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Séminaire INRA de Toulouse Mélisande ALBERT Travaux de thèse, Université Nice Sophia Antipolis, LJAD Vendredi 17 mars Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 1 / 28

Table of contents 1 Introduction of the neuroscience motivation Biological context and state of the art Statistical framework 2 Bootstrap approach Description of the method Consistency of the method Bootstrap test of independence 3 Permutation approach Description of the method Permutation test of independence Simulation study 4 Synchronization detection Multiple testing Simulation study Real Data Centered Test Statistic Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 2 / 28

1 Introduction of the neuroscience motivation Biological context and state of the art Statistical framework 2 Bootstrap approach 3 Permutation approach 4 Synchronization detection Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 3 / 28

Introduction of the biological context Transmission of the neuronal information The neuronal information is transmitted by the spikes/action potentials. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 4 / 28

Introduction of the biological context Synaptic integration excitation threshold of the neuron resting potential of the neuron various dendrites input signals Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 5 / 28

Introduction of the biological context Synaptic integration spike excitation threshold of the neuron resting potential of the neuron various dendrites input signals Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 5 / 28

Introduction of the biological context Synaptic integration spike excitation threshold of the neuron resting potential of the neuron various dendrites input signals [Grammont and Riehle (1999)]: neurons coordinate their activity at very precise moments. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 5 / 28

Introduction of the biological context Synchronization detection 0 100 200 300 Observation Raster plot 0.0 0.5 1.0 1.5 2.0 time in second Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 6 / 28

Introduction of the biological context Synchronization detection 0 100 200 300 Observation Modelling Raster plot Point processes =randomsetsofpoints Complex dependences 0.0 0.5 1.0 1.5 2.0 time in second Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 6 / 28

Introduction of the biological context Synchronization detection 0 100 200 300 Observation Modelling Raster plot Point processes =randomsetsofpoints Complex dependences 0.0 0.5 1.0 1.5 2.0 Synchronization time in second Tendency to co-activate Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 6 / 28

Introduction of the biological context Synchronization detection 0 100 200 300 Observation Modelling Raster plot Point processes =randomsetsofpoints Complex dependences 0.0 0.5 1.0 1.5 2.0 time in second Synchronization Coincidence Tendency to co-activate N 1 δ δ N 2 coïncidence coïncidence a b Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 6 / 28

Introduction of the biological context Synchronization detection 0 100 200 300 Observation Modelling Raster plot Point processes =randomsetsofpoints Complex dependences 0.0 0.5 1.0 1.5 2.0 time in second Synchronization Coincidence Tendency to co-activate = = N 1 δ δ N 2 coïncidence coïncidence a b Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 6 / 28

Introduction of the biological context Global approach On each time window: Observed total number of coincidences VS Expected under independence total number of coincidences Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 7 / 28

Introduction of the biological context Global approach On each time window: Observed total number of coincidences = occurence of a synchronization. Expected under independence total number of coincidences Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 7 / 28

Introduction of the biological context Global approach On each time window: independence test. Observed total number of coincidences = occurence of a synchronization. Expected under independence total number of coincidences Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 7 / 28

Introduction of the biological context Global approach On each time window: independence test. Observed total number of coincidences = occurence of a synchronization. Simultaneous application on each tim window: multiple tests. Expected under independence total number of coincidences Parametric methods (eg: Poisson point processes) not realistic in neuroscience. State of the art in neurosciences Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 7 / 28

Introduction of the biological context Global approach On each time window: independence test. Observed total number of coincidences = occurence of a synchronization. Simultaneous application on each tim window: multiple tests. Expected under independence total number of coincidences State of the art in neurosciences Parametric methods (eg: Poisson point processes) not realistic in neuroscience. Non-parametric approaches (eg: trial shuffling) not theoretically justified. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 7 / 28

Statistical framework Observation: X n =(X 1,...,X n ), where X i =(X 1 i, X 2 i )i.i.d. Aim at testing (H 0): X 1 X 2 against (H 1): X 1 X 2. X 1 1 X 2 1 X 1 2 X 2 2 X 1 3 X 2 3 0 1 X n:observedsample {X 1, X 2, X 3,...}i.i.d. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 8 / 28

Statistical framework Observation: X n =(X 1,...,X n ), where X i =(X 1 i, X 2 i )i.i.d. Aim at testing (H 0): X 1 X 2 against (H 1): X 1 X 2. Often, tests based on ϕ(x 1, x 2 ) ( dp(x 1, x 2 ) dp 1 (x 1 )dp 2 (x 2 ) ). State of the art: with n 3/2 sup v 1 V 1,v 2 V 2 i i ( ϕ(v 1,v 2 ) ( ) ( )) X 1 i, Xi 2 ϕ(v 1,v 2 ) X 1 i, X 2 i, [Blum, Kiefer, and Rosenblatt (1961)], for V 1 = V 2 = R and ϕ (v 1,v 2 )(x 1, x 2 )=1 {],v 1 ]} (x 1 ) 1 {],v 2 ]} (x 2 ), [Romano (1989)], bootstrap and permutation for V 1 and V 2 countable V.-C. classes of subsets, and ϕ (v 1,v 2 )(x 1, x 2 )=1 {v1 } (x 1 ) 1 {v2 } (x 2 ), [Van der Vaart and Wellner (1996)], bootstrap for V 1 and V 2 classes of real-valued functions, and ϕ (v 1,v 2 )(x 1, x 2 )=v 1 (x 1 ) v 2 (x 2 ). Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 8 / 28

Statistical framework Observation: X n =(X 1,...,X n ), where X i =(X 1 i, X 2 i )i.i.d. Aim at testing (H 0): X 1 X 2 against (H 1): X 1 X 2. Notion of (delayed) coincidence [Tuleau-Malot et al. (2014)] ϕ coinc δ counts the number of coincidences between two point processes: ( ϕ coinc δ X 1, X 2) = 1 { T S δ}. T X 1 S X 2 X 1 X 2 δ coincidence δ coincidence 0 1 Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 8 / 28

Statistical framework Observation: X n =(X 1,...,X n ), where X i =(X 1 i, X 2 i )i.i.d. Aim at testing (H 0): X 1 X 2 against (H 1): X 1 X 2. Often, tests based on ϕ(x 1, x 2 ) ( dp(x 1, x 2 ) dp 1 (x 1 )dp 2 (x 2 ) ). Unbiased estimator: 1 ( ( ) ( )) ϕ X 1 n(n 1) i, Xi 2 ϕ X 1 i, X 2 i. i i Let h ϕ ( (x 1, x 2 ), (y 1, y 2 ) ) = 1 2 ( ϕ(x 1, x 2 )+ϕ(y 1, y 2 ) ϕ(x 1, y 2 ) ϕ(y 1, x 2 ) ), x 1 x 2 x 1 y 2 y 1 y 2 y 1 x 2 0 1 ϕ(x 1, x 2 )+ϕ(y 1, y 2 ) 0 1 ϕ(x 1, y 2 ) ϕ(y 1, x 2 ) Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 8 / 28

Statistical framework Observation: X n =(X 1,...,X n ), where X i =(X 1 i, X 2 i )i.i.d. Aim at testing (H 0): X 1 X 2 against (H 1): X 1 X 2. Often, tests based on ϕ(x 1, x 2 ) ( dp(x 1, x 2 ) dp 1 (x 1 )dp 2 (x 2 ) ). Unbiased estimator: 1 ( ( ) ( )) ϕ X 1 n(n 1) i, Xi 2 ϕ X 1 i, X 2 i. i i Let h ϕ ( (x 1, x 2 ), (y 1, y 2 ) ) = 1 2 the previous estimator becomes ( ϕ(x 1, x 2 )+ϕ(y 1, y 2 ) ϕ(x 1, y 2 ) ϕ(y 1, x 2 ) ), 1 n(n 1) h ϕ (X i, X j). i j Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 8 / 28

Independence test Generalized test statistic Test statistic nun,h (X n)= n n(n 1) h (X i, X j), for h :(X X) 2 R well chosen such that { (A Cent for any X and X,i.i.d. withdistributionp 1 P 2 on X 2, ) E[h (X, X )] = 0, and that U n,h is non-degenerate. i j TCL for our test statistic under independence If U n,h is non-degenerate, then L ( nu n,h ; P 1 P 2) = N ( ) 0,σ 2. n + L ( nun,h ; Q ) denotes the distribution of nu n,h (Z n)forz n =(Z 1,...,Z n) sample of i.i.d. random variables with distribution Q. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 9 / 28

1 Introduction of the neuroscience motivation 2 Bootstrap approach Description of the method Consistency of the method Bootstrap test of independence 3 Permutation approach 4 Synchronization detection Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 10 / 28

Bootstrap approach Description Bootstrap approach inspired by Romano (1988) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. X 1 1 X 2 1 X 1 2 X 2 2 X 1 3 X 2 3 0 1 nun,h (X n): on original data Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 11 / 28

Bootstrap approach Description Bootstrap approach inspired by Romano (1988) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The bootstrap sample is X n = ( ( ) Xn,1,...,Xn,n) i.i.d. P 1 n Pn 2 1 = δ n X 1 i 1 i n ( 1 n 1 j n δ X 2 j ). X 1 1 X 2 1 X 1 2 X 2 2 X 1 3 X 2 3 0 1 nun,h (X n): on original data Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 11 / 28

Bootstrap approach Description Bootstrap approach inspired by Romano (1988) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The bootstrap sample is X n = ( ( ) Xn,1,...,Xn,n) i.i.d. P 1 n Pn 2 1 = δ n X 1 i 1 i n ( 1 n 1 j n δ X 2 j ). X 1 1 X 1 1 X 2 1 X 1 2 X 1 3 X 2 2 X 1 3 X 1 2 X 2 3 0 1 nun,h (X n): on original data 0 1 Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 11 / 28

Bootstrap approach Description Bootstrap approach inspired by Romano (1988) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The bootstrap sample is X n = ( ( ) Xn,1,...,Xn,n) i.i.d. P 1 n Pn 2 1 = δ n X 1 i 1 i n ( 1 n 1 j n δ X 2 j ). X 1 1 X 2 1 X 1 1 X 2 2 X 1 2 X 2 2 X 1 3 X 2 3 X 1 3 X 2 3 X 1 2 X 2 3 0 1 nun,h (X n): on original data 0 nun,h (X n ): on bootstrapped data 1 Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 11 / 28

Bootstrap approach Consistency of the method Theorem 1 Let h :(X X) 2 R such that (A Cent ), (A Cent (A Cent Emp ) Emp ) x 1 =(x1 1, x1 2 ),...,x n =(xn 1, xn 2 ) X 2, n h (( ) ( )) xi 1, x 2 i, x 1 j, x 2 j =0. i,i,j,j =1 Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 12 / 28

Bootstrap approach Consistency of the method Theorem 1 Let h :(X X) 2 R such that (A Cent ), (A Cent Emp ), (A Mmt brk ) for any X 1, X 2, X 3, X 4,i.i.d.withdistributionP on X 2, (A Mmt brk ) a, b, c, d {1, 2, 3, 4}, E [ h (( ( 2 Xa 1, Xb) 2, X 1 c, Xd))] 2 < +. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 12 / 28

Bootstrap approach Consistency of the method Theorem 1 Let h :(X X) 2 R such that (A Cent ), (A Cent Emp ), (A Mmt brk )and(a Cont ) There exists C subset of X 2 X 2,suchthat (A Cont (i) the kernel h is continuous in every (x, y) C ) for the topology induced by the Skorohod metric, (ii) (P 1 P 2 ) 2 (C) =1. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 12 / 28

Bootstrap approach Consistency of the method Theorem 1 Let h :(X X) 2 R such that (A Cent ), (A Cent Emp ), (A Mmt brk )and(a Cont ) hold, then P-a.s. in (X n) n. ( ( ) ( d W2 L nun,h ; Pn 1 Pn X 2 n, L nun,h ; P 1 P 2)) 0 n + cumulative distribution function 0.0 0.2 0.4 0.6 0.8 1.0 on data under (H0) on bootstrapped data 20 10 0 10 20 time Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 12 / 28

Bootstrap approach Consistency of the method Theorem 1 Let h :(X X) 2 R such that (A Cent ), (A Cent Emp ), (A Mmt brk )and(a Cont ) hold, then P-a.s. in (X n) n. ( ( ) ( d W2 L nun,h ; Pn 1 Pn X 2 n, L nun,h ; P 1 P 2)) 0 n + Main arguments: By Varadarajan [1958], Separability = P-a.s, Pn 1 Pn 2 = P 1 P 2. n + Skorohod s representation theorem. Continuity of h and LLN for U-statistics. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 12 / 28

Bootstrap approach Bootstrap independence test Bootstrap test of independence Φ + h,α = 1 { nun,h (X n)>q h,1 α (Xn) }, where q h,1 α(x n) is the (1-α)-quantile of L ( nun,h ; P 1 n P 2 n X n ). Theorem 2 Under similar assumptions as in theorem 1, Asymptotic size: Under (H 0), P ( Φ + h,α =1 ) α. n + Asymptotic power: For any alternative P such that E[h (X i, X i )] > 0, P ( Φ + h,α =1 ) n + 1. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 13 / 28

1 Introduction of the neuroscience motivation 2 Bootstrap approach 3 Permutation approach Description of the method Permutation test of independence Simulation study 4 Synchronization detection Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 14 / 28

Permutation approach Idea of permutation Permutation approach inspired by Hoeffding (1952) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. X 1 1 X 2 1 X 1 2 X 2 2 X 1 3 X 2 3 0 1 nun,h (X n): on original data Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 15 / 28

Permutation approach Idea of permutation Permutation approach inspired by Hoeffding (1952) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The permuted sample is, for Π n U (S n)independentofx n X Πn n = ( ) X Πn 1,...,Xn Πn, with X Π n i = ( Xi 1, X 2 Π n(i)). X 1 1 X 2 1 X 1 2 X 2 2 X 1 3 X 2 3 0 1 nun,h (X n): on original data Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 15 / 28

Permutation approach Idea of permutation Permutation approach inspired by Hoeffding (1952) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The permuted sample is, for Π n U (S n)independentofx n X Πn n = ( ) X Πn 1,...,Xn Πn, with X Π n i = ( Xi 1, X 2 Π n(i)). X 1 1 X 1 1 X 2 1 X 1 2 X 1 2 X 2 2 X 1 3 X 1 3 X 2 3 0 1 nun,h (X n): on original data 0 1 Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 15 / 28

Permutation approach Idea of permutation Permutation approach inspired by Hoeffding (1952) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The permuted sample is, for Π n U (S n)independentofx n X Πn n = ( ) X Πn 1,...,Xn Πn, with X Π n i = ( Xi 1, X 2 Π n(i)). X 1 1 X 2 1 X 1 1 X 2 2 X 1 2 X 2 2 X 1 2 X 2 3 X 1 3 X 2 3 X 1 3 X 2 1 0 1 nun,h (X n): on original data 0 1 nun,h ( X Π n n ) : on permuted data Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 15 / 28

Permutation approach Idea of permutation Permutation approach inspired by Hoeffding (1952) Given X n =(X i) 1 i n,wherex i = ( ) Xi 1, Xi 2 i.i.d. P in X X. The permuted sample is, for Π n U (S n)independentofx n X Πn n = ( ) X Πn 1,...,Xn Πn, with X Π n i = ( Xi 1, X 2 Π n(i)). Proposition Let Π n U(S n) X n. Under (H 0), X Πn n,andx n have the same distribution. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 15 / 28

Permutation approach Consistency of the method Theorem 1 Let h = h ϕ with ϕ : X 2 R satisfying (A Mmt ϕ ). { for any X with distribution P or P (A Mmt 1 P 2 on X 2, ϕ ) E [ ϕ 4 (X 1, X 2 ) ] < +. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 16 / 28

Permutation approach Consistency of the method Theorem 1 Let h = h ϕ with ϕ : X 2 R satisfying (A Mmt ϕ ). Then, if U n,h is non-degenerate, d W2 ( L ( nun,hϕ ( X Π n n ) Xn ), N ( 0,σ 2 )) P n + 0. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 16 / 28

Permutation approach Consistency of the method Theorem 1 Let h = h ϕ with ϕ : X 2 R satisfying (A Mmt ϕ ). Then, if U n,h is non-degenerate, d W2 ( L ( nun,hϕ ( X Π n n ) Xn ), N ( 0,σ 2 )) P n + 0. cumulative distribution function 0.0 0.2 0.4 0.6 0.8 1.0 gaussian distribution on permuted data 20 10 0 10 20 time Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 16 / 28

Permutation approach Consistency of the method Theorem 1 Let h = h ϕ with ϕ : X 2 R satisfying (A Mmt ϕ ). Then, if U n,h is non-degenerate, d W2 ( L ( nun,hϕ ( X Π n n ) Xn ), N ( 0,σ 2 )) P n + 0. Main arguments: Martingale difference array CLT ( weak convergence). Second order moments convergence. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 16 / 28

Permutation approach Permutation independence test Permutation test of independence Ψ + h,α = 1 { nun,h (X n)>q π h,1 α (Xn) }, where qh,1 α(x π n) is the (1-α)-quantile of L ( ( ) ) nun,h X Π n n Xn. Theorem 2 Under similar assumptions as in Theorem 1, Non-asymptotic level / Asymptotic size: Under (H 0), P ( Ψ + h,α =1 ) α. P ( Ψ + h ϕ,α =1 ) α. n + Asymptotic power: Under any alternative P such that E[h ϕ (X i, X i )] > 0, P ( Ψ + h ϕ,α =1 ) 1. n + Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 17 / 28

Simulation study Monte Carlo approach Monte Carlo approximation In the previous tests, replacing the exact quantiles qh,1 α(x n) and qh,1 α(x π n) by the corresponding empirical quantiles obtained from B(n) i.i.d. bootstrapped/permuted samples from X n,withb(n) + leads to the same theoretical results. n + Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 18 / 28

Simulation study Monte Carlo approach Monte Carlo approximation In the previous tests, replacing the exact quantiles qh,1 α(x n) and qh,1 α(x π n) by the corresponding empirical quantiles obtained from B(n) i.i.d. bootstrapped/permuted samples from X n,withb(n) + leads to the same theoretical results. n + Study on simulated and real data Implementation (R and C++) fast computation. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 18 / 28

Simulation study Parameters Simulated data Study of the level: homogeneous Poisson processes on [0.1, 0.2] with intensity λ = 60, inhomogeneous Poisson processes with intensity f λ (t) =t λt, λ =60. Study of the power: injection model X j = X j ind Xcom, for j =1, 2 with X 1 ind X 2 ind as above with parameter λ ind =54,andX com a common homogeneous Poisson process with intensity λ com =6. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 19 / 28

Simulation study Parameters Simulated data Study of the level: homogeneous Poisson processes on [0.1, 0.2] with intensity λ = 60, inhomogeneous Poisson processes with intensity f λ (t) =t λt, λ =60. Study of the power: injection model X j = X j ind Xcom, for j =1, 2 with X 1 ind X 2 ind as above with parameter λ ind =54,andX com a common homogeneous Poisson process with intensity λ com =6. n varies in {10, 20, 50, 100}, δ =0.01, B = 10000 step in the Monte Carlo approximation of the quantiles, Ntest = 5000 for the estimations of the size, and the power. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 19 / 28

Simulation study Results 0 0.05 0.1 0.15 n=10 n=20 n=50 n=100 0 0.2 0.4 0.6 0.8 1 n=10 n=20 n=50 n=100 CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS n=10 n=20 n=50 n=100 n=10 n=20 n=50 n=100 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 20 / 28

1 Introduction of the neuroscience motivation 2 Bootstrap approach 3 Permutation approach 4 Synchronization detection Multiple testing Simulation study Real Data Centered Test Statistic Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 21 / 28

Multiple testing Initial motivation Detect the synchronizations. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 22 / 28

Multiple testing Initial motivation Detect the synchronizations. Idea: simultaneously test independence on sliding time windows [a k, b k ], 0 a b 2 (H 0,k ):X 1 X 2 on [a k, b k ] against (H 1,k ):X 1 X 2 on [a k, b k ]. Aim: Control the m tests at a global level α. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 22 / 28

Multiple testing Initial motivation Detect the synchronizations. Idea: simultaneously test independence on sliding time windows [a k, b k ], 0 a b 2 (H 0,k ):X 1 X 2 on [a k, b k ] against (H 1,k ):X 1 X 2 on [a k, b k ]. Aim: Control the m tests at a global level α. The errors accumulate! Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 22 / 28

Multiple testing Initial motivation Detect the synchronizations. Idea: simultaneously test independence on sliding time windows [a k, b k ], 0 a b 2 (H 0,k ):X 1 X 2 on [a k, b k ] against (H 1,k ):X 1 X 2 on [a k, b k ]. Aim: Control the m tests at a global level α. The errors accumulate! Benjamini and Hochberg multiple testing procedure to control the False Discovery Rate (1995). Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 22 / 28

Multiple testing procedure Benjamini-Hochberg procedure Sort the p-values where p i corresponds to the test (H 0,i), p (1) p (m), Reject all null hypotheses { H (i) }1 i k where k = max {i ; p (i) i } m q. p (k+1) p (m) x α m x p (3) p (k) 3 p (1) 1 2 k k +1 m p (2) REJECT Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 23 / 28

Simulation study Parameters Independent Dependent 80 60 spont=60 h =-600.1 i->i [0,0.001] 40 20 h =0 i->j h =30.1 i->j [0,0.005] h =0 i->j h =-30.1 h =6.1 i->j [0,0.005] i->j [0,0.005] Injection Injection 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Poisson Hawkes A: Description of Experiment 1 Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 24 / 28

Simulation study Parameters Independent Dependent 80 60 spont=60 h =-600.1 i->i [0,0.001] 40 20 h =0 i->j h =30.1 i->j [0,0.005] h =0 i->j h =-30.1 h =6.1 i->j [0,0.005] i->j [0,0.005] Injection Injection 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Poisson Hawkes A: Description of Experiment 1 n =50, δ varies in {0.001, 0.002,...,0.04}, B = 10000 steps in the Monte Carlo approximation of the quantiles, m = 191 2 simultaneous tests. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 24 / 28

Simulation study Results MTGAUE Trial Shuffling delta 0.00 0.01 0.02 0.03 0.04 delta 0.00 0.01 0.02 0.03 0.04 0.0 0.5 1.0 1.5 2.0 time Permutation 0.0 0.5 1.0 1.5 2.0 time Bootstrap delta 0.00 0.01 0.02 0.03 0.04 delta 0.00 0.01 0.02 0.03 0.04 0.0 0.5 1.0 1.5 2.0 time 0.0 0.5 1.0 1.5 2.0 time Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 25 / 28

Real Data Experiment Start Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 26 / 28

Real Data Experiment 500 ms Start Preparatory signal Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 26 / 28

Real Data Experiment 500 ms delay Start Preparatory signal Response signal with random delay = { 600 ms with probability 0.3 1200 ms otherwise Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 26 / 28

Real Data Experiment 500 ms delay Start Preparatory signal Response signal { 600 ms with probability 0.3 with random delay = 1200 ms otherwise Keep only trials with a response signal at 1700 ms. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 26 / 28

Real Data 300 200 100 1 0 0 100 200 300 Results 0.0 0.5 1.0 1.5 2.0 0.0 time in second M A BERT 0.5 1.0 1.5 2.0 time in second m NRA T V m

Work still in progress and perspectives Conclusions and perspectives Asymptotic performances of the bootstrap approach. Asymptotic/non-asymptotic performances of the permutation approach. Choice of δ for the notion of coincidences? delta=0.001 delta=0.005 delta=0.01 delta=0.02 0 0.2 0.4 0.6 0.8 1 CLT B P GA TS CLT B P GA TS CLT B P GA TS CLT B P GA TS Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 28 / 28

Work still in progress and perspectives Conclusions and perspectives Asymptotic performances of the bootstrap approach. Asymptotic/non-asymptotic performances of the permutation approach. Choice of δ for the notion of coincidences? Theoretical justification of the multiple testing? Merci! Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 28 / 28

Centering issue for resampling approaches Resampling approach Test statistic based on the total number of coincidences with delay: C obs = C(X n)= n i=1 ϕ coinc δ ( ) X 1 i, Xi 2. General idea Reject independence when there are too many (resp. too few) coincidences compared to what is expected under independence. How to recreate the distribution under independence? Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 29 / 28

Centering issue for resampling approaches Resampling approach Test statistic based on the total number of coincidences with delay: C obs = C(X n)= n i=1 ϕ coinc δ ( ) X 1 i, Xi 2. General idea Reject independence when there are too many (resp. too few) coincidences compared to what is expected under independence. How to recreate the distribution under independence? Construct a new sample X n from the original one, i.e. X n,suchthat L ( C ( ) ) X Xn n L ( C ( )) X n, whether X n satisfies independence or not. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 29 / 28

The different resampling approaches Trial Shuffling, Full Bootstrap and Permutation Original data set Nature Randomness Computer randomness Surrogate data set built as either n=3 trials Permutation Trial-shuffling Pick n= 3 couples (i,j) with replacement in (1,2) (1,3) (2,1) (2,3) (3,1) (3,2) Full Bootstrap Pick n= 3 couples (i,j) with replacement in (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) Pick only 1 permutation given by 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1 in Unconditional distribution: all possible choices of both Nature and Computer randomness Conditional distribution: 1 fixed original data set (Nature randomness), all possible choices of Computer randomness Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 30 / 28

Conditional distributions of the number of coincidences How do they perform? L ( C ( X n ) Xn ) L ( C ( X n ))? Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 31 / 28

Conditional distributions of the number of coincidences How do they perform? L ( C ( X n ) Xn ) L ( C ( X n ))? Centering issue!!! Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 31 / 28

Centering trick In view of the statistical literature, it is not possible to estimate L ( C ( )) X n directly, BUT, L ( C ( X n ) E [ C ( X n ) Xn ] Xn ) L ( C ( ) [ ( )]) X n E C X n. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 32 / 28

Centering trick In view of the statistical literature, it is not possible to estimate L ( C ( )) X n directly, BUT, L ( C ( X n ) E [ C ( X n ) Xn ] Xn ) YET, E [ C ( )] X n is unknown... L ( C ( ) [ ( )]) X n E C X n. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 32 / 28

Centering trick In view of the statistical literature, it is not possible to estimate L ( C ( )) X n directly, BUT, L ( C ( X n ) E [ C ( X n ) Xn ] Xn ) YET, E [ C ( )] X n is unknown... Centering trick Let Ĉ 0(X n)= 1 n 1 and let i j ϕ coinc δ (X 1 i, X 2 j ), L ( C ( ) [ ( )]) X n E C X n. s.t. E [ Ĉ 0(X n) ] = E [ C ( X n )], U(X n)=c(x n) Ĉ 0(X n). Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 32 / 28

Centering trick In view of the statistical literature, it is not possible to estimate L ( C ( )) X n directly, BUT, L ( C ( X n ) E [ C ( X n ) Xn ] Xn ) YET, E [ C ( )] X n is unknown... Centering trick Let Ĉ 0(X n)= 1 n 1 and let i j ϕ coinc δ (X 1 i, X 2 j ), L ( C ( ) [ ( )]) X n E C X n. s.t. E [ Ĉ 0(X n) ] = E [ C ( X n )], U(X n)=c(x n) Ĉ 0(X n). Then L ( U ( X n ) E [ U ( X n ) Xn ] Xn ) L ( U ( X n )). Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 32 / 28

Centering trick In view of the statistical literature, it is not possible to estimate L ( C ( )) X n directly, BUT, L ( C ( X n ) E [ C ( X n ) Xn ] Xn ) YET, E [ C ( )] X n is unknown... Centering trick Let Ĉ 0(X n)= 1 n 1 and let i j ϕ coinc δ (X 1 i, X 2 j ), L ( C ( ) [ ( )]) X n E C X n. s.t. E [ Ĉ 0(X n) ] = E [ C ( X n )], U(X n)=c(x n) Ĉ 0(X n). Then L ( U ( X n ) E [ U ( X n ) Xn ] Xn ) L ( U ( X n )). with E [ U ( X TS n ) ] Xn U(X n) =, and n { E[U(X n ) X n]=0, E[U(X n ) X n]=0. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 32 / 28

Conditional distributions of the centered number of coincidences How do they perform? L ( U ( X n ) E [ U ( X n ) Xn ] Xn ) L ( U ( X n ))? Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 33 / 28

Conditional distributions of the centered number of coincidences How do they perform? L ( U ( X n ) E [ U ( X n ) Xn ] Xn ) L ( U ( X n ))? Critical value: (1 α)-quantile of L ( U ( X n ) E [ U ( X n ) Xn ] Xn ), Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 33 / 28

Conditional distributions of the centered number of coincidences How do they perform? L ( U ( X n ) E [ U ( X n ) Xn ] Xn ) L ( U ( X n ))? Critical value: (1 α)-quantile of L ( U ( X n ) E [ U ( X n ) Xn ] Xn ), with Monte Carlo approximation. Mélisande ALBERT Séminaire INRA Toulouse Vendredi 17 mars 33 / 28