Ray Richardson, Chief Technology Officer at Simularity at MLconf SEA - 5/01/15

© Copyright 2015 Simularity. All Rights Reserved

Ray Richardson, Founder & CTO | [email protected]

Practical Predictive Analytics on Time Series Data using SAX

MLConf Seattle, May 1, 2015

© Copyright 2015 Simularity. All Rights Reserved 2

Anomaly Detection

 A time series anomaly is simply an unusual subsequence of the series

 “Unusual” will be taken to mean “improbable” !  The degree of anomaly is isomorphic with the improbability of the

subsequence

!  Probability is not defined for Time Series

!  Probability can be defined for Symbols

 Mapping a time series to a symbol may allow us to assign a probability to the time series subsequence

 This involves mapping the time series subsequence to a symbol in some Symbol Space

© Copyright 2015 Simularity. All Rights Reserved 3 3

Symbolic Representation

 All data in a modern computer is in a Symbolic Representation !  Integers, Floating point numbers and Strings are all symbols, and are all

composed of bytes

 Anomaly detection requires a special kind of symbol – one from a Finite Symbol Space !  This means there are a finite number of symbols available


Finite Symbol Spaces

 For our purposes, a Finite Symbol Space is defined by 2 attributes !  An Alphabet, from which components are drawn !  A Symbol Length, defining the fixed number of components of the

symbol

 Thus, if we define the alphabet as a..d and a length of 4, a legitimate symbol might be abcd  Another legitimate symbol might be 10:15, where 10 is the row of a matrix and 15 is the column !  The size of the matrix must be constant

 Fixed point numbers are drawn from a Finite Symbol Space if there is a lower and upper bound


Why Finite Symbol Spaces?

 A Finite Symbol Space allows us to compute a (perhaps naïve) probability of seeing a particular symbol !  The number of possible symbols is al where a is the cardinality of the

alphabet and l is the length of the symbol !  Perhaps naïve due to the fact that some symbols may never appear

•  In some symbolic representations of time series aaaa and dddd represent the same series

 We can compute a probability of seeing a symbol if they are random – it’s the reciprocal of size of the symbol space


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


Normalizing Time Series

  A time series window can be put into a “normal form” called PAA (Piecewise Aggregate Approximation).   The PAA consists of K floating point values which represent the aggregate value of the times series over fixed time spans   Each value is the average of the readings that fall into each “box”

!  Each box is a time window with a start and end derived by segmenting the time series window into K windows


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

baabccbc


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


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


Building A SAX Word

 Convert the Time Series Window to a PAA of the length of the SAX word, and Z-normalize the PAA !  Which mean and standard deviation are used for normalization will

affect the outcome

 Compute the SAX letter by dividing the Standard Normal Distribution into K regions of equal area under the curve and assigning each component of the PAA a letter from the SAX Alphabet corresponding to the region indexed by the PAA value  Repeating for each value of the PAA yields a SAX word of equivalent length to the PAA

© Copyright 2015 Simularity. All Rights Reserved

How do we obtain SAX?

First convert the time series to PAA representation, then convert the PAA to symbols It takes linear time

0 20 40 60 80 100 120

C C

Slide by Eamonn Keogh and Jessica Lin. Used with permission.

0

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-

0 20 40 60 80 100 120

b b

b

a

c

c c

a

baabccbc


Encoding Magnitude And Slope

 The Magnitude and slope can be encoded in a SAX word  The Magnitude (mean) can be Z-normalized over the entire space of the time series, and divided into SAX letters !  These letters need not be from the same alphabet as the SAX word

which represents the shape, we just need to consider the alphabet size when computing the size of the Finite Symbol Space

 Slope can be encoded by dividing 180º into equal spaces, and assigning each space to a letter !  The slope can be determined by a number of methodologies


Computing The Anomaly

 We need a data structure, which uses SAX words as an index, and stores the number of times we have seen each SAX word, as well as the total number windows we’ve seen  Due to the fact that our SAX words are of a fixed length and alphabet, we know the total number of possible SAX words  Tries are one choice of data structure !  Allow for quick access

 Converting the SAX word to a number, which is an array index is another !  Requires exponentiation


Computing The Anomaly

 The procedure for examining a window !  Convert the window into a SAX word !  Lookup the current count for that SAX word and increment it !  Compute a metric which determines how anomalous the window is

using 3 values – The total number of windows, the number of instances of this SAX word, and the size of the Finite Symbol Space of SAX words

!  Compare the result of the metric with a predetermined threshold to decide whether or not this window is anomalous

 This procedure is repeated for constantly incoming Time Series Windows


The Metric

 Once we have determined the values, we need to turn them into a metric which tells us how anomalous a window is  The metric should discriminate !  We should be able to discriminate

between multiple levels of anomaly values

 The metric should be easy to compute !  Embedded applications may not have

complex math libraries which allow for complicated computation

 The metric should reflect the real world


The Metric – P-Values

 P-Values seem like a good metric !  Expressed as a probability, they have a connection to the real world

 Unfortunately, P-Values closely approach zero and one once the number of samples gets significant !  This makes it difficult to set an “anomaly threshold” !  This sets a hard criterion for an anomaly


The Metric – Log-Likelihood Ratio

 The Log-Likelihood ratio is perhaps a better choice of metric !  Scaling the ratio between -1.0 and 1.0 gives a manageable value !  Even extremely unlikely events can be discriminated

 Reversing the sign of the scaled log-likelihood ratio gives values that are easier to understand  Use the likelihood function for a binomial distribution !  The number of trials is the Total Windows !  The number of successes is the occurrence of this Window !  The Probability is the Symbol Probability

 The log likelihood is particularly useful as it accounts for the significance of the data i.e. the number of samples  Like P-Values, it requires a floating point library


The Metric – Rate Ratio

 The rate ratio is the number of times more likely the event is observed to have occurred, than would be predicted by random chance !  Smaller values mean more anomalous – less than 1 implies less likely than

chance !  The reciprocal of the rate ratio gives an anomaly score which increases !  Uses observed probabilities

 Doesn’t require math harder than division  Doesn’t account for significance – significance has to be accounted for by some other means


Other Means Of Symbolizing

 SAX may not always be the best way to reduce a window to a symbol !  SAX reduces resolution equally across all its members !  Tiny, but important variations will be lost

 Self Organizing Maps can also be used !  They require more computation, but don’t reduce resolution !  Self Organizing Maps can encode magnitude directly


Using Self Organizing Maps

  Self Organizing Maps (SOMs) are (typically) a grid of vectors, which can be thought of as weights or prototypes !  The SOM algorithm adjusts the prototypes based on training data

  To operate the SOM, a Window vector is compared to each of the prototypes – the best matching one “wins” and the symbol associated with the window is the row:column of the matching grid   The row:column is then used to index the count of how many times that prototype has been seen.   We now have the 3 values for computing the metric


Predicting Events

 A set of time series may be used to predict events !  We look for the correlation between the symbols representing the time

series windows and Events which happen in the future

 This can be used to categorize Events according to an Event Signature !  Event signatures imply outcomes at a particular time index


A Concrete Example

 The SMART data on hard drives can be used to predict failures !  Simularity used 53 of the sensors to test for anomalies and predict failures

 Information from nearly 400 hard drives was used to “train” the anomaly detector  Once trained, the system was used to identify Event Signatures which indicated failure  The time series in the system were reduced to SAX words, and correlated with a single event, failure (all that was known)  This can then be used to predict failure


Event Signatures For Failure Prediction

Notice there are two different event signatures for these failing drives


Credit

 This technique is similar, although not identical, to the TARZAN methodology outlined by Eamonn Keogh and Jessica Lin !  It and other work pertaining to SAX is available here:

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

 Self Organizing Maps were invented by Teuvo Kohonen http://www.cis.hut.fi/research/som-research/teuvo.html


Source Code

 Simularity maintains a GitHub repository of open-source software, including an implementation of SAX suitable for using with the techniques described here www.github.com/simularity/SAX

1160, Brickyard Cove Road, Suite 200 Point Richmond, CA 94801 United States

+ 1 678-488-8857

[email protected]

THANK YOU

@rayrichardson

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