SAX: a Novel Symbolic
Representation of Time Series
Presenter
Arif Bin Hossain
Authors
Jessica Lin
Eamonn Keogh
Li Wei
Stefano Lonardi
Slides incorporate materials kindly provided by Prof. Eamonn Keogh
Time Series
 A time series is a sequence of data points,
measured typically at successive times spaced at
uniform time intervals. [Wiki]
30
20
10
0
0
2000
4000
6000
 Example:
 Economic, Sales, Stock market forecasting
 EEG, ECG, BCI analysis
8000
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
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.
Data Mining Constraints
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
For example, suppose
you have one gig
of main memory
and want to do Kmeans clustering…
Bradley, M. Fayyad, & Reina: Scaling Clustering Algorithms to Large Databases. KDD 1998: 9-15
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
Some Common Approximation
Why Symbolic Representation?
• Reduce dimension
• Numerosity reduction
• Hashing
• Suffix Trees
• Markov Models
• Stealing ideas from text processing/ bioinformatics
community
Symbolic Aggregate ApproXimation (SAX)
• Lower bounding of Euclidean distance
• Lower bounding of the DTW distance
• Dimensionality Reduction
• Numerosity Reduction
baabccbc
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
How to obtain SAX?
 Step 1: Reduce dimension by PAA
Time series C of length n can be represented in a wdimensional space by a vector Ć = ć1,…ćw
 The ith element is calculated by

ci 

ni
w
w
n
j
c
j
n ( i 1) 1
w
Reduce dimension from 20 to 5. The 2nd element will be
5 8
C2 
Cj

20 j 5
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
0
20
40
60
80
100
120
How to obtain SAX?
 Step 2: Discretization
 Normalize Ć to have a Gaussian distribution
 Determine breakpoints that will produce a equal-sized areas
under Gaussian curve.
c
c
c
Words: 8
Alphabet: 3
b
b
a
0
20
b
a
40
60
80
100
baabccbc
120
Distance Measure
 Given 2 time series Q and C
 Euclidean distance

Distance after transforming the subsequence to PAA
Distance Measure
 Define MINDIST after transforming to symbolic
representation
 MINDIST lower bounds the true distance between
the original time series
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
Experimental Validation
 Clustering
 Hierarchical
 Partitional
 Classification
 Nearest neighbor
 Decision tree
 Motif discovery
Hierarchical Clustering
 Sample dataset consists 3 decreasing trend, 3
upward shift and 3 normal classes
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
Nearest Neighbor Classification
 SAX beats Euclidean distance due to the smoothing
effect of dimensional reduction
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
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
New Version: iSAX
 Use binary numbers for labeling the words
 Different alphabet size(cardinality)within a word
 Comparison of words with different cardinalities
Thank you
Questions?