Statistical Approach to Unsupervised Learning

1. Neural Computation0368-4149-01 Prof. Nathan Intrator Tuesday 16:00-19:00 Schreiber 7 Office hours: Wed 4-5 nin@tau.ac.il

2. Outline • Goals for neural learning - Unsupervised • Goals for statisical/computational learning • PCA • ICA • Exploratory Projection Pursuit • Search for non-Gaussian distributions • Practical implementations

3. Statistical Approach to Unsupervised Learning • Understanding the nature of data variability • Modeling the data (sometimes very flexible model) • Understanding the nature of the noise • Applying prior knowledge • Extracting features based on: • Prior knowledge • Class prediction • Unsupervised learning

4. Principal Component Analysis. Włodzisław Duch SCE, NTU, Singapore http://www.ntu.edu.sg/home/aswduch

5. n-dimensional vectors m-dimensional vectors m < n Neuronal Goal We look for axes which minimise projection errors and maximise the variance after projection Ex: transform from 2 to 1 dimension

6. more information (variance) rotate less information project Algorithm (cont’d) • Preserve as much of the variance as possible

7. Linear transformations – example 2D vectors X in a unit circle with mean (1,1);Y = A*X, A = 2x2 matrix The shape is elongated, rotated and the mean is shifted.

8. Invariant distances Euclidean distance is not invariant to general linear transformations This is invariant only for orthonormal matricesATA = Ithat make rigid rotations, without stretching or shrinking distances. Idea: standardize the data in some way to create invariant distances.

9. Data standardization For each vector componentX(j)T=(X1(j), ... Xd(j)),j=1 .. n calculate mean and std:n– number of vectors,d– their dimension Vector of mean feature values. Averages over rows.

10. Standard deviation Calculate standard deviation: Vector of mean feature values. Variance = square of standard deviation (std), sum of all deviations from the mean value. Transform X => Z, standardized data vectors

11. Std 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?

12. Terminology (Covariance) • How two dimensions vary from the mean with respect to each other • cov(X,Y)>0: Dimensions increase together • cov(X,Y)<0: One increases, one decreases • cov(X,Y)=0: Dimensions are independent

13. Terminology (Covariance Matrix) • Contains covariance values between all possible dimensions: • Example for three dimensions (x,y,z)(Always symetric): cov(x,x)variance of component x

14. 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)

15. Data standardization example Transformation Vector of mean feature values. Variance check it! For our exampleY=AX, assumingXmeans=1 and variances = 1 How to make this invariant?

16. Covariance matrix Variance (spread around mean value) + correlation between features. CXisd x d whereXisd x ndimensional matrix of vectors shifted to their means. Covariance matrix is symmetricCij = Cjiand positive definite. Diagonal elements are variances (square of std),si2 = Cii Pearson correlation coefficient Spherical distribution of data hasCij=I(unit matrix). Elongated ellipsoids: large off-diagonal elements, strong correlations between features.

17. Mahalanobis distance Linear combinations of features leads to rotations and scaling of data. Mahalanobis distance: is invariant to linear transformations:

18. 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 In matrix form,X, Yare dxn,Z, CX,CYaredxd C – symmetric, positive definite matrixXTCX > 0for||X||>0; its eigenvectors are orthonormal: its eigenvalues are all non-negative Z– matrix of orthonormaleigenvectors (becauseZis real+symmetric), transforms Xinto Y,with diagonalCY, i.e. decorrelated.

19. Matrix form Eigenproblem for C matrix in matrix form:

20. Principal components Z– principal components, of vectorsXtransformed using eigenvectors ofCX Covariance matrix of transformed vectors is diagonal => ellipsoidal distribution of data. PCA: 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. Smallli small variance  data change little in directionYi PCA minimizesCmatrix reconstruction errors: Zivectors for largeliare sufficient to get: because vectors for small eigenvalues will have very small contribution to the covariance matrix.

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

22. 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

23. 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.

24. 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!

25. 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.

26. 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.

27. 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?

28. 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.