Supervised Classification for Functional Data by Estimating the. Estimating the Density of Multivariate Depth

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1 Supervised Classification for Functional Data by Estimating the Density of Multivariate Depth Chong Ma University of South Carolina David B. Hitchcock University of South Carolina March 25, 2016

2 What is functional data? High-frequency measurements (depend on monitoring equipment). hand-writing data. spectral data. Smooth but complex processes. Repeated functional observations. Derivative information maybe useful.

3 Example for functional data Figure: Average daily temperature and precipitation records in 35 weather stations across Canada (classical and often-used)

4 Example for functional data Figure: Records of number of eggs laid by Mediterranean Fruit Fly (Ceratitis capitata) in each of 25 days (courtesy of H.-G. Mller).

5 Goals in Functional Data Analysis Estimation of distribution of functional data. Prediction and classification of response variable (scalers, vectors or functions) related to functional data. Relationship between derivatives of functions

6 Supervised classification of functional data aa ao Figure: 20 records of two phonemes in log-periodogram at 150 frequency points.

7 Supervised Classification Format {x i (t), y i }, i = 1,..., n and y i G = {1,..., G}. Given a training set of functional data with known group memberships, predict the group membership for a new functional observation. Toolbox Multivariate approaches: LDA, PDA, PCA,... Functional approaches: k-nearest neighbors, depth-based,...

8 Depth Notion Mahalanobis depth (Mahalanobis, 1935; Liu and Singh, 1993) MhD = {1 + (x µ) Σ 1 (x µ)} 1 Half-space depth (HD) (Tukey, 1975) HD F (x) = inf H {P F (H) : H is a closed half space in R d, x H} Simplicial depth (SD) (Liu, 1990) SD(F, x) = P F (x S(X 1,..., X d+1 ))

9 Tukey s Half Space Depth The halfspace depth of a point p with respect to S R d is defined as depth S (p) = min {q S a, q > a, p } a R d \{0} Figure: Tukey half space depth

10 Simplicial Depth Figure: Simplicial Depth

11 Modified Band Depth Figure: Modified Band Depth

12 Modified Band Depth Denote a functional observation {x(t), t T } by x, where T is compact. Without loss of generality, we set T as [0, 1]. i.i.d Assume x 1,..., x n F X, where F X is the cumulative distribution of the stochastic process X. The population version of modified band depth is MBD J (x) = where J MBD (j) (x) = j=2 A(x; X 1,..., X j ) = {t T : J E[λ r (A(x; X 1,..., X j ))] j=2 min X i(t) x(t) max X i(t)} i=1,...,j i=1,...,j and λ r (A(x; X 1,..., X j )) = λ(a(x; X 1,..., X j ))/λ(t )

13 Modified Band Depth The sample version of MBD (j) (x) is MBD (j) n (x) = ( ) n 1 λ r ({t T : j 1 i 1 < <i j n min x i(t) x(t) max x i(t)}) i=1,...,j i=1,...,j MBD n (j) (x) is consistent to MBD (j) (x). The computation load for MBD n (j) (x) is O(n J ).

14 Adjusted Modified Band Depth We propose an adjusted modified band depth for x (AMBD(x)) in order to make it more sense statistically. And we use bootstrapping method to estimate the population version of AMBD(x). AMBDJ (x) = MBD J (x) = E[λ r (A(x; X 1,..., X J ))] AMBD J n (x) = 1 nb nb i=1 λ r (A(x; x i1,..., x ij )) } is a bootstrapping sample of size J from the {x i1,..., x ij sample of n functional data, i = 1,..., nb. And we also have AMBDn J (x) a.s. = AMBD J (x) (see e.g. Cuevas, Febrero and Fraiman 2005).

15 Multivariate Depth The multivariate depth is formated for supervised classification problem. {x i (t), y i }, i = 1,..., n and y i G = {1,..., G}. Given a new functional observation x 0 F X0, assume F X0 {F X1,..., F XG }, the multivariate depth for x 0 is defined by MD(x 0 ) = (AMBD FX1 (x 0 ),..., AMBD FXG (x 0 )) where AMBD FXg (x 0 ) = E[λ r (A(x 0 ; X 1,..., X J i.i.d F Xg )) X 0 = x 0 ] MD(x 0 ) is a G-dimension observation from MD(X 0 ) which is a random vector related to F X0.

16 Multivariate Depth We assume MD(X 0 ) follows a multivariate normal distribution, i.e., MD(X 0 ) N G (µ 0, Σ 0 ). It implies that MD(X g ) N G (µ g, Σ g ) under the assumption of X 0 F X0 {F X1,..., F XG }. Given G groups of observed functional data, {x g 1,..., xg n g },g = 1,..., G, we are able to calculate the maximum likelihood estimators for µ g and Σ g. ˆµ g = 1 n g ˆΣ g = 1 n g n g MD(x g i ) (1) i=1 n g (MB(x g i ) ˆµ g )(MB(x g i ) ˆµ g ) (2) i=1

17 FID FID 98 FID FID FID 112 FID FID 91 FID FID 92 FID FID 87 FID 86 Forensic Casework Figure: 12 blue acrylic textile fiber absorbance spectral data

18 Forensic Casework MD DS TAD FM RP glm Figure: Misclassification rate for forensic fiber data.