Partitioning A Uniform Model for Data Mining

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Partitioning A Uniform Model for Data Mining

• Anne Denton, Qin Ding, William Jockheck, Qiang
  Ding and William Perrizo

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Motivation

• Databases and data warehouses are currently
  separate systems
• Why?
• Standard answer
• Details, details, details
• Our answer
• Fundamental issue of representation

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Relations Revisited

• R(A1, A2, , AN)
• Set of tuples
• Any choices at a fundamental level?
• Yes!
• Duality between
• Element-based representation
• Space-based representation

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Duality

• Element-based representation
• Standard representation of tuples with all their
  attributes

• Space-based representation
• The existence (count?) of a tuple is represented
  in its attribute space

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Similar Dualities in Physics

• Particles can be represented by the coordinates
  of their position
• More fundamental level
• Particle

• Particles can be 1 values in a grid of locations
• Field

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Space-Based Representation

• Consider standard tuples as vectors in the space
  of attribute domains
• Represent all possible attribute combinations as
  one bit
• 1 if data item is present
• 0 if it isnt
• Allowing counts could be useful for projections
  (?)

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Space-Based Representation as a Partition

• Partitions are mutually exclusive and
  collectively exhaustive sets of elements
• The Space-Based Representation partitions
  attribute space into two sets
• Data item present in database (1)
• Data item not present (0)

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Usefulness of Space-Based Representation

• No indexes needed instant value-based access
• Index locking becomes dimensional locking
• Aggregation very easy due to value-based ordering
• Selections become ands
• What experience do we have with space-based
  representations?

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Data Cube Representation

• One value (e.g., sales) given in the space of the
  key attributes
• Space-based with respect to key attributes
• Element-based with respect to non-key attributes

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Properties of the Domain Space

• Ideally space should have distance, norm, etc.
• Especially important for data mining
• Does that make sense for all domains?
• Can any domain be mapped to integer?

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Can all Domains be Mapped to Integer?

• Simplistic answer yes!
• All information in a computer is saved as bits
• Any sequence of bits can be interpreted as an
  integer
• Problems
• Order may be irrelevant, e.g., hair-color
• Order may be wrong, e.g., sign bit for int
• Spacing may vary, e.g., float (solution in paper
  intervalization)
• Domains may be very large, e.g., movies

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Categorical attributes (irrelevant order)

• We need more than one attribute for an
  appropriate representation
• Data mining solution
• 1 attribute per domain value
• Our solution
• 1 attribute per bit slice
• Values are corners of a Hypercube in
• log(Domain Size) dimensions
• Distances are given trough MAX metric

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Fundamental Partition(Space-Based Representation)

• d-dimensional representation
• d Number of attributes
• of represented points
• product of all d domain sizes
• Exponential in number of dimensions!
• We badly need compression!

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How Do We Handle Exponential Growth with d?

• How can we reduce of attributes, d?
• Review normalization
• We can decompose a relation into a set of
  relations each of which contains the entire key
  and one other attribute
• This decomposition is
• lossless
• dependency preserving (BCNF relations only)

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Compression for Non-Key Attributes

• Fundamental partition contains only one non-zero
  data-point in any non-key dimension
• Represent number by bit-slices
• Note
• This works for numerical and categorical
  attributes
• Original values can be regained by anding
• Example 5 (binary 101)
• bit 0 bit 1 bit 2

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Concept Hierarchies

• Bit sliced representation have significant
  benefits beyond compression
• Bit slices can be combined into concept
  hierarchies
• Highest level bit 0
• Next level bit 0 bit 1
• Next level bit 0 bit 1 bit 2

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Compression for Key Attributes

• Database state-independent compression could lead
  to information loss (counts gt 1)
• Database state-dependent compression
• Tree structure that eliminates pure subtrees gt
  P-trees

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Other Ideas

• Compression is better if attribute values are
  dense within their domain
• We could use extent domain
• Compression good
• Problems with insertion
• Reorganization of storage
• Index locking has to be reintroduced

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How Good is Compression?

• If all domains are dense, i.e. all values occur
• Size can easily be smaller than original relation
• If non-key attributes are sparse
• Not usually a problem good compression
• Problems only in extreme cases
• E.g., movies as attribute values!
• If key-attributes are sparse
• Larger potential for problems, but also large
  potential for benefit (see data cubes)

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Are Key-Attributes Usually Sparse?

• Many key attributes are dense (structure
  attributes as keys)
• Automatically generated IDs are usually
  sequential
• x and y in spatial data mining
• Time in data streams
• Keys in tables that represent relationships tend
  to be sparse (feature attributes as keys)
• Student / course offering / grade
• Data cubes!

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What Have We Gained?(Database Aspects)

• Data simultaneously acts as index
• No separate index locking
• (unless extent domain is used)
• All information saved as bit patterns
• Easy select
• Other database operations discussed in class

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Data Mining Benefits(Feature Attribute Keys)

• Direct mining possible on relations with feature
  attributes keys
• E.g., student / course offering / grade
• Rollup can be defined, etc.
• Clustering, classification, ARM can make use of
  proximity inherent in representation
• Bit-wise representation provides concept
  hierarchy for non-key attribute
• Tree structure provides concept hierarchy for key
  attributes

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Data Mining Benefits(Structure Attribute Keys)

• For relations with structure attribute keys data
  mining requires anding
• produces counts for feature attributes
• Bit-wise representation provides concept
  hierarchy for non-key attribute
• Duality
• Concept hierarchies in this representation map
  exactly to tree structure when the attribute is a
  key

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Mapping Concept HierarchiesBit Slices lt-gt Tree

• P-tree
• Take key attributes, e.g. x and y, and bit
  interleave them
• x 1 0 0 1
• y 1 1 0 1
• 1 1 0 1 0 0 1 1
• Two consecutive digits form a level in the P-tree
  or a level in a concept hierarchy

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How Could We Use That Duality?

• Join with other relations and project off key
  attributes
• Duality allows moving to space of non-key
  attributes (Meta P-trees)
• Can we do that?
• We lose uniqueness
• We can use 1 to represent 1 or more tuples
  (equivalent to relational algebra)
• Or we can introduce counts
• Can be useful for data mining
• Need for non-duplicate eliminating counts exists
  also in other applications

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How Do Hierarchies Benefit us in Databases?

• Multi-granularity Locking
• Subtrees form suitable units for storage in a
  block
• Fast value-based access!
• (Data represented as multilevel index)
• Access speed proportional to
• of levels in tree
• of bits for bit slices

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Summary

• Space-based representation has many benefits
• Value-based access and storage
• No separate index needed
• Rollups easy
• P-Trees
• Follow from systematic compression
• Benefits from concept hierarchies