The Chi-Square Test of Distance Correlation

Distance correlation has gained much recent attention in the statistics and machine learning community: the sample statistic is straightforward to compute, works for any metric or kernel choice, and equals 0 asymptotically if and only if independence. One major bottleneck is the testing process: the null distribution of distance correlation depends on the metric choice and marginal distributions, which cannot be readily estimated. To compute a p-value, the standard approach is to estimate the null distribution via permutation, which generally requires O(rn^2) time complexity for n samples and r permutations and too costly for big data applications. In this paper, we propose a chi-square distribution to approximate the null distribution of the unbiased distance correlation. We prove that the chi-square distribution either equals or well-approximates the null distribution, and always upper tail dominates the null distribution. The resulting distance correlation chi-square test does not require any permutation nor parameter estimation, is simple and fast to implement, works with any strong negative type metric or characteristic kernel, is valid and universally consistent for independence testing, and enjoys a similar finite-sample testing power as the standard permutation test. When testing one-dimensional data using Euclidean distance, the unbiased distance correlation testing runs in O(nlog(n)), rendering it comparable in speed to the Pearson correlation t-test. The results are supported and demonstrated via simulations.

The Exact Equivalence of Independence Testing and Two-Sample Testing

Testing independence and testing equality of distributions are two tightly related statistical hypotheses. Several distance and kernel-based statistics are recently proposed to achieve universally consistent testing for either hypothesis. On the distance side, the distance correlation is proposed for independence testing, and the energy statistic is proposed for two-sample testing. On the kernel side, the Hilbert-Schmidt independence criterion is proposed for independence testing and the maximum mean discrepancy is proposed for two-sample testing. In this paper, we show that two-sample testing are special cases of independence testing via an auxiliary label vector, and prove that distance correlation is exactly equivalent to the energy statistic in terms of the population statistic, the sample statistic, and the testing p-value via permutation test. The equivalence can be further generalized to K-sample testing and extended to the kernel regime. As a consequence, it suffices to always use an independence statistic to test equality of distributions, which enables better interpretability of the test statistic and more efficient testing.

A Consistent Independence Test for Multivariate Time-Series

A fundamental problem in statistical data analysis is testing whether two phenomena are related. When the phenomena in question are time series, many challenges emerge. The first is defining a dependence measure between time series at the population level, as well as a sample level test statistic. The second is computing or estimating the distribution of this test statistic under the null, as the permutation test procedure is invalid for most time series structures. This work aims to address these challenges by combining distance correlation and multiscale graph correlation (MGC) from independence testing literature and block permutation testing from time series analysis. Two hypothesis tests for testing the independence of time series are proposed. These procedures also characterize whether the dependence relationship between the series is linear or nonlinear, and the time lag at which this dependence is maximized. For strictly stationary auto-regressive moving average (ARMA) processes, the proposed independence tests are proven valid and consistent. Finally, neural connectivity in the brain is analyzed using fMRI data, revealing linear dependence of signals within the visual network and default mode network, and nonlinear relationships in other regions. This work opens up new theoretical and practical directions for many modern time series analysis problems.

mgcpy: A Comprehensive High Dimensional Independence Testing Python Package

With the increase in the amount of data in many fields, a method to consistently and efficiently decipher relationships within high dimensional data sets is important. Because many modern datasets are high-dimensional, univariate independence tests are not applicable. While many multivariate independence tests have R packages available, the interfaces are inconsistent, most are not available in Python. mgcpy is an extensive Python library that includes many state of the art high-dimensional independence testing procedures using a common interface. The package is easy-to-use and is flexible enough to enable future extensions. This manuscript provides details for each of the tests as well as extensive power and run-time benchmarks on a suite of high-dimensional simulations previously used in different publications. The appendix includes demonstrations of how the user can interact with the package, as well as links and documentation.

Estimating Information-Theoretic Quantities with Random Forests

Information-theoretic quantities, such as mutual information and conditional entropy, are useful statistics for measuring the dependence between two random variables. However, estimating these quantities in a non-parametric fashion is difficult, especially when the variables are high-dimensional, a mixture of continuous and discrete values, or both. In this paper, we propose a decision forest method, Conditional Forests (CF), to estimate these quantities. By combining quantile regression forests with honest sampling, and introducing a finite sample correction, CF improves finite sample bias in a range of settings. We demonstrate through simulations that CF achieves smaller bias and variance in both low- and high-dimensional settings for estimating posteriors, conditional entropy, and mutual information. We then use CF to estimate the amount of information between neuron class and other ceulluar feautres.

Sparse Representation Classification via Screening for Graphs

The sparse representation classifier (SRC) is shown to work well for image recognition problems that satisfy a subspace assumption. In this paper we propose a new implementation of SRC via screening, establish its equivalence to the original SRC under regularity conditions, and prove its classification consistency for random graphs drawn from stochastic blockmodels. The results are demonstrated via simulations and real data experiments, where the new algorithm achieves comparable numerical performance but significantly faster.

Decision Forests Induce Characteristic Kernels

Decision forests are popular tools for classification and regression. These forests naturally produce proximity matrices measuring how often each pair of observations lies in the same leaf node. Recently it has been demonstrated that these proximity matrices can be thought of as kernels, connecting the decision forest literature to the extensive kernel machine literature. While other kernels are known to have strong theoretical properties, such as being characteristic kernels, no similar result is available for any decision forest based kernel. We show that a decision forest induced proximity can be made into a characteristic kernel, which can be used within an independence test to obtain a universally consistent test. We therefore empirically evaluate this kernel on a suite of 12 high-dimensional independence test settings: the decision forest induced kernel is shown to typically achieve substantially higher power than other methods.

Random Projection Forests

Ensemble methods---particularly those based on decision trees---have recently demonstrated superior performance in a variety of machine learning settings. We introduce a generalization of many existing decision tree methods called "Random Projection Forests" (RPF), which is any decision forest that uses (possibly data dependent and random) linear projections. Using this framework, we introduce a special case, called "Lumberjack", using very sparse random projections, that is, linear combinations of a small subset of features. Lumberjack obtains statistically significantly improved accuracy over Random Forests, Gradient Boosted Trees, and other approaches on a standard benchmark suites for classification with varying dimension, sample size, and number of classes. To illustrate how, why, and when Lumberjack outperforms other methods, we conduct extensive simulated experiments, in vectors, images, and nonlinear manifolds. Lumberjack typically yields improved performance over existing decision trees ensembles, while mitigating computational efficiency and scalability, and maintaining interpretability. Lumberjack can easily be incorporated into other ensemble methods such as boosting to obtain potentially similar gains.

From Distance Correlation to Multiscale Graph Correlation

Understanding and developing a correlation measure that can detect general dependencies is not only imperative to statistics and machine learning, but also crucial to general scientific discovery in the big data age. In this paper, we establish a new framework that generalizes distance correlation --- a correlation measure that was recently proposed and shown to be universally consistent for dependence testing against all joint distributions of finite moments --- to the Multiscale Graph Correlation (MGC). By utilizing the characteristic functions and incorporating the nearest neighbor machinery, we formalize the population version of local distance correlations, define the optimal scale in a given dependency, and name the optimal local correlation as MGC. The new theoretical framework motivates a theoretically sound Sample MGC and allows a number of desirable properties to be proved, including the universal consistency, convergence and almost unbiasedness of the sample version. The advantages of MGC are illustrated via a comprehensive set of simulations with linear, nonlinear, univariate, multivariate, and noisy dependencies, where it loses almost no power in monotone dependencies while achieving better performance in general dependencies, compared to distance correlation and other popular methods.