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arxiv: 2605.19861 · v1 · pith:TEXFHD7Unew · submitted 2026-05-19 · 📊 stat.ME

Stationary subspace analysis for spatial data

Perttu Saarela , Klaus Nordhausen , Jaakko Pere , Anne M. Ruiz This is my paper

Pith reviewed 2026-05-20 01:57 UTC · model grok-4.3

classification 📊 stat.ME
keywords stationary subspace analysisspatial datablind source separationnonstationaritygeneralized eigenvalue problemspatial statisticsjoint diagonalization
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The pith

Spatially indexed data can be decomposed into stationary and nonstationary latent components by solving generalized eigenvalue problems on first- and second-order spatial statistics.

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

The paper extends stationary subspace analysis to spatial settings by defining spatial stationary subspace analysis. Three estimation procedures are derived that target different forms of nonstationarity through spatial mean differences and covariances. Each procedure reduces to a generalized eigenvalue problem whose solutions yield the unmixing matrix. When several forms of nonstationarity occur together, the procedures are fused by approximate joint diagonalization to improve recovery. A data-augmentation scheme is added to estimate the dimension of the nonstationary subspace from the data itself.

Core claim

Under a linear mixing model, the unmixing matrix that isolates the stationary subspace is recovered by solving generalized eigenvalue problems constructed from first- and second-order spatial statistics; the combined joint-diagonalization version yields superior separation, and a data-augmentation procedure estimates the unknown dimension of the nonstationary subspace.

What carries the argument

Generalized eigenvalue problems derived from spatial first- and second-order moments that isolate directions of nonstationarity, solved separately and then fused by approximate joint diagonalization.

If this is right

  • When the dimension of the nonstationary subspace is supplied, the latent components are recovered reliably.
  • The joint-diagonalization combination outperforms any single procedure when multiple nonstationarities are present.
  • The same estimation steps transfer directly to ordinary time-series settings without spatial indexing.
  • The data-augmentation dimension estimator removes the need to know the subspace size in advance.

Where Pith is reading between the lines

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

  • The framework could be tested on geospatial sensor networks or satellite imagery to isolate stable background patterns from localized changes.
  • The same eigenvalue construction might be adapted to graph-structured data where adjacency replaces spatial distance.
  • If the dimension estimator proves consistent, it could serve as a plug-in for other blind-source-separation techniques that currently require this parameter.

Load-bearing premise

The observed spatial data arise from a linear mixture of latent components whose nonstationarity appears in first- and second-order spatial statistics.

What would settle it

Generate spatial data from known stationary and nonstationary sources with a given subspace dimension; if the estimated unmixing matrix fails to recover components whose spatial statistics match the assumed stationarity, the separation claim is falsified.

Figures

Figures reproduced from arXiv: 2605.19861 by Anne M. Ruiz, Jaakko Pere, Klaus Nordhausen, Perttu Saarela.

Figure 1
Figure 1. Figure 1: Example of the importance of choosing the right partition for a spatial signal that is nonstationary in mean. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of the importance of choosing the right partition for a spatial signal that is nonstationary in variance. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Partitions for parameters used for generating the nonstationary signal [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows that all the presented methods find nonstationarities in the mean. Based on how the signals are defined, SPSSASIR struggles slightly on the 2-by-2 partition but outperforms other methods when granularity is increased. Setting 2 The stationary part s(u) is as defined above. The nonstationary part n(u) consists of three signals ni , where each signal is generated by scaling a stationary signal y with a… view at source ↗
Figure 5
Figure 5. Figure 5: Setting 2: nonstationarity in variance Note that this also causes some nonstationarity in the spatial covariance [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Setting 3: nonstationarity in spatial covariance. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Setting 4: all three types of nonstationarity are present. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Rank estimation for Setting 4 using augmented noise dimensions [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two maps of the measurement locations for the Kola data. The measurement locations are marked with a gray [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Plot of rank estimators f, Φ, g on the Kola data with the estimated rank qˆ = 5, highlighted in blue on the g ladle [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Nonstationary components and one stationary component of [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison between SPSSACOR (default) and scaled SPSSACOR (scaled) on data from Setting 3 from Section 3. 0.0 0.2 0.4 0.6 0.8 1.0 Proportion spSSAsir spSSAsave spSSAcor spSSAcomb spSSAsir spSSAsave spSSAcor spSSAcomb spSSAsir spSSAsave spSSAcor spSSAcomb spSSAsir spSSAsave spSSAcor spSSAcomb r=1 r=5 r=10 r=15 qˆ 1 2 3 4 5 6 7 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Rank estimation for Setting 1 using augmented noise dimensions [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Rank estimation for Setting 2 using augmented noise dimensions [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Rank estimation for Setting 3 using augmented noise dimensions [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
read the original abstract

Stationary subspace analysis (SSA) is a blind source separation framework that decomposes linearly mixed multivariate data into stationary and nonstationary components. We extend SSA to spatially indexed data by introducing spatial stationary subspace analysis (spSSA), which explicitly accounts for spatial dependence. We propose three estimation procedures for the unmixing matrix based on first- and second-order spatial statistics. Each procedure targets a different type of nonstationarity and can be formulated as the solution to a generalized eigenvalue problem. To address situations where multiple forms of nonstationarity are present simultaneously, we combine the three procedures using approximate joint diagonalization. Simulation studies demonstrate that this combined approach yields superior separation performance. When the dimension of the nonstationary subspace is known, the proposed methods reliably recover the latent stationary and nonstationary components. However, determining this dimension remains a fundamental challenge in SSA, for which no generally accepted solution currently exists. Building on our estimation procedures, we propose a novel data augmentation approach to estimate the dimension of the nonstationary subspace and demonstrate its effectiveness through simulation studies. The proposed methodology is easily transferable to time series settings, making it of broader methodological interest.

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 paper extends stationary subspace analysis (SSA) to spatially indexed data via spatial SSA (spSSA). It introduces three estimation procedures for the unmixing matrix, each based on distinct first- or second-order spatial statistics and cast as a generalized eigenvalue problem. These are combined via approximate joint diagonalization (AJD) when multiple nonstationarity types coexist. A data-augmentation scheme is proposed to estimate the dimension of the nonstationary subspace. Simulation studies are reported to show reliable recovery of latent stationary and nonstationary components when this dimension is known and superior separation performance for the combined AJD approach. The methods are noted to transfer to time-series settings.

Significance. If the central claims hold, the work supplies a practical, computationally tractable extension of SSA to spatial settings that could be useful in geostatistics, environmental monitoring, and spatial econometrics. Formulating the procedures as generalized eigenproblems and employing AJD for joint handling of multiple nonstationarity sources are technically attractive features. The dimension-estimation proposal directly addresses a long-standing open problem in SSA. The explicit transferability remark to time series broadens potential impact.

major comments (2)
  1. [Simulation studies] Simulation studies: the reported reliable recovery and superiority of the combined AJD approach are demonstrated only under data generated from the assumed linear mixing model in which nonstationarity is present in the first- and second-order spatial moments. No experiments are described in which nonstationarity is confined to higher-order moments (e.g., spatially varying skewness or kurtosis) while first- and second-order statistics remain constant; in that regime the three eigenproblems would treat the components as stationary and the unmixing matrix would fail to recover the true partition. This boundary case is load-bearing for the identifiability claim.
  2. [Estimation procedures] Estimation procedures and AJD combination: when the three generalized eigenproblems are solved separately and then jointly diagonalized, it is not shown how the procedure behaves if one or more of the targeted spatial statistics exhibits no nonstationarity. The paper should clarify whether the joint diagonalization step remains stable or introduces spurious directions in such cases.
minor comments (2)
  1. The abstract states that the methodology is 'easily transferable to time series settings'; a short dedicated subsection or illustrative example would make this claim concrete rather than promissory.
  2. Notation for the spatial covariance and variogram operators should be introduced with explicit definitions before the generalized eigenvalue problems are stated, to improve readability for readers outside spatial statistics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and describe the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [Simulation studies] Simulation studies: the reported reliable recovery and superiority of the combined AJD approach are demonstrated only under data generated from the assumed linear mixing model in which nonstationarity is present in the first- and second-order spatial moments. No experiments are described in which nonstationarity is confined to higher-order moments (e.g., spatially varying skewness or kurtosis) while first- and second-order statistics remain constant; in that regime the three eigenproblems would treat the components as stationary and the unmixing matrix would fail to recover the true partition. This boundary case is load-bearing for the identifiability claim.

    Authors: Our procedures are explicitly constructed around first- and second-order spatial statistics, and the simulation design follows the model assumptions under which nonstationarity is expressed through changes in these moments. We do not claim identifiability or recovery when nonstationarity appears solely in higher-order moments; such regimes lie outside the scope of the present first- and second-order extension. To make this boundary explicit, we will add a short discussion of the modeling assumptions together with a targeted simulation that demonstrates the expected failure when only higher-order nonstationarity is present. revision: yes

  2. Referee: [Estimation procedures] Estimation procedures and AJD combination: when the three generalized eigenproblems are solved separately and then jointly diagonalized, it is not shown how the procedure behaves if one or more of the targeted spatial statistics exhibits no nonstationarity. The paper should clarify whether the joint diagonalization step remains stable or introduces spurious directions in such cases.

    Authors: When a given spatial statistic is stationary, the associated matrix is already (approximately) diagonal. The AJD step seeks a single unmixing matrix that simultaneously diagonalizes all supplied matrices; a matrix that is already diagonal contributes no additional constraint and should not generate spurious directions. We have not, however, provided explicit verification of this behavior. In the revision we will insert a brief theoretical remark on the stability of AJD under partial nonstationarity and include a small simulation study in which one or two of the three statistics are stationary while the others are not. revision: yes

Circularity Check

0 steps flagged

No circularity: spSSA estimation procedures derived independently from spatial statistics

full rationale

The paper formulates three distinct generalized eigenvalue problems directly from first- and second-order spatial statistics to recover the unmixing matrix separating stationary and nonstationary subspaces under a linear mixing model. The combined approach via approximate joint diagonalization and the data-augmentation method for estimating nonstationary dimension are presented as novel extensions, with performance claims supported by simulations generated under the stated model assumptions. No step reduces by construction to a fitted parameter, self-referential prediction, or load-bearing self-citation chain; the derivation remains self-contained against external spatial statistics benchmarks without tautological equivalence to inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on the standard linear mixing assumption of SSA and treats the nonstationary dimension as an unknown to be estimated rather than introducing new free parameters or entities.

free parameters (1)
  • dimension of nonstationary subspace
    Unknown a priori; estimated via the proposed data augmentation approach rather than fitted directly to target results.
axioms (1)
  • domain assumption Data is a linear mixture of stationary and nonstationary latent components
    Core modeling assumption inherited from SSA and extended to spatial indexing.

pith-pipeline@v0.9.0 · 5734 in / 1213 out tokens · 49295 ms · 2026-05-20T01:57:12.110117+00:00 · methodology

discussion (0)

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