Direct Learning: Linear Classification. Donglin Zeng, Department of Biostatistics, University of North Carolina
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1 Direct Learning: Linear Classification
2 Logistic regression models for classification problem We consider two class problem: Y {0, 1}. The Bayes rule for the classification is I(P(Y = 1 X = x) > 1/2) so it is natural to estimate P(Y = 1 X = x) using training data. Standard statistical model is logistic regression model assuming P(Y = 1 X) = exp{β 0 + X T β} 1 + exp{β 0 + X T β} then the resulting prediction rule is I(β 0 + x T β > 0). Shrinkage can also be introduced into this regression, for example, Glasso min n i=1 { Y i (β 0 + X T i β) + log(1 + exp{β 0 + X T i β}) } +λ β L1.
3 Perceptron model for classification A more direct method (nothing to do with statistics) is to directly search a hyperplane separating two class data (perceptron model). This algorithm aims to find a separating hyperplane by minimizing the distance of misclassified points to the decision boundary: i M(2Y i 1)(β 0 + X T i β). Stochastic gradient decent is used to find the solution ( ) ( ) ( ) β β (2Yi 1)X + ρ i, 2Y i 1 β 0 β 0 where ρ is the learning and i is taken from M one by one. The algorithm may not have unique solutions and may cycle solutions when data are not separable. A more reliable algorithm is warranted.
4 Discriminant analysis The idea behind discriminant analysis is to compare the distributions of feature variables for each class and identify the best rule to discriminate these distributions. Let f k (x) be the density of X in Y = k and π k is the prevalence of Y = k, k {0, 1}. Clearly, P(Y = 1 X = x) = f 1 (x)π 1 f 1 (x)π 1 + f 0 (x)π 0. Thus, the Bayes rule is { I log f 1 (x) log f 0 (x) + log π } 1 > 0. π 0
5 Discriminant analysis under multivariate normality Assume f k (x) N(µ k, Σ) for k {0, 1} (homogeneous variance). The prediction rule becomes I(x T Σ 1 (µ 1 µ 0 ) 1 2 µt 1 Σ 1 µ µt 0 Σ 1 µ 0 + log π 1 π 0 > 0). The decision boundary is a linear function of X so this analysis is called linear discriminant analysis. Assume f k (x) N(µ k, Σ k ). Then prediction rule becomes ( I 1 2 xt Σ 1 1 x xt Σ 1 0 x + xt Σ 1 1 µ 1 x T Σ 1 0 µ 0) 1 2 µt 1 Σ 1 µ µt 0 Σ 1 µ 0 + log π ) 1 > 0. π 0 The decision boundary is a quadractic function of X so this analysis is called quadractic discriminant analysis.
6 Example of LDA
7 Example of QDA
8 Direct Learning: Nonlinear Prediction
9 Nonlinear prediction rules Nonlinear prediction rules can be obtained if we include high-order interactions in addition to X, or replace X with some basis functions in feature space. Commonly used and flexible basis functions includes splines and wavelets. Splines are piecewise polynomials and can consist of B-splines and natural cubic splines etc. For splines, we need to define a sequence of knots, degrees of polynomials and smoothness at knots. Splines can also be represented by polynomials and truncated polynomials, where truncations occur at interior knots: x k, (x ξ 1 ) k +, (x ξ 2 ) k +,...
10 Recursive algorithm for B-spline construction
11 R code of constructing B-splines
12 South African heart disease data
13 Smooth splines They are also piecewise polynomials. However, instead of specifying knots in advance, smoothing spline approximation is obtained by minimizing the following penalized least squares: n (Y i f (X i )) 2 + λ {f (t)} 2 dt. i=1 Note that λ regularizes the smoothness of f (t). The solution is a linear combination of natural cubic splines whose knots are placed on X 1,..., X n. Tuning is performed via Cp, CV or GCV: GCV = 1 n ( n i=1 Y i f λ (X i ) 1 S λ (i, i) ) 2, where S λ is some projection matrix on the space spanned by the cubic splines.
14 Approximation in Reproducing Kernel Hilbert Space (RKHS) Smoothing splines can be treated as a special case of the approximation in RKHS. For a RKHS, denoted by H K, a kernel function K(x, y) (x, y R p ) is essential (satisfying symmetry and positivity conditions). Then the RKHS consists of any linear combinations of the form { } H k = f (x) = m α m K(x, y m ) Key result: there exists a sequence of basis functions, φ j (x), such that K(x, y) = j=1 γ jφ j (x)φ j (y) (eigen-expansion of K(x, y)) for γ j 0, j γ2 j <..
15 RKHS: continue For f H K, f (x) = j c jφ j (x) and f (x) = j c jφ j (x). We define an inner product as < f, f > HK = j c j c j /γ j. Important property of RKHS: < K(x, ), K(y, ) > HK = K(x, y) (reproducing)
16 Practical usefulness of RKHS Regularization problem n min L(Y i, f (X i )) + λ f 2 H K i=1 has a solution with form f (x) = n i=1 α ik(x, X i ). Thus, we need to solve min α n L(Y i, j i=1 K(X i, X j )α j ) + λα T Kα, where K is matrix of (K(X i, X j )). Advantages are (1) no need of knowledge about basis functions; (2) the optimization is independent of the dimensionality of X. How large is RKHS: for a Gaussian kernel, K(x, y) = exp{ x y /σ 2 }, the resulting RKHS can approximate any function if σ is close to 0 enough.
17 Wavelet approximation Wavelets can be used to approximate possibly irregular function/surfaces; to localize and identify such accumulation of wavelets (small waves) through both frequency and spatial resolution. They are extensively used in data compression, turbulence analysis, image & signal processing and statistical estimation. Comparatively, Fourier spectrum analysis is a global frequency decomposition approach. Alternative approximation such as blocked Fourier analysis, Splines, local polynomial approximation etc. also have local adaptivity; however, wavelet analysis is more elegant and mathematically consistent.
18 Wavelet construction We start with some father wavelet φ(x) = I(x (0, 1]). Example: Haar system: Consider V 0 = {f L 2 (R) : f = k c ki (k,k+1] (x)} then {φ 0k = φ(x k)} is ONB of V 0. Let V 1 = {f (x) : f (x) = h(2x) for some h V 0 } then {φ 1k = 2φ 0k (2x) = 2φ(2x k)} is ONB of V 1. Generally, let V j = {f (x) : f (x) = h(2 j x) for some h V 0 } then {φ jk = 2 j/2 φ(2 j x k)} is ONB of V j for any j....v 1 V 0 V 1...; j=0 V j is dense in L 2 (R); V j+1 = V j W j, where {ψ jk = 2 j/2 ψ(2 j x k)} is ONB of W j with ψ(x) = I [0,1/2] (x) + I [1/2,1] (x) (mother wavelet).
19 Multiresolution analysis based on wavelets First, Second, L 2 (R) = V j W j W j+1 W j+2. V 0 = W 1 W 2 W j V j. In other words, any f (x) L 2 (R) can be recovered using lower resolution projection (V j ) and additional details from higher resolutions (W j,, W 1 ). This is called the multiresolution analysis of f. Other choices of father wavelets: Daubechies wavelets, Coiflet wavelets, Symmlets wavelets (smoother than Haart wavelets).
20 Wavelet 1D demo 0.02 Analyzed signal Discrete Transform, absolute coefficients. 5 4 level Absolute Values of Ca,b Coefficients for a = scales a time (or space) b
21 Wavelet 2D demo X Reconstructed image a0. a1 h1 v1 d1 a2 h2 v2 d2
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