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

© Copyright 2015 Simularity. All Rights Reserved
Ray Richardson, Founder & CTO | ray@simularity.com
Practical Predictive Analytics on
Time Series Data using SAX
MLConf Seattle, May 1, 2015

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

33
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

44
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

55
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

66
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

77
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

88
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

99
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

1010
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

1111
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

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
--
0 20 40 60 80 100 120
b
b b
a
cc c
a
baabccbc

1313
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

1414
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

1515
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

1616
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

1717
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

1818
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

1919
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

2020
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

2121
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

2222
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

2323
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

2424
Event Signatures For Failure Prediction
Notice there
are two
different
event
signatures
for these
failing drives

2525
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

2626
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
ray@simularity.com
THANK YOU
@rayrichardson

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

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