# Discovery of spatial patterns with Hidden Markov random field

Use HMRF to dissect spatial patterns in smFISH data

## Introduction

`smfishHmrf`

package provides tools for spatial pattern discovery using multivariate Gaussian models and hidden Markov random field fitted by expectation-maximization. We focus on seqFISH data.

single-molecule (sm)FISH is used to spatially profile cells at single-cell resolution. seqFISH, a type of smFISH utilizing sequential multiplexed barcoding schemes, can profile hundreds of genes’ transcript count for individual cells with high sensitivity.

Using gene expression from seqFISH, we can infer the spatial pattern that might exist in the spatial profile. This is analogous to image segmentation in computer vision. In the single-cell imaging, spatial pattern looks like a contiguous domain of cells, sharing expression of genes, and may suggest something about the cell’s local environment. This spatial state of the cell is unknown and needs to be identified from seqFISH data.

As the spatial states are assumed to be independent, we can use a mixture of multivariate Gaussian distribution to model the observed expression with parameters estimated by the EM algorithm. Since nearby cells tend to be of the same state, a **Markov random field model** (in this case **Pott’s model**) can be used to capture the spatial similarity of cells by making homogeneous the relationship between neighboring cells.

Typical package (such as ref1) runs HMRF on magnetic resonance imaging (MRI) data and is restricted to 1-channel (grayscale) and fixed to 4-neighbor regular grid of pixels. In our case, the package `smfishHmrf`

is more general:

It extends simple 1D Gaussian to multivariate Gaussian with multidimensional mean and covariance matrices estimated by EM.

As the cells do not fall on a regular grid, we adapt to a local neighborhood graph instead of a grid. This allows for a more flexible specification of number of neighbors and neighbor structure.

To illustrate the method, we have tested it on mouse brain visual cortex seqFISH imaged cells.