1. SAX: a Novel Symbolic Representation of Time Series
Authors
Jessica Lin
Eamonn Keogh
Li Wei
Stefano Lonardi
Presenter
Arif Bin Hossain
Slides incorporate materials kindly provided by Prof. Eamonn Keogh

2. Time Series
 A time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. [Wiki]
Example:
Economic, Sales, Stock market forecasting
EEG, ECG, BCI analysis
2000
4000
6000
8000 10 20 30

3. Problems
Join: Given two data collections, link items occurring in each
Annotation: obtain additional information from given data
Query by content: Given a large data collection, find the k most similar objects to an object of interest.
Clustering: Given a unlabeled dataset, arrange them into groups by their mutual similarity

4. Problems (Cont.)
Classification: Given a labeled training set, classify future unlabeled examples
Anomaly Detection: Given a large collection of objects, find the one that is most different to all the rest.
Motif Finding: Given a large collection of objects, find the pair that is most similar.

5. Data Mining Constraints
For example, suppose you have one gig of main memory and want to do K-means clustering…
Clustering ¼ gig of data, 100 sec
Clustering ½ gig of data, 200 sec
Clustering 1 gig of data, 400 sec
Clustering 1.1 gigs of data, few hours
Bradley, M. Fayyad, & Reina: Scaling Clustering Algorithms to Large Databases. KDD 1998: 9-15

6. Generic Data Mining
Create an approximation of the data, which will fit in main memory, yet retains the essential features of interest
Approximately solve the problem at hand in main memory
Make (hopefully very few) accesses to the original data on disk to confirm the solution

7. Some Common Approximation

8. Why Symbolic Representation?
Reduce dimension
Numerosity reduction
Hashing
Suffix Trees
Markov Models
Stealing ideas from text processing/ bioinformatics community

9. Symbolic Aggregate ApproXimation (SAX)
Lower bounding of Euclidean distance
Lower bounding of the DTW distance
Dimensionality Reduction
Numerosity Reduction
baabccbc

10. SAX
Allows a time series of arbitrary length n to be reduced to a string of arbitrary length w (w<<n)
Notations C
A time series C = c1, ….., cn Ć
A Piecewise Aggregate Approximation of a time series Ć = ć1,…ćw Ĉ
A symbolic representation of a time series
Ĉ = ĉ1, …, ĉw w
Number PAA segments representing C a
Alphabet size

11. How to obtain SAX? Step 1: Reduce dimension by PAA
Time series C of length n can be represented in a w-dimensional space by a vector Ć = ć1,…ćw
The ith element is calculated by
Reduce dimension from 20 to 5. The 2nd element will be

12. How to obtain SAX? Data is divided into w equal sized frames.
Mean value of the data falling within a frame is calculated
Vector of these values becomes the PAA C C 20 40 60 80
100
120

13. How to obtain SAX? baabccbc Step 2: Discretization c b a
Normalize Ć to have a Gaussian distribution
Determine breakpoints that will produce a equal-sized areas under Gaussian curve. - 20 40 60 80
100
120 b a c
Words: 8
Alphabet: 3
baabccbc

14. Distance Measure Given 2 time series Q and C Euclidean distance
Distance after transforming the subsequence to PAA

15. Distance Measure
Define MINDIST after transforming to symbolic representation
MINDIST lower bounds the true distance between the original time series

16. Numerosity Reduction Subsequences are extracted by a sliding window
Sequences are mostly repetitive subsequence
Sliding window finds aabbcc
If the next sequence is also aabbcc, just store the position
This optimization depends on the data, but typically yields a reduction factor of 2 or 3
Space shuttle telemetry with subsequence length 32

17. Experimental Validation
Clustering
Hierarchical
Partitional
Classification
Nearest neighbor
Decision tree
Motif discovery

18. Hierarchical Clustering
Sample dataset consists 3 decreasing trend, 3 upward shift and 3 normal classes

19. Partitional Clustering (k-means)
Assign each point to one of k clusters whose center is nearest
Each iteration tries to minimize the sum of squared intra-clustered error

20. Nearest Neighbor Classification
SAX beats Euclidean distance due to the smoothing effect of dimensional reduction

21. Decision Tree Classification
Since decision trees are expensive to use with high dimensional dataset, Regression Tree [Geurts.2001] is a better approach for data mining on time series

22. Motif Discovery
Implemented the random projection algorithm of Tompa and Buhler [ICMB2001]
Hashing subsequenced into buckets using a random subset of their features as a key

23. New Version: iSAX Use binary numbers for labeling the words
Different alphabet size(cardinality)within a word
Comparison of words with different cardinalities

24. Thank you
Questions?