An important component of most data analyses is data exploration for the purpose of finding "good models'' on which to base statistical inference. The problem is that data exploration tends to invalidate statistical inference. We show by way of example how a simple variable screening exercise can reduce confidence interval coverage from nominally 95% to nearly 0%. We further illustrate with published examples how data exploration in practice tends to be informal and ad hoc, both in research and in teaching. In this talk we examine two forms of data exploration that are common in regression contexts: selection of covariates, and selection of transformations. We provide a solution to the problem of valid post-selection inference through a reduction to large-scale family-wise error control. The main novelty of the solution is that it applies (1) without any model assumptions (2) to random covariates and (3) to a diverging number of covariates. Furthermore, the solution can be generalized to many other post-selection inference problems. Justifications are asymptotic with rates similar to those of the literature on high-dimensional regression but without common assumptions of linearity, homoskedasticity, Gaussianity, and sparsity.

This talk is based on joint work with “Larry's group,” including the late Larry Brown, Andreas Buja, Junhui Cai, Edward George, and Linda Zhao.

Trend filtering is a modern approach to nonparametric regression that is more adaptive to local smoothness than splines or basis procedures while being computationally simpler. Current theoretical and empirical analysis of trend filtering focuses on estimating a function corrupted by Gaussian noise, but our work extends this technique to general exponential family distributions. This extension is motivated by the need to study massive, gridded climate data derived from polar-orbiting satellites. We present inference algorithms tailored to large problems, theoretical results for general loss functions, and principled methods for tuning parameter selection without excess computation.

Change point detection is a popular tool for identifying locations in a data sequence where an abrupt change occurs in the data distribution and has been widely studied for Euclidean data. Modern data very often is non-Euclidean, for example distribution valued data or network data. Change point detection is a challenging problem when the underlying data space is a metric space where one does not have basic algebraic operations like addition of the data points and scalar multiplication.

In this talk, I propose a method to infer the presence and location of change points in the distribution of a sequence of independent data taking values in a general metric space. Change points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fréchet mean or Fréchet variance or both. The proposed method is based on comparisons of Fréchet variances before and after putative change point locations. First, I will establish that under the null hypothesis of no change point the limit distribution of the proposed scan function is the square of a standardized Brownian Bridge. It is well known that such convergence is rather slow in moderate to high dimensions. For more accurate results in applications, I will provide a theoretically justified bootstrap-based scheme for testing the presence of change points. Next, I will show that when a change point exists, (1) the proposed test is consistent under contiguous alternatives and (2) the estimated location of the change-point is consistent. All of the above results hold for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. I will illustrate the efficacy of the proposed approach in empirical studies and in real data applications with sequences of maternal fertility distributions and email communication networks. Finally, I will talk about some future extensions and other related research directions, for instance, when one has samples of dynamic metric space data. This talk is based on joint work with Prof. Hans-Georg Müller.

In this talk I will give two perspectives on the estimation task of low-rank random graphs. Specifically, I will focus on estimating the latent positions in random dot product graphs. One perspective is the global estimation task. The minimax lower bound for global estimation of the latent positions is established, and this minimax lower bound is achieved by a Bayes procedure, referred to as the posterior spectral embedding. The other perspective is the local estimation task. We define local efficiency in estimating each individual latent position, propose a novel one-step estimator that takes advantage of the curvature information of the likelihood function (i.e., derivatives information) of the sampling model, and show that this estimator is locally efficient. The previously widely adopted adjacency spectral embedding is proven to be locally inefficient due to the ignorance of the Bernoulli likelihood function of the sampling model. Simulation examples and the analysis of a real-world Wikipedia graph dataset are provided to demonstrate the usefulness of the proposed methods.

Causal inference from observational data is a vital problem, but it comes with strong assumptions. Most methods assume that we observe all confounders, variables that affect both the causal variables and the outcome variables. But whether we have observed all confounders is a famously untestable assumption. We describe the deconfounder, a way to do causal inference from observational data allowing for unobserved confounding.

How does the deconfounder work? The deconfounder is designed for problems of multiple causal inferences: scientific studies that involve many causes whose effects are simultaneously of interest. The deconfounder uses the correlation among causes as evidence for unobserved confounders, combining unsupervised machine learning and predictive model checking to perform causal inference. We study the theoretical requirements for the deconfounder to provide unbiased causal estimates, along with its limitations and tradeoffs. We demonstrate the deconfounder on real-world data and simulation studies.

In modern psychological and biomedical research with diagnostic purposes, scientists often formulate the key task as inferring the fine-grained latent information under structural constraints. These structural constraints usually come from the domain experts’ prior knowledge or insight. The emerging family of Structured Latent Attribute Models (SLAMs) accommodate these modeling needs and have received substantial attention in psychology, education, and epidemiology. SLAMs bring exciting opportunities and unique challenges. In particular, with high-dimensional discrete latent attributes and structural constraints encoded by a design matrix, one needs to balance the gain in the model’s explanatory power and interpretability, against the difficulty of understanding and handling the complex model structure.

In the first part of this talk, I present identifiability results that advance the theoretical knowledge of how the design matrix influences the estimability of SLAMs. The new identifiability conditions guide real-world practices of designing diagnostic tests and also lay the foundation for drawing valid statistical conclusions. In the second part, I introduce a statistically consistent penalized likelihood approach to selecting significant latent patterns in the population. I also propose a scalable computational method. These developments explore an exponentially large model space involving many discrete latent variables, and they address the estimation and computation challenges of high-dimensional SLAMs arising from large-scale scientific measurements. The application of the proposed methodology to the data from an international educational assessment reveals meaningful knowledge structure of the student population.

Making informed decisions about model adequacy has been an outstanding issue for regression models with discrete outcomes. Standard residuals such as Pearson and deviance residuals for such outcomes often show a large discrepancy from the hypothesized pattern even under the true model and are not informative especially when data are highly discrete. To fill this gap, we propose a surrogate empirical residual distribution function for general discrete (e.g. ordinal and count) outcomes that serves as an alternative to the empirical Cox-Snell residual distribution function. When at least one continuous covariate is available, we show asymptotically that the proposed function converges uniformly to the identity function under the correctly specified model, even with highly discrete (e.g. binary) outcomes. Through simulation studies, we demonstrate empirically that the proposed surrogate empirical residual distribution function is highly effective for various diagnostic tasks, since it is close to the hypothesized pattern under the true model and significantly departs from this pattern with model misspecification.
 

Motivated by problems in data clustering and semi-supervised learning, we establish general conditions under which families of nonparametric mixture models are identifiable by introducing a novel framework for clustering overfitted parametric (i.e. misspecified) mixture models. These conditions generalize existing conditions in the literature, allowing for general nonparametric mixture components. Notably, our results avoid imposing assumptions on the mixture components, and instead impose regularity assumptions on the underlying mixing measure. After a discussion of some statistical aspects of this problem, we will discuss two applications of this framework. First, we extend classical model-based clustering to nonparametric settings and develop a practical algorithm for learning nonparametric mixtures. Second, we analyze the sample complexity of semi-supervised learning (SSL) and introduce new assumptions based on the mismatch between a mixture model learned from unlabeled data and the true mixture model induced by the (unknown) class conditional distributions. Under these assumptions, we establish an \Omega(K\log K) labeled sample complexity bound without imposing parametric assumptions, where K is the number of classes. These results suggest that even in nonparametric settings it is possible to learn a near-optimal classifier using only a few labeled samples.

Mia Hubert, Peter J. Rousseeuw and Wannes Van den Bossche Department of Mathematics, KU Leuven, Belgium

Multivariate data are typically represented by a rectangular matrix (table) in which the rows are the objects (cases) and the columns are the variables (measurements). When there are many variables one often reduces the dimension by principal component analysis (PCA), which in its basic form is not robust to outliers. Much research has focused on handling rowwise outliers, i.e. rows that deviate from the majority of the rows in the data (for instance, they might belong to a diff t population). In recent years also cellwise outliers are receiving attention. These are suspicious cells (entries) that can occur anywhere in the table. Even a relatively small proportion of outlying cells can contaminate over half the rows, which causes rowwise robust methods to break down (Alqallaf et al., 2009).

In this paper a new PCA method is constructed which combines the strengths of two existing robust methods, DetectDeviatingCells (Rousseeuw and Van den Bossche, 2019) and ROBPCA, in order to be robust against both cellwise and rowwise outliers. At the same time, the algorithm can cope with missing values. As of yet it is the only PCA method that can deal with all three problems simultaneously. Its name MacroPCA stands for PCA allowing for Missings And Cellwise & Rowwise Outliers. Several simulations and real data sets illustrate its robustness. New residual maps are introduced, which help to determine which variables are responsible for the outlying behavior. The method is well-suited for online process control. The function MacroPCA has been incorporated in the R package cellWise (Raymaekers et al., 2019), which also contains a vignette with real data examples.

References

Alqallaf, F., Van Aelst, S., Yohai, V., and Zamar, R.H. (2009), Propagation of outliers in multivariate data, The Annals of Statistics, 37, 311–331.

Raymaekers, J., Rousseeuw, P.J., Van den Bossche, W., and Hubert, M. (2019). cellWise: Analyzing Data with Cellwise Outliers. R package, CRAN.

Rousseeuw, P.J., and Van den Bossche, W. (2019). Detecting Deviating Data Cells. Technometrics, 60, 123–145.