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Robust Data Modeling Methods for Handling Outliers and Missing Data in Statistical Analysis

This seminar tackles techniques for handling outliers and missing data in statistical models, focusing on applications in finance, engineering, and sciences. It delves into regression, matrix factorization, and robust linear regression for high-dimensional problems. The discussion covers robust algorithms like Random Sample Consensus (RANSAC) and Bayesian Sparse Robust Regression (BSRR), along with their performance in empirical studies. The seminar explores facial age estimation using robust regression techniques and concludes with insights into Robust Relevance Vector Machine (RVM) models.

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Robust Data Modeling Methods for Handling Outliers and Missing Data in Statistical Analysis

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  1. Handling Outliers and Missing Data in Statistical Data Models KaushikMitra Date: 17/1/2011 ECSU Seminar, ISI

  2. Statistical Data Models • Goal: Find structure in data • Applications • Finance • Engineering • Sciences • Biological • Wherever we deal with data • Some examples • Regression • Matrix factorization • Challenges: Outliers and Missing data

  3. Outliers Are Quite Common Google search results for `male faces’

  4. Need to Handle Outliers Properly Removing salt-and-pepper (outlier) noise Noisy image Gaussian filtered image Desired result

  5. Missing Data Problem Missing tracks in structure from motion Completing missing tracks Completed tracks by a sub-optimal method Desired result Incomplete tracks

  6. Our Focus • Outliers in regression • Linear regression • Kernel regression • Matrix factorization in presence of missing data

  7. Robust Linear Regression for High Dimension Problems

  8. What is Regression? • Regression • Find functional relation between y and x • x: independent variable • y: dependent variable • Given • data: (yi,xi) pairs • Model y = f(x, w)+n • Estimate w • Predict y for a new x

  9. Robust Regression • Real world data corrupted with outliers • Outliers make estimates unreliable • Robust regression • Unknown • Parameter, w • Outliers • Combinatorial problem • N data and k outliers • C(N,k) ways

  10. Prior Work • Combinatorial algorithms • Random sample consensus (RANSAC) • Least Median Squares (LMedS) • Exponential in dimension • M-estimators • Robust cost functions • local minima

  11. Robust Linear Regression model • Linear regression model : yi=xiTw+ei • ei, Gaussian noise • Proposed robust model: ei=ni+si • ni, inlier noise (Gaussian) • si, outlier noise (sparse) • Matrix-vector form • y=Xw+n+s • Estimate w, s x1T x2T . . xNT n1 n2 . . nN s1 s2 . . sN w1 w2 . wD y1 y2 . . yN + + =

  12. Simplification • Objective (RANSAC): Find w that minimizes the number of outliers • Eliminate w • Model: y=Xw+n+s • Premultiple by C: CX=0, N ≥ D • Cy=CXw+Cs+Cn • z=Cs+g • g Gaussian • Problem becomes: • Solve for s -> identify outliers -> LS -> w

  13. Relation to Sparse Learning • Solve: • Combinatorial problem • Sparse Basis Selection/ Sparse Learning • Two approaches : • Basis Pursuit (Chen, Donoho, Saunder 1995) • Bayesian Sparse Learning (Tipping 2001)

  14. Basis Pursuit Robust regression (BPRR) • Solve • Basis Pursuit Denoising (Chen et. al. 1995) • Convex problem • Cubic complexity : O(N3) • From Compressive Sensing theory (Candes 2005) • Equivalent to original problem if • s is sparse • C satisfy Restricted Isometry Property (RIP) • Isometry: ||s1 - s2|| = ||C(s1 – s2)|| • Restricted: to the class of sparse vectors • In general, no guarantees for our problem

  15. Bayesian Sparse Robust Regression (BSRR) • Sparse Bayesian learning technique (Tipping 2001) • Puts a sparsity promoting prior on s : • Likelihood : p(z/s)=Ν(Cs,εI) • Solves the MAP problem p(s/z) • Cubic Complexity : O(N3)

  16. Setup for Empirical Studies • Synthetically generated data • Performance criteria • Angle between ground truth and estimated hyper-planes

  17. Vary Outlier Fraction • BSRR performs well in all dimensions • Combinatorial algorithms like RANSAC, MSAC, LMedS not practical in high dimensions Dimension = 2 Dimension = 32 Dimension = 8

  18. Facial Age Estimation • Fgnet dataset : 1002 images of 82 subjects • Regression • y : Age • x: Geometric feature vector

  19. Outlier Removal by BSRR • Label data as inliers and outliers • Detected 177 outliers in 1002 images • Leave-one-out testing

  20. Summary for Robust Linear Regression • Modeled outliers as sparse variable • Formulated robust regression as Sparse Learning problem • BPRR and BSRR • BSRR gives the best performance • Limitation: linear regression model • Kernel model

  21. Robust RVM Using Sparse Outlier Model

  22. Relevance Vector Machine (RVM) • RVM model: • : kernel function • Examples of kernels • k(xi, xj) = (xiTxj)2 : polynomial kernel • k(xi, xj) = exp( -||xi - xj||2/2σ2) : Gaussian kernel • Kernel trick: k(xi,xj) = ψ(xi)Tψ(xj) • Map xi to feature space ψ(xi)

  23. RVM: A Bayesian Approach • Bayesian approach • Prior distribution : p(w) • Likelihood : • Prior specification • p(w) : sparsity promoting prior p(wi) = 1/|wi| • Why sparse? • Use a smaller subset of training data for prediction • Support vector machine • Likelihood • Gaussian noise • Non-robust : susceptible to outliers

  24. Robust RVM model • Original RVM model • e, Gaussian noise • Explicitly model outliers, ei= ni + si • ni, inlier noise (Gaussian) • si, outlier noise (sparse and heavy-tailed) • Matrix vector form • y = Kw + n + s • Parameters to be estimated: w and s

  25. Robust RVM Algorithms • y = [K|I]ws + n • ws = [wTsT]T : sparse vector • Two approaches • Bayesian • Optimization

  26. Robust Bayesian RVM (RB-RVM) • Prior specification • w and s independent : p(w, s) = p(w)p(s) • Sparsity promoting prior for s: p(si)= 1/|si| • Solve for posterior p(w, s|y) • Prediction: use w inferred above • Computation: a bigger RVM • ws instead of w • [K|I] instead of K

  27. Basis Pursuit RVM (BP-RVM) • Optimization approach • Combinatorial • Closest convex approximation • From compressive sensing theory • Same solution if [K|I] satisfies RIP • In general, can not guarantee

  28. Experimental Setup

  29. Prediction : Asymmetric Outliers Case

  30. Image Denoising • Salt and pepper noise • Outliers • Regression formulation • Image as a surface over 2D grid • y: Intensity • x: 2D grid • Denoised image obtained by prediction

  31. Salt and Pepper Noise

  32. Some More Results RVM RB-RVM Median Filter

  33. Age Estimation from Facial Images • RB-RVM detected 90 outliers • Leave-one-person-out testing

  34. Summary for Robust RVM • Modeled outliers as sparse variables • Jointly estimated parameter and outliers • Bayesian approach gives very good result

  35. Limitations of Regression • Regression: y = f(x,w)+n • Noise in only “y” • Not always reasonable • All variables have noise • M = [x1x2 … xN] • Principal component analysis (PCA) • [x1x2 … xN] = ABT • A: principal components • B: coefficients • M = ABT: matrix factorization (our next topic)

  36. Matrix Factorization in the presence of Missing Data

  37. Applications in Computer Vision • Matrix factorization: M=ABT • Applications: build 3-D models from images • Geometric approach (Multiple views) • Photometric approach (Multiple Lightings) Structure from Motion (SfM) Photometric stereo

  38. Matrix Factorization • Applications in Vision • Affine Structure from Motion (SfM) • Photometric stereo • Solution: SVD • M=USVT • Truncate S to rank r • A=US0.5, B=VS0.5 xij yij M = = CST Rank 4 matrix M = NST, rank = 3

  39. Missing Data Scenario • Missed feature tracks in SfM • Specularities and shadow in photometric stereo Incomplete feature tracks

  40. Challenges in Missing Data Scenario • Can’t use SVD • Solve: • W: binary weight matrix, λ: regularization parameter • Challenges • Non-convex problem • Newton’s method based algorithm (Buchanan et. al. 2005) • Very slow • Design algorithm • Fast (handle large scale data) • Flexible enough to handle additional constraints • Ortho-normality constraints in ortho-graphic SfM

  41. Proposed Solution • Formulate matrix factorization as a low-rank semidefinite program (LRSDP) • LRSDP: fast implementation of SDP (Burer, 2001) • Quasi-Newton algorithm • Advantages of the proposed formulation: • Solve large-scale matrix factorization problem • Handle additional constraints

  42. Low-rank Semidefinite Programming (LRSDP) • Stated as: • Variable: R • Constants • C: cost • Al, bl: constants • Challenge • Formulating matrix factorization as LRSDP • Designing C, Al, bl

  43. Matrix factorization as LRSDP: Noiseless Case • We want to formulate: • As: • LRSDP formulation: C identity matrix, Al indicator matrix

  44. Affine SfM • Dinosaur sequence • MF-LRSDP gives the best reconstruction 72% missing data

  45. Photometric Stereo • Face sequence • MF-LRSDP and damped Newton gives the best result 42% missing data

  46. Additional Constraints:Orthographic Factorization • Dinosaur sequence

  47. Summary • Formulated missing data matrix factorization as LRSDP • Large scale problems • Handle additional constraints • Overall summary • Two statistical data models • Regression in presence of outliers • Role of sparsity • Matrix factorization in presence of missing data • Low rank semidefinite program

  48. Thank you! Questions?

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