TY - JOUR

T1 - Conditional independence testing in Hilbert spaces with applications to functional data analysis

AU - Lundborg, Anton Rask

AU - Shah, Rajen D.

AU - Peters, Jonas

N1 - Publisher Copyright: © 2022 The Authors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) published by John Wiley & Sons Ltd on behalf of Royal Statistical Society.

PY - 2022

Y1 - 2022

N2 - We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises testing the significance of X in a regression model of scalar response Y on functional regressors X and Z. We show, however, that even in the idealised setting where additionally (X, Y, Z) has a Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed and we argue that a convenient way of specifying these assumptions is based on choosing methods for regressing each of X and Y on Z. We propose a test statistic involving inner products of the resulting residuals that is simple to compute and calibrate: type I error is controlled uniformly when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models and testing for edges in a functional graphical model for EEG data.

AB - We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises testing the significance of X in a regression model of scalar response Y on functional regressors X and Z. We show, however, that even in the idealised setting where additionally (X, Y, Z) has a Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed and we argue that a convenient way of specifying these assumptions is based on choosing methods for regressing each of X and Y on Z. We propose a test statistic involving inner products of the resulting residuals that is simple to compute and calibrate: type I error is controlled uniformly when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models and testing for edges in a functional graphical model for EEG data.

KW - function-on-function regression

KW - functional graphical model

KW - significance testing

KW - truncated functional linear model

KW - uniform type I error control

U2 - 10.1111/rssb.12544

DO - 10.1111/rssb.12544

M3 - Journal article

AN - SCOPUS:85139761307

VL - 84

SP - 1821

EP - 1850

JO - Journal of the Royal Statistical Society, Series B (Statistical Methodology)

JF - Journal of the Royal Statistical Society, Series B (Statistical Methodology)

SN - 1369-7412

IS - 5

ER -