The Binary Space Partitioning-Tree Process

The Mondrian process represents an elegant and powerful approach for space partition modelling. However, as it restricts the partitions to be axis-aligned, its modelling flexibility is limited. In this work, we propose a self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the Mondrian process. The BSP-Tree process is an almost surely right continuous Markov jump process that allows uniformly distributed oblique cuts in a two-dimensional convex polygon. The BSP-Tree process can also be extended using a non-uniform probability measure to generate direction differentiated cuts. The process is also self-consistent, maintaining distributional invariance under a restricted subdomain. We use Conditional-Sequential Monte Carlo for inference using the tree structure as the high-dimensional variable. The BSP-Tree process's performance on synthetic data partitioning and relational modelling demonstrates clear inferential improvements over the standard Mondrian process and other related methods

Analyzing in situ gene expression in the mouse brain with image registration, feature extraction and block clustering

Background: Many important high throughput projects use in situ hybridization and may require the analysis of images of spatial cross sections of organisms taken with cellular level resolution. Projects creating gene expression atlases at unprecedented scales for the embryonic fruit fly as well as the embryonic and adult mouse already involve the analysis of hundreds of thousands of high resolution experimental images mapping mRNA expression patterns. Challenges include accurate registration of highly deformed tissues, associating cells with known anatomical regions, and identifying groups of genes whose expression is coordinately regulated with respect to both concentration and spatial location. Solutions to these and other challenges will lead to a richer understanding of the complex system aspects of gene regulation in heterogeneous tissue. Results: We present an end-to-end approach for processing raw in situ expression imagery and performing subsequent analysis. We use a non-linear, information theoretic based image registration technique specifically adapted for mapping expression images to anatomical annotations and a method for extracting expression information within an anatomical region. Our method consists of coarse registration, fine registration, and expression feature extraction steps. From this we obtain a matrix for expression characteristics with rows corresponding to genes and columns corresponding to anatomical sub-structures. We perform matrix block cluster analysis using a novel row-column mixture model and we relate clustered patterns to Gene Ontology (GO) annotations. Conclusion: Resulting registrations suggest that our method is robust over intensity levels and shape variations in ISH imagery. Functional enrichment studies from both simple analysis and block clustering indicate that gene relationships consistent with biological knowledge of neuronal gene functions can be extracted from large ISH image databases such as the Allen Brain Atlas [1] and the Max-Planck Institute [2] using our method. While we focus here on imagery and experiments of the mouse brain our approach should be applicable to a variety of in situ experiments

Matrix tile analysis

Many tasks require finding groups of elements in a matrix of numbers, symbols or class likelihoods. One approach is to use efficient bi- or tri-linear factorization techniques including PCA, ICA, sparse matrix factorization and plaid analysis. These techniques are not appropriate when addition and multiplication of matrix elements are not sensibly defined. More directly, methods like biclustering can be used to classify matrix elements, but these methods make the overlyrestrictive assumption that the class of each element is a function of a row class and a column class. We introduce a general computational problem, ‘matrix tile analysis ’ (MTA), which consists of decomposing a matrix into a set of non-overlapping tiles, each of which is defined by a subset of usually nonadjacent rows and columns. MTA does not require an algebra for combining tiles, but must search over an exponential number of discrete combinations of tile assignments. We describe a loopy BP (sum-product) algorithm and an ICM algorithm for performing MTA. We compare the effectiveness of these methods to PCA and the plaid method on hundreds of randomly generated tasks. Using doublegene-knockout data, we show that MTA finds groups of interacting yeast genes that have biologically-related functions.