient tree Use symbol “t” to represent a zero-ed subtree

18. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Error based zeroing Example: threshold = 2  result in 8 symbols to encode

19. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Error based zeroing Example: threshold = 4  results in 6 symbols to encode

20. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Important Properties Error bound additivity  Multihop network transmission  Multiple time series aggregation Patch-ability  Exploiting varying compressibility of input stream along timeline  Smoothing error range of output stream

21. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Numerical evaluation Data set  Real world data from TAO project (http://www.pmel.noaa.gov/tao)http://www.pmel.noaa.gov/tao  Including air temperature and subsurface temperature at different depths Air temperature characteristics

22. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Adaptive Compression : Max normalized error

23. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Adaptive Compression: smoothed max normalized error

24. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Preservation of statistical interpretation How well to preserve multivariate correlationship? Cross correlation between variables x and y is defined as: Where d is temporal delay between x and y.

25. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Data sets Subsurface temperatures at depths 25m and 50m

26. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Cross relation under different compression ratios

27. Networks Lab @ UC Davis lijian@cs.ucdavis.edu Conclusion Rate adaptive compression scheme Improve error bound, achieving soft guarantee Preservation of multivariate correlationship