1. Time Series Data Analysis - II
Yaji Sripada

2. In this lecture you learn
Structural representations of time series
SAX
Computing SAX
Data analysis using SAX
Visualization using SAX
Dept. of Computing Science, University of Aberdeen

3. Dept. of Computing Science, University of Aberdeen
Introduction
Time series exhibit an internal structure
Elements of this structure have domain specific meanings
E.g. the spikes on the gas turbine data (from last lecture) have domain specific meaning
The structural elements of a time series are usually approximations (abstractions) of the original data
Experts in any domain reason in terms of these abstractions and not in terms of the original time series
Understanding time series = understanding their structure
Dept. of Computing Science, University of Aberdeen

4. Several structural representations
Time series can be represented in terms of
Linear segments (we already saw this last week)
Aggregate Approximations (will study in this lecture)
Non-linear segments (Not in this course)
Wavelets (involve complex mathematics – not in this course)
And many more
The primary motivation behind creating the above structural representations is time series data mining
Dept. of Computing Science, University of Aberdeen

5. Which structure is the most useful?
All these structural representations are useful
may be more used in some application domains than others
A good representation exhibits meaningful structure
But meaning is attributed to a structure based on domain knowledge and user tasks
This means, select a representation that helps easy computation of meaning
Our approach to selecting the right representation
Based on the domain KA we learn the trends and patterns that are meaningful
Select one or more representations that facilitate the computation of required trends and patterns
Dept. of Computing Science, University of Aberdeen

6. Symbolic Aggregate Approximation (SAX)
A recently developed symbolic representation of time series is claimed to facilitate easy pattern computation
is the main SAX page
We introduced this representation in the last lecture
We study how to create this representation in this lecture because it allows
Novel data analysis of time series and
Novel visualization of time series
We will study briefly data analysis and visualization with SAX
The above link has all the required details for further study
Dept. of Computing Science, University of Aberdeen

7. Dept. of Computing Science, University of Aberdeen
Creating SAX
Input
Real valued time series (blue curve)
Output
Symbolic representation of the input time series (red string)
Process
First convert the input series into piecewise aggregate approximation (PAA) representation (grey steps)
Then convert the PAA into a string of symbols (red string)
PAA
Input Series
SAX
baabccbc
Dept. of Computing Science, University of Aberdeen

8. Dept. of Computing Science, University of Aberdeen
Example Data
Time
Depth 20
4.2 40
9.2 60
14.8 80 15
100 17
120 18
140
19.7
160 20
180
20.8
200
21.3
220
21.6
240
20.6
260
16.9
280
12.8
Dept. of Computing Science, University of Aberdeen

9. Dept. of Computing Science, University of Aberdeen
Creating PAA
Normalize the input time series
Subtract the mean from each value and divide the deviation with standard deviation
Divide input time series of length n into w portions of equal length
w is the parameter that controls the length of PAA and therefore the length of SAX
If w is large you have a detailed (fine) PAA and a detailed SAX
If w is small you have an abstract (coarse) PAA and an abstract SAX
Choice of w should be based on the application requirements
Dept. of Computing Science, University of Aberdeen

10. Dept. of Computing Science, University of Aberdeen
Creating PAA (2)
Two cases
n/w is a whole number
Simple case of each portion having n/w number of values from the input time series
n/w is a fraction
Complicated case because you cannot assign equal number of whole numbered values from the input series to w equal sized portions
Our example data has n = 14
If w = 3, then n/w is a fraction
The length of each portion is 14/3 =
Each portion should have values from the original time series
Dept. of Computing Science, University of Aberdeen

11. Dept. of Computing Science, University of Aberdeen
Creating PAA (3)
We use the following scheme to achieve values in each portion
The following is the list of indexes of the 14 values in a input series
The first portion will have values at 1, 2, 3, and 4
We need more to complete this portion
We achieve this by inserting times the 5th value
The remaining times the 5th value is inserted into the second portion
Dept. of Computing Science, University of Aberdeen

12. Dept. of Computing Science, University of Aberdeen
Creating PAA (4)
Using the above scheme our three lists are
4.2, 9.2, 14.8, 15 and *17
0.3333*17, 18, 19.7, 20, 20.8, *21.3
0.6667*21.3, 21.6, 20.6, 16.9, 12.8
(Note: here we have shown the values from the un-normalized input series)
Each of the above sublists have equal portions from the input series
Next for each of the sublists compute the average (mean)
In our case, three sublists will each have an average value
PAA is simply a vector of these average values
{avg1, avg2, avg3}
{ , , } for our example (using normalized values)
Dept. of Computing Science, University of Aberdeen

13. Dept. of Computing Science, University of Aberdeen
Properties of PAA
PAA is simple to compute (as can be seen from the previous slides)
Achieves dimensionality reduction
From 14 values our input series is reduced to 3 values
Any similarities computed on the PAA will be true on input series as well
Lower bounding distance
Very useful property for a structural representation
Allows data analysis to be performed on the approximate representation rather than the original series
Dept. of Computing Science, University of Aberdeen

14. Dept. of Computing Science, University of Aberdeen
Symbol Mapping
In this step, each average value from the PAA vector is replaced by a symbol from an alphabet
An alphabet size, a of 5 to 8 is recommended
a,b,c,d,e
a,b,c,d,e,f
a,b,c,d,e,f,g
a,b,c,d,e,f,g,h
Given an average value we need a symbol
This is achieved by using the normal distribution from statistics
Because our input series is normalized we can use normal distribution as the data model
We divide the area under the normal distribution into ‘a’ equal sized areas where a is the alphabet size
Each such area is bounded by breakpoints
Dept. of Computing Science, University of Aberdeen

15. Symbol mapping - breakpoints
Breakpoints for different alphabet sizes can be structured as a lookup table
When a=3
Average values below are replaced by ‘A’
Average values between and 0.43 are replaced by ‘B’
Average values above 0.43 are replaced by ‘C’
Using this table, SAX for our input series is ‘ADD’
a=3
a=4
a=5 b1
-0.43
-0.67
-0.84 b2
0.43
-0.25 b3
0.67
0.25 b4
0.84
Dept. of Computing Science, University of Aberdeen

16. SAX Computation – in pictures20 40 60 80
100
120
This slide taken from Eamonn’s Tutorial on SAX - 20 40 60 80
100
120 b a c
baabccbc
Dept. of Computing Science, University of Aberdeen

17. Data Analysis using SAX
A general approach is to convert time series into SAX
Use SAX representations to train Markov models (details not here) on normal data
The model captures the probabilities of normal patterns
The trained models are then used to test incoming data for known and unknown patterns
Dept. of Computing Science, University of Aberdeen

18. Visualization using SAXb c d
Mark Frequencies
Given a SAX representation
count the frequencies of patterns (substrings) of required length and
use them to color code a mosaic for visualizing time series
For example, given ‘baabccbc’ as the SAX representation
We calculate the frequencies of substrings of length 1 and represent them in a mosaic
Visualizations for substrings of length>1 are possible (please refer to the SAX site) 2 3
0.67 1
Normalize
Color code cells
Dept. of Computing Science, University of Aberdeen

19. Dept. of Computing Science, University of Aberdeen
Summary
Structural representations help in understanding time series through
Data analysis + Visualization
SAX is claimed to be a landmark representation of time series
Symbolic and therefore allows use of discrete data structures and their corresponding algorithms for analysis
Also helps with visualization
Dept. of Computing Science, University of Aberdeen