Projective Mapping: A noniteratve method for the Layout of Multidimensional Data

1
Projective Mapping A non-iteratve method for the
Layout of Multidimensional Data

• Dr. Karina Assiter
• Wentworth Institute of Technology
• SCI2002

2
Outline

• Background
• Projective Mapping
• Experimental Results
• Conclusions
• References

3
Background

• Data generation and Data understanding
• Data mining example
• Objective of Layout
• Layout Methods
• MDS Optimization
• Faithfulness in Mapping
• Subjective Layout

4
Data generation and data understanding

• Data generation
• Provides Raw material
• Results in large, high-dimensional data sets
• Amount in worlds databases doubles every 20
  months
• Many attributes for each sample
• Data understanding
• Fuels business growth (Competitive advantage)
• Insights and patterns used to predict the future
• Techniques
• Clustering
• Classifying
• Visualizing (layout)

5
Data Mining Example

• Iris Dataset - R.A. Fisher (1930s)
• 50 samples
• Three types of plants (setosa, versicolor,
  virginica)
• Four attributes (Sepal length/width,Petal
  length/width)
• Techniques
• Classify new plants (learn rules)
• If petal-length lt 2.45 then Iris-setosa
• Cluster items that fall together
• If we did not know the type of Iris
• Visualize the dataset (layout)

6
Objective of Layout

• Create a mapping F from h samples in an
  n-dimensional sample set S to representative
  points in an m-dimensional representation set R.

7
Layout Methods

• Multidimensional scaling (MDS)
• Position points in R so that the distances match
  (as closely as possible) the set of
  dissimilarities in S.
• Self-organizing maps
• Mapping from an input data space S onto a
  two-dimensional array of nodes R
• Similar samples in S mapped to nearby nodes in R
• Subjective Layout methods
• Samples placement in R depends upon its
  relationship with user defined (and positioned)
  anchors
• Inter-sample relationships may or may not be
  preserved.

8
Illustration of MDS
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MDS Optimization Methods

• Move points around in R to minimize error
  function
• Error function
• How accurately proximity values in S are related
  to distances in R
• Example E2 1 ? (d(Pi , Pj )
  d(Qi , Qj )) 2 ? d (Pi ,
  Pj ) iltj d(Pi , Pj ) iltj
• General characteristics
• Run time complexity of O(h2)
• h is number of samples
• Iterative (continue until stopping condition)

10
Faithfulness in Mapping

• A mapping F S -gt R of an x-dimensional
  subspace of an n-dimensional space into an
  m-dimensional reference space is faithful if the
  mapping preserves proximities precisely so that
  d(Pi,Pj) d( F(Pi ), F(Pj )), when Pi and Pj are
  in S.

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Illustration of Faithfulness
S
R
F
P2
Q2
P1
Q1
P3
P4
Q3
Q4
12
Subjective Layout

• A samples mapping in R depends upon its
  relationship with user defined (and positioned)
  anchors
• Inter-object relationships may or may not be
  preserved.
• Classified as
• Query-relative (traditional)
• Anchors in T are queries (keywords, mailing
  lists, etc,)
• Point-relative
• Anchors in T are representatives of samples from
  S

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Illustration of Subjective Layout
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Projective Mapping

• Overview
• Illustration of placing a sample
• Explain reference set T
• Method Details
• Determine Sense of P in P

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Projective Mapping Overview

• Samples in a dataset S are mapped into a
  representation set R based on the centroid of
  their geometric relationships to a set T of
  pre-positioned references

16
Illustration of placing a sample
S
R
Sk
Rk
J
Sj
J
Rj
L1
L1
L2
L2
Qi
P
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Reference Set T

• k samples selected from S
• S1, S2, Sk
• Mapped into R
• Arbitrarily
• User positions
• Computed
• Optimization method that preserves distances

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Method Details
I. Determine plane P onto which we wish to
project w1, w2) II. For each vector, P, in S
A. For each valid pair of references,
Sj and Sk 1. Determine sense (left
or right) of projected P in P (explained in
next slide) 2. In S, find unit vector B
perpendicular to L1 through P 3. In
S, get t1 and t2 from
J Sj (Sk Sj) t1 J
P B t2       4. In R, find unit
vector B perpendicular to L1, depending on
sense B ( Rk1 - Rj1 , Rj0 - Rk0
) Sense was left B ( Rj1
Rk1 , Rk0 Rj0 ) Sense was right
5. Use t1 to determine J in R
J Rj (Rk Rj ) t1
6. Determine scale factor from ratio between
line Sk Sj in S and line Rk - Rj in R
sf ( Rk-Rj / Sk-Sj  
7. Calculate Qi based on t2 and B
Qi J (B t2 sf)
B. Average results from each pair to get
position Q.
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Determine Sense of P in P
B-coord of Sk (-2,1)
B-coord of Sj (2,-1)
B-coord of P (-1,1)
W2 (0,1,0)
w1 (1,0,0)
Plane P to project onto (1,0,0), (0,1,0)
Equation of line between B-coord Sj and
B-coord Sk of 2x 4y 0 So
2(-1) 4(1) gt 0 left side of line
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Experimental Results

• Example I
• Embedded two-dimensional data
• Example II
• Lines to Lines
• Example III
• Rotation
• Example IV
• Identity mapping
• Example V (Optional)
• Cube series

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Example I Embedded two-dimensional data
z
S
Samples in S (triangles) x y z P0
.5 0 0 P1 0 0 .5 P2 .5 0
-.5 P3 1 0 0 P4 0 0 -2 P5 -1
0 0
S1 at (0,0,1)
S2 at (-1,0,0)
S0 at (1,0,0)
x
x
y
S3 at (0,0,-1)
R
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Example II Lines to lines
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Example III Rotation
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Example IV Identity Mapping
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Example V Cube series (Optional)
Front face, horizontal plane
Front face T, Plane between horizontal and 45
Front face T, Plane between 45 and vertical
Front face T, Plane is vertical
Diagonal T, vertical plane
Diagonal T, horizontal plane
Diagonal T, Planes between horizontal and 45
Diagonal T, Planes between 45 and vertical
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Run-time Complexity

• Inserting one sample
• Depends on k
• The number of references in T
• Upper bound of O(k2)
• k! All ways to select k items 2
  ways
• (k-2)! 2! 
• Inserting h samples
• Upper bound of O(k2 h).

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Comparing Methods
Hypercube samples 00000-11111.
Projective mapping References in S
00000-00011 Associated references in R
00-11. The plane for the projection was
00001,00010.
Projective Mapping on the Hypercube
Sammon Mapping on the Hypercube
Iris dataset - 150 samples - three clusters of
flowers.
Sammon Mapping on the Iris dataset
Projective Mapping on the Iris dataset
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Conclusions

• Benefits of PM
• Drawbacks of PM
• Uses for Projective mapping
• Future Work

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Benefits of PM

• Fast O (k2h)
• Non-iterative
• Guaranteed upper-bound
• Adaptive
• Point-relative Subjective
• In one and two-dimensions
• Faithful mapping of references applied to samples
• Linear transformation of references applied to
  samples
• Structure preserving (with distortion)
• Works with fallible and sparse datasets
• Creates consistent layouts
• Not domain specific

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Drawbacks of PM

• With high-dimensional data
• Distortion occurs
• Plane selection is hard to optimize
• Faithfulness requires Euclidean distances

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Uses for Projective Mapping

• Visualization to discover
• Data Dimensionality
• Data Structure
• Relationship between a subset of references and
  all other samples

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Future work

• Generalize for
• N-dimensional to n-dimensional mapping
• Vary for testing
• Plane selection
• Data dimensionality
• Reference set
• Selection
• Size
• Visualize as slide show
• multiple-views of dataset

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Selected References

• Assiter, K.A. (2001) Projective Mapping A
  non-iterative method for the layout of
  multidimensional data. Dissertation. Tufts
  University, Medford, MA, 2001.
• Chalmers, M. and P. Chitson (1992). Bead
  Explorations in Information Visualization. SIGIR
  '92 Proceedings of the Fifteenth annual
  International ACM SIGIR Conference on Research
  and Development in Information Retrieval,
  Denmark, ACM Press.
• Cox, T. F. and M. A. Cox (1994). Multidimensional
  Scaling. Monographs on Statistics and Applied
  Probability. London, Chapman Hall
• Fairchild, K. M., S. E. Poltrock and G. W. Furnas
  (1988). SemNet Three-Dimensional Graphic
  Representations of Large Knowledge Bases.
  Cognitive Science and its Applications for
  Human-Computer Interaction. R. Guindon.
  Hillsdale, New Jersey, Lawrence Erlbaum
  Associates 201-233.
• Kruskal, J. B. (1964a). Multidimensional Scaling
  by optimizing goodness of fit to a non-metric
  hypothesis. Psychometrika 29(1) 1-27. Reprinted
  in Key Texts in Multidimensional Scaling, P.M.
  Davies and A.P.M Coxon, Eds. Heinemann
  Educational Books, Exeter, N.H.., 1982, pp 59-83.
• Kruskal, J. B. (1964b). Non-metric
  multidimensional Scaling A numerical method.
  Psychometrika 29(2) 115--129, Reprinted in Key
  Texts in Multidimensional Scaling, P.M. Davies
  and A.P.M Coxon, Eds. Heinemann Educational
  Books, Exeter, N.H.., 1982, pp 59-83.
• Olsen, K. A. (1993). Visualization of a document
  Collection The Vibe System. Information
  processing and management 29(1) 69-81, Pergamon
  Press Ltd, 1993.
• Sammon, J. W. (1969). A Nonlinear mapping for
  Data Structure Analysis. IEEE Transactions on
  Computers 18(5) 401-409, May 1969.

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End
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Multidimensional Scaling

• Metric
• Preserve actual proximities
• Types
• Classical (PVA)
• Least squares
• Optimization method with global error function
• Non-metric
• Preserve rank order of proximities
• Optimization method with global error function
• Spring model methods
• Attractive and repulsive forces between objects
  act to either bring them together or push them
  apart
• Optimization with local error function

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Notation
 S N-dimensional sample set Sj, Sk
Projection pair of references in S J Point
where P is projected perpendicularly to L2 L1
Line that goes through Sj and Sk L2 Line
that goes through J and P P Sample in S to
be mapped t1 Placement of J along L1
(time parameter in linear equation) t2
Distance between J and P (not a time
parameter)   R Two-dimensional
representation set Rj, Rk Projection
pair of references in R J Intersection
point between L1 and L2 L1 Line that
goes through Rj and Rk L2 Line that goes
through J and Qi Qi Result of Two point
n-dimensional projective mapping