Tests for separability in nonparametric covariance operators of random surfaces

Transcription

1 Tests for separability in nonparametric covariance operators of random surfaces Shahin Tavakoli (joint with John Aston and Davide Pigoli) April 19, 2016

2 Analysis of Multidimensional Functional Data Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

3 Functional Data Analysis Spatial Statistics Typical Framework 1 Want to probe the law of X, a random function in a Hilbert space (X L 2 ([0, 1] d, R)) 2 Observe independent and identically distributed realizations X 1,..., X n First order structure The mean function µ(t) = E [X(t)], t [0, 1] d, t X(t) is the parametrization. Second order structure The covariance function, resp. operator c(t, s) = cov (X(t), X(s)), t, s [0, 1] d, Cf(t) = c(t, s)f(s)ds, f L 2 ([0, 1] d, R). [0,1] d Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

4 FDA: Covariance is Crucial Karhunen-Loève expansion (basis for fpca) Let X be a random element of L 2 ([0, 1] d, R) and assume E X 2 <, and let C = n 1 λnϕn 2 ϕ n, be the SVD decomposition. The Karhunen-Loève (KL) expansion of X is ( ) X µ = ϕ n 2 ϕ n X = ξ nϕ n, in L 2, n=1 n=1 where Eξ n = 0, E [ξ nξ m] = δ n,mλ n, and ξ n = X µ, ϕ n. q 2 E X ξ nϕ n = λ n n=1 n>q Yields a separation of variables: {stochastic}+{functional} Uncorrelates: the ξ ns are uncorrelated, and the ϕ ns are orthogonal Optimal linear finite dimensional approximations (fpca µ + q n=1 ξnϕn) technology transfer from multivariate analysis: (e.g. Modeling, Inference, Classification) In inference problems, eigenfunctions form the natural basis for regularization Smoothness: each ϕ j accounts for λ j / n 1 λn of the total variation Consistent estimation of eigenstructure of the covariance operator (Dauxois, Pousse, Romain, 1982) Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

5 FDA: Estimation of the Covariance In practice, the covariance is unknown, but can be estimated by the sample covariance, Ĉ n = n 1 n ( Xi X ) ( 2 Xi X ), i=1 given an i.i.d. sample X 1,..., X n. Asymptotics (as n ) Consistency a.s. Ĉn C in trace norm, d CLT n(ĉn C) N(0, Γ), w.r.t. Hilbert Schmidt norm, [( ) ( )] Γ = E (X µ) 2(X µ) C (X µ) 2(X µ) C. Eigenstructure of C consistently estimated as a by-product! 2 Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

6 FDA: Estimation of the Covariance In practice, the covariance is unknown, but can be estimated by the sample covariance, Ĉ n = n 1 n ( Xi X ) ( 2 Xi X ), i=1 given an i.i.d. sample X 1,..., X n. Issues Computational complexity If X L 2 ([0, 1] 2, R) is represented by m = p q basis functions (think m-dimensional vector), then the covariance is O(m 2 )-dimensional. Statistical Accuracy Compared to the number of unknown parameters, the sample size is order of magnitudes smaller. E.g. p = q = 100 = unknown parameters, but usually n 100. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

7 Workaround: Separability assumption Definition If X L 2 ([ S, S] d [0, T ], R) is random, with covariance then the covariance is separable if C(s, t, s, t ) = cov(x(s, t), X(s, t )) (0.1) C(s, t, s, t ) = C 1 (s, s )C 2 (t, t ) Under separability assumption, Estimation of C is much faster and accurate. Eigenfunctions of C given by ϕ i (s)ψ j (t), where ϕ i, ψ j are the eigenfunctions of C 1, resp. C 2. Even if separability does not hold, working under this assumption provides Biased estimator of covariance, but with lower variance. still provides a valid basis, but not optimal one. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

8 Covariance: Memory size (in R) If X is represented by m = p p basis functions (think m-dimensional vector), then the covariance is O(m 2 )-dimensional. GB 1e 05 1e+01 1e+07 Slice of fmri full fmri Computational Limit Structural MRI Hard drive (1TB) RAM (16GB) full covariance p Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

9 Covariance: Memory size (in R) If X is represented by m = p p basis functions (think m-dimensional vector), then the covariance is O(m 2 )-dimensional. GB 1e 05 1e+01 1e+07 Slice of fmri full fmri Computational Limit Structural MRI Hard drive (1TB) RAM (16GB) full covariance separable covariance p Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

10 Example X 1,..., X n i.i.d. X R In general, the covariance has 5050 degrees of freedom. Under separability, this reduces to 110 degrees of freedom! Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

11 What this talk is about Eigenstructure of the covariance is crucial in Functional Data. But it is not computable in some cases. Separability can help. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

12 What this talk is about Eigenstructure of the covariance is crucial in Functional Data. But it is not computable in some cases. Separability can help. Is separability a valid assumption? How can we test the separability assumption? without computing the sample full covariance. without parametric assumptions on the covariance (or marginal covariances). Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

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14 Mathematical Framework Tensor product of Hilbert Spaces L 2 ([ S, S] d [0, T ], R) = L 2 ([ S, S] d, R) L 2 ([0, T ], R) which corresponds to saying that X(s, t) = f i (s)g i (t) i 1 General formulation of the problem: X H 1 H 2 is a random element. If E X 2 <, then its covariance operator C is trace-class: C S 1 (H 1 H 2 ). Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

15 Separability Given operators C 1 S 1 (H 1 ) and C 2 S 1 (H 2 ), we define C 1 C 2 S 1 (H 1 H 2 ) on simple tensors: ( C1 C 2 ) (u v) = C1 u C 2 v, u H 1, v H 2. Hence, ( ) C1 C 2 (X(s, t)) = C 1 (f i (s)) C 2 (g i (t)). i 1 Separability (think independence ) An operator C S 1 (H 1 H 2 ) is called separable if C = C 1 C 2, for some C 1 S 1 (H 1 ), C 2 S 1 (H 2 ). Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

16 Separability Given operators C 1 S 1 (H 1 ) and C 2 S 1 (H 2 ), we define C 1 C 2 S 1 (H 1 H 2 ) on simple tensors: ( C1 C 2 ) (u v) = C1 u C 2 v, u H 1, v H 2. Hence, ( ) C1 C 2 (X(s, t)) = C 1 (f i (s)) C 2 (g i (t)). i 1 Separability (think independence ) An operator C S 1 (H 1 H 2 ) is called separable if C = C 1 C 2, for some C 1 S 1 (H 1 ), C 2 S 1 (H 2 ). How could we approximate an arbitrary C S 1 (H 1 H 2 ) by a separable operator? Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

17 Partial Traces Partial Trace ( Think marginal distribution ) We denote by Tr 1 : S 1(H 1 H 2) S 1(H 2) the unique continuous linear operator satisfying Tr 1 ( A1 A 2 ) = Tr (A1) A 2, A 1 S 1(H 1), A 2 S 1(H 2), Likewise, Tr 2 : S 1(H 1 H 2) S 1(H 1). L 2 version If H 1 H 2 = L 2 ([ S, S] d [0, T ], R) = L 2 ([ S, S] d, R) L 2 ([0, T ], R), and C(s, t, s, t ) is a trace-class operator on that space, Tr 1(C)(t, t ) = C(s, t, s, t )ds s [ S,S] d Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

18 Partial Traces and Separable Approximation Characterization of separability An operator C S 1 (H 1 H 2 ) is separable if and only if Tr(C)C = Tr 2 (C) Tr 1 (C). Separable approximation We define the separable approximation of C S 1 (H 1 H 2 ) by Sep(C) = Tr 2 (C) Tr 1 (C) Tr(C) = Tr 2 (C) Tr(C) Tr 1 (C) Tr(C), provided Tr(C) 0. Notice that Sep is not linear, but continuous at C s.t. Tr(C) 0. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

19 Testing separability Let H 0 : C = C 1 C 2, and Ĉn be the sample covariance operator based on X 1,..., X n. Full Test D n = )) n (Ĉn Sep (Ĉn. But this requires computing the full sample covariance! Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

20 Testing separability Let H 0 : C = C 1 C 2, and Ĉn be the sample covariance operator based on X 1,..., X n. Full Test D n = )) n (Ĉn Sep (Ĉn. But this requires computing the full sample covariance! Can be avoided by looking at projections along the eigenfunctions of Sep(Ĉn)! Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

21 Testing separability Let H 0 : C = C 1 C 2, and Ĉn be the sample covariance operator based on X 1,..., X n. Full Test D n = )) n (Ĉn Sep (Ĉn. But this requires computing the full sample covariance! Can be avoided by looking at projections along the eigenfunctions of Sep(Ĉn)! Projected tests For integers r, s 1, let T n(r, s) = D n(û r ˆv s), û r ˆv s = ( ) 1 n 2 n Xk X n, v i 2 û j ˆλrˆγ s, n k=1 where (ˆλ r, û r) is the r-th eigenvalue/eigenfunction pair of Tr 2 (Ĉn) and (ˆγs, ˆvs) is the s-th eigenvalue/eigenfunction pair of Tr 1 (Ĉn). Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

22 Asymptotics Assumption Assume X H. There exists an orthonormal basis (e i ) i 1 H such that i=1 ( E X, e j 4) 1/4 <. (Equivalent to -summability of eigenvalues if X is Gaussian). Theorem Let X H = H 1 H 2 be a random element satisfying the Assumption. Then if the covariance operator C of X is separable, and Tr(C) 0, D n = )) d n (Ĉn Sep (Ĉn Z, n, where Z is a Gaussian random element of S 1 (H 1 H 2 ) with mean 0 and covariance... Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

23 Asymptotic Covariance of D n [ [ ] [ ]] E Tr (A 1 A 2 )Z Tr (B 1 B 2 )Z = [ ] Tr (A B)Γ + Tr [BC] [ ] Tr (A Id H )Γ Tr(C) Tr[B [( ) 1C 1 ] Tr A (Id H1 B 2 ) Tr[C 1 ] Tr[B [ ] 2C 2 ] (B Tr (Id H 1 Id H2 ))Γ Tr[C 2 ] { Tr[A 2C 2 ] Tr[C 2 ] Tr[B 2C 2 ] Tr Tr[C 2 ] [( ) ] Tr (A 1 Id H2 ) B Γ Tr[B 2C 2 ] Tr[C 2 ] { ] Γ + Tr[AC] Tr[C] [( (A 1 Id H2 ) (B 1 Id H2 ) Tr [( ) ] Tr A (B 1 Id H2 ) Γ [ ] B)Γ (Id H + Tr[BC] Tr[Γ] Tr[C] Tr[B [ ] 1C 1 ] (Id } Tr (Id H H1 B 2 ))Γ Tr[C 1 ] + Tr[BC] Tr[C] ) ] Γ Tr[B [( ) ] 1C 1 ] Tr (A 1 Id H2 ) (Id } H1 B 2 ) Γ Tr[C 1 ] { Tr[A 1C 1 ] Tr[C 1 ] Tr[B 2C 2 ] Tr Tr[C 2 ] [( ) ] Tr (Id H1 A 2 ) B Γ [( (Id H1 A 2 ) (B 1 Id H2 ) + Tr[BC] Tr[C] ) ] Γ Tr[B [( ) 1C 1 ] Tr (Id H1 A 2 ) (Id H1 B 2 ) Γ] }. Tr[C 1 ] ) ] Tr [((A 1 Id H2 ) Id H Γ ) ] Tr [((Id H1 A 2 ) Id H Γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

24 Asymptotics of the projections Corollary Under the conditions of previous Theorem, if I {(i, j) : i, j 1} is a finite set of indices such that λ rγ s > 0 for each (r, s) I, then d (T N (r, s)) (r,s) I N(0, Σ), as N, ) where Σ = (Σ (r,s),(r,s ) (r,s),(r,s is given by ) I Σ (r,s),(r,s ) = β rsr s + αrs β r s + α r s βr s + α rs β r s + α r s β rs Tr(C) + αrsα r s β Tr(C) 2 + λrλ r β s s Tr(C 1 ) 2 + γsγ s β r r Tr(C 2 ) 2 λr β r s s + λ r β rs s Tr(C 1 ) γs β r s r + γ s β rsr Tr(C 2 ) ( αrs γs β r Tr(C) Tr(C 2 ) + λ r β s ) α ( ) r s γs βr λr β s + Tr(C 1 ) Tr(C) Tr(C 2 ) Tr(C 1 ) where µ = E [X], α rs = λ rγ s, β ijkl = E [ X µ, ui v j 2 X µ, uk v l 2], and denotes summation over the corresponding index, i.e. β r jk = i 1 β rijk. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

25 Gaussian Setting Corollary Let T N (r, s) = D N (û r ˆv s), û r ˆv s,, where r, s 1 are fixed. Assume the conditions of Theorem hold, and that X is Gaussian. If I {(i, j) : i, j 1} is a finite set of indices such that λ rγ s > 0 for each (r, s) I, then (T N (r, s)) (r,s) I d N(0, Σ), as N. where Σ (r,s),(r,s ) = 2λrλ r γsγ ( ) s Tr(C) 2 δ rr Tr(C 1 ) 2 + C (λr + λ r ) Tr(C 1) ( ) δ ss Tr(C 2 ) 2 + C (γs + γ s ) Tr(C 2), and δ ij = 1 if i = j, and zero otherwise. In particular, notice that Σ itself is separable. Notice that each Σ (r,s),(r,s ) can be consistently estimated, thus we can construct an asymptotically χ 2 test for separability. I Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

26 Testing separability in practice Tests Base tests on functionals of D n = n (Ĉn Sep T n(r, s) = D n(û r ˆv s), û r ˆv s : G n(i) = (r,s) I T 2 n(r, s), G a n(i) = (r,s) I T n(r, 2 s)/ˆσ 2 (r, s) G n(i) = ˆΣ 1/2 T/2 L,I Tn(I)ˆΣ R,I 2, (Ĉn )). Recall that D n 2 2 = i,j 1 Dn(e i f j ) 2, where (e i ) i 1 is an orthonormal basis of H 1, and (f j ) j 1 is an orthonormal basis of H 2. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

27 Testing separability in practice Tests Base tests on functionals of D n = n (Ĉn Sep T n(r, s) = D n(û r ˆv s), û r ˆv s : G n(i) = (r,s) I T 2 n(r, s), G a n(i) = (r,s) I T n(r, 2 s)/ˆσ 2 (r, s) G n(i) = ˆΣ 1/2 T/2 L,I Tn(I)ˆΣ R,I 2, (Ĉn )). Recall that D n 2 2 = i,j 1 Dn(e i f j ) 2, where (e i ) i 1 is an orthonormal basis of H 1, and (f j ) j 1 is an orthonormal basis of H 2. Asymptotics + Parametric Bootstrap Empirical Bootstrap G n(i) X X Ga n (I) X X Gn(I), X X X D n 2 2 X X Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

28 Testing separability in practice Tests Base tests on functionals of D n = n (Ĉn Sep T n(r, s) = D n(û r ˆv s), û r ˆv s : G n(i) = (r,s) I T 2 n(r, s), G a n(i) = (r,s) I T n(r, 2 s)/ˆσ 2 (r, s) G n(i) = ˆΣ 1/2 T/2 L,I Tn(I)ˆΣ R,I 2, (Ĉn )). Recall that D n 2 2 = i,j 1 Dn(e i f j ) 2, where (e i ) i 1 is an orthonormal basis of H 1, and (f j ) j 1 is an orthonormal basis of H 2. Asymptotics + Parametric Bootstrap Empirical Bootstrap G n(i) X X Ga n (I) X X Gn(I), X X X D n 2 2 X X R package covsep available on CRAN & Github Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

29 Gaussian Scenario; I = {(1, 1)} Empirical power Empirical power N=10 N=25 N=50 N= γ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

30 Gaussian Scenario; I = {(r, s) : r, s = 1, 2} Empirical power Empirical power N=10 N=25 N=50 N= γ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

31 Application: Acoustic Phonetic Data Data: Log-spectrograms of recorded speech for 23 speakers (k) of 5 different Romance languages (L), pronouncing 10 different words (i) (numbers from 1 to 10): Rik L (ω, t) = SL ik (ω, t) (1/n n i i) Sik L (ω, t). k=1 We consider the residuals of same language as replicates from the same acoustic process : R L (ω, t) 81 points frequency, 100 points in the time. = full covariance structure with about degrees of freedom Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

32 Separability of Covariance for Acoustic Phonetic Data P-value for separability of the covariance of each language cov ( R L (ω, t), R L (ω, t ) ) = C L 1 (ω, ω )C L 2 (t, t ). I French Italian Portuguese American Spanish Iberian Spanish I < < < < I I < Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

33 Conclusions Tools for testing separability without computing the full covariance operator. No parametric assumptions! Hilbert space setup: applicable for multidimensional or multivariate functional data! Implemented in R package covsep (CRAN & Github) Can be generalized to higher dimensions, i.e. random element of H 1 H 2 H 3 with covariance C = C 1 C 2 C 3... (have fun!) Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

34 Thank you for your attention References [1] Gneiting, T. (2002), Nonseparable, stationary covariance functions for space-time data, Journal of the American Statistical Association 97(458), [2] Lu, N. & Zimmerman, D. L. (2005), The likelihood ratio test for a separable covariance matrix, Statistics & probability letters 73(4), [3] Fuentes, M. (2006), Testing for separability of spatial-temporal covariance functions, Journal of statistical planning and inference 136(2), [4] Mitchell, M. W., Genton, M. g. & Gumpertz, M. L. (2005), Testing for separability of space-time covariances, Environmetrics 16(8), [5] Aston, J. & Pigoli, D. & Tavakoli, S.(2015), Tests for separability in nonparametric covariance operators of random surfaces, arxiv: , under revision. [6] Tavakoli, S. (2016), covsep: Tests for Determining if the Covariance Structure of 2-Dimensional Data is Separable. R package version URL: Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

35 Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

36 References (Fourier analysis of Functional Time Series) Panaretos, V. M. & Tavakoli, S. (2013a), Cramér-Karhunen-Loève Representation and Harmonic Principal Component Analysis of Functional Time Series, Stochastic Processes and their Applications, 123 (7). Panaretos, V. M. & Tavakoli, S. (2013b), Fourier Analysis of Stationary Processes in Function Space, Annals of Statistics, 41 (2). Tavakoli, S. & Panaretos, V. M. (2015), Detecting and Localizing Differences in Functional Time Series Dynamics: A Case Study in Molecular Biophysics, to appear in JASA Application & Case Studies. Shahin Tavakoli (2015). ftsspec: Spectral Density Estimation and Comparison for Functional Time Series. R package version Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

37 Testing separability in practice Recall that T n (r, s) = D n (û r ˆv s ), û r ˆv s. Base tests on functionals of D n = )) n (Ĉn Sep (Ĉn : G n (r, s) = T 2 n(r, s) G n (r, s) = T 2 n(r, s)/ˆσ 2 (r, s) G n (I) = (r,s) I G n(r, s) G n (I) = (r,s) I G n (r, s) Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

38 Testing separability in practice Recall that T n (r, s) = D n (û r ˆv s ), û r ˆv s. Base tests on functionals of D n = )) n (Ĉn Sep (Ĉn : G n (r, s) = T 2 n(r, s) G n (r, s) = T 2 n(r, s)/ˆσ 2 (r, s) G n (I) = (r,s) I G n(r, s) G n (I) = (r,s) I G n (r, s) Devil s advocate: What if the departure from separability is outside of this subspace? Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

39 Testing separability in practice Recall that T n (r, s) = D n (û r ˆv s ), û r ˆv s. Base tests on functionals of D n = )) n (Ĉn Sep (Ĉn : G n (r, s) = T 2 n(r, s) G n (r, s) = T 2 n(r, s)/ˆσ 2 (r, s) G n (I) = (r,s) I G n(r, s) G n (I) = (r,s) I G n (r, s) Devil s advocate: What if the departure from separability is outside of this subspace? D n 2 2 = i,j 1 D n(e i f j ) 2, where (e i ) i 1 is an orthonormal basis of H 1, and (f j ) j 1 is an orthonormal basis of H 2. Hilbert Schmidt Test (HS) Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

40 Parametric Bootstrap Approximations Given X 1,..., X n, I. compute X, Sep II. For b = 1,..., B, X b i (Ĉn ), and H n = H n (X 1,..., X n ). 1 Create bootstrap ( samples )) X b = {X1, b..., Xn}, b where i.i.d. F X, Sep (Ĉn. 2 Compute H b n = H n (X b ), III. Compute the estimated bootstrap p-value p = 1 B B 1 {H b n >H n}, b=1 where 1 {A} = 1 if A is true, and zero otherwise. Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

41 Empirical Bootstrap Approximations Given X = {X 1,..., X n ) }, I. Compute Sep (Ĉn and H n = H n (X 1,..., X n ). II. For b = 1,..., B, 1 Create the bootstrap sample X b = {X1, b..., Xn} b by drawing with repetition from X 1,..., X n. 2 For each bootstrap sample, compute b n = n (X, X b ). III. Compute the estimated bootstrap p-value p = 1 B 1 B { b n >H n}, b=1 where 1 {A} = 1 if A is true, and zero otherwise. H n n (X, X ) G n (I) G n (I) D n 2 2 D n D n 2 2. (i,j) I (T n(i, j) T n (i, j)) 2. (i,j) I (T n(i, j) T n (i, j)) 2 /ˆσ 2 (i, j). Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

42 Simulation Studies Generate X 1,, X n R 32 7 with mean zero and covariance C = C (γ), where C (γ) (i 1, j 1, i 2, j 2) = (1 γ)c 1(i 1, i 2)c 2(j 1, j 2) { 1 + γ ( j 1 j 2 ) exp ( } i 1 i 2 32 )2 ( j 1 j 2, ) γ [0, 1]; i 1, i 2 = 1,..., 32; j 1, j 2 = 1,..., 7. γ = 0, 0.01,..., 0.1; n = 10, 25, 50, 100; B = 1000 The X where simulated with Gaussian distribution (Gaussian scenario) and multivariate Student t distribution with 6 degrees of freedom (non-gaussian distribution). Compare G n(i), G n(i), D n 2 2 for different sets of indices I. Replicated 1000 each combination of parameters. level α = 5%. About 280 CPU-days of simulations! Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

43 Simulation Studies c 1, c 2 are given by Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

44 Gaussian Scenario; I = {(1, 1)} N=10 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

45 Gaussian Scenario; I = {(1, 1)} N=25 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

46 Gaussian Scenario; I = {(1, 1)} N=50 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

47 Gaussian Scenario; I = {(1, 1)} N=100 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

48 Non-Gaussian Scenario; I = {(1, 1)} N=10 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

49 Non-Gaussian Scenario; I = {(1, 1)} N=25 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

50 Non-Gaussian Scenario; I = {(1, 1)} N=50 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

51 Non-Gaussian Scenario; I = {(1, 1)} N=100 CLT ParBoot ParBoot* EmpBoot EmpBoot* HS ParBoot HS EmpBoot Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

52 Increasing projection subspaces (Gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=10 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

53 Increasing projection subspaces (Gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=25 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

54 Increasing projection subspaces (Gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=50 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

55 Increasing projection subspaces (Gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=100 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

56 Increasing projection subspaces (non-gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=10 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

57 Increasing projection subspaces (non-gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=25 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

58 Increasing projection subspaces (non-gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=50 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48

59 Increasing projection subspaces (non-gaussian Scenario) I 1 = {(1, 1)}, I 2 = {(r, s) : r, s = 1, 2}, I 3 = {(r, s) : r = 1,..., 4; s = 1,..., 10}. level/power N=100 I_1 I_2 I_ γ Shahin Tavakoli (Cambridge) Testing Separability April 19, / 48