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 Dept. of Computing Science, University of Aberdeen

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

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? 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

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

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

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

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

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 = 4.66667 Each portion should have 4.66667 values from the original time series Dept. of Computing Science, University of Aberdeen

Dept. of Computing Science, University of Aberdeen 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

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 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

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

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

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’ 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

SAX Computation – in pictures 20 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

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

Visualization using SAX b 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

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