IMPORTANT%20EXTREMA%20OF%20TIME%20SERIES:%20THEORY%20AND%20APPLICATIONS

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IMPORTANT EXTREMA OF TIME SERIESTHEORY AND
APPLICATIONS
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Introduction

• Time series
• Definition Sequence of values measured at equal
  intervals
• Examples stock prices, weather data,
  electrocardiograms, etc.
• Compression using Important extrema
• Importance levels
• Indexing based importance
• Distance computation based indexing trees
• Pattern retrieval based on indexing trees

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Previous Work

• Previous work Specific
• Indexing and fast retrieval of time series
  Pratt, 2001 Pratt and Fink, 2002 Fink and
  Pratt, 2003
• Compression
• Use Important points
• Identify important points using ration between
  values of points
• Use the parameter R to control the compression
• Do not have the direct correlation between
  compression rate and R
• Similarity measures and pattern retrieval
• Define leg/extended leg as segment between two
  consecutive important points
• Index the legs with their ration
• Retrieve the legs that are similar to the
  prominent leg of the pattern
• Find the similarity between pattern and the
  series who has leg similar to prominent leg
• Output the series whose similarity is within the
  given threshold

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Basic Concepts

• Related structures and algorithms
• Stacks and linked lists
• Red-black trees
• Order statistics
• Time series representation
• Series structure
• full-size number of points in the original
  series
• cmpr-size number of points in the compressed
  series
• points array or red-black tree of all points

• Point structure
• index index of the point in the original series
• value value of the point
• type end-point, minimum, maximum or
  non-extremal
• next next point in the compressed series
• prev previous point in the compressed series
• side strict, left, right, or flat
• imp importance of point

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Basic Concepts..

• Distance measures
• Distance between real values
• Definition two-argument function that satisfy
  the following properties
• For every value a, dist(a,a) 0
• For every two values a and b, dist(a,b)
  dist(b,a)
• For every three values a, b, and c, if altbltc,
  then dist(a,b) lt dist(a,c) and dist(b,c) lt
  dist(a,c)
• Examples
• a-b, a-b/ab

• Distance composition
• Definition real-value function f(d1,,dq) on
  non-negative real arguments such that f(0,,0)
  0 and f is monotonically increasing on each of
  its argument
• Example
• If w1, ,wq are weights and dist1, ,distq are
  distance functions then w1.dist1.wq.distq is a
  composition.

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Basic Concepts..

• Distance between time series
• Definition For two equal-length series, a1,an
  and b1,,bn, and a distance function dist for
  real values, the corresponding lr distance
  between the series is lr (1/n . Eni1 (dist(ai
  ,bi))r)1/r
• Example In the below picture the l1 distance
  between series is 2.0 and l2 distance is 2.1
• Advantage of using distance functions
• Flexibility of choosing different distance
  functions and their compositions.

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• Questions ?

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Important Points

• Extrema
• We define extremum as a minimum or maximum in a
  series
• Formal definition of a minimum The point ai of a
  time series a1,,an is a strict minimum if
  ailtai-1 and ailtai1
• Example
• Strict, left, right, and flat extrema

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Important Points..

• Extrema..
• Compression by extracting all extrema
• Example
• Algorithm
• Space complexity constant
• Time complexity O(n) n number of points in
  the series
• Can process live series

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Important Points..

• Important Extrema
• Higher compression by selecting only certain
  important extrema.
• Control the compression rate using the parameter
  R
• Formal definition of important minimum The point
  ai of a time series a1,,an is a strict important
  minimum if
• If ai is minimum among ail,,air, and
• dist(ai,ail) gt R and dist(ai,air) gt R
• Examples of Strict, left, right, and flat
  important minima

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Important Points..

• Important Extrema..
• Compression by extracting important extrema
• Example
• Algorithm
• Space complexity constant
• Time complexity O(n) n number of points in
  the series
• Can process live series

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Important Points..

• Derivative series
• Compression based on changes in slope
• Example
• Algorithm
• The algorithm for important extrema can modified
  for derivative series

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Important Points..

• Importance Levels
• Idea Assign numerical importance to the extrema
  and use the importance for compression, indexing,
  pattern retrieval
• DefinitionIf a point is a strict (left, right,
  flat) extremum for compression with some value of
  R, then its strict (left, right, flat) importance
  is the maximal value of R for which it is a
  strict (left, right, flat) extremum.
• Example with distance function a-b

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Important Points..

• Importance Levels
• strict, left, right, and flat importance
• Algorithm to assign importance
• One pass through the series
• Space complexity O(m) m number of extrema
  in the series
• Time complexity O(n) n number of points in
  the series

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Important Points..

• Compression rate
• rate number of points removed during the
  compression
• Problem Select important extrema according to a
  given rate
• Example The compression rate in below series is
  60 since we have selected eight of 20 points

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Important Points..

• Compression rate..
• Three-pass algorithm
• Makes three-passes through the series
• Space complexity O(m) m number of extrema
  in the series
• Time complexity O(n) n number of points in
  the series
• One-pass algorithm
• Makes one pass and uses red-black tree to keep
  only desired number of important extrema.
• Space complexity O(m) m number of extrema
• Time complexity O(nm.lgs) n number of
  points s number of points in the compressed
  series
• Dependency on the distance functions

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• Questions ?

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Indexing Trees

• Idea
• Index the series based on importance for fast
  retrieval of compressed series
• Indexing based on importance
• Can retrieve s points in O(s) time
• Need to sort the retrieved points, which takes
  O(s.lgs)
• Augmented indexing structure
• Example series

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Indexing Trees..

• Augmented indexing structure ..
• Structure
• Left superior nearest extremum to the left in
  the original series with strictly greater
  importance
• Right Superior nearest extremum to the right
  with equal or greater importance

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Indexing Trees..

• Algorithms
• Sorted retrieval
• Space and Time complexity O(s) s number of
  points to be retrieved
• Building augmented tree
• Space complexity O(m) m number of extrema
• Time complexity O(m.lgm)
• Range Tree
• Problem retrieve important points of a given
  segment ail,,air
• Idea
• Use Range tree
• Index the points in the series by position as
  well as importance
• Space complexity of building the range tree
  O(m)
• Time complexity of building the range tree O(m.
  lgm)
• Time complexity of retrieval O(s. lgs lgm)

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Distance Computation

• Distance range
• Problem find the distance range between two
  compressed series
• Idea
• Each leg in the series can be bounded by a
  rectangle as shown
• Bounding rectangle represents the bounds of all
  the points in that segment
• Find the distance range between the series by
  finding the distance range between the bounding
  rectangles
• Algorithm
• Time complexity O(s) s number of points in
  the compressed series
• Space complexity constant if the compressed
  series are in file
• O(s) if we
  need to retrieve compressed series
• from
  tree

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Distance Computation..

• Approximate Distance
• Problem find the distance between two series
  with a given a approximation
• Algorithm
• Uses pre-computed indexing trees
• Generated highly compressed versions to start. If
  the distance range is not within the given
  accuracy, it increases the number of points by a
  factor of 2
• Time and space complexity O(s) s number of
  points required for a given accuracy
• Threshold Test
• Problem determine whether the distance between
  two series is smaller than the given threshold
• Algorithm
• Same idea as in Approximate distance
• Checks whether the lower bound is greater than
  the threshold or the upper bound is less than the
  threshold

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Pattern Retrieval

• Range Query
• Fetch all the series that are within a given
  range from the pattern
• Nearest neighbor
• Fetch the closest series from the pattern
• Multiple neighbors
• Fetch given number of closest series to the
  pattern
• Complexity of the algorithms
• Time complexity
• Best case O(N) N number of series in
  database
• Worst case O(n.N) n number of point in each
  series
• Space complexity
• Best case O(s.N) s number of points required
  to perform the threshold test

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Conclusions

• Distance measures
• Strict, left, right, and flat important extrema
• Extrema in derivative series
• Importance levels
• compression, indexing, and pattern retrieval
  techniques using importance level
• Future work
• Experiments
• Investigating the usefulness

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• Questions ?

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Acknowledgements

• Eugene Fink
• Colleagues and Management at Nielsen Media
  Research
• Colleagues at Cognizant Technology Solutions, Ltd
• Family
• Parents Mohan Reddy Gandhi and Sulochana Gandhi
• Brother Sharat Gandhi
• Wife Madhuri Gandhi