Bayesian Circle Segmentation with Application to DNA Copy Number Alteration Detection

Junfeng Liu,Harry Yang,Harner Harner

doi:10.6339/jds.2008.06(1).390

Abstract

Several statistical approaches have been proposed to consider circumstances under which one universal distribution is not capable of fit ting into the whole domain. This paper studies Bayesian detection of mul tiple interior epidemic/square waves in the interval domain, featured by two identical statistical distributions at both ends. We introduce a simple dimension-matching parameter proposal to implement the sampling-based posterior inference for special cases where each segmented distribution on a circle has the same set of regulating parameters. Molecular biology research reveals that, cancer progression may involve DNA copy number alteration at genome regions and connection of two biologically inactive chromosome ends results in a circle holding multiple epidemic/square waves. A slight modification of a simple novel Bayesian change point identification algo rithm, random grafting-pruning Markov chain Monte Carlo (RGPMCMC), is proposed by adjusting the original change point birth/death symmetric transition probability with a differ-by-one change point number ratio. The algorithm performance is studied through simulations with connection to DNA copy number alteration detection, which promises potential applica tion to cancer diagnosis at the genome level.

Highlights

Change point models usually incorporate either a single series of observations where change points are taken as “separations” of neighboring distinct segments described by individual statistical distributions, or across multiple serials of signals where change points are taken as physical locations in a continuous one-dimensional linear space (Liu et al, 2006)
Sampling-based approaches for change point detection in the literature are the reversible jump Markov chain Monte Carlo (RJMCMC) by Green (1995), the continuous time birth/death process MCMC by Stephens (2000), the product partition model based Bayesian algorithm by Loschi, Cruz and Arellano-Valle (2005) which stems from Yao (1984) and Barry and Hartigan (1992, 1993), non-MCMC based recursive sampling algorithm by Fearnhead (2005) and others
Sen and Srivastava (1975) tested whether the means of independent sequential random variables are the same or there is a shift after some point; Olshen et al (2004) developed a frequentist sequential circular binary segmentation (CBS) algorithm to detect multiple interior waves in a linear domain, which is essentially a circular domain by connecting two ends with identical statistical properties

Summary

Introduction

Change point models usually incorporate either a single series of observations where change points are taken as “separations” of neighboring distinct segments described by individual statistical distributions, or across multiple serials of signals where change points are taken as physical locations in a continuous one-dimensional linear space (Liu et al, 2006). Considering measure based probability and assuming segment Cj is randomly selected for change point (with index ∗) birth proposal, i.e., the new candidate segment starting location t∗ ∈(tj,tj+1)=Cj, and A is any Borel measurable set within segment Cj. The change point (segment) birth proposal probability given current Kold segments is the product of 1/Kold and A/(segment Cj length), where the former one is the probability of selecting segment Cj out of current Kold ones, the latter one is the probability of landing in set A conditional on segment Cj, i.e., Proposition 2 Under stochastic “+/−” move type in the parameter sampling process, birth/death proposal has transition probability only proportional to the ratio of two differ-by-one segment numbers. In view of detailed balance proposal, there is a positive probability that the chain lies in any small neighborhood after one sampling iteration to meet the aperiodicity; the chain can move from any value to any other value in steps of one at a time to establish the irreducibility

Simulation Study

Discussion