Transcript pca

Neuronal Goal
We look for axes which minimise projection errors and maximise
the variance after projection
n-dimensional
m-dimensional
vectors
vectors
Ex:
m<n
transform from 2 to 1 dimension
Algorithm (cont’d)

Preserve as much of the variance as possible
more information (variance)
rotate
less
information
project
Linear transformations – example
2D vectors X in a unit circle with mean (1,1); Y = A*X, A = 2x2 matrix
 Y1   2 1  X 1 
 Y    1 1  X 
 2 
 2 
The shape is elongated, rotated and the mean is shifted.
Invariant distances
Euclidean distance is not invariant to general linear transformations
Y  AX
1
Y Y
2
2



  X   X   A A  X   X  
1
 Y Y
2
1
2
T
T
Y   Y
1
T
1
2
2
This is invariant only for orthonormal matrices ATA = I
that make rigid rotations, without stretching or shrinking distances.
Idea: standardize the data in some way to create invariant distances.
Data standardization
For each vector component X(j)T=(X1(j), ... Xd(j)), j=1 .. n
calculate mean and std: n – number of vectors, d – their dimension
1 n ( j)
1 n ( j)
Xi   X i ; X  X
n j 1
n j 1
X (1)
X (2)
X( n )
X1
X 1(1)
X 1(2)
X 1( n )
X2
X
(1)
2
(2)
2
(n)
2
Xd
X d(1)
X
X d(2)
X
X d( n )
Vector of mean
feature values.
Averages over
rows.
Standard deviation
Calculate standard deviation:
1 n ( j)
Xi   X i
n j 1

1 n
( j)
i 
X
 Xi

i
n  1 j 1
2
Vector of mean feature values.

2
Variance = square of standard
deviation (std), sum of all
deviations from the mean value.
Transform X => Z, standardized data vectors
Zi( j )   X i( j )  X i   i
Std data
Std data: zero mean and unit variance.


1 n ( j) 1 n
Zi   Z i   X i( j )  X i  i  0
n j 1
n j 1

2
Z ,i

1 n
( j)

Z
 Zi

i
n  1 j 1

2

1 n
( j)

X
 Xi

i
n  1 j 1

2
i2  1
Standardize data after making data transformation.
Effect: data is invariant to scaling only (diagonal transformation).
Distances are invariant, data distribution is the same??
How to make data invariant to any linear transformations?
Terminology (Covariance)
• How two dimensions vary from the mean with respect to each other
n
cov( X , Y ) 
(X
i 1
i
 X )(Yi  Y )
(n  1)

cov(X,Y) > 0: Dimensions increase together
cov(X,Y) < 0: One increases, one decreases

cov(X,Y) = 0: Dimensions are independent

Terminology (Covariance Matrix)
• Contains covariance values between all possible dimensions:
•
C nxn  (cij | cij  cov( Dimi , Dim j ))
Example for three dimensions (x,y,z) (Always symetric):
 cov( x, x) cov( x, y ) cov( x, z ) 


C   cov( y, x) cov( y, y ) cov( y, z ) 
 cov( z, x) cov( z, y ) cov( z, z ) 


cov(x,x)  variance of component x
Properties of the Cov matrix
• Can be used for creating a distance that
is not sensitive to linear transformation
• Can be used to find directions which
maximize the variance
• Determines a Gaussian distribution
uniquely (up to a shift)
Data standardization example
For our example Y=AX, assuming X means=1 and variances = 1
 Y1   2 1  X 1 
 Y    1 1  X 
 2 
 2 
Transformation
Vector of mean
feature values.
1
 3  2 1 1
X  Y 
 1
1
2
1
1
 
  
 
1
 5
σ X    σ Y     Diag  AA T 
1
 2
1
Y Y
 2
2

1
 X X
2

T

AT A X    X 
1
Variance
check it!
2

How to make this
invariant?
Covariance matrix
Variance (spread around mean value) + correlation between features.

1 n
(k )
Cij 
X
 Xi

i
n  1 k 1
 X
(k )
j
 X j ; i, j  1 d
CX is d x d
T
1
1
(k )
(k )
T
CX 
X

X
X

X

XX




n  1 k 1
n 1
n
where X is d x n dimensional matrix of vectors shifted to their means.
Covariance matrix is symmetric Cij = Cji and positive definite.
Diagonal elements are variances (square of std), i2 = Cii
Pearson correlation coefficient
rij  Cij  i j  [1, 1]
Spherical distribution of data has Cij=I (unit matrix).
Elongated ellipsoids: large off-diagonal elements, strong correlations
between features.
Mahalanobis distance
Linear combinations of features leads to rotations and scaling of data.
Y  AX; Y  AX; CY  AC X A T
X
Mahalanobis distance:
is invariant to linear transformations:
1
Y Y
2
2
CY


  X   X  
1
 Y Y
2
1
2
1
2
 X
X

2
CX
 XTCX1X

T
CY1 Y 1  Y  2 
T
T 1

A  A  CX1A 1  A X 1  X  2 


2
CX
T


Principal components
How to avoid correlated features?
Correlations  covariance matrix is non-diagonal !
Solution: diagonalize it, then use transformation that makes it
diagonal to de-correlate features. Z are the eigen vectors of Cx
Y  Z X; CX Z  i Z ; CX Z  ZΛ
T
(i )
(i )
CY  ZT CX Z  ZT ZΛ  Λ
In matrix form,
X, Y are dxn,
Z, CX, CY are
dxd
C – symmetric, positive definite matrix XTCX > 0 for ||X||>0;
its eigenvectors are orthonormal:
Z(i )T  Z( j )  ij
its eigenvalues are all non-negative
Z – matrix of orthonormal eigenvectors (because Z is real+symmetric),
transforms X into Y, with diagonal CY, i.e. decorrelated.
Matrix form
Eigenproblem for C matrix in matrix form:
 C11 C12
C
 21 C22


 Cd 1 Cd 2
 Z11 Z12
Z
 21 Z 22


 Zd1 Zd 2
C1d  Z11 Z12
C2 d 
 Z 21 Z 22


Cdd  Z d 1 Z d 2
Z1d  1 0
Z 2 d 
 0 2


Z dd  0 0
C X Z  ZΛ
Z1d 
Z 2 d 



Z dd 
0
0 


d 
Principal components
PCA: old idea, C. Pearson (1901), H. Hotelling 1933
Y  Z X;
T
CY  ZT C X Z  Λ
Z – principal components, of vectors X
transformed using eigenvectors of CX
Covariance matrix of transformed
vectors is diagonal => ellipsoidal
distribution of data.
Result: PC are linear combinations of all features, providing new
uncorrelated features, with diagonal covariance matrix = eigenvalues.
Small i  small variance  data change little in direction Yi
PCA minimizes C matrix reconstruction errors:
ZΛZ
Zi vectors for large i are sufficient to get:
because vectors for small eigenvalues will have very
small contribution to the covariance matrix.
T
 CX
Two components for visualization
Diagonalization methods: see Numerical Recipes, www.nr.com
New coordinate system:
axis ordered according to variance
= size of the eigenvalue.
First k dimensions account for
k
Vk 

i 1
i
d

i 1
i
fraction of all variance (please note that i are variances);
frequently 80-90% is sufficient for rough description.
Solving for Eigenvalues & Eigenvectors
• Vectors x having same direction as Ax are called
eigenvectors of A (A is an n by n matrix).
• In the equation Ax=x,  is called an eigenvalue
of A.
• Ax=x  (A-I)x=0
• How to calculate x and :
– Calculate det(A-I), yields a polynomial (degree n)
– Determine roots to det(A-I)=0, roots are eigenvalues 
– Solve (A- I) x=0 for each  to obtain eigenvectors x
PCA properties
PC Analysis (PCA) may be achieved by:
• transformation making covariance matrix diagonal
• projecting the data on a line for which the sums of squares of
distances from original points to projections is minimal.
• orthogonal transformation to new variables that have stationary
variances
True covariance matrices are usually not known, estimated from data.
This works well on single-cluster data; more complex structure may
require local PCA, separately for each cluster.
PC is useful for: finding new, more informative, uncorrelated features;
reducing dimensionality: reject low variance features,
reconstructing covariance matrices from low-dim data.
PCA Wisconsin example
Wisconsin Breast Cancer data:
• Collected at the University of Wisconsin Hospitals, USA.
• 699 cases, 458 (65.5%) benign (red), 241 malignant (green).
• 9 features: quantized 1, 2 .. 10, cell properties, ex:
Clump Thickness, Uniformity of Cell Size, Shape, Marginal
Adhesion, Single Epithelial Cell Size, Bare Nuclei,
Bland Chromatin, Normal
Nucleoli, Mitoses.
2D scatterograms do not show
any structure no matter which
subspaces are taken!
Example cont.
PC gives useful information
already in 2D.
Taking first PCA component of
the standardized data:
If (Y1>0.41) then benign
else malignant
18 errors/699 cases = 97.4%
Transformed vectors are not
standardized, std’s are below.
Eigenvalues converge
slowly, but classes are
separated well.
PCA disadvantages
Useful for dimensionality reduction but:
•
Largest variance determines which components are used, but
does not guarantee interesting viewpoint for clustering data.
•
The meaning of features is lost when linear combinations are
formed.
Analysis of coefficients in Z1 and other important eigenvectors may
show which original features are given much weight.
PCA may be also done in an efficient way by performing singular
value decomposition of the standardized data matrix.
PCA is also called Karhuen-Loève transformation.
Many variants of PCA are described in A. Webb, Statistical pattern
recognition, J. Wiley 2002.
Exercise (will be part of Ex. 1)
• How would you calculate efficiently the PCA
of data where the dimensionality d is much
larger than the number of vector observations
n?
2 skewed distributions
PCA transformation for 2D data:
First component will be chosen along
the largest variance line, both clusters
will strongly overlap, no interesting
structure will be visible.
In fact projection to orthogonal axis to
the first PCA component has much
more discriminating power.
Discriminant coordinates should be
used to reveal class structure.
Projection Pursuit
Projection Pursuit (PP)
PCA and FDA are linear, PP may be linear or non-linear.
Find interesting “criterion of fit”, or “figure of merit” function,
that allows for low-dim (usually 2D or 3D) projection.
Y( j )T  Y1( j ) ,Y2( j )   f  X( j ) ; W  ;
I (Y; W)  I  f  X; W  
General transformation
with parameters W.
Index of “interestingness”
Interesting indices may use a priori knowledge about the problem:
1.
2.
3.
mean nearest neighbor distance – increase clustering of Y(j)
maximize mutual information between classes and features
find projection that have non-Gaussian distributions.
The last index does not use a priori knowledge; it leads to the
Independent Component Analysis (ICA).
ICA features are not only uncorrelated, but also independent.
Kurtosis
ICA is a special version of PP, recently very popular.
Gaussian distributions of variable Y are characterized by 2 parameters:
mean value:
Y  E{Y }
variance:
 Y2  E{Y  E (Y )}2
These are the first 2 moments of distribution; all higher are 0 for G(Y).
One simple measure of non-Gaussianity of projections is the
4-th moment (cumulant) of the distribution, called kurtosis, measures
“skewedness” of the distribution. For E{Y}=0 kurtosis is:
k 4 Y   E Y
4
  3 E Y 
2
2
Super-Gaussian distribution: long tail, peak at zero,
k4(y)>0, like binary image data.
sub-Gaussian distribution is more flat and has
k4(y)<0, like speech signal data.
Correlation and independence
Variables are statistically independent if their joint probability
distribution is a product of probabilities for all variables:
p  X 1, X 2
n
X n    pi  X i 
i 1
Features Yi, Yj are uncorrelated if covariance is diagonal, or:
E YiY j   E Yi  E Y j 
Uncorrelated features are orthogonal.
Statistically independent features Yi, Yj for any functions give:




E f1 Yi  f 2 Y j   E  f1 Yi  E f 2 Y j 
This is much stronger condition than correlation; in particular the
functions may be powers of variables; any non-Gaussian distribution
after PCA transformation will still have correlated features.
PP/ICA example
Example: PCA and PP based on maximal kurtosis: note nice
separation of the blue class.
Some remarks
• Many formulations of PP and ICA methods exist.
• PP is used for data visualization and dimensionality reduction.
• Nonlinear projections are frequently considered, but solutions are
more numerically intensive.
• PCA may also be viewed as PP, max (for standardized data):

W  arg max E  W X 
(1)
W 1
T
2

Index I(Y;W) is based here
on maximum variance.
Other components are found in the space orthogonal to W1T X
W(k )
2


k 1





 arg max E  W T   I   W ( i ) W T( i )  X   
W 1
i 1
   


Same index is used, with projection on space orthogonal to k-1 PCs.
How do we find multiple Projections
• Statistical approach is complicated:
– Perform a transformation on the data to
eliminate structure in the already found
direction
– Then perform PP again
• Neural Comp approach: Lateral
Inhibition
ICA demos
• ICA has many applications in signal and image analysis.
• Finding independent signal sources allows for separation of
signals from different sources, removal of noise or artifacts.
Observations X are a linear mixture W of unknown sources Y
X  WTY
Both W and Y are unknown! This is a blind separation problem.
How can they be found?
If Y are Independent Components and W linear mixing the problem
is similar to FDA or PCA, only the criterion function is different.
Play with ICALab PCA/ICA Matlab software for signal/image
analysis:
http://www.bsp.brain.riken.go.jp/page7.html
ICA demo: images & audio
Example from Cichocki’s lab,
http://www.bsp.brain.riken.go.jp/page7.html
X space for images:
take intensity of all pixels  one vector per image, or
take smaller patches (ex: 64x64), increasing # vectors
• 5 images: originals, mixed, convergence of ICA iterations
X space for signals:
sample the signal for some time Dt
• 10 songs: mixed samples and separated samples
Self-organization
PCA, FDA, ICA, PP are all inspired by statistics, although some neuralinspired methods have been proposed to find interesting solutions,
especially for their non-linear versions.
• Brains learn to discover the structure of signals: visual, tactile,
olfactory, auditory (speech and sounds).
• This is a good example of unsupervised learning: spontaneous
development of feature detectors, compressing internal information
that is needed to model environmental states (inputs).
• Some simple stimuli lead to complex behavioral patterns in animals;
brains use specialized microcircuits to derive vital information from
signals – for example, amygdala nuclei in rats sensitive to
ultrasound signals signifying “cat around”.
Models of self-organizaiton
SOM or SOFM (Self-Organized Feature Mapping) – self-organizing
feature map, one of the simplest models.
How can such maps develop spontaneously?
Local neural connections: neurons interact strongly with those
nearby, but weakly with those that are far (in addition inhibiting
some intermediate neurons).
History:
von der Malsburg and Willshaw (1976), competitive learning, Hebb
mechanisms, „Mexican hat” interactions, models of visual systems.
Amari (1980) – models of continuous neural tissue.
Kohonen (1981) - simplification, no inhibition; leaving two essential
factors: competition and cooperation.
Computational Intelligence:
Methods and Applications
Lecture 8
Projection Pursuit &
Independent Component Analysis
Włodzisław Duch
SCE, NTU, Singapore
Google: Duch
Computational Intelligence:
Methods and Applications
Lecture 6
Principal Component Analysis.
Włodzisław Duch
SCE, NTU, Singapore
http://www.ntu.edu.sg/home/aswduch