A statistical method designed to use location as part of the analysis has picked out distinct patterns in human brain tissue and African vegetation records. The approach combines probabilistic principal component analysis with a spatial prior, a statistical expectation about how locations relate, and is designed for datasets with many spatial locations.
The preprint reports tests on synthetic simulations, one postmortem human dorsolateral prefrontal cortex section and GIMMS NDVI observations for sub-Saharan Africa. The brain and Africa results are exploratory model comparisons; they do not establish biological or geographic causes.
A map-aware version of a familiar tool
Probabilistic principal component analysis reduces a large set of measurements to a smaller set of underlying factors. Here, the model puts spatial dependence directly on orthogonal loading directions. Its maximum a posteriori, or MAP, rule is derived from an eigendecomposition of the empirical covariance combined with a spatial covariance prior scaled by sample size.
To handle large maps, the paper uses a minorization-maximization step inside an expectation-maximization algorithm, shortened to MM-EM. The estimator is designed to have computational complexity linear in the number of spatial locations. A separate validation procedure randomly holds out locations and uses their predictive loss to choose the spatial prior covariance parameterization.
The model can also use spatial dependence that changes across a map and differs by direction, rather than assuming one relationship everywhere.
Synthetic tests focused on scale
The synthetic study tested stationary, smoothly nonstationary and circular-shape covariance settings. It used domains of 400 or 2,500 locations; sample sizes of 200, 500, 1,000, 5,000 or 10,000 observations; two noise-variance settings; three factors; and 20 replicates per setting.
The evaluation assessed errors in the model’s estimated loadings, latent structure and noise variance. In a separate single-core timing study, each dataset had 3,000 samples or features and three factors. Computation time increased linearly with the number of spatial locations, a result limited to the tested simulation configuration.
Brain-tissue maps separated layers and white matter
One postmortem human dorsolateral prefrontal cortex section, identified as section 151675, supplied 3,592 in-tissue Visium spots and 33,538 gene features. The expression matrix was normalized, log-transformed and mean-centered for each gene.
The spatial orthogonal factor model, or SOFM, used seven factors and randomly held out 400 spots. PCA and Laminae-PCA used seven components. In filtered maps, the first SOFM factor was strongest in white matter, the second was associated with layers 1 and 6, and the third with much of layer 5. Additional factors showed contrasts between layers.
In the filtered analysis, SOFM Factor 7 appeared non-laminar, and HBB had its largest negative coefficient. In a separate analysis of 3,000 highly variable genes, Factor 6 had HBB as its largest positive coefficient and IGKC as its largest negative coefficient. Factor 7 ranked HBB and IGKC as the top two negative-coefficient genes.
Estimated spatial length scales varied across the tissue and appeared strongest in the first cortical layer. The first three SOFM factors had larger per-factor contributions to empirical covariance than the corresponding Laminae-PCA factors. The map and gene interpretations were exploratory, and coefficient sign alone does not show enrichment or depletion.
A second test used vegetation records across Africa
The environmental application used GIMMS NDVI vegetation observations for sub-Saharan conterminous Africa. The short analysis covered one year, 239,318 spatial locations and 24 temporal observations. Its comparison used PCA on a full 40-year series with 984 temporal observations.
Estimated dependence was stronger in the Sahel and sub-humid tropics in both directions, and weaker across the transition zones between the Sahara and Sahel and between the Sahel and humid tropics. The first component of both models captured about 50 percent of total variance and represented opposing seasonal dynamics between the northern and southern hemispheres. Later components explained much less variance.
Compared with one-year PCA and checked against 40-year PCA, the single-year SOFM identified subregions near South Sudan and Mozambique that the one-year PCA missed. SOFM captured those areas in its third factor. The comparison does not establish that the pattern is a ground-truth climate feature or that spatial regularization caused its recovery.
What the analysis leaves open
The brain result comes from one tissue section, so generalizability across sections and donors was not established. The DLPFC factor and gene interpretations remained exploratory, with formal gene-level inference left for future work.
The theorem characterizes the MAP loading directions, while MAP results for the other model parameters, including L and the noise variance, were left for future research.
The Africa analysis used a single-year spatial model with a 40-year PCA benchmark rather than an external ground-truth outcome. Its recovered regions therefore remain a comparison between model outputs, not proof of definitive climate boundaries or causal effects.
A preprint with code available
The manuscript is arXiv preprint version 1, dated 25 August 2026. The paper reports publicly accessible SOFM code and a pip installation command.
Paper data and sources
Original title: Spatially orthogonal factor models for spatial transcriptomics and remote sensing data
Authors: Dan Cunha, Lukas M. Weber, Mark A. Friedl, Luis Carvalho
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text