How Many Independent Samples Does a Satellite Image Contain? Generalization Bounds for Spatially Dependent Data

Robin Young

stat.MLOct 6, 2026

How Many Independent Samples Does a Satellite Image Contain? Generalization Bounds for Spatially Dependent Data

Authors: Robin Young

Organizations: Department of Computer Science and Technology University of Cambridge, UK

Abstract

Machine learning classifiers for remote sensing imagery are typically evaluated as though every pixel were an independent sample. Spatial autocorrelation violates this assumption, since neighboring pixels carry redundant information which inflates sample sizes. How many independent samples does a satellite image actually contain? For an $n \times n$ image whose spatial correlation persists over a range of $r$ pixels, the effective sample size is $Θ(n^2/r^2)$ , not $n^2$ . We prove this as a finite-sample upper bound for classifiers on spatially correlated data, and show via a matching lower bound that the rate is tight, and no algorithm can do better. We extend the results to images with directional correlation and spatially varying correlation structure. Our result justifies spatial cross-validation since block holdout with separation proportional to the correlation range achieves optimal generalization guarantees, while random holdout can underestimate confidence interval widths by a factor proportional to $r$ . We validate the theory on synthetic data and satellite image tiles from three sensors (Landsat 8, Sentinel-2, and Sentinel-1).

Figures & tables

Fig. 1: Graphical illustration of the blocking construction in Thm. 1 .

Fig. 2: Simulation validation of the results. (A) Random holdout error (red) diverges from generalization error (green) as the correlation range $r$ increases. Shaded area is the optimism gap. (B) Empirical effective sample size (blue) scales log-log slope $=-1.78$ vs. theory $-2$ . (C) Validation optimism decays with distance from the training boundary, reaching zero at $d\approx r$ (vertical line). The gray curve shows $\propto e^{-d/r}$ . All experiments use $n{=}500$ , $d{=}6$ spectral bands, exponential covariance.

Region	Tile	Sensor	Scale	$\hat{r}$ (px)	$\hat{r}$ (m)	$n_{\mathrm{test}}$	$n_{\mathrm{eff}}$
Iowa	IA-L1	Landsat 8	30m	15.0	449	145684	1930
	IA-L2	Landsat 8	30m	25.0	751	139876	610
	IA-L3	Landsat 8	30m	13.0	390	147744	2687
	IA-L4	Landsat 8	30m	23.2	696	141620	1861
	IA-L5	Landsat 8	30m	5.6	168	149865	11915
	IA-L6	Landsat 8	30m	20.8	625	142002	2463

TABLE I: Summary of correlation ranges and effective sample sizes across all study tiles.

\hat{r}

: fitted exponential range from the NDVI (or VV) variogram.

n_{\mathrm{eff}}

: block-bootstrap effective sample size at block size

b=\hat{r}

Fig. 3: Block-bootstrap effective sample size as a function of block size $b$ for four tiles spanning different regions.

Tile	$\hat{r}_{1}$ (px)	$\hat{r}_{2}$ (px)	Ratio	$r_{1}r_{2}$	Waste
IA-S1a	17	12	1.4x	208	27%
IA-S1b	75	75	1.0x	5563	0%
CO-S1	60	10	5.9x	604	83%
CB-S1	9	73	8.2x	643	88%
AZ-S1	9	21	2.3x	185	57%
NE-S1	47	36	1.3x	1696	22%

TABLE II: Directional correlation ranges from Sentinel-1 SAR.

\hat{r}_{1}

\hat{r}_{2}

: N–S and E–W exponential ranges. Ratio:

\max/\min

of the two ranges. Waste: fraction of effective samples lost by using isotropic blocking with

r_{\max}

instead of the anisotropic

r_{1}r_{2}

Tile	$K$	$\bar{r}$ (px)	$r_{k}$ range	$n_{\mathrm{eff}}^{\mathrm{global}}$	$n_{\mathrm{eff}}^{\mathrm{strat}}$ / $n_{\mathrm{eff}}^{\mathrm{global}}$
IA-L1	6	14.0	3.2–17.6	1161	2.4x
CO-L1	5	17.9	4.6–32.1	730	4.3x
CO-L2	5	28.2	3.4–28.8	295	5.2x
CO-L3	6	17.1	11.3–60.3	802	0.4x
CO-S2	4	21.2	2.2–17.7	522	2.2x
CB-L1	5	50.3	6.6–32.4	96	3.5x

TABLE III: Stratified vs. global effective sample size for six heterogeneous tiles.

\bar{r}

: global variogram range.

r_{k}

range: min–max of per-stratum ranges (NLCD classes with

\geq 500

pixels).

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Fig. S1: Effective sample size versus correlation range for binary and four-class classification at $n=500$ .

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Beyond the Performance Illusion: Structure-Aware Stratified Partitioning and Curriculum Distributionally Robust Optimization for Spatially Correlated Domains

Prathamesh Patil, Arpit Jain, Aswanth Krishnan

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Partitioning Balanced Learning Spatio-Temporal Correlations Distributionally-Robust Optimization Artificial Intelligence Systems

Open paper AskPDF arXiv

Performance evaluation in AI systems commonly assumes that random dataset splits produce independent and identically distributed (i.i.d.) subsets. We show that this assumption often breaks down in spatiotemporally correlated domains such as aerial surveillance, precision agriculture, and medical imaging, leading to two systematic failures: data leakage, where correlated samples span training and validation splits and inflate performance estimates, and hidden stratification, where errors on minority subpopulations are obscured by aggregate metrics. To address these issues, we propose a unified evaluation and training framework for spatially correlated data. We introduce Structure-Aware Stratified Partitioning (SASP), which constructs validation splits that reduce spatiotemporal leakage while preserving meaningful class balance, and Curriculum Distributionally Robust Optimization (CDRO), a curriculum-based relaxation of distributionally robust training that stabilizes optimization under these stricter splits. Across multiple benchmarks, this combination yields consistently improved generalization, more reliable confidence calibration, and exposes failure modes that remain hidden under conventional random-split evaluation.

Abstract

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