Theoretical Foundations of Equitability and the Maximal Information Coefficient

Theoretical Foundations of Equitability and the Maximal Information Coefficient

Yakir A. Reshef, David N. Reshef, Pardis C. Sabeti, Michael Mitzenmacher
(Submitted on 21 Aug 2014)

The maximal information coefficient (MIC) is a tool for finding the strongest pairwise relationships in a data set with many variables (Reshef et al., 2011). MIC is useful because it gives similar scores to equally noisy relationships of different types. This property, called {\em equitability}, is important for analyzing high-dimensional data sets.
Here we formalize the theory behind both equitability and MIC in the language of estimation theory. This formalization has a number of advantages. First, it allows us to show that equitability is a generalization of power against statistical independence. Second, it allows us to compute and discuss the population value of MIC, which we call MIC_*. In doing so we generalize and strengthen the mathematical results proven in Reshef et al. (2011) and clarify the relationship between MIC and mutual information. Introducing MIC_* also enables us to reason about the properties of MIC more abstractly: for instance, we show that MIC_* is continuous and that there is a sense in which it is a canonical “smoothing” of mutual information. We also prove an alternate, equivalent characterization of MIC_* that we use to state new estimators of it as well as an algorithm for explicitly computing it when the joint probability density function of a pair of random variables is known. Our hope is that this paper provides a richer theoretical foundation for MIC and equitability going forward.
This paper will be accompanied by a forthcoming companion paper that performs extensive empirical analysis and comparison to other methods and discusses the practical aspects of both equitability and the use of MIC and its related statistics.


RNA-Seq Mapping Errors When Using Incomplete Reference Transcriptomes of Vertebrates

RNA-Seq Mapping Errors When Using Incomplete Reference Transcriptomes of Vertebrates
Alexis Black Pyrkosz, Hans Cheng, C. Titus Brown
(Submitted on 11 Mar 2013)

Whole transcriptome sequencing is increasingly being used as a functional genomics tool to study non- model organisms. However, when the reference transcriptome used to calculate differential expression is incomplete, significant error in the inferred expression levels can result. In this study, we use simulated reads generated from real transcriptomes to determine the accuracy of read mapping, and measure the error resulting from using an incomplete transcriptome. We show that the two primary sources of count- ing error are 1) alternative splice variants that share reads and 2) missing transcripts from the reference. Alternative splice variants increase the false positive rate of mapping while incomplete reference tran- scriptomes decrease the true positive rate, leading to inaccurate transcript expression levels. Grouping transcripts by gene or read sharing (similar to mapping to a reference genome) significantly decreases false positives, but only by improving the reference transcriptome itself can the missing transcript problem be addressed. We also demonstrate that employing different mapping software does not yield substantial increases in accuracy on simulated data. Finally, we show that read lengths or insert sizes must increase past 1kb to resolve mapping ambiguity.

Equitability, mutual information, and the maximal information coefficient

Equitability, mutual information, and the maximal information coefficient
Justin B. Kinney, Gurinder S. Atwal
(Submitted on 31 Jan 2013)

Reshef et al. recently proposed a new statistical measure, the “maximal information coefficient” (MIC), for quantifying arbitrary dependencies between pairs of stochastic quantities. MIC is based on mutual information, a fundamental quantity in information theory that is widely understood to serve this need. MIC, however, is not an estimate of mutual information. Indeed, it was claimed that MIC possesses a desirable mathematical property called “equitability” that mutual information lacks. This was not proven; instead it was argued solely through the analysis of simulated data. Here we show that this claim, in fact, is incorrect. First we offer mathematical proof that no (non-trivial) dependence measure satisfies the definition of equitability proposed by Reshef et al.. We then propose a self-consistent and more general definition of equitability that follows naturally from the Data Processing Inequality. Mutual information satisfies this new definition of equitability while MIC does not. Finally, we show that the simulation evidence offered by Reshef et al. was artifactual. We conclude that estimating mutual information is not only practical for many real-world applications, but also provides a natural solution to the problem of quantifying associations in large data sets.

Identifying and Mapping Cell-type Specific Chromatin Programming of Gene Expression

Identifying and Mapping Cell-type Specific Chromatin Programming of Gene Expression
Troels T. Marstrand, John D. Storey
(Submitted on 11 Oct 2012)

A problem of substantial interest is to systematically map variation in chromatin structure to gene expression regulation across conditions, environments, or differentiated cell types. We developed and applied a quantitative framework for determining the existence, strength, and type of relationship between high-resolution chromatin structure in terms of DNaseI hypersensitivity (DHS) and genome-wide gene expression levels in 20 diverse human cell lines. We show that ~25% of genes show cell-type specific expression explained by alterations in chromatin structure. We find that distal regions of chromatin structure (e.g., +/- 200kb) capture more genes with this relationship than local regions (e.g., +/- 2.5kb), yet the local regions show a more pronounced effect. By exploiting variation across cell-types, we were capable of pinpointing the most likely hypersensitive sites related to cell-type specific expression, which we show have a range of contextual usages. This quantitative framework is likely applicable to other settings aimed at relating continuous genomic measurements to gene expression variation.

Best Practices for Scientific Computing

Outside our usual remit, but likely of interest to many of our readers. See here for online peer review.

Best Practices for Scientific Computing
D. A. Aruliah, C. Titus Brown, Neil P. Chue Hong, Matt Davis, Richard T. Guy, Steven H. D. Haddock, Katy Huff, Ian Mitchell, Mark Plumbley, Ben Waugh, Ethan P. White, Greg Wilson, Paul Wilson
(Submitted on 1 Oct 2012)
Scientists spend an increasing amount of time building and using software. However, most scientists are never taught how to do this efficiently. As a result, many are unaware of tools and practices that would allow them to write more reliable and maintainable code with less effort. We describe a set of best practices for scientific software development that have solid foundations in research and experience, and that improve scientists’ productivity and the reliability of their software.