SAX-TimeSeries

Copyright © 2014 Oracle and/or its affiliates. All rights reserved. | Oracle Confidential – Internal/Restricted/Highly Restricted 1
Symbolic Representations of Time Series
- Nikita

Copyright © 2014 Oracle and/or its affiliates. All rights reserved. |
Time Series
 A time series is a sequence of pairs
- Each pair consists of a Time Index and a Value
- The Time Index may be implied if there is a constant difference
between values
 The time series can be segmented into “Windows” which represent
the time series between 2 Time Indices
 Symbols can represent Windows. Because symbols in a Finite
Symbol Space have a probability, we can think of the probability of a
time series. Symbols are easy to store and manipulate – each
symbol can be represented as an integer
Oracle Confidential – Internal/Restricted/Highly Restricted 2
0 2000 4000 6000 8000
0
10
20
30

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

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

The Symbolic Representation Of Time Series
A number of algorithms exist to represent time series as symbols in a Finite
Symbol Space
 These algorithms are often though of as “Feature Reducers”
Self Organizing Maps are a traditional form of Feature Reducer
SAX (Symbolic Aggregate approXimation) is another, designed specifically for
time series
There are many other ways to reduce a time series to symbol
 As long as the symbol is drawn from a Finite Symbol Space, the technique
described here will work

What is SAX?
 SAX is a methodology for reducing a time series window to a symbol
 The technique was developed by Dr. Eamonn Keogh et al. at the University of
California at Riverside in the early 2000’s
 It has since drawn a great deal of attention in the world of time series analysis
 Allows a time series of arbitrary length n to be reduced to a string of arbitrary
length w (w<<n)
 SAX is the first symbolic representation for time series that allows for
dimensionality reduction and indexing with a lower-bounding distance measure.

What is lower bounding?
 Lower bounding means that for all Q and S, we have DLB(Q’,S’) <= D(Q,S).

What’s a SAX Word?
A SAX word is the symbol generated by the SAX algorithm
It is defined by a SAX Alphabet and a length
 The SAX Alphabet is traditionally represented by letters, and its components
are referred to as “SAX Letters”
 The size of the alphabet is typically small – this is particularly important for
anomaly detection
When we write out a description of a SAX word, we typically use a string like
representation, such as “abcdefg”
 SAX letters don’t have to be letters – implementations often use numbers
based at zero, however, we often display them as letters

Symbolic Aggregate ApproXimation
Lower bounding of Euclidean distance
Dimensionality Reduction
Numerosity Reduction
baabccbc

Normalization of Time Series
 Normalization to Zero Mean and Unit of Energy.
 The procedure ensures, that all elements of the input vector are transformed
into the output vector whose mean is approximately 0 while the standard
deviation is in a range close to 1. The formula behind the transform is shown
below:
 Z-normalization is an essential preprocessing step which allows an algorithm to
focus on the structural similarities/dissimilarities rather than on the amplitude.
In order to make meaningful comparisons between two time series, both must
be normalized.

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
0
--
0 20 40 60 80 100 120
bb
b
a
c
c
c
a

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
 

i
ij
jn
w
i
w
n
w
n
cc
1)1(

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
0
--
0 20 40 60 80 100 120
bb
b
a
c
c
c
a
baabccbc
Words: 8
Alphabet: 3

Gaussian distribution
 Most "natural" distributions
 A Gaussian process uses lazy learning and a measure of the similarity between
points (this is the kernel function) to predict the value for an unseen point from
training data
Ref : https://www.isixsigma.com/tools-templates/normality/tips-recognizing-and-transforming-
non-normal-data/

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
baabccbcCˆ
babcaccaQˆ
  

w
i iiw
n
cqdistCQMINDIST 1
2
)ˆ,ˆ()ˆ,ˆ(
dist() can be implemented using a
table lookup.

Novelty Detection
 Fault detection
 Interestingness detection
 Anomaly detection
 Surprisingness detection

SAX-TimeSeries

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