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Modeling Recurrent Distributions in Streams using Possible Worlds Michael Geilke, Andreas Karwath, and Stefan Kramer Johannes Gutenberg-Universitt Mainz, Germany October 20, 2015 Modeling Recurrent Distributions in Streams using Possible

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Modeling Recurrent Distributions in Streams using Possible Worlds

Michael Geilke, Andreas Karwath, and Stefan Kramer

Johannes Gutenberg-UniversitΓ€t Mainz, Germany

October 20, 2015

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Modeling Recurrent Distributions in Streams using Possible Worlds

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Modeling Recurrent Distributions in Streams using Possible Worlds

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Modeling Recurrent Distributions in Streams using Possible Worlds

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Modeling Recurrent Distributions in Streams using Possible Worlds

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EDDO

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Modeling Recurrent Distributions in Streams using Possible Worlds

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𝑔 EDDO

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Modeling Recurrent Distributions in Streams using Possible Worlds

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𝑔 EDDO

Query 1 [marginalize]

What is the data distribution of the sensors in the living room?

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Modeling Recurrent Distributions in Streams using Possible Worlds

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𝑔 EDDO

Query 2 [setting hard evidence]

Two residents are in the living room. What is the probability that they watch TV?

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Modeling Recurrent Distributions in Streams using Possible Worlds

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𝑔 EDDO POEt Alg 1 Alg 2 Out 1 Out 2 Itemsets

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Modeling Recurrent Distributions in Streams using Possible Worlds

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Modeling Recurrent Distributions in Streams using Possible Worlds

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Modeling Recurrent Distributions in Streams using Possible Worlds

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𝑔 EDDO

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Modeling Recurrent Distributions in Streams using Possible Worlds

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Recurrences

  • day and night
  • working days and weekends
  • seasons
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Modeling Recurrent Distributions in Streams using Possible Worlds

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Recurrences

  • pattern could be more complex
  • may only affect a part of the house
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Modeling Recurrent Distributions in Streams using Possible Worlds

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Goal: a representation that

  • is constantly updated
  • is representing current and historical data distributions,
  • is able to represent recurrences
  • provides a query mechanism
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Modeling Recurrent Distributions in Streams using Possible Worlds

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do all of that in an online fashion Tasks for proposed method

  • 1. recognize regions of drift
  • 2. represent density of data stream segments
  • 3. identify recurrences on the density level
  • 4. identify recurrences between parts of different densities
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Modeling Recurrent Distributions in Streams using Possible Worlds

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do all of that in an online fashion Tasks for proposed method

  • 1. recognize regions of drift
  • 2. represent density of data stream segments
  • 3. identify recurrences on the density level
  • 4. identify recurrences between parts of different densities
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Modeling Recurrent Distributions in Streams using Possible Worlds

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do all of that in an online fashion Tasks for proposed method

  • 1. recognize regions of drift
  • 2. represent density of data stream segments
  • 3. identify recurrences on the density level
  • 4. identify recurrences between parts of different densities
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Modeling Recurrent Distributions in Streams using Possible Worlds

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do all of that in an online fashion Tasks for proposed method

  • 1. recognize regions of drift
  • 2. represent density of data stream segments
  • 3. identify recurrences on the density level
  • 4. identify recurrences between parts of different densities
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Modeling Recurrent Distributions in Streams using Possible Worlds

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do all of that in an online fashion Tasks for proposed method

  • 1. recognize regions of drift
  • 2. represent density of data stream segments
  • 3. identify recurrences on the density level
  • 4. identify recurrences between parts of different densities
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Modeling Recurrent Distributions in Streams using Possible Worlds

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do all of that in an online fashion Tasks for proposed method

  • 1. recognize regions of drift
  • 2. represent density of data stream segments
  • 3. identify recurrences on the density level
  • 4. identify recurrences between parts of different densities
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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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Window-based approach

  • extension of an approach by Dries and RΓΌckert

A B C

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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Window-based approach

  • extension of an approach by Dries and RΓΌckert
  • compute density values with current estimate f

A B C 𝑔 𝑔

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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Window-based approach

  • extension of an approach by Dries and RΓΌckert
  • compute density values with current estimate f
  • perform drift detection with Wilcoxon rank-sum test

A B C 𝑔 𝑔

Wilcoxon

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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Window-based approach

  • extension of an approach by Dries and RΓΌckert
  • compute density values with current estimate f
  • perform drift detection with Wilcoxon rank-sum test
  • update f with clean instances only

A B C 𝑔

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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A B C

Window-based approach

  • extension of an approach by Dries and RΓΌckert
  • compute density values with current estimate f
  • perform drift detection with Wilcoxon rank-sum test
  • update f with clean instances only

𝑔

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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A B C 𝑔

Window-based approach

  • extension of an approach by Dries and RΓΌckert
  • compute density values with current estimate f
  • perform drift detection with Wilcoxon rank-sum test
  • update f with clean instances only
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Modeling Recurrent Distributions in Streams using Possible Worlds

Recognize Regions of Drift

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Window-based approach

  • extension of an approach by Dries and RΓΌckert
  • compute density values with current estimate f
  • perform drift detection with Wilcoxon rank-sum test
  • update f with clean instances only

C

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Densities

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C

Recurrent or new?

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Densities

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C

𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

Recurrent or new?

  • compare with pool of existing density

estimates

  • use statistical test we proposed earlier
  • reactivate estimate if one is found
  • initialize a new one otherwise

𝑔

𝑗

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Densities

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C

𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

Recurrent or new?

  • compare with pool of existing density

estimates

  • use statistical test we proposed earlier
  • reactivate estimate if one is found
  • initialize a new one otherwise

𝑔

𝑗

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Densities

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C

𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

Recurrent or new?

  • compare with pool of existing density

estimates

  • use statistical test we proposed earlier
  • reactivate estimate if one is found
  • initialize a new one otherwise

𝑔

𝑗

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Densities

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C

𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

Recurrent or new?

  • compare with pool of existing density

estimates

  • use statistical test we proposed earlier
  • reactivate estimate if one is found
  • initialize a new one otherwise

𝑔

𝑗

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Densities

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C

𝑔

1

𝑔

2

𝑔

3

𝑔

4

𝑔

5

Recurrent or new?

  • compare with pool of existing density

estimates

  • use statistical test we proposed earlier
  • reactivate estimate if one is found
  • initialize a new one otherwise

Wilcoxon

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Modeling Recurrent Distributions in Streams using Possible Worlds

Recurrences of Density Parts

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𝑔 π‘Œ1, π‘Œ2, … , π‘Œ8 = 𝑔

1 π‘Œ1, π‘Œ3, π‘Œ8 βˆ™ 𝑔 2 π‘Œ2, π‘Œ4, π‘Œ5 βˆ™ 𝑔 3 π‘Œ6 βˆ™ 𝑔 4 π‘Œ7

Introduction of modules If the 𝑔

𝑗 cannot be decomposed any further, then 𝑔 1, 𝑔 2, 𝑔 3, 𝑔 4

are called the modules of 𝑔.

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p1 p2 p3 p4

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Modeling Recurrent Distributions in Streams using Possible Worlds

Query Mechanism

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p1 p2 p3 p4

  • probabilistic extension of

possible worlds semantics

  • requires density estimators

supporting inference tasks

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Modeling Recurrent Distributions in Streams using Possible Worlds

Query Mechanism

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p1 p2 p3 p4

Query 3 [over multiple worlds] Given world W, what is the probability that the resident will switch

  • n the light in the office room?
  • probabilistic extension of

possible worlds semantics

  • requires density estimators

supporting inference tasks

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Modeling Recurrent Distributions in Streams using Possible Worlds

Evaluation: Modules

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  • evaluation on

synthetic and real-world datasets

  • without modules performance is

better in many cases, but only slightly

  • more explicit representation that

enables detection of recurrences

Datasets Synthetic Bayesian networks with different numbers of nodes, different numbers of instances different numbers of variable groups Real-World Electricity Shuttle Waterlevel Covertype

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Modeling Recurrent Distributions in Streams using Possible Worlds

Evaluation: Recurrences

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Recurrences Densities Modules 416 1259 100% 78% distance distance

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Modeling Recurrent Distributions in Streams using Possible Worlds

Conclusions and Future Work

  • framework to model recurrent densities and

recurrent parts of the densities

  • online estimator
  • extension of possible worlds semantics for

query mechanism

Future Work:

  • more sophisticated modeling of density parts (conditional)
  • recycling of modules
  • implementation of query mechanism

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Modeling Recurrent Distributions in Streams using Possible Worlds

Thank you for your attention

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