Differentially private (DP) mechanisms protect individual-level information by introducing randomness into the statistical analysis procedure. Despite the availability of numerous DP tools, there remains a lack of general techniques for conducting statistical inference under DP. We examine a DP bootstrap procedure that releases multiple private bootstrap estimates to infer the sampling distribution and construct confidence intervals (CIs). Our privacy analysis presents new results on the privacy cost of a single DP bootstrap estimate, applicable to any DP mechanisms, and identifies some misapplications of the bootstrap in the existing literature. Using the Gaussian-DP (GDP) framework (Dong et al.,2022), we show that the release of $B$ DP bootstrap estimates from mechanisms satisfying $(\mu/\sqrt{(2-2/\mathrm{e})B})$-GDP asymptotically satisfies $\mu$-GDP as $B$ goes to infinity. Moreover, we use deconvolution with the DP bootstrap estimates to accurately infer the sampling distribution, which is novel in DP. We derive CIs from our density estimate for tasks such as population mean estimation, logistic regression, and quantile regression, and we compare them to existing methods using simulations and real-world experiments on 2016 Canada Census data. Our private CIs achieve the nominal coverage level and offer the first approach to private inference for quantile regression.

Recent work has shown that sparse graphs containing many triangles cannot be reproduced using a finite-dimensional representation of the nodes, in which link probabilities are inner products. Here, we show that such graphs can be reproduced using an infinite-dimensional inner product model, where the node representations lie on a low-dimensional manifold. Recovering a global representation of the manifold is impossible in a sparse regime. However, we can zoom in on local neighbourhoods, where a lower-dimensional representation is possible. As our constructions allow the points to be uniformly distributed on the manifold, we find evidence against the common perception that triangles imply community structure.

We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig's projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural-network-based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.

The quality of air is closely linked with the life quality of humans, plantations, and wildlife. It needs to be monitored and preserved continuously. Transportations, industries, construction sites, generators, fireworks, and waste burning have a major percentage in degrading the air quality. These sources are required to be used in a safe and controlled manner. Using traditional laboratory analysis or installing bulk and expensive models every few miles is no longer efficient. Smart devices are needed for collecting and analyzing air data. The quality of air depends on various factors, including location, traffic, and time. Recent researches are using machine learning algorithms, big data technologies, and the Internet of Things to propose a stable and efficient model for the stated purpose. This review paper focuses on studying and compiling recent research in this field and emphasizes the Data sources, Monitoring, and Forecasting models. The main objective of this paper is to provide the astuteness of the researches happening to improve the various aspects of air polluting models. Further, it casts light on the various research issues and challenges also.

Geostatistics, the analysis of geocoded data, is traditionally based on stochastic process models which offer a coherent way to model data at any finite collection of locations while ensuring the generalizability of inference to the entire region.Gaussian processes (GP) with a mean function capturing effects of covariates and the covariance function encoding the spatial dependence, is a staple for geostatistical analysis, offering theoretical guarantees and practical benefits. GP are flexible enough to model any smooth spatial surface, and can be specified parsimoniously with covariance functions using a very small set of parameters. The spatial covariance parameters offer insights into the smoothness and spatial properties of the response process (Stein, 1999). The finite dimensional realizations of a GP are multivariate Gaussian, thereby offering estimates of the mean and covariance parameters via convenient maximization of the Gaussian likelihood, and predictions at new locations by using conditional Gaussian distributions (see, e.g., Banerjee et al., 2014; Cressie and Wikle, 2015, for detailed exposition on GP models for spatial and spatio-temporal data). Also, computational roadblocks to using GP for large spatial data have been greatly mitigated by recent advances (see, Heaton et al., 2019, for a recent review of scalable GP approaches). The mean function of a Gaussian process is often modeled as a linear regression on the covariates. The growing popularity and accessibility of machine learning algorithms such as neural networks, random forests, gradient boosted trees, capable of modeling complex non-linear relationships has heralded a paradigm shift. Practitioners are increasingly shunning models with parametric assumptions like linearity in favor of these machine learning approaches that can capture non-linearity and high-order interactions in a data-driven manner. The field of spatial statistics has not been insulated from this machine learning revolution.

The machine learning (ML) paradigm has gained much popularity today. Its algorithmic models are employed in every field, such as natural language processing, pattern recognition, object detection, image recognition, earth observation and many other research areas. In fact, machine learning technologies and their inevitable impact suffice in many technological transformation agendas currently being propagated by many nations, for which the already yielded benefits are outstanding. From a regional perspective, several studies have shown that machine learning technology can help address some of Africa's most pervasive problems, such as poverty alleviation, improving education, delivering quality healthcare services, and addressing sustainability challenges like food security and climate change. In this state-of-the-art paper, a critical bibliometric analysis study is conducted, coupled with an extensive literature survey on recent developments and associated applications in machine learning research with a perspective on Africa. The presented bibliometric analysis study consists of 2761 machine learning-related documents, of which 89% were articles with at least 482 citations published in 903 journals during the past three decades. Furthermore, the collated documents were retrieved from the Science Citation Index EXPANDED, comprising research publications from 54 African countries between 1993 and 2021. The bibliometric study shows the visualization of the current landscape and future trends in machine learning research and its application to facilitate future collaborative research and knowledge exchange among authors from different research institutions scattered across the African continent.

Background: The use of machine learning (ML) in mental health (MH) research is increasing, especially as new, more complex data types become available to analyze. By systematically examining the published literature, this review aims to uncover potential gaps in the current use of ML to study MH in vulnerable populations of immigrants, refugees, migrants, and racial and ethnic minorities. Methods: In this systematic review, we queried Google Scholar for ML-related terms, MH-related terms, and a population of a focus search term strung together with Boolean operators. Backward reference searching was also conducted. Included peer-reviewed studies reported using a method or application of ML in an MH context and focused on the populations of interest. We did not have date cutoffs. Publications were excluded if they were narrative or did not exclusively focus on a minority population from the respective country. Data including study context, the focus of mental healthcare, sample, data type, type of ML algorithm used, and algorithm performance was extracted from each. Results: Our search strategies resulted in 67,410 listed articles from Google Scholar. Ultimately, 12 were included. All the articles were published within the last 6 years, and half of them studied populations within the US. Most reviewed studies used supervised learning to explain or predict MH outcomes. Some publications used up to 16 models to determine the best predictive power. Almost half of the included publications did not discuss their cross-validation method. Conclusions: The included studies provide proof-of-concept for the potential use of ML algorithms to address MH concerns in these special populations, few as they may be. Our systematic review finds that the clinical application of these models for classifying and predicting MH disorders is still under development.

The training of neural networks is a complex, high-dimensional, non-convex and noisy optimization problem whose theoretical understanding is interesting both from an applicative perspective and for fundamental reasons. A core challenge is to understand the geometry and topography of the landscape that guides the optimization. In this work, we employ standard Statistical Mechanics methods, namely, phase-space exploration using Langevin dynamics, to study this landscape for an over-parameterized fully connected network performing a classification task on random data. Analyzing the fluctuation statistics, in analogy to thermal dynamics at a constant temperature, we infer a clear geometric description of the low-loss region. We find that it is a low-dimensional manifold whose dimension can be readily obtained from the fluctuations. Furthermore, this dimension is controlled by the number of data points that reside near the classification decision boundary. Importantly, we find that a quadratic approximation of the loss near the minimum is fundamentally inadequate due to the exponential nature of the decision boundary and the flatness of the low-loss region. This causes the dynamics to sample regions with higher curvature at higher temperatures, while producing quadratic-like statistics at any given temperature. We explain this behavior by a simplified loss model which is analytically tractable and reproduces the observed fluctuation statistics.

In recent years, reinforcement learning (RL) has acquired a prominent position in the space of health-related sequential decision-making, becoming an increasingly popular tool for delivering adaptive interventions (AIs). However, despite potential benefits, its real-life application is still limited, partly due to a poor synergy between the methodological and the applied communities. In this work, we provide the first unified survey on RL methods for learning AIs, using the common methodological umbrella of RL to bridge the two AI areas of dynamic treatment regimes and just-in-time adaptive interventions in mobile health. We outline similarities and differences between these two AI domains and discuss their implications for using RL. Finally, we leverage our experience in designing case studies in both areas to illustrate the tremendous collaboration opportunities between statistical, RL, and healthcare researchers in the space of AIs.

In the interest of interpretability, computation, and circumventing the statistical curse of dimensionality plaguing high dimensional regression, structure is often assumed on the true regression function. Indeed, it might plausibly be argued that sparse linear regression is the distinguishing export of modern statistics. Despite its popularity, circumstances may call for more flexibility to capture nonlinear effects of the covariates. Striking a balance between flexibility and structure, Hastie and Tibshirani [19] proposed generalized additive models (GAMs) as a natural extension to the vaunted linear model. In a GAM, the regression function admits an additive decomposition of univariate (nonlinear) component functions. However, as in the linear model, the sample size must outpace the dimension for consistent estimation. Following modern statistical instinct, a sparse additive model is compelling [28, 32, 34, 37, 38, 47]. The regression function admits an additive decomposition of univariate functions for which only a small subset are nonzero; it is the combination of a GAM and sparsity.