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arxiv: 2605.18339 · v1 · pith:OOKRR3K5new · submitted 2026-05-18 · 📊 stat.ME · math.ST· stat.TH

Compositional Periodic Spline Approximation for Circular Density Data in Bayes Spaces

Jitka Machalov\'a , Jana Heckenbergerov\'a , Karel Hron This is my paper

Pith reviewed 2026-05-20 00:09 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords circular densitiesBayes spacesperiodic splinescentered log-ratio transformationfunctional data analysiscompositional data analysissmoothing splineswind direction
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The pith

Centered log-ratio transformation enables periodic spline approximation for circular densities in Bayes spaces.

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

This paper proposes a framework to approximate circular density data by using compositional periodic splines in Bayes spaces. Densities are mapped via the centered log-ratio transformation into a subspace of L2 functions. This mapping keeps the relative nature of the distributions and their periodic structure, so that tools from functional data analysis can be used directly. The authors build periodic splines using coefficients under a zero-integral constraint and supply matrix forms for smoothing and penalized splines to make estimation efficient. Application to wind direction data shows the method gives smooth estimates and allows functional regression.

Core claim

Applying the centered log-ratio transformation represents densities in a subspace of the standard L2 space, enabling functional data analysis tools while preserving relative nature and periodic structure. A coefficient-based construction of periodic splines with zero-integral constraint is developed with matrix formulations for smoothing and penalized splines for efficient estimation. The method applied to long-term wind direction data yields smooth interpretable density estimates and supports statistical analysis including functional regression.

What carries the argument

The centered log-ratio transformation to a Bayes space subspace combined with coefficient-based periodic splines under zero-integral constraint, which allows matrix-based smoothing and penalized estimation while maintaining compositional properties.

If this is right

  • Smooth density estimates are obtained for circular data such as wind directions.
  • Functional regression and other analyses become possible on the density functions.
  • Matrix formulations support efficient and practical implementation of the estimators.
  • Extensions to more complex density-valued data are feasible.

Where Pith is reading between the lines

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

  • The method may offer advantages over standard circular density estimators by enforcing periodicity and relative scale through the compositional transform.
  • It could be applied to other periodic phenomena like daily activity patterns or seasonal wind variations for better modeling.
  • Future work might combine this with machine learning for high-dimensional circular data analysis.
  • A practical test would involve real-world datasets from ecology or meteorology to assess accuracy in capturing multimodal circular distributions.

Load-bearing premise

The centered log-ratio transformation preserves the relative compositional nature and periodic structure of the densities sufficiently to allow undistorted application of standard functional data analysis tools.

What would settle it

Simulate data from a known circular density, estimate it with the proposed splines, and compare the recovered density to the truth using a compositional distance measure; large discrepancies would falsify the claim that properties are preserved.

Figures

Figures reproduced from arXiv: 2605.18339 by Jana Heckenbergerov\'a, Jitka Machalov\'a, Karel Hron.

Figure 1
Figure 1. Figure 1: Linear plot (left) and raw circular data plot (right) of wind directions recorded at Pardubice Airport in January 2014 Wind direction datasets typically contain angles measured in degrees clockwise from the north. Graphical representations of such circular data are shown in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Linear histogram (left) with the cut-point at 0 (north) and raw circular data plot with a rose diagram (right) for the wind directions recorded at Pardubice Airport in January 2014 as tp,0 = 1 n n ∑ j=1 z p j = 1 n n ∑ j=1 e ipθj = 1 n n ∑ j=1 (cos pθj +isin pθj) = ap +ibp, where ap = 1 n n ∑ j=1 cos pθj , bp = 1 n n ∑ j=1 sin pθj . The complex number tp,0 defines the p-th mean resultant vector in the comp… view at source ↗
Figure 3
Figure 3. Figure 3: Raw circular data plot together with bold arrow identifying the mean resultant vector for the wind directions recorded at Pardubice Airport in January 2014 3.2. Circular Distributions By analogy with random variables on the real line, a circular probability distribution can be specified via its distribution function. However, two important complications arise due to the periodic nature of circular variable… view at source ↗
Figure 4
Figure 4. Figure 4: Raw circular data plot with three von Mises kernel density estimates with bandwidths of 10 (dotted), 40 (solid) and 120 (dashed) for the wind directions recorded at Pardubice Airport in January 2014 symmetric about its mean direction, with a normalizing constant involving a modified Bessel function. In recent years, kernel density estimation has become a popular tool for graphical representation of circula… view at source ↗
Figure 5
Figure 5. Figure 5: Histogram for January in the years 2014 and 2022 5.1. Periodic spline approximation To numerically and graphically illustrate the construction of periodic spline approxima￾tions of wind direction distributions, we employed datasets covering the period 2014– [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Wind rose for January in the years 2014 and 2022 2023. Each dataset corresponds to a single month and consists of hourly measurements collected over all days of that month. For each monthly dataset, the histogram of wind di￾rection was first transformed into relative frequencies in order to approximate the proba￾bility density function. These densities were then mapped from the Bayes space B2 (I) to the Hi… view at source ↗
Figure 7
Figure 7. Figure 7: Boxplots of smoothing or penalization parameter (left) and SSE values (right) for approximation variants (a)-(d) Figures 8 and 9 provide representative examples for March 2018 and September 2023, corresponding to the largest and smallest values of the SSE among all months and all variants, respectively. In [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Splines in L2 0 (0,2π) for March 2018 (left) and September 2023 (right) [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cubic spline in B2 (0,2π) for March 2018 (left) and September 2023 (right) towards lower values, which visually affects the result in B2 (0,2π). Sometimes, it may be useful to plot the smoothing splines in polar coordinates. For example, for January 2014 and 2022, the splines are shown in [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Splines in polar coordinates B2 (0,2π) for January 2014 (left) and 2022 (right) Another way to compare the differences between spline variants is through graphical output [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: All four variants of splines in L2 0 (0,2π) for January 2014 (left) and whole year 2014 (right) 5.2. Statistical analysis of wind direction dataset Let us display all 120 curves (10 years with monthly resolution) in the space L 2 0 (0,2π) for variant (a), which was selected as the best according to the average value of the SSE criterion [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Dataset of 120 periodic smoothing splines (January 2014 - December 2023) in the space L2 0 (0,2π) [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Average and standard deviation curves of dataset (upper left and right); dataset augmented by average in L2 0 (0,2π) resp. B2 (0,2π) (bottom left resp. right) The dependence of wind direction distribution on time is examined using FDA re￾gression, where the explanatory variable is the month index i = 1,...,120. Let us build [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Functional time regression parameters in L2 0 (0,2π) resp. B2 (0,2π) (upper left resp. right); intercept clr( ˆβ0(t)) in L2 0 (0,2π) (bottom left) and time parameter clr( ˆβ1(t)) in L2 0 (0,2π) (bottom right) side, the confidence band for the time parameter clr( ˆβ1(t)) is considerably wider. This leads to the conclusion that the annual distribution of wind direction does not depend linearly on time and d… view at source ↗
Figure 15
Figure 15. Figure 15: Bootstrap confidence bands for functional time regression parameters in L2 0 (0,2π) (intercept clr( ˆβ0(t)) (left) and time parameter clr( ˆβ1(t)) (right)) [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Functional wind speed regression parameters in L2 0 (0,2π) resp. B2 (0,2π) (upper left resp. right); intercept clr( ˆβ0(t)) in L2 0 (0,2π) (bottom left) and wind speed parameter clr( ˆβ1(t)) in L2 0 (0,2π) (bottom right) [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Bootstrap confidence bands for functional wind speed regression parameters in L 2 0 (0,2π) (intercept clr( ˆβ0(t)) (left) and wind speed parameter clr( ˆβ1(t)) (right)) Similarly to the time regression, [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Prediction of direction for wind speed ranging from 7 to 19 km/hr in L2 0 (0,2π) (left) and corresponding wind direction densities in B2 (0,2π) (right). 6. Conclusion This paper developed a compositional periodic spline framework for the approxima￾tion and analysis of circular density data in Bayes spaces. The proposed methodology combines the geometric structure of B2 (I) with periodic spline smoothing i… view at source ↗
read the original abstract

This paper proposes a novel framework for the approximation and analysis of circular density data using compositional periodic splines within Bayes spaces with the Hilbert space structure. By applying the centered log-ratio transformation, densities are represented in a subspace of the standard $L^2$ space of real-valued functions, which enables the use of functional data analysis tools while preserving the relative nature of distributions and their periodic structure. A coefficient-based construction of periodic splines with a zero-integral constraint is developed, together with matrix formulations for both smoothing splines and penalized splines, allowing efficient estimation and implementation. The methodology is applied to long-term wind direction data, where it provides smooth and interpretable density estimates and supports further statistical analysis, including functional regression. The results demonstrate the practical relevance of the proposed approach and its potential for extensions to more complex density-valued data.

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

1 major / 3 minor

Summary. The paper proposes a framework for approximating and analyzing circular density data via compositional periodic splines in Bayes spaces. It applies the centered log-ratio transformation to represent densities in the zero-integral subspace of L², enabling standard functional data analysis tools while preserving periodicity and relative information. A coefficient-based construction of periodic splines enforcing the zero-integral constraint is developed, together with matrix formulations for smoothing splines and penalized splines to support efficient estimation. The approach is illustrated on long-term wind direction data, yielding smooth density estimates and supporting functional regression.

Significance. If the constructions are rigorously established, the work offers a coherent extension of functional data analysis to compositional circular data, with computational advantages from the matrix methods and direct enforcement of the integral constraint. The wind-direction application demonstrates utility for directional distributions. Credit is given for the explicit adaptation of FDA techniques to the constrained periodic setting and for avoiding post-hoc adjustments to the zero-integral condition.

major comments (1)
  1. The central claim that the clr map and coefficient-based splines allow direct application of FDA tools without distorting periodic or compositional properties would be strengthened by an explicit verification (e.g., in the spline-construction section) that the zero-integral constraint is preserved exactly under the smoothing and penalty operators for arbitrary smoothing parameters.
minor comments (3)
  1. Abstract: the phrase 'Bayes spaces with the Hilbert space structure' should be accompanied by the explicit inner product once the transformation is introduced.
  2. Application section: quantitative metrics (e.g., integrated squared error or cross-validated log-likelihood) comparing the spline estimates to circular kernel density estimators would better support the claim of practical relevance.
  3. The matrix formulations for penalized splines should include the explicit expression for the penalty matrix (or its null-space projection) to facilitate reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation of minor revision. The single major comment is addressed below; we have revised the manuscript to incorporate an explicit verification as suggested.

read point-by-point responses

  1. Referee: The central claim that the clr map and coefficient-based splines allow direct application of FDA tools without distorting periodic or compositional properties would be strengthened by an explicit verification (e.g., in the spline-construction section) that the zero-integral constraint is preserved exactly under the smoothing and penalty operators for arbitrary smoothing parameters.

    Authors: We agree that making this preservation explicit strengthens the central claim. The coefficient-based periodic splines are constructed from a basis that lies exactly in the zero-integral subspace (Section 3), and both the smoothing-spline and penalized-spline estimators are obtained from linear systems whose matrices are formed by inner products within that same subspace. Consequently, for any smoothing parameter the solution coefficient vector automatically satisfies the linear constraint that enforces zero integral. In the revised manuscript we have added a short proposition (new Proposition 3.1) together with its proof in the spline-construction section, verifying that the estimated functions remain in the zero-integral subspace for arbitrary smoothing parameters. This confirms that the clr-transformed estimates can be treated with standard FDA tools without post-hoc adjustments or distortion of the periodic or compositional structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central construction begins with the standard centered log-ratio (clr) map, which sends positive periodic densities to the zero-integral subspace of L2 while preserving periodicity pointwise; this is an external, independently verifiable property of the clr transformation rather than a result derived inside the paper. From this embedding the authors introduce a coefficient-based basis for periodic splines that exactly enforces the integral-zero constraint via linear conditions on the coefficients, then obtain the usual matrix representations for smoothing and penalized splines by direct substitution of the constrained basis into the standard quadratic forms. None of these steps reduces by the paper's own equations to a fitted parameter renamed as a prediction, nor does any load-bearing premise rest on a self-citation whose content is itself unverified; the subsequent application to wind-direction data is presented as an illustration, not as a self-referential validation. The derivation therefore remains self-contained against external benchmarks of spline theory and compositional data analysis.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from compositional data analysis and spline theory; one likely free parameter is the smoothing penalty weight, chosen during estimation. No new invented entities are introduced.

free parameters (1)
  • smoothing penalty parameter
    Used in the penalized spline formulation to control smoothness versus fit; its value is determined during estimation on the wind data.
axioms (1)
  • domain assumption Centered log-ratio transformation maps positive densities to a subspace of L2 functions while preserving relative information and periodicity.
    Invoked to justify moving circular densities into standard functional data analysis tools.

pith-pipeline@v0.9.0 · 5683 in / 1475 out tokens · 76003 ms · 2026-05-20T00:09:18.118430+00:00 · methodology

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