Spatial Hidden Markov Model

The model assumes each spatial location (spot or cell) belongs to an unobserved discrete state representing a functional tissue domain. The observed gene expression vector at a location is treated as a multivariate emission conditioned on this hidden state. The goal is to infer the most probable sequence of hidden states across the entire tissue by balancing the likelihood of the observed expression data with the spatial coherence of the state assignments.

Unlike temporal HMMs, spatial HMMs enforce state consistency through a Potts model or Markov Random Field (MRF) prior on the hidden states. This prior penalizes neighboring locations that are assigned to different states, encouraging spatially smooth domain boundaries. The strength of this penalty is controlled by an interaction parameter that determines how much weight is given to spatial coherence versus the gene expression likelihood.

Parameter estimation is typically performed via the Expectation-Maximization (EM) algorithm. In the E-step, the model computes the posterior probability of each hidden state at every location given the current parameter estimates. In the M-step, the emission distribution parameters and spatial interaction parameters are updated to maximize the expected complete-data log-likelihood. This iterative process converges to a local optimum that defines the spatial domains.

The gene expression data for each hidden state is commonly modeled as a multivariate Gaussian distribution. This captures the mean expression profile and the gene-gene covariance structure within each domain. For count-based data, alternatives include Poisson or Negative Binomial emissions. The choice of emission distribution directly impacts the model's ability to accurately represent the technical noise characteristics of the spatial transcriptomics platform.

The primary application is unsupervised tissue segmentation. By applying a spatial HMM to a spatial transcriptomics dataset, the inferred hidden states directly correspond to anatomically meaningful regions. This reveals functional tissue architecture without requiring prior histological annotations. The resulting domains can be visualized as a spatial map of state assignments, highlighting cortical layers in the brain, tumor microenvironments, or developmental zones.

Exact inference in spatial HMMs is computationally intractable due to the complex dependencies in the MRF prior. Practical implementations rely on variational Bayes or mean-field approximations to make inference scalable to datasets with tens of thousands of spatial locations. These methods decouple the dependencies by assuming a factorized posterior distribution, enabling efficient iterative updates that scale linearly with the number of spatial locations.

The unsupervised identification of tissue regions with coherent gene expression profiles and histology. While Spatial HMMs infer domains through latent state transitions, other approaches use graph-based clustering or non-negative matrix factorization. The goal is partitioning a tissue into anatomically meaningful zones—such as cortical layers in the brain or tumor microenvironments—without prior annotation.

A statistical measure of the degree to which a variable's values at nearby locations are more similar than expected by random chance. This is the foundational assumption behind Spatial HMMs: that neighboring spots share hidden states. Key metrics include:

• Moran's I: Quantifies overall clustering of expression patterns
• Geary's C: More sensitive to local differences
• Spatial HMMs explicitly model this dependency through transition probabilities between adjacent locations

A data structure where each spatial location is a node, and edges connect neighboring locations based on a distance threshold or k-nearest neighbors. This graph serves as the structural backbone for Spatial HMMs, defining which locations can influence each other's hidden state transitions. Construction methods include:

• Delaunay triangulation on cell centroids
• Fixed-radius neighbor graphs for regular grids
• k-NN graphs with k typically set to 6-8 for 2D tissues

A computational process that estimates the proportions of different cell types within each spatial transcriptomics spot by separating the mixed gene expression signal. While Spatial HMMs identify tissue domains, deconvolution reveals the cellular composition within those domains. Common methods include:

• Reference-based: Uses single-cell RNA-seq signatures to estimate proportions
• Reference-free: Blind source separation via NMF or Bayesian models
• Integration with HMM-derived domains enables spatial niche analysis

A computational method that orders cells based on their spatial coordinates and gene expression profiles to reconstruct dynamic biological processes in situ. Unlike Spatial HMMs which partition space into discrete domains, trajectory inference models continuous gradients of change. Applications include:

• Mapping developmental lineages within embryos
• Reconstructing tumor invasion fronts
• Identifying differentiation gradients in adult tissues
• Often combined with RNA velocity for directional information

The computational alignment of multiple tissue images or spatial datasets into a common coordinate system to enable integrative cross-modality analysis. Spatial HMMs applied to registered data can identify domains conserved across:

• Serial tissue sections from the same sample
• Multi-modal data (e.g., transcriptomics + proteomics)
• Biological replicates for robust domain calling
• Atlas-level integration against reference tissue maps

Key techniques include landmark-based, intensity-based, and deep learning approaches.