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Marginal Coordinate Test for Fréchet Regression with Random Objects
Jiaye Chen, Rui Qiu, Roulin Wang, Zhou Yu
TL;DR
The paper addresses calibrated marginal coordinate testing for whether an original predictor remains informative after accounting for other predictors in metric-space response regression. It develops a semi-supervised U-statistic-based procedure and establishes null calibration, bootstrap validity, consistency, local power, and simultaneous false discovery rate control.
Problem
Existing methods do not directly provide a calibrated marginal coordinate test for whether an original predictor remains informative after conditioning on the remaining predictors in metric-space response regression.
Method
The paper develops a semi-supervised procedure using an unlabeled sample for predictor conditional-mean estimation and an independent labeled sample to construct a pairwise U-statistic.
Results
The method has a weighted centered chi-square null limit, valid wild bootstrap, consistency against fixed detectable alternatives, local power under mean-element alternatives, and asymptotic FDR control through truncated p-to-e calibration and e-BH.
Takeaways & Limitations
FMCT provides marginal coordinate testing for regression with Euclidean predictors and random-object responses under the paper's stated conditions.
Takeaways & Limitations
Nonrejection of the tested null does not establish conditional independence, and the taxi-flow analysis reports that weather is conditionally irrelevant to taxi flows.
Abstract
from arXiv · showhide
We develop a marginal coordinate test for regression with Euclidean predictors and a random-object response in a separable metric space. The goal is to test whether a predictor provides additional information about the response conditional on the remaining predictors. In a semi-supervised design, an unlabeled sample is used to estimate predictor conditional means, while an independent labeled sample is reserved for inference. The resulting residuals are combined with a product-space kernel to form a kernel conditional mean dependence (KCMD) U-statistic without requiring a response residual. The primary identity-based test targets a necessary conditional mean restriction, while a multiple-transformation extension probes broader alternatives. We establish a weighted centered chi-square null limit, wild bootstrap validity, consistency against fixed detectable alternatives, and local power under mean-element alternatives. For simultaneous inference, truncated p-to-e calibration combined with e-BH provides asymptotic false discovery rate control under general dependence. Simulations with Euclidean and non-Euclidean responses, together with a New York City taxi-flow analysis, illustrate the method.
1 Introduction
The paper develops a calibrated marginal coordinate test for whether a predictor remains informative about a metric-space response after conditioning on other predictors. Its semi-supervised, product-space construction preserves response geometry and conditioning variables while supporting asymptotic inference and simultaneous FDR control.
- Motivation: Existing methods do not directly provide a calibrated marginal coordinate test for general metric-space responses after accounting for other predictors.Earlier procedures target scalar or Euclidean responses, estimate subspaces, or require additional conditional operator estimation.
- Target: The target hypothesis tests whether Y is conditionally independent of coordinate Xj given the remaining predictors X−j.This concerns the full conditional law rather than marginal association, screening, or a specified conditional mean.
- Construction: Transformations of Xj are residualized against X−j, and their conditional association with (Y, X−j) is assessed using a product-space kernel and response distances.The complete integrable transformation class characterizes the coordinate hypothesis exactly, while characteristic kernels convert the restrictions into scalar criteria.
- Construction: The resulting construction avoids central subspace estimation and response-side regression while retaining both response geometry and conditioning variables.This provides a metric-compatible route to the same coordinate hypothesis studied in sufficient dimension reduction.
- Extensions: The primary FMCT uses the identity transformation, while finite transformation families broaden detectable alternatives at the cost of additional nuisance estimation.A finite family recovers the complete active set only under an appropriate coordinate-exhaustiveness condition.
- Inference: An unlabeled predictor sample estimates nuisance regressions, an independent labeled sample constructs the statistic, and wild bootstrap calibration supports inference.The paper also combines bootstrap p-values with truncated p-to-e calibration and e-BH for asymptotic FDR control under general or arbitrary dependence.
2 Population characterization
The section characterizes conditional independence through transformations of each predictor coordinate and a KCMD criterion, then distinguishes exact population benchmarks from finite practical specifications.
- Exact population characterization: The complete integrable transformation class yields an exact characterization of whether Xj is conditionally independent of Y given X−j.The characterization uses conditional moment restrictions for every admissible transformation.
- Finite specifications: A fixed transformation can refute the null when nonzero, but a zero value is inconclusive because it examines only one aspect of conditional behavior.The complete transformation class is therefore a population benchmark rather than a literal implementation.
- Exact population characterization: Residualizing g(Xj) against X−j removes predictor-side explained variation without requiring a response residual.The conditioning set must retain both Y and X−j; averaging over X−j alone can miss departures that vary with the remaining predictors.
- KCMD representation: Under negative-type response metrics, a Hilbert embedding and characteristic product-space kernel make KCMD zero exactly when the relevant conditional mean restriction holds.The framework covers Euclidean responses, Wasserstein distributions, SPD matrices with Log-Cholesky distance, and unit spheres with geodesic distance.
- Finite specifications: A finite transformation family recovers the active coordinates when it is coordinate exhaustive, but no fixed finite family is automatically exhaustive over all alternatives.Restricting the family cannot create a false active coordinate, but it can miss an active one; coordinate exhaustiveness restores the population equivalence.
- Identity and multiple transformations: The identity transformation provides a direct, interpretable single-transformation baseline, while multiple families can include polynomials, slicing indicators, trigonometric components, and B-splines.The population criteria avoid estimating a central subspace, its structural dimension, or a response-side regression.
3 Semi-supervised coordinate inference
The procedure uses unlabeled predictors to estimate nuisance conditional means and an independent labeled sample to construct a coordinatewise KCMD U-statistic. It provides asymptotic calibration through a nonpivotal null limit and wild bootstrap, with extensions to power analysis and simultaneous FDR control.
- Target and construction: The method targets Cook’s marginal coordinate hypothesis using an auxiliary conditional mean dependence criterion whose rejection refutes the scientific null.The identity specification detects alternatives in the identity-detectable class, including the scientific marginal coordinate hypothesis when the singleton identity family is coordinate exhaustive.
- Target and construction: An unlabeled predictor sample estimates fj(X−j)=E(Xj|X−j), while an independent labeled sample supplies residuals and pairwise kernel values for inference.This separation avoids the additional U-centering used in related procedures.
- Asymptotic theory: Conditional on the nuisance-estimation sample, the statistic is an order-two U-statistic whose oracle kernel is degenerate under the null.The estimated-versus-oracle difference is controlled by nuisance-estimation rates and sample-size conditions.
- Asymptotic theory: Under the null, the statistic converges to a weighted sum of centered chi-square variables, with weights determined by an unknown integral operator.Because the limit is nonpivotal, the procedure approximates its quantiles using a wild bootstrap.
- Bootstrap calibration: Wild-bootstrap quantiles consistently approximate null critical values, yielding coordinatewise tests that are asymptotically exact at level α.The bootstrap uses independent multipliers satisfying mean-zero, unit-variance, and finite-fourth-moment conditions.
- Simultaneous inference: Calibrated e-values combined with e-BH control the limiting false discovery rate without imposing a particular dependence structure.The construction first establishes asymptotic validity of calibrated e-values and then applies e-BH.
5 Numerical experiments
The experiments evaluate FMCT for coordinatewise and simultaneous testing with Euclidean and non-Euclidean responses. Across settings, active-coordinate power generally increases with labeled-sample size while null-coordinate rejection frequencies remain near nominal levels, with some mild overrejection.
- Euclidean responses: FMCT-NN and FMCT-Lasso achieve power above 0.9 for all four active coordinates at n2=100 in the linear-response example.Their null-coordinate rejection frequencies are broadly comparable to those of the competing methods.
- Euclidean responses: FMCT-NN and FMCT-Lasso detect nonlinear effects from X3 and X4, with power increasing from approximately 0.92 at n2=100 to 1 at n2=200.Their average power is slightly below Wn-Lasso at n2=100 but becomes comparable at larger sample sizes.
- Euclidean responses: T NL-lasso rejects X3 and X4 near the nominal level in the nonlinear example, indicating little power against these effects.FMCT null-coordinate rejection frequencies are generally close to nominal, with occasional mild overrejection.
- Metric-space responses: For distributional responses, power increases with n2 for every active coordinate, although X5 is more difficult to detect.Null-coordinate rejection frequencies remain generally close to the nominal level across reported sample sizes.
- Metric-space responses: SPD-matrix responses show a similar improvement in power, while the spherical setting has high power at the two larger sample sizes.Null-coordinate rejection frequencies are mostly close to nominal, with occasional mild overrejection.
- Metric-space responses: Across the three response geometries, FMCT detects predictor contributions and performs better as more labeled observations become available.The experiments cover probability distributions, SPD matrices, and spherical data.
- Simultaneous testing: In simultaneous Euclidean testing, both FMCT variants keep empirical FDR below nominal levels for every reported sample size.FMCT-NN has high power in Example 6, whereas FMCT-Lasso power increases steadily with n2; Wn-Lasso attains power one but exceeds nominal empirical FDR.
6 Real data analysis
The New York City taxi analysis applies FMCT to hourly flow-network responses and service and weather predictors in a semi-supervised design. Coordinatewise testing identifies several service variables, while simultaneous testing selects only average fare and cash; weather nonselection is not evidence of conditional irrelevance.
- Data and design: The semi-supervised split masks responses for 1,204 observations and retains 212 labeled observations for inference.This gives an unlabeled-to-labeled ratio of approximately 17:3.
- Simultaneous results: Under simultaneous testing, only average fare and cash are selected at both FDR levels.No weather variable is selected in the simultaneous analysis.
- Interpretation and limitations: The analysis reports strongest conditional evidence for fare and cash after adjustment, but it does not support a causal interpretation.Weather nonselection may partly reflect the coarser daily resolution of those predictors and does not establish conditional irrelevance.
- Interpretation and limitations: Because primary FMCT uses g(x) = x, it may miss contributions expressed only through conditional variances or other nonlinear features.A richer transformation family can assess sensitivity to such structures.
7 Conclusion
The paper develops FMCT for marginal coordinate testing with Euclidean predictors and metric-space responses, with theory for calibration, power, and simultaneous inference. Its primary test targets a necessary conditional mean restriction, so rejection supports evidence against conditional independence while nonrejection does not establish it.
- Contribution and scope: FMCT addresses marginal coordinate testing for Euclidean predictors and a metric-space response, with guarantees for null calibration, bootstrap validity, power, and simultaneous inference under suitable conditions.Full active-set recovery additionally requires coordinate exhaustiveness.
- Targeted hypothesis: The scientific null Y ⟂⟂ X_j | X_−j implies the auxiliary conditional mean restriction targeted by the primary statistic.The primary test therefore focuses on a necessary conditional mean condition rather than directly establishing the full null from nonrejection.
- Targeted hypothesis: Rejection provides evidence against the scientific null, whereas nonrejection does not establish conditional independence.This distinction limits how the primary test's results should be interpreted.
- Asymptotic theory: Under stated conditions, the null limit is a weighted sum of centered chi-square variables, with wild bootstrap validity and consistency against fixed detectable alternatives.The paper also characterizes power under local mean-element alternatives.
- Simultaneous inference: Truncated p-to-e calibration combined with e-BH yields asymptotic false discovery rate control for simultaneous inference.Prespecified finite transformation families can probe broader alternatives with analogous bootstrap calibration and asymptotic theory.
- Future directions: Future work includes data-adaptive transformation families and uniform bootstrap and FDR guarantees when the number of predictors grows.These are stated scope extensions beyond the current guarantees.