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Time Series Data Analysis - II. Yaji Sripada. In this lecture you learn. Structural representations of time series SAX Computing SAX Data analysis using SAX Visualization using SAX. Introduction. Time series exhibit an internal structure

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Time series data analysis ii l.jpg

Time Series Data Analysis - II

Yaji Sripada


In this lecture you learn l.jpg

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


Introduction l.jpg

Introduction

  • Time series exhibit an internal structure

    • Elements of this structure have domain specific meanings

    • E.g. a scuba dive is composed of

      • one or more descent segments,

      • one or more bottom segments and

      • finally one or more ascent segments in that order

      • These segments have specific meaning in the domain of scuba diving

    • 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


Several structural representations l.jpg

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


Which structure is the most useful l.jpg

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


S ymbolic a ggregate appro x imation sax l.jpg

Symbolic Aggregate Approximation (SAX)

  • A recently developed symbolic representation of time series is claimed to facilitate easy pattern computation

  • http://www.cs.ucr.edu/~eamonn/SAX.htm 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


Creating sax l.jpg

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


Example data l.jpg

Example Data

Dept. of Computing Science, University of Aberdeen


Creating paa l.jpg

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


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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 = 4.66667

      • Each portion should have 4.66667 values from the original time series

Dept. of Computing Science, University of Aberdeen


Creating paa 3 l.jpg

Creating PAA (3)

  • We use the following scheme to achieve 4.6667 values in each portion

  • The following is the list of indexes of the 14 values in a input series

    1 2 3 4 5 6 7 8 9 10 11 12 13 14

  • The first portion will have values at 1, 2, 3, and 4

  • We need 0.6667 more to complete this portion

  • We achieve this by inserting 0.6667 times the 5th value

  • The remaining 0.3333 times the 5th value is inserted into the second portion

Dept. of Computing Science, University of Aberdeen


Creating paa 4 l.jpg

Creating PAA (4)

  • Using the above scheme our three lists are

    • 4.2, 9.2, 14.8, 15 and 0.6667*17

    • 0.3333*17, 18, 19.7, 20, 20.8, 0.3333*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}

    • {-0.9338,0.53135,0.34767} for our example (using normalized values)

Dept. of Computing Science, University of Aberdeen


Properties of paa l.jpg

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


Symbol mapping l.jpg

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


Symbol mapping breakpoints l.jpg

Symbol mapping - breakpoints

  • Breakpoints for different alphabet sizes can be structured as a lookup table

  • When a=3

    • Average values below -0.43 are replaced by ‘A’

    • Average values between -0.43 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’

Dept. of Computing Science, University of Aberdeen


Sax computation in pictures l.jpg

c

c

c

b

b

b

a

a

-

-

0

0

40

60

80

100

120

20

SAX Computation – in pictures

C

C

0

20

40

60

80

100

120

This slide taken from Eamonn’s Tutorial on SAX

baabccbc

Dept. of Computing Science, University of Aberdeen


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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


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Visualization using SAX

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)

Normalize

Color code cells

Dept. of Computing Science, University of Aberdeen


Summary l.jpg

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


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