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.
Geodesic distance is the shortest path between two points in a Riemannian manifold. Manifold learning algorithms, such as Isomap, seek to learn a manifold that preserves geodesic distances. However, such methods operate on the ambient dimensionality, and are therefore fragile to noise dimensions. We developed an unsupervised random forest method (URerF) to approximately learn geodesic distances in linear and nonlinear manifolds with noise. URerF operates on low-dimensional sparse linear combinations of features, rather than the full observed dimensionality. To choose the optimal split in a computationally efficient fashion, we developed a fast Bayesian Information Criterion statistic for Gaussian mixture models. We introduce geodesic precision-recall curves which quantify performance relative to the true latent manifold. Empirical results on simulated and real data demonstrate that URerF is robust to high-dimensional noise, where as other methods, such as Isomap, UMAP, and FLANN, quickly deteriorate in such settings. In particular, URerF is able to estimate geodesic distances on a real connectome dataset better than other approaches.
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.
We introduce GraSPy, a Python library devoted to statistical inference, machine learning, and visualization of random graphs and graph populations. This package provides flexible and easy-to-use algorithms for analyzing and understanding graphs with a scikit-learn compliant API. GraSPy can be downloaded from Python Package Index (PyPi), and is released under the Apache 2.0 open-source license. The documentation and all releases are available at https://neurodata.io/graspy.
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.
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.
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.
Understanding the relationships between different properties of data, such as whether a connectome or genome has information about disease status, is becoming increasingly important in modern biological datasets. While existing approaches can test whether two properties are related, they often require unfeasibly large sample sizes in real data scenarios, and do not provide any insight into how or why the procedure reached its decision. Our approach, "Multiscale Graph Correlation" (MGC), is a dependence test that juxtaposes previously disparate data science techniques, including k-nearest neighbors, kernel methods (such as support vector machines), and multiscale analysis (such as wavelets). Other methods typically require double or triple the number samples to achieve the same statistical power as MGC in a benchmark suite including high-dimensional and nonlinear relationships - spanning polynomial (linear, quadratic, cubic), trigonometric (sinusoidal, circular, ellipsoidal, spiral), geometric (square, diamond, W-shape), and other functions, with dimensionality ranging from 1 to 1000. Moreover, MGC uniquely provides a simple and elegant characterization of the potentially complex latent geometry underlying the relationship, providing insight while maintaining computational efficiency. In several real data applications, including brain imaging and cancer genetics, MGC is the only method that can both detect the presence of a dependency and provide specific guidance for the next experiment and/or analysis to conduct.
Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering - clustering the vertices of a graph based on their spectral embedding - is commonly approached via K-means (or, more generally, Gaussian mixture model) clustering composed with either Laplacian or Adjacency spectral embedding (LSE or ASE). Recent theoretical results provide new understanding of the problem and solutions, and lead us to a 'Two Truths' LSE vs. ASE spectral graph clustering phenomenon convincingly illustrated here via a diffusion MRI connectome data set: the different embedding methods yield different clustering results, with LSE capturing left hemisphere/right hemisphere affinity structure and ASE capturing gray matter/white matter core-periphery structure.