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arxiv: 2508.04074 · v3 · pith:TUF7KQ2Onew · submitted 2025-08-06 · 📊 stat.AP

Matrix Factorization-Based Solar Spectral Irradiance Missing Data Imputation with Uncertainty Quantification

Yuxuan Ke , Xianglei Huang , Odele Coddington , Yang Chen This is my paper

Pith reviewed 2026-05-21 23:47 UTC · model grok-4.3

classification 📊 stat.AP
keywords solar spectral irradiancemissing data imputationmatrix factorizationuncertainty quantificationconformal predictiontime series reconstructionclimate data analysis
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The pith

Low-rank matrix factorization with autoregressive regularization and periodic detrending imputes missing solar spectral irradiance data while producing calibrated uncertainty intervals.

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

The paper shows that daily solar spectral irradiance measurements form a matrix with significant gaps from instrument issues, yet this matrix has exploitable low-rank structure along with periodic solar cycles and temporal correlations across wavelengths. By embedding these features into a matrix factorization model via a two-stage fitting process and conformal prediction, the method recovers the missing entries more accurately than standard alternatives. A sympathetic reader would care because reliable gap-filled SSI records with uncertainty estimates support better analysis of solar variability and its role in climate processes. The approach is tested on both synthetic data and actual TSIS-1 SIM observations, confirming gains in accuracy, interval calibration, and speed.

Core claim

The proposed low-rank matrix factorization incorporates autoregressive temporal regularization, periodic spline detrending, and cross-spectral covariance information. Implemented as a two-stage procedure to handle scattered and extended missingness separately and fitted by alternating optimization, the model is paired with a distribution-free conformal prediction procedure for uncertainty. Synthetic experiments and real-data comparisons demonstrate that this structure-aware reconstruction outperforms Gaussian process regression, linear time series smoothing, and prior matrix-completion methods in imputation accuracy while delivering calibrated intervals of practical length.

What carries the argument

Low-rank matrix factorization augmented with autoregressive temporal regularization, periodic spline detrending, and cross-spectral covariance, applied in a two-stage procedure and paired with conformal prediction for intervals.

If this is right

  • Reconstructed SSI values achieve higher accuracy than those from Gaussian process regression or standard matrix completion on both synthetic and real TSIS-1 data.
  • The conformal intervals are calibrated and of practical length, making the output directly usable in climate studies.
  • Alternating optimization renders the procedure computationally efficient relative to competing methods.
  • The two-stage design separately addresses random scattered gaps and long instrument-downtime blocks.

Where Pith is reading between the lines

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

  • The same factorization-plus-regularization structure could be tested on other periodic multi-channel time series that exhibit low-rank behavior across channels.
  • If the cross-spectral covariance terms prove dominant, the method might be simplified by dropping the autoregressive term on datasets with weaker temporal dependence.
  • The conformal prediction step could be replaced by parametric alternatives if future work establishes that the residuals follow a stable distribution.

Load-bearing premise

The observed SSI measurements admit an approximately low-rank structure that is adequately captured by the chosen factorization after the periodic detrending and autoregressive terms are included, and that the two-stage procedure does not bias the estimates for the patterns of missingness actually present in the data.

What would settle it

On a held-out subset of real TSIS-1 measurements with artificially introduced gaps matching the observed missingness patterns, the conformal intervals would fail to achieve the nominal coverage rate (for example, covering far fewer than 95 percent of the true values at the 95 percent level).

Figures

Figures reproduced from arXiv: 2508.04074 by Odele Coddington, Xianglei Huang, Yang Chen, Yuxuan Ke.

Figure 1
Figure 1. Figure 1: Missingness in SSI data. (A) Observed SSI data. White entries represent missing [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Loss functions diagram. through a two-step approach. In the first step, we concentrate on imputing the scattered missingness with the cross-sectional variance-covariance structure and splines detrending in the observed space. This involves estimating both the model parameters and the missing values using an EM algorithm. In the second step, given the estimated values for the scattered missingness from the … view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of SIAP. Step 1 jointly estimates the observed space detrending (mod￾ule 1), low-rank matrix factorization (module 2) and cross-sectional covariance (module 3). Step 2 models the temporal dynamics where module 4 is latent space detrending and mod￾ule 5 is temporal smoothing using AR regulariation. Be = (eb1, . . . , ebn) ⊤ = B − ΦΘ. 1 |{j:(i,j)∈Ωc}| P {j:(i,j)∈Ωc} 1 n Xij ∈ Cbijo . In Section … view at source ↗
Figure 4
Figure 4. Figure 4: Simulation study of relative MRAE margin with varying missingness ratio. The [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The average coverage rate of uncertainty intervals per wavelength (i.e., row). The [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Zoomed-in integrated SSI imputations in 300-400nm. The points represent down [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
read the original abstract

The solar spectral irradiance (SSI) depicts the spectral distribution of solar energy flux reaching the top of the Earth's atmosphere. Daily SSI measurements constitute a matrix with spectrally (rows) and temporally (columns) resolved solar energy flux measurements. The most recent SSI measurements have been made by NASA's Total and Spectral Solar Irradiance Sensor-1 (TSIS-1) Spectral Irradiance Monitor (SIM) since March 2018. This data has considerable missing data due to both random factors and instrument downtime, a periodic trend related to the Sun's cyclical magnetic activity, and varying degrees of correlation among the spectra, some approaching unity. We propose a low-rank matrix factorization method for SSI reconstruction that incorporates autoregressive temporal regularization, periodic spline detrending, and cross-spectral covariance information. The method is implemented as a two-stage procedure designed to address scattered missingness and extended downtime missingness, respectively, and is fitted using efficient alternating optimization algorithms. We further accompany the reconstructed SSI values with a distribution-free interval estimation procedure based on conformal prediction. Through synthetic experiments and real-data analyses, we compare this method with Gaussian process regression, linear time series smoothing, and existing matrix-completion approaches in terms of imputation accuracy, interval coverage, interval length, and computational efficiency. The results show that exploiting the periodic, temporal, and cross-spectral structure of SSI substantially improves reconstruction performance and yields calibrated uncertainty intervals, producing a reconstructed SSI data product suitable for downstream climate science studies.

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 proposes a low-rank matrix factorization method for imputing missing values in daily solar spectral irradiance (SSI) data from TSIS-1 SIM, incorporating autoregressive temporal regularization, periodic spline detrending, and cross-spectral covariance. It uses a two-stage procedure (first for scattered missingness, second for extended downtime), fitted via alternating optimization, and supplies distribution-free uncertainty intervals via conformal prediction. Synthetic and real-data experiments compare the approach to Gaussian process regression, linear smoothing, and prior matrix-completion methods on accuracy, coverage, interval length, and efficiency, claiming substantial gains from exploiting periodic, temporal, and cross-spectral structure.

Significance. If the central claims hold, the work supplies a practical, uncertainty-aware imputation tool for a high-value geophysical dataset used in climate studies. The combination of low-rank factorization with domain-specific regularizers and conformal intervals is a reasonable fit for the problem; the reported gains over standard baselines and the emphasis on calibrated intervals are strengths that could support downstream applications if bias from the two-stage procedure is shown to be negligible.

major comments (2)
  1. [Section 3 (two-stage procedure)] The two-stage procedure (first stage for scattered missingness with spline detrending, second stage for extended blocks) risks systematic bias when downtime intervals align with the ~11-year solar cycle. Because the stages are optimized separately rather than under a joint objective, residual periodic components not fully removed by the first-stage spline can be absorbed into the low-rank factors or AR terms in the second stage. The manuscript should include a targeted simulation or diagnostic (e.g., recovery error stratified by phase of the solar cycle) to demonstrate that this interaction does not produce detectable bias under the observed missingness patterns.
  2. [Section 4 (synthetic experiments)] The claim that the factorization plus AR and spline terms yields an approximately low-rank structure well-matched to SSI is central but rests on empirical performance rather than a direct check. The paper should report the effective rank of the observed data matrix after periodic detrending and the sensitivity of reconstruction error to the chosen factorization rank (listed among the free parameters).
minor comments (2)
  1. [Section 5] Notation for the conformal prediction intervals should be introduced explicitly (e.g., how the nonconformity scores are computed from the matrix factorization residuals) rather than left implicit in the uncertainty quantification section.
  2. [Section 6] The real-data analysis would benefit from a table or figure showing the fraction and temporal distribution of missing entries (scattered vs. block) to allow readers to assess how representative the test cases are of the actual TSIS-1 record.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make to strengthen the presentation and address the concerns.

read point-by-point responses

  1. Referee: [Section 3 (two-stage procedure)] The two-stage procedure (first stage for scattered missingness with spline detrending, second stage for extended blocks) risks systematic bias when downtime intervals align with the ~11-year solar cycle. Because the stages are optimized separately rather than under a joint objective, residual periodic components not fully removed by the first-stage spline can be absorbed into the low-rank factors or AR terms in the second stage. The manuscript should include a targeted simulation or diagnostic (e.g., recovery error stratified by phase of the solar cycle) to demonstrate that this interaction does not produce detectable bias under the observed missingness patterns.

    Authors: We agree that the interaction between the two-stage procedure and the solar cycle merits explicit verification. The periodic spline in the first stage is designed to remove the dominant ~11-year component before the second stage operates on the residuals. To directly address the concern, we will add a targeted simulation in the revised manuscript: synthetic SSI matrices will be generated with missingness blocks aligned to different phases of the solar cycle, and we will report recovery error and bias stratified by cycle phase. This diagnostic will confirm that residual bias remains negligible under the missingness patterns observed in the TSIS-1 SIM record. revision: yes

  2. Referee: [Section 4 (synthetic experiments)] The claim that the factorization plus AR and spline terms yields an approximately low-rank structure well-matched to SSI is central but rests on empirical performance rather than a direct check. The paper should report the effective rank of the observed data matrix after periodic detrending and the sensitivity of reconstruction error to the chosen factorization rank (listed among the free parameters).

    Authors: We concur that a direct quantification of effective rank and rank sensitivity would strengthen the justification for the low-rank model. In the revised manuscript we will report the singular-value spectrum of the detrended data matrix to document its effective rank. We will also include a sensitivity analysis showing reconstruction error (and interval coverage) as a function of the factorization rank k over a range centered on the value used in the main experiments, thereby confirming that performance is robust to modest changes in this hyperparameter. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on external benchmarks

full rationale

The paper defines a low-rank matrix factorization procedure with autoregressive regularization, periodic spline detrending, and cross-spectral terms, implemented via two-stage alternating optimization. All performance claims (imputation accuracy, interval calibration) are evaluated against independent baselines (Gaussian process regression, linear smoothing, prior matrix-completion methods) on both synthetic and real-data hold-outs. No load-bearing step reduces by construction to a fitted quantity or self-citation; the derivation chain is self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach rests on domain assumptions about data structure rather than new physical postulates; free parameters are the factorization rank and regularization weights that must be chosen or tuned.

free parameters (2)
  • factorization rank
    Low-rank dimension chosen to approximate the SSI matrix; value not stated in abstract.
  • regularization weights
    Strengths of autoregressive temporal term and periodic spline term; fitted or cross-validated.
axioms (2)
  • domain assumption SSI measurements exhibit low-rank structure across spectra and time
    Invoked to justify matrix factorization as the core reconstruction engine.
  • domain assumption Solar magnetic activity produces periodic trends that can be removed by splines
    Used to justify the periodic detrending component.

pith-pipeline@v0.9.0 · 5803 in / 1443 out tokens · 46373 ms · 2026-05-21T23:47:20.860719+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

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    impose sparse assumption on the precision matrices by including only the nearest neighbors of each node in the graph, which reduce the computation complexity to O(mn log(mn)). Meanwhile, Gardner et al. (2018) explores conjugate gradient techniques to compute a linear solve LA−1R given positive definite matrix A and left, right matrices L, R, which can be ...

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    41 Here, y1 ∼ N (ξ, Λ), {xt} is the observed processand {yt} is the latent or state process. Lag- p vector autoregressive (VAR) series can be reformulated as a VAR(1) series by stacking the lagged variables vertically, resulting in a higher-dimensional representation. For example, a MARSS model with maximum time lag p without dimension reduction is specif...

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    The black curve represents the CSIM observations, while the gray curve represents the TSIS-1 observations. TSIS-1 data has 2104 spectral channels ranging from 200.015nm to 2399.011nm, while CSIM data has 2343 channels ranging from 210.014nm to 2596.299nm. In the time dimension, the two data sets have some overlap, which enables us to evaluate our imputati...

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