Bayesian Modeling of Spatial Molecular Profiling Data via Gaussian Process

Recent advances in spatially variable gene detection in spatial transcriptomics

With the emergence of advanced spatial transcriptomic technologies, there has been a surge in research papers dedicated to analyzing spatial transcriptomics data, resulting in significant contributions to our understanding of biology. The initial stage of downstream analysis of spatial transcriptomic data has centered on identifying spatially variable genes (SVGs) or genes expressed with specific spatial patterns across the tissue. SVG detection is an important task since many downstream analyses depend on these selected SVGs. Over the past few years, a plethora of new methods have been proposed for the detection of SVGs, accompanied by numerous innovative concepts and discussions. This article provides a selective review of methods and their practical implementations, offering valuable insights into the current literature in this field.

Single-cell omics: experimental workflow, data analyses and applications

Cells are the fundamental units of biological systems and exhibit unique development trajectories and molecular features. Our exploration of how the genomes orchestrate the formation and maintenance of each cell, and control the cellular phenotypes of various organismsis, is both captivating and intricate. Since the inception of the first single-cell RNA technology, technologies related to single-cell sequencing have experienced rapid advancements in recent years. These technologies have expanded horizontally to include single-cell genome, epigenome, proteome, and metabolome, while vertically, they have progressed to integrate multiple omics data and incorporate additional information such as spatial scRNA-seq and CRISPR screening. Single-cell omics represent a groundbreaking advancement in the biomedical field, offering profound insights into the understanding of complex diseases, including cancers. Here, we comprehensively summarize recent advances in single-cell omics technologies, with a specific focus on the methodology section. This overview aims to guide researchers in selecting appropriate methods for single-cell sequencing and related data analysis.

Categorization of 31 computational methods to detect spatially variable genes from spatially resolved transcriptomics data

In the analysis of spatially resolved transcriptomics data, detecting spatially variable genes (SVGs) is crucial. Numerous computational methods exist, but varying SVG definitions and methodologies lead to incomparable results. We review 31 state-of-the-art methods, categorizing SVGs into three types: overall, cell-type-specific, and spatial-domain-marker SVGs. Our review explains the intuitions underlying these methods, summarizes their applications, and categorizes the hypothesis tests they use in the trade-off between generality and specificity for SVG detection. We discuss challenges in SVG detection and propose future directions for improvement. Our review offers insights for method developers and users, advocating for category-specific benchmarking.

An interpretable Bayesian clustering approach with feature selection for analyzing spatially resolved transcriptomics data

Recent breakthroughs in spatially resolved transcriptomics (SRT) technologies have enabled comprehensive molecular characterization at the spot or cellular level while preserving spatial information. Cells are the fundamental building blocks of tissues, organized into distinct yet connected components. Although many non-spatial and spatial clustering approaches have been used to partition the entire region into mutually exclusive spatial domains based on the SRT high-dimensional molecular profile, most require an ad hoc selection of less interpretable dimensional-reduction techniques. To overcome this challenge, we propose a zero-inflated negative binomial mixture model to cluster spots or cells based on their molecular profiles. To increase interpretability, we employ a feature selection mechanism to provide a low-dimensional summary of the SRT molecular profile in terms of discriminating genes that shed light on the clustering result. We further incorporate the SRT geospatial profile via a Markov random field prior. We demonstrate how this joint modeling strategy improves clustering accuracy, compared with alternative state-of-the-art approaches, through simulation studies and 3 real data applications.

iIMPACT: integrating image and molecular profiles for spatial transcriptomics analysis

Current clustering analysis of spatial transcriptomics data primarily relies on molecular information and fails to fully exploit the morphological features present in histology images, leading to compromised accuracy and interpretability. To overcome these limitations, we have developed a multi-stage statistical method called iIMPACT. It identifies and defines histology-based spatial domains based on AI-reconstructed histology images and spatial context of gene expression measurements, and detects domain-specific differentially expressed genes. Through multiple case studies, we demonstrate iIMPACT outperforms existing methods in accuracy and interpretability and provides insights into the cellular spatial organization and landscape of functional genes within spatial transcriptomics data.

Differential gene expression analysis of spatial transcriptomic experiments using spatial mixed models

Spatial transcriptomics (ST) assays represent a revolution in how the architecture of tissues is studied by allowing for the exploration of cells in their spatial context. A common element in the analysis is delineating tissue domains or "niches" followed by detecting differentially expressed genes to infer the biological identity of the tissue domains or cell types. However, many studies approach differential expression analysis by using statistical approaches often applied in the analysis of non-spatial scRNA data (e.g., two-sample t-tests, Wilcoxon's rank sum test), hence neglecting the spatial dependency observed in ST data. In this study, we show that applying linear mixed models with spatial correlation structures using spatial random effects effectively accounts for the spatial autocorrelation and reduces inflation of type-I error rate observed in non-spatial based differential expression testing. We also show that spatial linear models with an exponential correlation structure provide a better fit to the ST data as compared to non-spatial models, particularly for spatially resolved technologies that quantify expression at finer scales (i.e., single-cell resolution).

Joint Bayesian estimation of cell dependence and gene associations in spatially resolved transcriptomic data

Recent technologies such as spatial transcriptomics, enable the measurement of gene expressions at the single-cell level along with the spatial locations of these cells in the tissue. Spatial clustering of the cells provides valuable insights into the understanding of the functional organization of the tissue. However, most such clustering methods involve some dimension reduction that leads to a loss of the inherent dependency structure among genes at any spatial location in the tissue. This destroys valuable insights of gene co-expression patterns apart from possibly impacting spatial clustering performance. In spatial transcriptomics, the matrix-variate gene expression data, along with spatial coordinates of the single cells, provides information on both gene expression dependencies and cell spatial dependencies through its row and column covariances. In this work, we propose a joint Bayesian approach to simultaneously estimate these gene and spatial cell correlations. These estimates provide data summaries for downstream analyses. We illustrate our method with simulations and analysis of several real spatial transcriptomic datasets. Our work elucidates gene co-expression networks as well as clear spatial clustering patterns of the cells. Furthermore, our analysis reveals that downstream spatial-differential analysis may aid in the discovery of unknown cell types from known marker genes.

Bayesian hidden mark interaction model for detecting spatially variable genes in imaging-based spatially resolved transcriptomics data

Recent technology breakthroughs in spatially resolved transcriptomics (SRT) have enabled the comprehensive molecular characterization of cells whilst preserving their spatial and gene expression contexts. One of the fundamental questions in analyzing SRT data is the identification of spatially variable genes whose expressions display spatially correlated patterns. Existing approaches are built upon either the Gaussian process-based model, which relies on ad hoc kernels, or the energy-based Ising model, which requires gene expression to be measured on a lattice grid. To overcome these potential limitations, we developed a generalized energy-based framework to model gene expression measured from imaging-based SRT platforms, accommodating the irregular spatial distribution of measured cells. Our Bayesian model applies a zero-inflated negative binomial mixture model to dichotomize the raw count data, reducing noise. Additionally, we incorporate a geostatistical mark interaction model with a generalized energy function, where the interaction parameter is used to identify the spatial pattern. Auxiliary variable MCMC algorithms were employed to sample from the posterior distribution with an intractable normalizing constant. We demonstrated the strength of our method on both simulated and real data. Our simulation study showed that our method captured various spatial patterns with high accuracy; moreover, analysis of a seqFISH dataset and a STARmap dataset established that our proposed method is able to identify genes with novel and strong spatial patterns.

Recent advances in differential expression analysis for single-cell RNA-seq and spatially resolved transcriptomic studies

Differential expression (DE) analysis is a necessary step in the analysis of single-cell RNA sequencing (scRNA-seq) and spatially resolved transcriptomics (SRT) data. Unlike traditional bulk RNA-seq, DE analysis for scRNA-seq or SRT data has unique characteristics that may contribute to the difficulty of detecting DE genes. However, the plethora of DE tools that work with various assumptions makes it difficult to choose an appropriate one. Furthermore, a comprehensive review on detecting DE genes for scRNA-seq data or SRT data from multi-condition, multi-sample experimental designs is lacking. To bridge such a gap, here, we first focus on the challenges of DE detection, then highlight potential opportunities that facilitate further progress in scRNA-seq or SRT analysis, and finally provide insights and guidance in selecting appropriate DE tools or developing new computational DE methods.

Navigating the landscapes of spatial transcriptomics: How computational methods guide the way

Spatially resolved transcriptomics has been dramatically transforming biological and medical research in various fields. It enables transcriptome profiling at single-cell, multi-cellular, or sub-cellular resolution, while retaining the information of geometric localizations of cells in complex tissues. The coupling of cell spatial information and its molecular characteristics generates a novel multi-modal high-throughput data source, which poses new challenges for the development of analytical methods for data-mining. Spatial transcriptomic data are often highly complex, noisy, and biased, presenting a series of difficulties, many unresolved, for data analysis and generation of biological insights. In addition, to keep pace with the ever-evolving spatial transcriptomic experimental technologies, the existing analytical theories and tools need to be updated and reformed accordingly. In this review, we provide an overview and discussion of the current computational approaches for mining of spatial transcriptomics data. Future directions and perspectives of methodology design are proposed to stimulate further discussions and advances in new analytical models and algorithms. This article is categorized under: RNA Methods > RNA Analyses in Cells RNA Evolution and Genomics > Computational Analyses of RNA RNA Export and Localization > RNA Localization.

Niche-DE: niche-differential gene expression analysis in spatial transcriptomics data identifies context-dependent cell-cell interactions

Existing methods for analysis of spatial transcriptomic data focus on delineating the global gene expression variations of cell types across the tissue, rather than local gene expression changes driven by cell-cell interactions. We propose a new statistical procedure called niche-differential expression (niche-DE) analysis that identifies cell-type-specific niche-associated genes, which are differentially expressed within a specific cell type in the context of specific spatial niches. We further develop niche-LR, a method to reveal ligand-receptor signaling mechanisms that underlie niche-differential gene expression patterns. Niche-DE and niche-LR are applicable to low-resolution spot-based spatial transcriptomics data and data that is single-cell or subcellular in resolution.

Bayesian Hidden Mark Interaction Model for Detecting Spatially Variable Genes in Imaging-Based Spatially Resolved Transcriptomics Data

Recent technology breakthroughs in spatially resolved transcriptomics (SRT) have enabled the comprehensive molecular characterization of cells whilst preserving their spatial and gene expression contexts. One of the fundamental questions in analyzing SRT data is the identification of spatially variable genes whose expressions display spatially correlated patterns. Existing approaches are built upon either the Gaussian process-based model, which relies on ad hoc kernels, or the energy-based Ising model, which requires gene expression to be measured on a lattice grid. To overcome these potential limitations, we developed a generalized energy-based framework to model gene expression measured from imaging-based SRT platforms, accommodating the irregular spatial distribution of measured cells. Our Bayesian model applies a zero-inflated negative binomial mixture model to dichotomize the raw count data, reducing noise. Additionally, we incorporate a geostatistical mark interaction model with a generalized energy function, where the interaction parameter is used to identify the spatial pattern. Auxiliary variable MCMC algorithms were employed to sample from the posterior distribution with an intractable normalizing constant. We demonstrated the strength of our method on both simulated and real data. Our simulation study showed that our method captured various spatial patterns with high accuracy; moreover, analysis of a seqFISH dataset and a STARmap dataset established that our proposed method is able to identify genes with novel and strong spatial patterns.

Reconstructing Spatial Transcriptomics at the Single-cell Resolution with BayesDeep

Spatially resolved transcriptomics (SRT) techniques have revolutionized the characterization of molecular profiles while preserving spatial and morphological context. However, most next-generation sequencing-based SRT techniques are limited to measuring gene expression in a confined array of spots, capturing only a fraction of the spatial domain. Typically, these spots encompass gene expression from a few to hundreds of cells, underscoring a critical need for more detailed, single-cell resolution SRT data to enhance our understanding of biological functions within the tissue context. Addressing this challenge, we introduce BayesDeep, a novel Bayesian hierarchical model that leverages cellular morphological data from histology images, commonly paired with SRT data, to reconstruct SRT data at the single-cell resolution. BayesDeep effectively model count data from SRT studies via a negative binomial regression model. This model incorporates explanatory variables such as cell types and nuclei-shape information for each cell extracted from the paired histology image. A feature selection scheme is integrated to examine the association between the morphological and molecular profiles, thereby improving the model robustness. We applied BayesDeep to two real SRT datasets, successfully demonstrating its capability to reconstruct SRT data at the single-cell resolution. This advancement not only yields new biological insights but also significantly enhances various downstream analyses, such as pseudotime and cell-cell communication.

Benchmarking computational methods to identify spatially variable genes and peaks

Spatially resolved transcriptomics offers unprecedented insight by enabling the profiling of gene expression within the intact spatial context of cells, effectively adding a new and essential dimension to data interpretation. To efficiently detect spatial structure of interest, an essential step in analyzing such data involves identifying spatially variable genes. Despite researchers having developed several computational methods to accomplish this task, the lack of a comprehensive benchmark evaluating their performance remains a considerable gap in the field. Here, we present a systematic evaluation of 14 methods using 60 simulated datasets generated by four different simulation strategies, 12 real-world transcriptomics, and three spatial ATAC-seq datasets. We find that spatialDE2 consistently outperforms the other benchmarked methods, and Moran's I achieves competitive performance in different experimental settings. Moreover, our results reveal that more specialized algorithms are needed to identify spatially variable peaks.

A SELECTIVE REVIEW OF RECENT DEVELOPMENTS IN SPATIALLY VARIABLE GENE DETECTION FOR SPATIAL TRANSCRIPTOMICS

With the emergence of advanced spatial transcriptomic technologies, there has been a surge in research papers dedicated to analyzing spatial transcriptomics data, resulting in significant contributions to our understanding of biology. The initial stage of downstream analysis of spatial transcriptomic data has centered on identifying spatially variable genes (SVGs) or genes expressed with specific spatial patterns across the tissue. SVG detection is an important task since many downstream analyses depend on these selected SVGs. Over the past few years, a plethora of new methods have been proposed for the detection of SVGs, accompanied by numerous innovative concepts and discussions. This article provides a selective review of methods and their practical implementations, offering valuable insights into the current literature in this field.

Spatial transcriptomics in development and disease

The proper functioning of diverse biological systems depends on the spatial organization of their cells, a critical factor for biological processes like shaping intricate tissue functions and precisely determining cell fate. Nonetheless, conventional bulk or single-cell RNA sequencing methods were incapable of simultaneously capturing both gene expression profiles and the spatial locations of cells. Hence, a multitude of spatially resolved technologies have emerged, offering a novel dimension for investigating regional gene expression, spatial domains, and interactions between cells. Spatial transcriptomics (ST) is a method that maps gene expression in tissue while preserving spatial information. It can reveal cellular heterogeneity, spatial organization and functional interactions in complex biological systems. ST can also complement and integrate with other omics methods to provide a more comprehensive and holistic view of biological systems at multiple levels of resolution. Since the advent of ST, new methods offering higher throughput and resolution have become available, holding significant potential to expedite fresh insights into comprehending biological complexity. Consequently, a rapid increase in associated research has occurred, using these technologies to unravel the spatial complexity during developmental processes or disease conditions. In this review, we summarize the recent advancement of ST in historical, technical, and application contexts. We compare different types of ST methods based on their principles and workflows, and present the bioinformatics tools for analyzing and integrating ST data with other modalities. We also highlight the applications of ST in various domains of biomedical research, especially development and diseases. Finally, we discuss the current limitations and challenges in the field, and propose the future directions of ST.

Harnessing computational spatial omics to explore the spatial biology intricacies

Spatially resolved transcriptomics (SRT) has unlocked new dimensions in our understanding of intricate tissue architectures. However, this rapidly expanding field produces a wealth of diverse and voluminous data, necessitating the evolution of sophisticated computational strategies to unravel inherent patterns. Two distinct methodologies, gene spatial pattern recognition (GSPR) and tissue spatial pattern recognition (TSPR), have emerged as vital tools in this process. GSPR methodologies are designed to identify and classify genes exhibiting noteworthy spatial patterns, while TSPR strategies aim to understand intercellular interactions and recognize tissue domains with molecular and spatial coherence. In this review, we provide a comprehensive exploration of SRT, highlighting crucial data modalities and resources that are instrumental for the development of methods and biological insights. We address the complexities and challenges posed by the use of heterogeneous data in developing GSPR and TSPR methodologies and propose an optimal workflow for both. We delve into the latest advancements in GSPR and TSPR, examining their interrelationships. Lastly, we peer into the future, envisaging the potential directions and perspectives in this dynamic field.

SMASH: Scalable Method for Analyzing Spatial Heterogeneity of genes in spatial transcriptomics data

In high-throughput spatial transcriptomics (ST) studies, it is of great interest to identify the genes whose level of expression in a tissue covaries with the spatial location of cells/spots. Such genes, also known as spatially variable genes (SVGs), can be crucial to the biological understanding of both structural and functional characteristics of complex tissues. Existing methods for detecting SVGs either suffer from huge computational demand or significantly lack statistical power. We propose a non-parametric method termed SMASH that achieves a balance between the above two problems. We compare SMASH with other existing methods in varying simulation scenarios demonstrating its superior statistical power and robustness. We apply the method to four ST datasets from different platforms uncovering interesting biological insights.

nnSVG for the scalable identification of spatially variable genes using nearest-neighbor Gaussian processes

Feature selection to identify spatially variable genes or other biologically informative genes is a key step during analyses of spatially-resolved transcriptomics data. Here, we propose nnSVG, a scalable approach to identify spatially variable genes based on nearest-neighbor Gaussian processes. Our method (i) identifies genes that vary in expression continuously across the entire tissue or within a priori defined spatial domains, (ii) uses gene-specific estimates of length scale parameters within the Gaussian process models, and (iii) scales linearly with the number of spatial locations. We demonstrate the performance of our method using experimental data from several technological platforms and simulations. A software implementation is available at https://bioconductor.org/packages/nnSVG .

Spatial Transcriptomics Arena (STAr): an Integrated Platform for Spatial Transcriptomics Methodology Research

The emerging field of spatially resolved transcriptomics (SRT) has revolutionized biomedical research. SRT quantifies expression levels at different spatial locations, providing a new and powerful tool to interrogate novel biological insights. An essential question in the analysis of SRT data is to identify spatially variable (SV) genes; the expression levels of such genes have spatial variation across different tissues. SV genes usually play an important role in underlying biological mechanisms and tissue heterogeneity. Currently, several computational methods have been developed to detect such genes; however, there is a lack of unbiased assessment of these approaches to guide researchers in selecting the appropriate methods for their specific biomedical applications. In addition, it is difficult for researchers to implement different existing methods for either biological study or methodology development. Furthermore, currently available public SRT datasets are scattered across different websites and preprocessed in different ways, posing additional obstacles for quantitative researchers developing computational methods for SRT data analysis. To address these challenges, we designed Spatial Transcriptomics Arena (STAr), an open platform comprising 193 curated datasets from seven technologies, seven statistical methods, and analysis results. This resource allows users to retrieve high-quality datasets, apply or develop spatial gene detection methods, as well as browse and compare spatial gene analysis results. It also enables researchers to comprehensively evaluate SRT methodology research in both simulated and real datasets. Altogether, STAr is an integrated research resource intended to promote reproducible research and accelerate rigorous methodology development, which can eventually lead to an improved understanding of biological processes and diseases. STAr can be accessed at https://lce.biohpc.swmed.edu/star/ .

SRTsim: spatial pattern preserving simulations for spatially resolved transcriptomics

Spatially resolved transcriptomics (SRT)-specific computational methods are often developed, tested, validated, and evaluated in silico using simulated data. Unfortunately, existing simulated SRT data are often poorly documented, hard to reproduce, or unrealistic. Single-cell simulators are not directly applicable for SRT simulation as they cannot incorporate spatial information. We present SRTsim, an SRT-specific simulator for scalable, reproducible, and realistic SRT simulations. SRTsim not only maintains various expression characteristics of SRT data but also preserves spatial patterns. We illustrate the benefits of SRTsim in benchmarking methods for spatial clustering, spatial expression pattern detection, and cell-cell communication identification.

Statistical Power Analysis for Designing Bulk, Single-Cell, and Spatial Transcriptomics Experiments: Review, Tutorial, and Perspectives

Gene expression profiling technologies have been used in various applications such as cancer biology. The development of gene expression profiling has expanded the scope of target discovery in transcriptomic studies, and each technology produces data with distinct characteristics. In order to guarantee biologically meaningful findings using transcriptomic experiments, it is important to consider various experimental factors in a systematic way through statistical power analysis. In this paper, we review and discuss the power analysis for three types of gene expression profiling technologies from a practical standpoint, including bulk RNA-seq, single-cell RNA-seq, and high-throughput spatial transcriptomics. Specifically, we describe the existing power analysis tools for each research objective for each of the bulk RNA-seq and scRNA-seq experiments, along with recommendations. On the other hand, since there are no power analysis tools for high-throughput spatial transcriptomics at this point, we instead investigate the factors that can influence power analysis.

A guidebook of spatial transcriptomic technologies, data resources and analysis approaches

Advances in transcriptomic technologies have deepened our understanding of the cellular gene expression programs of multicellular organisms and provided a theoretical basis for disease diagnosis and therapy. However, both bulk and single-cell RNA sequencing approaches lose the spatial context of cells within the tissue microenvironment, and the development of spatial transcriptomics has made overall bias-free access to both transcriptional information and spatial information possible. Here, we elaborate development of spatial transcriptomic technologies to help researchers select the best-suited technology for their goals and integrate the vast amounts of data to facilitate data accessibility and availability. Then, we marshal various computational approaches to analyze spatial transcriptomic data for various purposes and describe the spatial multimodal omics and its potential for application in tumor tissue. Finally, we provide a detailed discussion and outlook of the spatial transcriptomic technologies, data resources and analysis approaches to guide current and future research on spatial transcriptomics.

A Primer on Preprocessing, Visualization, Clustering, and Phenotyping of Barcode-Based Spatial Transcriptomics Data

Recent developments in spatially resolved transcriptomics (ST) have resulted in a large number of studies characterizing the architecture of tissues, the spatial distribution of cell types, and their interactions. Furthermore, ST promises to enable the discovery of more accurate drug targets while also providing a better understanding of the etiology and evolution of complex diseases. The analysis of ST brings similar challenges as seen in other gene expression assays such as scRNA-seq; however, there is the additional spatial information that warrants the development of suitable algorithms for the quality control, preprocessing, visualization, and other discovery-enabling approaches (e.g., clustering, cell phenotyping). In this chapter, we review some of the existing algorithms to perform these analytical tasks and highlight some of the unmet analytical challenges in the analysis of ST data. Given the diversity of available ST technologies, we focus this chapter on the analysis of barcode-based RNA quantitation techniques.

SpatialCorr identifies gene sets with spatially varying correlation structure

Recent advances in spatially resolved transcriptomics technologies enable both the measurement of genome-wide gene expression profiles and their mapping to spatial locations within a tissue. A first step in spatial transcriptomics data analysis is identifying genes with expression that varies spatially, and robust statistical methods exist to address this challenge. While useful, these methods do not detect spatial changes in the coordinated expression within a group of genes. To this end, we present SpatialCorr, a method for identifying sets of genes with spatially varying correlation structure. Given a collection of gene sets pre-defined by a user, SpatialCorr tests for spatially induced differences in the correlation of each gene set within tissue regions, as well as between and among regions. An application to cutaneous squamous cell carcinoma demonstrates the power of the approach for revealing biological insights not identified using existing methods.

Belayer: Modeling discrete and continuous spatial variation in gene expression from spatially resolved transcriptomics

Spatially resolved transcriptomics (SRT) technologies measure gene expression at known locations in a tissue slice, enabling the identification of spatially varying genes or cell types. Current approaches for these tasks assume either that gene expression varies continuously across a tissue or that a tissue contains a small number of regions with distinct cellular composition. We propose a model for SRT data from layered tissues that includes both continuous and discrete spatial variation in expression and an algorithm, Belayer, to learn the parameters of this model. Belayer models gene expression as a piecewise linear function of the relative depth of a tissue layer with possible discontinuities at layer boundaries. We use conformal maps to model relative depth and derive a dynamic programming algorithm to infer layer boundaries and gene expression functions. Belayer accurately identifies tissue layers and biologically meaningful spatially varying genes in SRT data from the brain and skin.

Spatial components of molecular tissue biology

Methods for profiling RNA and protein expression in a spatially resolved manner are rapidly evolving, making it possible to comprehensively characterize cells and tissues in health and disease. To maximize the biological insights obtained using these techniques, it is critical to both clearly articulate the key biological questions in spatial analysis of tissues and develop the requisite computational tools to address them. Developers of analytical tools need to decide on the intrinsic molecular features of each cell that need to be considered, and how cell shape and morphological features are incorporated into the analysis. Also, optimal ways to compare different tissue samples at various length scales are still being sought. Grouping these biological problems and related computational algorithms into classes across length scales, thus characterizing common issues that need to be addressed, will facilitate further progress in spatial transcriptomics and proteomics.

Spatial Statistical Models: An Overview under the Bayesian Approach

Spatial documentation is exponentially increasing given the availability of Big Data in the Internet of Things, enabled by device miniaturization and data storage capacity. Bayesian spatial statistics is a useful statistical tool to determine the dependence structure and hidden patterns in space through prior knowledge and data likelihood. However, this class of modeling is not yet well explored when compared to adopting classification and regression in machine-learning models, in which the assumption of the spatiotemporal independence of the data is often made, that is an inexistent or very weak dependence. Thus, this systematic review aims to address the main models presented in the literature over the past 20 years, identifying the gaps and research opportunities. Elements such as random fields, spatial domains, prior specification, the covariance function, and numerical approximations are discussed. This work explores the two subclasses of spatial smoothing: global and local.