Statistical Depth Function
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1 Machine Learning Journal Club, Gatsby Unit June 27, 2016
2 Outline L-statistics, order statistics, ranking. Instead of moments: median, dispersion, scale, skewness,... New visualization tools: depth contours, sunburst plot. Testing: symmetry. Depth function properties.
3 L-statistics, ranking
4 L-statistics Given: x 1,...,x n R samples. Order statistics: x [1]... x [n]. Linear combination of order statistics. Rank: position of x i.
5 L-statistics: examples Dispersion: 1 range: x [n] x [1]. 2 interquartile range: x [3n/4] x [n/4] = Q 3 Q 1.
6 L-statistics: examples Dispersion: 1 range: x [n] x [1]. 2 interquartile range: x [3n/4] x [n/4] = Q 3 Q 1. Location: median: x [n/2]. α-trimmed mean = middle of the (1 2α)-th fraction of observations, (1 α)n 1 x [i]. (1 2α)n i=αn+1
7 L-statistics Typical application: robust estimation. Question: Similar notions in R d?
8 L-statistics Typical application: robust estimation. Question: Similar notions in R d? Critical: x [1]... x [n].
9 L-statistics Typical application: robust estimation. Question: Similar notions in R d? Critical: x [1]... x [n]. Idea: center-outward ordering.
10 Median
11 Motivation: median F: continuous c.d.f. on R. median: point splitting the probability mass to solution of F(m) = 1 2.
12 Motivation: median F: continuous c.d.f. on R. median: point splitting the probability mass to solution of F(m) = 1 2. alternative view: m := argmaxd(x) := 2F(x)[1 F(x)] x R
13 Motivation: median F: continuous c.d.f. on R. median: point splitting the probability mass to solution of F(m) = 1 2. alternative view: m := argmaxd(x) := 2F(x)[1 F(x)] x R = P X1,X 2 F(x [X 1,X 2 ]).
14 Motivation: median F: continuous c.d.f. on R. median: point splitting the probability mass to solution of F(m) = 1 2. alternative view: m := argmaxd(x) := 2F(x)[1 F(x)] x R = P X1,X 2 F(x [X 1,X 2 ]). Moving away from m, D(x) decreases monotonically to 0.
15 Let us extend the median to R d! Simplitical depth [Liu, 1990]: SD(x;F) := P F (x (X 1,...,X d+1 )).
16 Let us extend the median to R d! Simplitical depth [Liu, 1990]: SD(x;F) := P F (x (X 1,...,X d+1 )). Half-space depth [Tukey, 1975]: HD(x;F) := inf x H R d P F(X H) minimum probability of any closed halfspace containing x.
17 Let us extend the median to R d! Simplitical depth [Liu, 1990]: SD(x;F) := P F (x (X 1,...,X d+1 )). Half-space depth [Tukey, 1975]: HD(x;F) := inf x H R d P F(X H) minimum probability of any closed halfspace containing x. median: m = argmax x R d D(x;F).
18 Side-note: these quantities are also computable depth R-package: simplicial, half-space, Oja s depth. Ref. (half-space depth): [Dyckerhoff and Mozharovskyi, 2016].
19 Side-note: these quantities are also computable depth R-package: simplicial, half-space, Oja s depth. Ref. (half-space depth): [Dyckerhoff and Mozharovskyi, 2016]. Simplicial depth: x = n α i x i, i=1 n α i = 1 i=1 linear equation. If α i 0 ( i) Yes.
20 Example: SD depth-induced contours normal exponential center-outward ordering. high-depth: center, low-depth: outlyingness. expansion of contours: scale (dispersion), skewness, kurtosis.
21 Location Median/center: deepest point.
22 Location Median/center: deepest point. Other robust location statistics: idea: assign smaller weights to more outlier data. given: w : [0,1] R 0 nonincreasing weight function L w = n i=1 x [i]w(i/n) n j=1 w(j/n).
23 Location Median/center: deepest point. Other robust location statistics: idea: assign smaller weights to more outlier data. given: w : [0,1] R 0 nonincreasing weight function L w = n i=1 x [i]w(i/n) n j=1 w(j/n). Specifically: w(t) = I(t 1/n): sample median. w(t) = I(t 1 α): 100α%-trimmed mean.
24 Location Median/center: deepest point. Other robust location statistics: idea: assign smaller weights to more outlier data. given: w : [0,1] R 0 nonincreasing weight function L w = n i=1 x [i]w(i/n) n j=1 w(j/n). Specifically: w(t) = I(t 1/n): sample median. w(t) = I(t 1 α): 100α%-trimmed mean. No expectation requirement!
25 Dispersion, scale Idea-1: similar to location, the dispersion S = n ( )( ) T i=1 x[i] x [1] x[i] x [1] w(i/n) n j=1 w(j/n).
26 Dispersion, scale Idea-1: similar to location, the dispersion S = n ( )( ) T i=1 x[i] x [1] x[i] x [1] w(i/n) n j=1 w(j/n). Scale: det(s). Equivariance under affine tranformations: y i = Ax i +b S y = AS x A T.
27 Dispersion, scale Idea-1: similar to location, the dispersion S = n ( )( ) T i=1 x[i] x [1] x[i] x [1] w(i/n) n j=1 w(j/n). Scale: det(s). Equivariance under affine tranformations: y i = Ax i +b S y = AS x A T. No moment constraints!
28 Dispersion: idea-2 Idea Dispersion := speed how the data depth decreases. p-th (empirical) central region: scale curve: C n,p := conv { x [1],...,x [np] }. S n (p) := vol(c n,p ). faster growing of p S n (p) = larger scale of the distribution.
29 Scale curve example
30 Skewness: departure from spherical symmetry Spherical symmetry around c: X c = U(X c), U: orthogonal.
31 Skewness: departure from spherical symmetry Spherical symmetry around c: X c = U(X c), U: orthogonal. Strategy: take the smallest sphere containing C n,p, determine the fraction of the data within this sphere, plot it as function of p ( p).
32 Skewness: departure from spherical symmetry Spherical symmetry around c: X c = U(X c), U: orthogonal. Strategy: take the smallest sphere containing C n,p, determine the fraction of the data within this sphere, plot it as function of p ( p). Sperical symmetry (0,0) (1,1) linear curve, i.e. area of the gap = 0.
33 Spherical symmetric/skewed: demo Spherical (area = 0.08) Asymmetric (area = 0.2)
34 Sunburst plot
35 Sunburst plot = 2D boxplot For a given depth function: identify the center, central 50% points ( IQR), ray from non-central points to center ( whiskers). normal exponential
36 Testing
37 Symmetries X R d is centrally symmetric around c: X c distr = c X.
38 Symmetries X R d is centrally symmetric around c: X c distr = c X. X c angularly symmetric around c: X c is C-sym. around 0. 2
39 Symmetries X R d is centrally symmetric around c: X c distr = c X. X c angularly symmetric around c: X c is C-sym. around 0. 2 half-space symmetric around c: P(X H) 1 2, for H c, closed half-space.
40 Symmetries X R d is centrally symmetric around c: X c distr = c X. X c angularly symmetric around c: X c is C-sym. around 0. 2 half-space symmetric around c: P(X H) 1 2, for H c, closed half-space. Note: C-symmetry A-symmetry H-symmetry. A-symmetry H-symmetry: in the discrete case, continuous:. d = 1: all = standard symmetry.
41 Testing: A-symmetry idea SD is maximized at the center of symmetry, = 1. 2 d F: absolutely continuous, SD(c ;F) 1, = F: A-sym. 2 d around c.
42 Testing: A-symmetry idea SD is maximized at the center of symmetry, = 1. 2 d F: absolutely continuous, SD(c ;F) 1, = F: A-sym. 2 d around c. c: hypothesized center. Decision: if 1 2 d SD(c;F n ) is large reject H 0.
43 Testing: A-symmetry idea SD is maximized at the center of symmetry, = 1. 2 d F: absolutely continuous, SD(c ;F) 1, = F: A-sym. 2 d around c. c: hypothesized center. 1 Decision: if SD(c;F 2 d n ) is large reject H 0. Under the hood: 1 SD(c;F 2 d n ): degenerate U-statistic, n [ 1 SD(c;F 2 d n ) ] -sum of χ 2 -s.
44 Depth function properties
45 Desirable properties F:=Borel probability measures on R d. D( ; ) : R d F R 0 bounded, 1 affine invariance: D(Ax+b;F AX+b ) = D(x;F X ) A : invertible,b,x.
46 Desirable properties F:=Borel probability measures on R d. D( ; ) : R d F R 0 bounded, 1 affine invariance: D(Ax+b;F AX+b ) = D(x;F X ) A : invertible,b,x. 2 maximality at center: For F with center c, D(c;F) = sup x R d D(x;F).
47 Desirable properties F:=Borel probability measures on R d. D( ; ) : R d F R 0 bounded, 1 affine invariance: D(Ax+b;F AX+b ) = D(x;F X ) A : invertible,b,x. 2 maximality at center: For F with center c, D(c;F) = sup x R d D(x;F). 3 monotonicity relative to the deepest point: For F with deepest point c, D(x;F) D(αx+(1 α)c;f), α [0,1].
48 Desirable properties F:=Borel probability measures on R d. D( ; ) : R d F R 0 bounded, 1 affine invariance: D(Ax+b;F AX+b ) = D(x;F X ) A : invertible,b,x. 2 maximality at center: For F with center c, D(c;F) = sup x R d D(x;F). 3 monotonicity relative to the deepest point: For F with deepest point c, 4 vanishing at infinity: D(x;F) D(αx+(1 α)c;f), α [0,1]. D(x;F) x 0, F.
49 Categorization of depth functions [Zuo and Serfling, 2000] Vanishing at infinity: useful for establishing sup D(x;F n ) D(x;F) n 0 (F-a.s.). x
50 Categorization of depth functions [Zuo and Serfling, 2000] Properties of HD and SD: half-space (HD): 1-4 (H-sym.) simplicial (SD): continuous distributions: 1-4 (A-sym.) discrete: 2/3 might fail.
51 Types D A (x;f) = E F [h A (x;x 1 ;...;X r )] example SD, 1 D B (x;f) = 1+E F [h B (x;x 1 ;...;X r )], D C (x;f) = 1 1+O(x;F), D D (x;f) = inf P F(X C) example HD. x C C h A (x;x 1,...,x r ): bounded, closeness of x to x i -s. h B (x;x 1,...,x r ): unbounded, distance of x from x i -s. O(x; F): unbounded outlyingness function. C: closedness properties.
52 Types D A (x;f) = E F [h A (x;x 1 ;...;X r )] example SD, 1 D B (x;f) = 1+E F [h B (x;x 1 ;...;X r )], D C (x;f) = 1 1+O(x;F), D D (x;f) = inf P F(X C) example HD. x C C h A (x;x 1,...,x r ): bounded, closeness of x to x i -s. h B (x;x 1,...,x r ): unbounded, distance of x from x i -s. O(x; F): unbounded outlyingness function. C: closedness properties. Note: D A (x;f n ): U/V-statistic.
53 Type B examples Let Σ = cov(f). In simplicial volume depth (D SVD α), D L2: ( ) α vol[ (x,x 1,...,x d )] h(x;x 1,...,x d ) =,α > 0, det(σ) Properties: h(x;x 1 ) = x x 1 Σ 1, z M = z T Mz. D SVD α: 1-4 (C-sym.), D L2: 1-4 (A-sym.).
54 Type C example Projection depth: worst case outlyingness w.r.t. 1d-median, for any 1d projection. u T x med(u T X) O(x;F) = sup u 2 =1 MAD(u T,X F, X) MAD(Y) = med( Y med(y) ). Properties: 1-4 (H-sym.)
55 Summary Depth functions: simplicial, half-space,... They define: center-outward ordering, ranks, L-statistics, symmetry test. location (median,...), dispersion, scale, skewness, kurtosis (no moments), sunburst plot.
56 Thank you for the attention!
57 Dyckerhoff, R. and Mozharovskyi, P. (2016). Exact computation of the halfspace depth. Technical report, University of Cologne. Liu, R. (1990). On a notion of data depth based on random simplices. The Annals of Statistics, 18: Tukey, J. (1975). Mathematics and picturing data. In International Congress on Mathematics, pages Zuo, B. Y. and Serfling, R. (2000). General notions of statistical depth function. The Annals of Statistics, 28:
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