Transcript Slides - Network Protocols Lab

Dimensionality Reduction
CS 685: Special Topics in Data Mining
Spring 2008
Jinze Liu
The UNIVERSITY
of Mining,
KENTUCKY
CS685 : Special
Topics in Data
UKY
Overview
• What is Dimensionality Reduction?
– Simplifying complex data
– Using dimensionality reduction as a Data Mining
“tool”
– Useful for both “data modeling” and “data analysis”
– Tool for “clustering” and “regression”
• Linear Dimensionality Reduction Methods
– Principle Component Analysis (PCA)
– Multi-Dimensional Scaling (MDS)
• Non-Linear Dimensionality Reduction
– Next week
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What is Dimensionality Reduction?
• Given N objects, each with M measurements, find
the best D-dimensional parameterization
• Goal: Find a “compact parameterization” or
“Latent Variable” representation
D
~
p


x

f
(
p
)
Given N examples of xi   find
where
M
• Underlying assumptions to DimRedux
– Measurements over-specify data, M > D
– The number of measurements exceed the number of
“true” degrees of freedom in the system
– The measurements capture all of the significant variability
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Uses for DimRedux
• Build a “compact” model of the data
– Compression for storage, transmission, & retrieval
– Parameters for indexing, exploring, and organizing
– Generate “plausible” new data
• Answer fundamental questions about data
– What is its underlying dimensionality?
How many degrees of freedom are exhibited?
How many “latent variables”?
– How independent are my measurements?
– Is there a projection of my data set where
important relationships stand out?
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DimRedux in Data Modeling
• Data Clustering - Continuous to
Discrete
– The curse of dimensionality: the
sampling density is proportional to N1/p.
– Need a mapping to a lower-dimensional
space that preserves “important”
relations
• Regression Modeling –
Continuous to Continuous
– A functional model that
generates input data
– Useful for interpolation
• Embedding Space
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Today’s Focus
• Linear DimRedux methods
– PCA – Pearson (1901); Hotelling (1935)
– MDS – Torgerson (1952), Shepard (1962)
• “Linear” Assumption
~
x  Mp
where the matrix
M is parameters
mxd
– Data is a linear function
of the
(latent
variables)
x( )  x1  ( 1 )x2
" line  like"
x( , )  x1  x2  ( 1   )x3
" plane like"
– Data lies on a linear (Affine) subspace
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PCA: What problem does it solve?
• Minimizes “least-squares” (Euclidean) error
– The D-dimensional model provided by PCA has the
smallest Euclidean error of any D-parameter linear
model.
 n ~

min  ( xi xi )2 
 i1

~x
where i is the model predicted by the D-
dimensional PCA.
• Projects data s.t. the variance is maximized
• Find an optimal “orthogonal” basis set for
describing the given data
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Principle Component Analysis
• Also known to engineers as the Karhunen-Loéve
Transform (KLT)
• Rotate data points to align successive axes with
directions of greatest variance
– Subtract mean from data
– Normalize variance along each direction, and reorder
according
to the variance magnitude from high to low
– Normalized variance direction = principle component
• Eigenvectors of system’s Covariance Matrix
1 n
T
C
(
x


)
(
x


)
 i
i
n 1 i
( C i I) ei  0
permute to order eigenvectors in descending order
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Simple PCA Example
• Simple 3D example
>> x = rand(2, 500);
>> z = [1,0; 0,1; -1,-1] * x + [0;0;1] * ones(1, 500);
>> m = (100 * rand(3,3)) * z + rand(3, 500);
>> scatter3(m(1,:), m(2,:), m(3,:), 'filled');
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Simple PCA Example (cont)
>> mm = (m- mean(m')' * ones(1, 500));;
>> [E,L] = eig(cov(mm ‘ ));
>> E
E=
0.8029 -0.5958 0.0212
0.1629 0.2535 0.9535
0.5735 0.7621 -0.3006
>> L
L=
172.2525
0
0
0 116.2234
0
0
0 0.0837
>> newm = E’ * (m - mean(m’)’' * ones(1, 500));
>> scatter3(newm(1,:), newm(2,:), newm(3,:), 'filled');
axis([-50,50, -50,50, -50,50]);
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Simple PCA Example (cont)
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PCA Applied to Reillumination
• Illumination can be modeled as an additive
linear system.
R xy (wi )
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Simulating New Lighting
• We can simulate the appearance of a model under new illumination by
combining images taken from a set of basis lights
•
We can then capture real-world lighting and use it to modulate our basis lighting
functions
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Problems
• There are too many basis lighting functions
– These have to be stored in order to use them
– The resulting lighting model can be huge, in particular
when representing high frequency lighting
– Lighting differences can be very subtle
• The cost of modulation is excessive
– Every basis image must be scaled and added together
– Each image requires a high-dynamic range
• Is there a more compact representation?
– Yes, use PCA.
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PCA Applied to Illumination
• More than 90%
variance is
captured in the
first five principle
components
• Generate new
illumination by
combining only 5
basis images
V0
for n lights
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Results Video
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Results Video
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Results Video
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MDS: What problem does it solve?
• Takes as input a dissimilarity matrix M,
containing pairwise dissimilarities between
N-dimensional data points
• Finds the best D-dimensional linear
parameterization compatible with M
• (in other words, outputs a projection of
data in D-dimensional space where the
pairwise distances match the original
dissimilarities as faithfully as possible)
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Multidimensional Scaling (MDS)
• Dissimilarities can be metric or non-metric
• Useful when absolute measurements are
unavailable; uses relative measurements
• Computation is invariant to dimensionality of
data
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An example: map of the US
• Given only the distance between a bunch of cities
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An example: map of the US
• MDS finds suitable coordinates for the points of the
specified dimension.
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MDS Properties
• Parameterization is not unique –
Axes are meaningless
– Not surprising since Euclidean transformations and
reflections preserve distances between points
• Useful for visualizing relationships in high
dimensional data.
– Define a dissimilarity measure
– Map to a lower-dimensional space using MDS
• Common preprocess before cluster analysis
– Aids in understanding patterns and relationships in
data
• Widely used in marketing and psychometrics
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Dissimilarities
• Dissimilarities are distance-like quantities
that satisfy the following conditions:
1)  ij  0
2)  ii  0
3)  ij   ji
(self - similarity )
(symmetry)
• A dissimilarity is metric if, in addition, it
satisfies:
k ij  ik  kj
“The triangle inequality”
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Relating MDS to PCA
• Special case: when distances are Euclidean
• PCA = eigendecomposition of covariance
matrix MTM
• Convert the pair-wise distance matrix to
the covariance matrix
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How to get MTM from Euclidean Pair-wise
Distances
Law of cosines
i
d ki
d ij

Definition of a dot product
j
k
d kj
• Eigendecomposition on b to get VSVT
• VS1/2 = matrix of new coordinates
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Algebraically…
n
n
n
n


1 1
1
1
Bij  2  n  Dik  n  Dkj  Dij  n2  Dkl 
k 1
k 1 l 1
 k 1

The
distance
between
points
pi and pj
The *Row Average* the
average distance that a
given point is from pi
The
“Matrix Average”
The *Column Average* the
average distance that a
given point is from pj
So we “centered” the
matrix
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MDS Mechanics
• Given a Dissimilarity matrix, D, the MDS model
is computed as follows:
B   21 HD2H
(B   i I) ei  0
• Where, H, the so called “centering” matrix, is
a scaled identity matrix computed as follows:
i
H  (1 n1)I
• MDS coordinates given by (in order of
decreasing :
pi  ( 1 e1,i , 2 e2,i ,....)
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MDS Stress
• The residual variance of B (i.e. the sum of the remaining
eigenvalues) indicate the goodness of fit for the selected
d-dimensional model
• This term is often called MDS “stress”
• Examining the residual variance gives an indication of the
inherent dimensionality
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Reflectance Modeling Example
The top row of white, grey,
and black balls have the same
“physical” reflectance
parameters, however, the
bottom row is “perceptually”
more consistent.
• From Pellacini, et. al. “Toward a Psychophysically-Based
Light Reflection Model for Image Synthesis,” SIGGRAPH
2000
• Objective – Find a perceptually meaningful
parameterization for reflectance modeling
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Reflectance Modeling Example
• User Task – Subjects were presented with 378 pairs of
rendered spheres an asked to rate their difference in
“glossiness” on a scale of 0 (no difference) to 100.
• A dissimilarity 27 x 27 dissimilarity matrix was
constructed and MDS applied
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Reflectance Modeling Example
• Parameters of a
2D embedding
space were
determined
• Two axes of
“gloss” were
established
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Limitations of Linear methods
• What if the data does not lie
within a linear subspace?
• Do all convex combinations of
the measurements generate
plausible data?
• Low-dimensional non-linear
Manifold embedded in a higher
dimensional space
• Next time: Nonlinear
Dimensionality Reduction
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Summary
• Linear dimensionality reduction tools are widely
used for
– Data analysis
– Data preprocessing
– Data compression
• PCA transforms the measurement data s. t.
successive directions of greatest variance are
mapped to orthogonal axis directions (bases)
– An D-dimensional embedding space
(parameterization) can be established by modeling the
data using only the first d of these basis vectors
– Residual modeling error is the sum of the remaining
eigenvalues
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Summary (cont)
• MDS finds a d-dimensional parameterization that
best preserves a given dissimilarity matrix
– Resulting model can be Euclidean transformed to align
data with a more intuitive parameterization
– An D-dimensional embedding spaces
(parameterization) are established by modeling the
data using only the first d coordinates of the scaled
eigenvectors
– Residual modeling error (MDS stress) is the sum of the
remaining eigenvalues
– If Euclidean metric dissimilarity matrix is used for MDS
the resulting d-dimensional model will match the PCA
weights for the same dimensional model
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