Supplementary Material to Wasserstein Covariance for Multiple Random Densities

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1 Biometrika (8, xx, x, pp. 6 Prited i Great Britai Supplemetary Material to Wasserstei Covariace for Multiple Radom Desities BY ALEXANDER PETEEN Departmet of Statistics ad Applied Probability, Uiversity of Califoria, Sata Barbara, Califoria 96, U.S.A. 5 peterse@pstat.ucsb.edu AND HANS-GEORG MÜLLER Departmet of Statistics, Uiversity of Califoria, Davis, Califoria 9566, U.S.A. hgmueller@ucdavis.edu SUMMARY Sectio S. demostrates a valid desity estimator with the properties required by Assumptio i the paper. Sectio S. cotais proofs of Theorems ad, while Sectio S. cotais a additioal theoretical result. Sectio S. provides additioal visualizatios for the coectivity ad mortality aalyses. S.. MODIFIED KERNEL DENSITY ESTIMATOR 5 A desity estimator ˆf that satisfies Assumptio is defied as follows. Assume oe has a i.i.d. sample W,..., W N f. Let K be a bouded pdf o [, ], symmetric about. We use a adapted versio of the modified kerel desity estimator (Peterse ad Müller 6 m ( / Wr u m ˆf(u = K w(u, h h r= r= D f K for u D f = [a f, b f ] ad elsewhere. The weight fuctio w(u, h = ( Wr v (a f uh K(vdv}, af u a f + h, (bf uh K(vdv}, bf h u b f,, otherwise, h } w(v, hdv, ( is desiged to remove the boudary bias of the stadard kerel desity estimator, which is a major issue due to the compact support of he desities, while simultaeously maitaiig the itegrated squared error rate ad rederig a estimate that is oegative ad itegrates to oe. This leads to the followig propositio. PROPOSITION. With D as i Assumptio, assume further that sup sup max f(u, /f(u, f (u } < M <. u D f f D C 6 Biometrika Trust

2 5 A. PETEEN AND H. G. MÜLLER For geeric f D, give a i.i.d. sample W r f, (r =,..., N ad assumig D f is kow, the ˆf give i ( satisfies Assumptio with b N = N /, usig a badwidth h = tn / for ay t >. Proof. Propositio of Peterse ad Müller (6 immediately implies that sup E d ( ˆf, } f = O (N / f D uder the give assumptios, where d is the L (I metric ad I is the iterval i Assumptio. Choose C > such that I [ C/, C/] ad let F ad ˆF be the cdfs of f ad ˆf. It is clear that d (F, ˆF Cd (f, ˆf, ad that (F is uiformly bouded by M over f D. By the mea value theorem, ad usig the chage of variables u = ˆF (t, d W (f, ˆf = F (t ˆF (t} dt 5 M [ D f F (u ˆF (u} f(udu ] + F (u ˆF } } (u ˆf(u f(u du D f M C(MC + d (f, ˆf by Cauchy-Schwarz, ad the result follows. S.. PROOFS OF THEOREMS AND Throughout the proofs, we use the otatios f, g = f(ug(udu for the stadard ier product o L [, ], f = ( f, f / for the correspodig L orm ad [,] for the stadard L ([, ] orm. Let X i (t = X i (t,..., X ip (t} be the vector of quatile fuctios for subject i, ad set ν(t = E X (t}, ν(t = X i(t, ad Xi c(t = X i(t ν(t. Lettig U i (s, t = Xi c(sxc i (t ad T (s, t = ν(s ν(s} ν(t ν(t}, set C jk (s, t = U i (s, t} jk T (s, t} jk, s, t. Proof of Theorem. Defie Σ = } U i (t, t T (t, t dt ad, for s, t, defie the four dimesioal array D(s, t with elemets D(s, t} jklm = cov [U i (s, s} jk, U i (t, t} lm ].

3 ] Sice E [U i (t, t} jk T (t, tdt, Wasserstei Covariace = C jk (t, t, if R i = U i(t, tdt, the E(R i = Σ. With τ = Σ Σ = ( R i Σ τ. ( Let S jklm = cov(r i jk, (R i lm } = D(s, t} jklmds dt, so that ( R i Σ C weakly, where C is a zero-mea p p Gaussia matrix with covariace structure S. Next, with F deotig the Frobeius orm, it is easy to show that τ F p j= ˆν j ν j. Sice each 5 summad is O p (, this proves that τ = o p ( /. Hece, ( Σ Σ C weakly. Next, due to Assumptio, the fuctios X ij ad ˆX ij are uiformly bouded for all i ad j, ad this boud clearly carries over for the quatile averages ν j = X ij ad ˆν j = ˆX ij. Hece, for some positive costat C, (ˆΣ ( Σ jk jk = ( ˆX ij ˆν j, ˆX ik ˆν k X ij ν j, X ik ν k 5 ( = ˆX ij ˆν j, ˆX ik X ik + ˆX ij X ij, X ik ν k ( ˆX ij ˆν j, ˆX ik X ik + ˆX ij X ij, X ik ν k ( C ˆX ik X ik + ˆX ij X ij } = d W ( ˆf ik, f ik + d W ( ˆf ij, f ij. The result the follows from Assumptios ad. 55 Proof of Theorem. Adoptig the same otatios as i the proof of Theorem, let R ijk = (R i jk ad Σ (y j, y k = R ijk τ jk,

4 A. PETEEN AND H. G. MÜLLER 6 where τ jk = T jk(t, tdt ca be show to satisfy p p j,k= τ jk = O p(. Usig Assumptios ad, we immediately obtai Σ Σ = O + p [,] p Σ (y j, y k Σ } (y j, y k j,k= } = O + p p Σ (y j, y k R ijk. j,k= Observig E(R ijk = Σ (y j, y k ad var(r ijk = G(y j, y k, where G(y, z = cov X(s; c yx(s; c z, X(t; c yx(t; c z} dsdt, [,] Assumptio 5 implies E p p Σ (y j, y k so that Fially, j,k= Σ Σ by Assumptios ad, so that completig the proof. [,] = O p(, as claimed. } R ijk = p p j,k= max Σ (y j, y k ˆΣ (y j, y k } = O p (b N j,k=,...,p Σ ˆΣ = O p(b N, [,] G(y j, y k = O(, 65 S.. THEOREM THEOREM. Uder Assumptios ad, the covariace fuctio estimates satisfy } Ĉjk (s, t C jk (s, t ds dt = Op ( + b N. Proof of Theorem. Write U ijk (s, t = U i (s, t} jk. Sice E U ijk (s, t} = C jk (s, t, Assumptio implies E ( X j <, whece ( Cjk E U ijk = O(. [,] Furthemore, with T jk (s, t = T (s, t} jk, T jk = ν [,] j ν j ν k ν k = O p (,

5 Wasserstei Covariace 5 so that C jk C jk [,] = O p (. Lastly, uder Assumptio, there exists a costat C such that C } jk Ĉjk C d W (f ij, [,] ˆf ij + d W (f ik, ˆf ik = O p (b N by applyig Assumptio, which proves the result. S.. ADDITIONAL FIGURES Fig. 5. Slices of covariace surface estimates Ĉjk(s, t, j < k, for s =.5,.5,.75, correspodig to top, middle, ad bottom rows, respectively. The left, middle, ad right colums correspod to ormal, mild cogitive impairmet, ad Alzheimer s subjects.

6 6 A. PETEEN AND H. G. MÜLLER Fig. 6. Estimated covariace (top row ad correlatio (bottom row matrices of regioal homogeeity scores for ormal (left, mild cogitive impairmet (middle ad Alzheimer s (right subjects. Labels: ad (left ad right middle frotal, ad (left ad right parietal, ad (left ad right middle temporal, (medial superior frotal, (medial prefrotal, (posterior cigulate/precueus ad (right supramargial. Positive (egative values are draw i black (white ad larger circles correspod to larger absolute values.

7 Wasserstei Covariace 7 Fig. 7. Estimated regioal homogeeity correlatio submatrices correspodig to lateral hub pairs for ormal (left ad Alzheimer s (right subjects, after reorderig of hubs by hierarchical clusterig usig Ward s criterio. Rectagles idicate the groupigs whe three clusters are used. Labels correspod to those i Figure 6 ad further explaatios ca be foud i the captio of Figure Fig. 8. Wasserstei mea desity surfaces for mortality betwee 98 ad 5, for Easter Europea (left ad other (right coutries.