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arxiv: 2604.21498 · v1 · submitted 2026-04-23 · 📊 stat.ME · stat.AP

Analyzing directional errors in spatial orientation using nonparametric circular regression with mixed covariates

Mario Francisco-Fern\'andez , Andrea Meil\'an-Vila This is my paper

Pith reviewed 2026-05-09 21:56 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords nonparametric circular regressionmixed covariatesbootstrap bandwidth selectionspatial orientationangular errorsproduct kerneldirectional errors
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The pith

A product-kernel nonparametric circular regression with bootstrap bandwidth selection reveals nonlinear patterns in directional errors from spatial tasks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a nonparametric method to analyze signed angular errors in spatial orientation experiments involving participants with different vision levels under multiple sensory conditions. It uses a product-kernel estimator that handles both continuous and categorical predictors to capture how these factors influence directional accuracy. Asymptotic bias and variance are derived for the estimator, but practical bandwidth selection relies on a new bootstrap criterion based on the cosine loss. Simulations show this selector offers a good bias-variance balance and stable results compared to cross-validation and rule-of-thumb methods. Applied to the data, the models display distinct nonlinear effects by condition and include bootstrap confidence bands for uncertainty assessment.

Core claim

We propose a nonparametric circular regression framework using a product-kernel estimator for mixed continuous and categorical covariates to model signed angular errors. Asymptotic bias and variance expressions are derived, yet a bootstrap bandwidth selection criterion tailored to the cosine loss is introduced for practical use. When applied to spatial updating data from blind, low-vision, and sighted participants across five sensory conditions, the method identifies nonlinear, condition-specific patterns and provides simultaneous bootstrap confidence bands, with simulations confirming a favorable bias-variance trade-off and stable inference.

What carries the argument

The product-kernel estimator for nonparametric circular regression, which multiplies separate kernels for continuous and categorical covariates to estimate the conditional mean of the circular response variable.

If this is right

  • The framework reveals nonlinear, condition-specific patterns in the spatial updating data.
  • Uncertainty is quantified using simultaneous bootstrap confidence bands.
  • The proposed bootstrap selector achieves a favorable bias-variance trade-off in simulations.
  • It yields stable inference relative to cross-validation and rule-of-thumb approaches.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar nonparametric approaches might apply to other circular response problems in fields like biology or robotics.
  • The bootstrap selection strategy could be adapted for other kernel-based estimators in circular statistics.
  • These models may help design interventions that account for sensory-specific error patterns in orientation tasks.

Load-bearing premise

That the bootstrap bandwidth criterion produces reliable smoothing parameters for the product-kernel estimator in finite samples even with mixed covariates.

What would settle it

A new simulation study or data application in which the bootstrap selector produces worse mean integrated squared error or invalid coverage probabilities for the regression function compared to competing methods would falsify the favorable performance claim.

Figures

Figures reproduced from arXiv: 2604.21498 by Andrea Meil\'an-Vila, Mario Francisco-Fern\'andez.

Figure 1
Figure 1. Figure 1: Spatial-updating task used by Legge et al. (2016b). Participants are guided along a [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circular regression functions used in the simulation study: the left panel corresponds to [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of ∥Hmethod∥/∥HCASE∥ for regression function R1. sample sizes and regression complexities in the settings considered here, albeit at a higher computa￾tional cost. In large-scale applications where runtime is a primary constraint, the RoT selector can serve as a fast baseline (or initializer), often providing competitive fits when n is large. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of ∥Hmethod∥/∥HCASE∥ for regression function R2. 6 Analysis of the spatial orientation under sensory conditions We now apply the proposed mixed-covariate circular regression methodology to the spatial-updating data of Legge et al. (2016b), briefly described in the Introduction. The data are publicly available through the University of Minnesota Data Repository (DRUM) at http://hdl.handle.net/1… view at source ↗
Figure 5
Figure 5. Figure 5: Summaries of directional errors by sensory condition. Points on the outer ring are in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Circular error versus target distance, colored and shaped by sensory condition (semi [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Estimated circular regression curves and simultaneous confidence bands for the estimated [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Circular residual boxplots by sensory condition for the bootstrap estimator [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Estimated circular regression surfaces using a single global bandwidth [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
read the original abstract

Spatial orientation is a fundamental cognitive skill that relies on sensory information to update perceived direction. Understanding how sensory conditions influence directional accuracy is important for both cognitive science and the design of assistive technologies. We analyze experimental data in which blind, low-vision, and sighted participants performed spatial updating tasks under five sensory conditions, with signed angular error as the response. To model these data, we propose a nonparametric circular regression framework that accommodates both continuous and categorical predictors via a product-kernel estimator. Bandwidth selection is crucial in this setting, yet developing practical data-driven methods remains challenging. We derive asymptotic bias and variance expressions for the estimator, though these results do not directly lead to a feasible plug-in bandwidth selector. To address this, we develop a bootstrap bandwidth selection criterion tailored to the cosine loss and compare it with cross-validation and rule-of-thumb approaches in simulation studies. Applied to the spatial updating data, the proposed framework reveals nonlinear, condition-specific patterns and quantifies uncertainty via simultaneous bootstrap confidence bands. Across the scenarios considered, the proposed bootstrap selector achieves a favorable bias-variance trade-off and yields stable inference relative to the competing methods. An implementation is available in the R package circMixedReg.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a nonparametric circular regression model based on a product-kernel estimator that accommodates mixed continuous and categorical covariates. Asymptotic bias and variance expressions are derived for the estimator under a cosine loss, but these do not yield a feasible plug-in bandwidth rule. A bootstrap bandwidth selector tailored to the cosine loss is therefore proposed and benchmarked against cross-validation and rule-of-thumb methods in simulations. The framework is applied to signed angular error data from spatial-updating experiments involving blind, low-vision, and sighted participants across five sensory conditions, producing nonlinear, condition-specific fits together with simultaneous bootstrap confidence bands. An R package (circMixedReg) is provided.

Significance. If the bootstrap selector is reliable for the mixed-covariate product kernel, the work supplies a practical, data-driven tool for circular regression that respects the directional nature of the response while delivering uncertainty quantification. The real-data application demonstrates how the method can uncover condition-specific nonlinear patterns in cognitive experiments, and the open implementation supports reproducibility.

major comments (2)
  1. [Bootstrap bandwidth selection] Bootstrap bandwidth selection section: No theorem is provided establishing consistency or rate of convergence for the bootstrap criterion under the product-kernel estimator with mixed covariates, even though the paper explicitly states that the derived asymptotics (bias/variance) do not produce a plug-in rule. This is load-bearing for the central claim of stable inference and reliable confidence bands in the application.
  2. [Simulation studies] Simulation studies section: The simulation designs are not shown to match the real-data sample size, category balance across sensory conditions, or error distribution; without this calibration it is unclear whether the reported favorable bias-variance trade-off for the bootstrap selector extends to the spatial-updating application.
minor comments (2)
  1. [Abstract] Abstract: the statement that the bootstrap selector 'achieves a favorable bias-variance trade-off' should be supported by explicit numerical summaries (e.g., average ISE or coverage rates) from the simulation tables.
  2. [Methods] Methods: the precise form of the product kernel for the mixed (continuous + categorical) case should be written out explicitly before the asymptotic results are stated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Bootstrap bandwidth selection] Bootstrap bandwidth selection section: No theorem is provided establishing consistency or rate of convergence for the bootstrap criterion under the product-kernel estimator with mixed covariates, even though the paper explicitly states that the derived asymptotics (bias/variance) do not produce a plug-in rule. This is load-bearing for the central claim of stable inference and reliable confidence bands in the application.

    Authors: We agree that a formal consistency result for the bootstrap bandwidth selector would strengthen the theoretical foundation. The current manuscript presents the bootstrap criterion as a practical alternative when plug-in rules are unavailable and supports its use through extensive simulations, but does not include an explicit theorem. In the revision we will add a theorem establishing consistency (and the associated rate) of the bootstrap selector for the product-kernel estimator with mixed covariates, under standard regularity conditions on the circular density, the kernel functions, and the covariate distributions. revision: yes

  2. Referee: [Simulation studies] Simulation studies section: The simulation designs are not shown to match the real-data sample size, category balance across sensory conditions, or error distribution; without this calibration it is unclear whether the reported favorable bias-variance trade-off for the bootstrap selector extends to the spatial-updating application.

    Authors: We concur that closer calibration of the simulations to the empirical features of the spatial-updating data would make the performance comparison more relevant. The existing simulations explore a range of sample sizes and covariate configurations, yet they do not explicitly replicate the observed n, the category frequencies across the five sensory conditions, or the specific error distribution seen in the real data. We will revise the simulation section to include additional designs that match these characteristics and will report the resulting bias-variance trade-offs for the bootstrap selector under those calibrated settings. revision: yes

Circularity Check

0 steps flagged

No circularity: asymptotics acknowledged as insufficient for plug-in selector; bootstrap criterion introduced as independent practical surrogate

full rationale

The paper explicitly derives asymptotic bias and variance for the product-kernel estimator with mixed covariates but states that these expressions do not produce a feasible plug-in bandwidth selector. It therefore introduces a separate bootstrap bandwidth criterion based on the cosine loss, which is then benchmarked against cross-validation and rule-of-thumb selectors in simulation studies before being applied to the spatial-updating data. No equation or step reduces a reported quantity (such as the selected bandwidth, the estimated regression function, or the simultaneous confidence bands) to a fitted parameter or self-citation by construction; the bootstrap procedure operates on the data independently of the asymptotic formulas and is externally validated through comparative simulations. The overall framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on standard kernel smoothing assumptions plus domain-specific circular data handling; bandwidths are treated as selected rather than free parameters.

free parameters (1)
  • bandwidth parameters
    Chosen via bootstrap, cross-validation or rule-of-thumb; central to the estimator but data-driven.
axioms (2)
  • domain assumption Signed angular error behaves as a circular random variable suitable for cosine loss
    Invoked in the regression setup and loss function for directional data.
  • domain assumption Product kernel structure separates continuous and categorical effects appropriately
    Used to accommodate mixed covariates in the nonparametric estimator.

pith-pipeline@v0.9.0 · 5509 in / 1272 out tokens · 36142 ms · 2026-05-09T21:56:21.455745+00:00 · methodology

discussion (0)

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Reference graph

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