Computational Intelligence: Computational Intelligence: Methods and ApplicationsMethods and Applications

Lecture 4 CI: simple visualization.

Source: Włodzisław Duch; Dept. of Informatics, UMK; Google: W Duch

2D projections: scatterplots 2D projections: scatterplots Simplest projections: use scatterplots, select only 2 features. Example: sugar – teeth decay.

If d=3 than d(d-1)/2=3 subsets in 2D are formed and sometimes displayed in one figure.

Each 2D point is an orthogonal projection from all other d-2 dimensions.

What to look for:

correlations between variables,

clustering of different objects.

Problem: for discrete values data points overlap.

Extreme case: binary data in many dimensions, all structure is hidden, each scatterogram shows 4 points.

Sugar example Sugar example

What conclusion can we draw?

Can there be alternative explanations?

Brain-body index example Brain-body index example

What conclusion can we draw?

Are whales and elephants smarter than man?

Are correlations sufficient to establish causes?

4 Gaussians in 8D, 4 Gaussians in 8D, XX1 1 vs. vs. XX224 Gaussians in 8D, 4 Gaussians in 8D, XX1 1 vs. vs. XX22

Scatterograms of 8D data in F1/F2 dimensions.

4 Gaussian distributions, each in 4D, have been generated, the red centered at (0,0,0,0), green at (1,1/2,1/3,1/4), yellow at 2(1,1/2,1/3,1/4) and blue at 3(1,1/2,1/3,1/4)

Demonstration of various projections using Ghostminer software.

4 Gaussians in 8D, 4 Gaussians in 8D, XX1 1 vs. vs. XX554 Gaussians in 8D, 4 Gaussians in 8D, XX1 1 vs. vs. XX55

What happened here?

All Xi vs. Xi+4 have this kind of plots.

How were the remaining 4 features generated?

Cars example Cars example Scatterograms for all feature pairs, data on cars with 3, 4, 5, 6 or 8 cylinders.

To detailed? We are interested in trends that can be seen in probability density functions.

Cluster all points that are close for cars with N cylinders. This may be done by adding Gaussian noise with a growing variance to each point.

See this on the movie:

Movie for cars.

Direct representation: GTDirect representation: GT

How to deal with more than 3D? We cannot see more dimensions.

Grand Tour: move between different 2D projections; implemented in XGobi, XLispStat, ExplorN software packages.

Ex: 7D data viewed as scatterplot in Grand Tour

More examples: http://www.public.iastate.edu/~dicook/JSS/paper/paper.html

Try to view 9D cube – most of the time looks like Gaussian cloud.

It may take time to “calibrate our eyes” to imagine high-D structure.

Direct representation: starDirect representation: starStar Plots, radar plots:

represent the value of each component in a “spider net”.

Useful to display single or a few vectors per plot, uses many plots.

Too many individual plots? Cluster similar ones, as in the car example.

x1

x2

x3x4

x5

Direct representation: starDirect representation: starWorking woman

population changes in different states, projections and reality.

Direct representations: ||Direct representations: ||Parallel coordinates: instead of perpendicular axes use parallel!

Many engineering applications, popular in bioinformatics.

Two clusters in 3D

See more examples at: http://www.nbb.cornell.edu/neurobio/land/PROJECTS/Inselberg/

Instead of creating perpendicular axes

put each coordinate on the horizontal x axis

and its value on the vertical y axis.

Point in N dim => line with N segments.

|| lines|| linesLines in parallel representation:

2D line 3D line 4D line

|| cubes|| cubesHypercubes in parallel representation:

2D (square) 3D cube: 8 vertices 8D: 256 vertices

|| spheres|| spheresHypercubes in parallel representation:

2D (circle) 3D (sphere) ... 8D: ???

Try some other geometrical figures and see what patterns are created.

|||| coordintates coordintatesRepresentation of 10-dim. line (x1,,... x10) t, car information data

Parallax software: http://www.kdnuggets.com/software/parallax/

IBM Visualization Data Explorer http://www.research.ibm.com/dx/

has Parallel Coordinates module

http://www.cs.wpi.edu/Research/DataExplorer/contrib/parcoord/

Financial analysis example.

More toolsMore tools

Statgraphics charting tools

Modeling and Decision Support Tools collected at the University of Cambridge (UK) are at:

http://www.ifm.eng.cam.ac.uk/dstools/

Book: T. Soukup, I. Davidson, Visual Data Mining-Techniques and Tools for Data Visualization and Mining. Wiley 2002

More tools:

http://www.is.umk.pl/~duch/CI.html#vis

Computational Intelligence: Computational Intelligence: Methods and ApplicationsMethods and Applications

Lecture 5 EDA and linear transformations.

Source: Włodzisław Duch; Dept. of Informatics, UMK; Google: W Duch

Chernoff facesChernoff facesHumans have specialized brain areas for face recognition.

For d < 20 represent each feature by changing some face elements.

Interesting applets:

http://www.cs.uchicago.edu/~wiseman/chernoff/

http://www.cs.unm.edu/~dlchao/flake/chernoff/ (Chernoff park)

http://kspark.kaist.ac.kr/Human%20Engineering.files/Chernoff/Chernoff%20Faces.htm

Fish viewFish viewOther shapes may also be used to visualized data, for example fish.

Ring visualizationRing visualization (SunBurst) (SunBurst)Show tree-like hierarchical representation in form of rings.

Other EDA techniquesOther EDA techniques

NIST Engineering Statistics Handbook has a chapter on

exploratory data analysis (EDA).

http://www.itl.nist.gov/div898/handbook/index.htm

Unfortunately many visualization programs are written for X-Windows only, are in Fortran, or S or R languages.

Sonification: data converted to sounds!

Example: sound of EEG data.

Java Voice

Think about potential applications! More: http://sonification.de/

http://en.wikipedia.org/wiki/Sonification

CI approach to visualizationCI approach to visualization

Scatterograms: project all data on two features.

Find more interesting directions to create projections.

Linear projections:

• Principal Component Analysis,

• Discriminant Component Analysis,

• Projection Pursuit – “define interesting” projections.

Non-linear methods – more advanced, some will appear later.

Statistical methods: multidimensional scaling.

Neural methods: competitive learning, Self-Organizing Maps.

Kernel methods, principal curves and surfaces.

Information-theoretic methods.

Distances in feature spacesDistances in feature spaces

Data vector, d-dimensions XT = (X1, ... Xd), YT = (Y1, ... Yd)

Distance, or metric function, is a 2-argument function that satisfies:

Distance functions measure (dis)similarity.

, 0; , ,

, , ,

d d d

d d d

X Y X Y X Y Y X

X Y X Z Z Y

1/ 2

2

21

11

d

i ii

d

i ii

X Y

X Y

X Y

X Y

Popular distance functions:

Euclidean distance (L2 norm)

Manhattan (city-block) distance

(L1 norm)

Two metric functionsTwo metric functions

Equidistant points in 2D:

Euclidean case: circle or sphere Manhattan case: square

Identical distance between two points X, Y: imagine that in 10 D !

X1

X2

X1

X2

L L X P Y P

X X

YY

All points in the shaded area have the same Manhattan distance to X and Y!

isotropic non-isotropic

Linear transformationsLinear transformations

2D vectors X in a unit circle with mean (1,1); Y = AX, A = 2x2 matrix

The shape and the mean of data distribution is changed.

Scaling (diagonal aii elements); rotation (off-diag), mirror reflection.

Distances between vectors are not invariant: ||Y1-Y2||≠||X1-X2||

1 11 12 1

2 21 22 2

Y a a X

Y a a X

Invariant distancesInvariant distances

Euclidean distance is not invariant to linear transformations Y = AX,

scaling of units has strong influence on distances.

How to select scaling/rotations for simplest description of data?

Orthonormal matrices: ATA = I, are inducing rigid rotations.

To achieve full invariance requires therefore standardization of data (scaling invariance) and should use covariance matrix.

Mahalanobis metric will replace ATA by inverse of the covariance matrix.

T21 2 1 2 1 2

T1 2 1 2T

Y Y Y Y Y Y

X X A A X X

Data standardizationData 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 1

1 1;

i

n nj j

ij j

X Xn n

X X Vector of mean

feature values.

Averages over rows.

(1) (2) ( )

(1) (2) ( )1 1 1 1

(1) (2) ( )2 2 2 2

(1) (2) ( )

n

n

n

nd d d d

X X X X

X X X X

X X X X

X X X

Standard deviationStandard deviation

Calculate standard deviation:

Why n1, not n ? If true mean was known it should be n, but if the mean

is calculated the formula with n1 converges to the true variance!

Transform X => Z, standardized data vectors:

( )

1

22 ( )

1

1

1

1

i

i i

nj

ij

nj

ij

X Xn

X Xn

Vector of mean feature values.

Variance = square of standard deviation (std), sum of all deviations from the mean value.

( ) ( )j ji i i iZ X X

Standardized dataStandardized data

Std data: zero mean and unit variance.

Standardize data after making data transformation.

Effect: data is invariant to scaling only; for diagonal transformations distances after standardization are invariant, are based on identical units.

Note: it does not mean that all data models are performing better!

How to make data invariant to any linear transformations?

,

( ) ( )

1 1

2 22 ( ) ( ) 2

1 1

1 10

1 11

1 1

i i

Z i i i

n nj j

i i ij j

n nj j

i i ij j

Z Z X Xn n

Z Z X Xn n

Std exampleStd example

Before std

After std

Mean and std are shown using a colored bar; minimum and max values may extend outside.

Some features (ex. yellow), have large values; some (ex: gray) have small values; this may depend on units used to measure them.

Standardized data have all mean 0 and =1, thus contribution from different features to similarity or distance calculation is comparable.

Computational Intelligence: Computational Intelligence: Methods and ApplicationsMethods and Applications

Lecture 6 Principal Component Analysis.

Source: Włodzisław Duch; Dept. of Informatics, UMK; Google: W Duch

Linear transformations – exampleLinear transformations – example2D vectors X uniformly distributed in a unit circle with mean (1,1);

Y = AX, A = 2x2 matrix

The shape is elongated, rotated and the mean is shifted.

1 1

2 2

2 1

1 1

Y X

Y X

Invariant distancesInvariant distances

Euclidean distance is not invariant to general linear transformations

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.

Y A X

2 T1 2 1 2 1 2

T1 2 1 2T

Y Y Y Y Y Y

X X A A X X

Data standardizationData 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 1

1 1;

i

n nj j

ij j

X Xn n

X X Vector of mean

feature values.

Averages over rows.

(1) (2) ( )

(1) (2) ( )1 1 1 1

(1) (2) ( )2 2 2 2

(1) (2) ( )

n

n

n

nd d d d

X X X X

X X X X

X X X X

X X X

Standard deviationStandard deviation

Calculate standard deviation:

Transform X => Z, standardized data vectors

( )

1

22 ( )

1

1

1

1

i

i i

nj

ij

nj

ij

X Xn

X Xn

Vector of mean feature values.

Variance = square of standard deviation (std), sum of all deviations from the mean value.

( ) ( )j ji i i iZ X X

Std dataStd data

Std data: zero mean and unit variance.

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?

,

( ) ( )

1 1

2 22 ( ) ( ) 2

1 1

1 10

1 11

1 1

i i

Z i i i

n nj j

i i ij j

n nj j

i i ij j

Z Z X Xn n

Z Z X Xn n

Data standardization exampleData standardization example

In slide 2 example Y=AX, assume all X means =1 and variances = 1

Transformation

Vector of mean

feature values.

Variance

check it!

1 3 2 1 1

1 2 1 1 1

X Y

1 1

2 2

2 1

1 1

Y X

Y X

2 2 T1 5Diag

1 2X

Yσ σ AA

2 T1 2 1 2 1 2T Y Y X X A A X X How to make this

invariant?

Covariance matrixCovariance matrix

Variance (spread around mean value) + correlation between features.

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

( ) ( )

1

T( ) ( ) T

1

1; , 1

1

1 1

1 1

i

nk k

ij i j jk

nk k

k

C X X X X i j dn

n n

XC X X X X XX

[ 1, 1]ij ij i jr C

Spherical distribution of data has Cij=I (unit matrix).

Elongated ellipsoids: large off-diagonal elements, strong correlations between features.

CX is d x d

CorrelationCorrelation

Correlation coefficient is linear and may be confusing …

Mahalanobis distanceMahalanobis distance

Linear combinations of features leads to rotations and scaling of data.

Mahalanobis distance defined as:

is invariant to linear transformations:

T; ; Y X Y AX Y AX C AC A

2 T1 2 1 2 1 21

T 11 2 1 2T T 1 1

21 2

Y

X

YC

X

C

Y Y Y Y C Y Y

X X A A C A A X X

X X

2 T 1

XXC

X X C X

Principal componentsPrincipal components

How to avoid correlated features?

Correlations covariance matrix is non-diagonal !

Solution: diagonalize it, then use the transformation that makes it

diagonal to de-correlate features.

C – symmetric, positive definite matrix XTCX > 0 for ||X||>0;

its eigenvectors are orthonormal:

its eigenvalues are all non-negative i ≥ 0

Z – matrix of orthonormal eigenvectors (because CX is real+symmetric),

transforms X into Y, with diagonal CY, i.e. decorrelated.

T ( ) ( )

T T

; ;i ii

X X

Y X

Y Z X C Z Z C Z ZΛ

C Z C Z Z ZΛ Λ

In matrix form, X, Y are dxn, Z, CX, CY are dxd

( )T ( )i jij Z Z

Matrix formMatrix form

Eigenproblem for C matrix in matrix form:X C Z ZΛ

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

11 12 1 1

21 22 2 2

1 2

0 0

0 0

0 0

d d

d d

d d dd d d dd

d

d

d d dd d

C C C Z Z Z

C C C Z Z Z

C C C Z Z Z

Z Z Z

Z Z Z

Z Z Z

Principal componentsPrincipal componentsPCA: old idea, C. Pearson (1901), H. Hotelling 1933

Result: PC are linear combinations of all features, providing new uncorrelated features, with diagonal covariance matrix = eigenvalues.

T

T

;

Y X

Y Z X

C Z C Z Λ

TXZΛZ C

Small i small variance data change little in direction Yi

PCA minimizes C matrix reconstruction errors:

Zi vectors for large i are sufficient to get:

because vectors for small eigenvalues will have very

small contribution to the covariance matrix.

Y – principal components, or vectors X transformed using eigenvectors of CX

Covariance matrix of transformed vectors is diagonal => ellipsoidal distribution of data.

Two components for visualizationTwo components for visualization

New coordinate system: axis ordered according to variance = size of the eigenvalue.

First k dimensions account for

1

1

k

ii

dk

ii

V

fraction of all variance (please note that i are variances); frequently 80-90% is sufficient for rough description.

Diagonalization methods: see Numerical Recipes, www.nr.com

PCA propertiesPCA 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 Y(W) – around max. variance change is minimal.

True covariance matrices are usually not known, they have to be estimated from data.

This works well on single-cluster data;

more complex structure may require local PCA: the PCA transformation should then be done separately for each cluster or neighborhood of a query vector X.

Some remarks on PCASome remarks on PCA

PC results obviously depend on the initial scaling of the features, therefore one should standardize the data first to make it independent of scaling or measurement units. Example: Heart data.

Assume that the data matrix X has been standardized, show that:

that is the mean stays as zero and the variance of principal components

is equal to the eigenvalues. Therefore rejecting Yi components with

small variance leads to small errors in reconstruction of X = ZY, where rejected components are replaced by zero values.

PC is useful for:

finding new, more informative, uncorrelated features;

reducing dimensionality: reject low variance features,

reconstructing original data from lower-dimensional projections.

20i i iY Y

PCA Wisconsin examplePCA Wisconsin exampleWisconsin 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.Example cont.PC gives useful information already in 2D.

Taking first PCA component of the standardized data:

If (Y1.41) then benign else malignant

18 errors/699 cases = 97.4%

Transformed vectors are not

standardized, std’s are below.

Eigenvalues decrease to zero slowly, but classes are well separated.

PCA disadvantagesPCA disadvantagesUseful 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.

2 skewed distributions2 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.