Temporal Data Mining

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Temporal Data Mining

• Claudio Bettini, X.Sean Wang and
  Sushil Jajodia
• Presented by Zhuang Liu

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Outline

• What is Data Mining?
• Formal Problem Definition
• TAG (Timed Automaton with Granularity)
• A Naive Solution
• Techniques for Improving Performance
• Experimental Results

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What is Data Mining

• Data Mining
• A non-trivial extraction of implicit, previously
  unknown potentially useful
  information from data
• Common Data Mining Techniques
• association-rule mining
• Sequential mining (Temporal mining)
• Clustering
• Classification
• Outlier detection

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Temporal Data Mining

• Finding time-related frequent patterns (frequent
  sub-sequences)
• which pairs of events occur frequently one week
  after another
• A simple example user may be interested in
  finding all those events that frequently follow
  within 2 business days of a rise of the IBM stock
  price.

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Definition

• Event Type (E)
• e.g. deposit to an account
• e.g. price increase of a specific stock
• Event e
• An event e is a pair e(E, t), where E is an
  event type and t is a positive integer, called
  the timestamp of e .
• Event Sequence
• An Event Sequence a finite set of events.
• Each event (E, t) appearing in an event
  sequence represents the occurrence of event type
  E at time t.

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Granularity

• Granularity is a mappingµfrom the set of the
  positive integers to subset of the time domain
  such that for all positive integers i and j with
  iltj
• (1) implies that
  each number in
• ??i? is less than all the numbers in
  ??j?, and
• (2) implies .
• Example year, month, week, day, business-day,
  business-week etc.

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TCG

• A temporal constraint with granularity (TCG)
  m,n? is a binary relation on positive integers.
  For positive integers t1 and t2, (t1, t2)
  satisfies m,n ? iff
• (1) t1 ? t2
• (2) and are both defined, and
  
• (3)
• Example TCG0,0day, 0,2hour, 1,1month

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Event Structure

• An event structure (with granularities) is a
  rooted directed acyclic graph (W,A,G), where W is
  a finite set of event variables, A ? W ? W andG
  is a mapping from A to the finite set of TCGs.
• Complex event type derived from S
• each variable associated with a specific
  event type.
• Complex event matching S
• each variable associated with a distinct
  event such that the event timestamps satisfy the
  time constraints.

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Example of Event Structure

• Assign the event types for x0 , x1, x2, x3, to
  be IBM-rise, IBM-earnings-report, HP-rise, and
  IBM-fall, respectively, we have a complex event
  type. This complex event type describes that the
  IBM earnings were reported one business day after
  the IBM stock rose, and in the same or the next
  week the IBM stock fell while the HP stock rose
  within 5 business days after the same rise of the
  IBM stock and within 8 hours before the same fall
  of the IBM stock.

1,1b-day
0,1week
0,8hours
0,5b-day
Figure 1 An event structure
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Formal Problem Definition

• An event-mining problem is a quadruple (S, ?, E0
  , ?),
• where S is an event structure, ? is the
  minimum confidence value, E0 an event type, and ?
  is a partial mapping which assigns a set of event
  types to some of the variables (expect root).
• An event-mining problem is the problem of finding
  all complex event types such that each occurs
  frequently in the input sequence and is derived
  from S by assigning E to the root and a specific
  event type to each of the other variables.
• Example (S, 0.8, IBM-rise, ?)

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TAG

• Timed Automaton with Granularities
• A basic component to test if a candidate complex
  event type appears frequent in a time sequence.
• A timed automaton with granularities is a 6-tuple
  ????, S, S0, C, T, F), where
• (1) ? is a finite set of input letters,
• (2) S is a finite set of states,
• (3) S0 ? S is a set of start states,
• (4) C is a finite set of clocks,
• (5) T ? S ? S ? ? ? 2C? ?(C) is a set of
  transitions,
• (6) F ? S is a set of accepting states.

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TAG

• ?(C) is the set of all the formulas called clock
  constraints.
• A transition (s, s, e, ?, ?) represents a
  transition from state s to state s on input
  symbol e. the set ? ? C gives the clocks to be
  reset with this transition. And ? is a clock
  constraint over C.
• Is essentially standard finite automata with some
  modifications.
• Each TAG maintains a set of clocks.
• Both input symbol and clock determine the next
  state.
• A run is an accepting run if the last state is in
  the set F. An event sequence is accepted by a TAG
  if there exists an accepting run.

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A Naïve Solution

• Consider all the event types that occur in the
  given event sequence, and consider all the
  complex types derived from the given event
  structure, one from each assignment of these
  event types to the variables. Each of these
  complex types is called a candidate complex type
  for the event-mining problem.
• For each candidate complex type, start the
  corresponding TAG at every occurrence of E0. That
  is, for each occurrence of E0 in the event
  structure, use the rest of the event sequence as
  the input to one copy of the TAG. By counting the
  number of TAGs reaching a final state, versus the
  number of occurrences of E0 , all the solutions
  of the event-mining problem will be derived.
• The number of candidate types is exponential in
  the number of event types occurring in the event
  structure. Too costly.

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Techniques to improve performance

• The performance of this algorithm can be improved
  by
• identifying the possible inconsistencies in the
  given event structure before starting the
  process,
• reducing the length of the sequence,
• reducing the number of times an automaton has to
  be started,
• reducing the number of different automata to be
  started,
• applying the naïve algorithm.

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Recognition of Inconsistent Event Structures

• A event structure is consistent if there exists a
  complex event that matches that event structure.
• If an event structure is inconsistent, it should
  be discarded even before the mining process
  starts.
• It is difficult to determine the consistency of
  event structures.
• Use approximated polynomial algorithms to check
  the consistency of event structures.

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Recognition of Inconsistent Event Structures

• If one of the constraints implied by the given
  ones is the empty one, i.e. unsatisfiable, the
  whole event structure is inconsistent.
• A TCG m, n? is logically implied by a TCG m,
  n ? if each pair (x, y) satisfying the second
  constraint, satisfies also the first one.
• For example, a TCG 1,2b-week can be converted
  into 3,18day or 0,1month, while it cannot
  be converted into 2,3week-end or 1,3week,
  since the resulting constraints are not implied
  by 1,2b-week.

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Reduction of the Event Sequence

• We can reduce the event sequence by
• exploiting the granularities.
• For example, if a discovery problem is defined on
  the sub-structure excluding variable x3, the
  input event sequence can be reduced discarding
  any event that does not occur in a business day.

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Reduction of the occurrences of the root

• The basic idea is to remove those occurrences of
  reference types which cannot be the root of a
  complex event matching the given structure.
• It is possible that for some occurrences of the
  reference types in the sequence, a constraint is
  unsatisfiable.
• Consider all the non-empty sets of explicit and
  implicit constraints on the pair of the root and
  each non-root node. Check if one of the
  constraints cannot be satisfied.
• For example, if no event occurs in the sequence
  in the next business day of an IBM-rise event,
  this particular reference event can be discarded.
  (No automaton is started for it.)

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Reduction of the occurrences of the root

• Let N be the number of occurrences of the
  reference event type in the sequence.
• Let N be the number of occurrences of reference
  events for which one of the constraints is
  unsatisfiable. These are reference events that
  are certainly not the root of a complex event
  satisfying the given event structure.
• If N/N 1-?, there cannot be any frequent
  complex event type and the empty set should be
  returned to the user.
• Otherwise, remove these occurrences of the
  reference type and modify ? into ? (? N) / (N-
  N) .

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Reduction of the Candidate Type

• Based on the property if a complex event type
  occurs frequently, then any of its sub-type
  should also occur frequently.
• In other words, if one assignment to two
  variables is not frequent, any candidate complex
  event type including this assignment wont be
  frequent. So we can remove these complex event
  type from the candidate complex event type.
• For each subset W of W, the induced approximated
  sub-structure of W is (W, A, G), where A
  consists of all pairs (X, Y) ? W ? W, such that
  there is a path from X to Y in S and there is at
  least one constraint on (X,Y).

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Reduction of the Candidate Type

• To find the solutions to the induced discovery
  problems is rather straightforward and simple in
  time complexity. Indeed, the induced
  sub-structure gives the distance from the root to
  the variable (in effect, two distances, namely
  the minimum distance and the maximum distance).
• For each occurrence of E0 , this distance
  translates into a window, i.e., a period of time
  during which the event for X must appear.
• Extend the sub-structure to more than one
  non-root variable. These variable form a chain in
  S.

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Experimental Results

• Closing prices of 439 stocks for 517 trading days
• Price changes are partitioned into 7 categories
  (- ?, -5), (-5, -3), (-3, 0), (0, 0), (0,
  3), (3, 5), (5, ?)
• Total number of event types is 2978. The number
  of event is 181089.
• The reference event type X0 the drop of IBM
  stock of less than 3. Minimum confidence value
  is 0.7. There is no other assignment to other
  variables.

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Experimental Results cont.
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Experimental Results cont.

• This experiment focuses on Step 4, namely
  reduction of the candidate complex event types by
  using sub-structures.
• The result shows that after using heuristics the
  number of candidate complex event types reduces
  significantly.

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Experimental Results cont.
The two frequent event combinations discovered in
the experiment
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References

• C. Bettini, Wang, X.S., Jajodia, S. and Jia-Ling,
  L. "Discovering Temporal Relationships with
  Multiple Granularities in Time Sequences". IEEE
  Transations on Knowledge and Data Engineering,
  Vol. 10 (2), 1998.
• C. Bettini, X. Wang, and S. Jajodia. A General
  Framework for Time Granularity and its
  Application to Temporal Reasoning. Annals of
  Mathematics and Artificial Intelligence, Vol. 22
  (1-2), pages 29-58, Baltzer Science Publishers,
  1998.
• C. Bettini, X. S. Wang, and S. Jajodia. Testing
  complex temporal relationships involving multiple
  granularities and its application to data mining.
  In Proceedings of the Fifteenth ACM
  SIGACT-SIGMODSIGART Symposium on Principles of
  Database Systems (PODS'96), pages 68-78,
  Montreal, Canada, June 1996
• C. Bettini, X. Sean Wang, and S. Jajodia. Mining
  temporal relationships with multiple
  granularities in time sequences. Data Engineering
  Bulletin, 2132--38, 1998.

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• Thank you
• Question?