Under the banner of "big data," the detection and classification of structure in extremely large, high-dimensional, data sets are two of the central statistical challenges of our times. Among the most intriguing new approaches to this challenge is "TDA," or "topological data analysis," one of the primary aims of which is providing nonmetric, but topologically informative, preanalyses of data which make later, more quantitative, analyses feasible. While TDA rests on strong mathematical foundations from topology, in applications, it has faced challenges due to difficulties in handling issues of statistical reliability and robustness, often leading to an inability to make scientific claims with verifiable levels of statistical confidence. We propose a methodology for the parametric representation, estimation, and replication of persistence diagrams, the main diagnostic tool of TDA. The power of the methodology lies in the fact that even if only one persistence diagram is available for analysis-the typical case for big data applications-the replications permit conventional statistical hypothesis testing. The methodology is conceptually simple and computationally practical, and provides a broadly effective statistical framework for persistence diagram TDA analysis. We demonstrate the basic ideas on a toy example, and the power of the parametric approach to TDA modeling in an analysis of cosmic microwave background (CMB) nonhomogeneity.

We introduce a new model for planar point point processes, with the aim of capturing the structure of point interaction and spread in persistence diagrams. Persistence diagrams themselves are a key tool of TDA (topological data analysis), crucial for the delineation and estimation of global topological structure in large data sets. To a large extent, the statistical analysis of persistence diagrams has been hindered by difficulties in providing replications, a problem that was addressed in an earlier paper, which introduced a procedure called RST (replicating statistical topology). Here we significantly improve on the power of RST via the introduction of a more realistic class of models for the persistence diagrams. In addition, we introduce to TDA the idea of bagplotting, a powerful technique from non-parametric statistics well adapted for differentiating between topologically significant points, and noise, in persistence diagrams.Outside the setting of TDA, our model provides a setting for fashioning point processes, in any dimension, in which both local interactions between the points, along with global restraints on the general point cloud, are important and perhaps competing.

The coronavirus disease 2019 (COVID-19) induced a lockdown that has resulted in a sharp reduction in air and motor traffic and industrial activities. This in turn led to a reduction in air pollution around the world. It is important to quantify the extent of that reduction in order to estimate the weight of the impact of air and motor traffic and industrial activities over the total variation of air quality. An assessment of the extent of air pollution is aimed at facilitating the crafting of policies toward the reduction of pollution and the improvement in air quality. The aim of this paper is to evaluate the impact of the COVID-19 outbreak on air pollution in Israel. Particularly, we focus on Haifa and Greater Tel-Aviv (Gush-Dan), two regions with high air pollution, while examining different types of air monitoring stations. The period to which we refer to is March 8, 2020, to May 2, 2020. The results reveal two main findings: (1) During the COVID-19 lockdown, pollution emissions decreased relative to the same period in 2019. The biggest reduction was observed in NO x , which, on average, was 41%. Surprisingly, ground-level ozone ( O 3 ) increased, and appeared to behave similarly to the ozone weekend effect. (2) The total percentage variation in pollution emission that was explained by the lockdown was at most 26%. By adding the meteorological conditions (which included measures of wind direction, wind speed, and temperature) as a factor in addition to the lockdown effect, this percent increased to 47%.

The Cox regression model is a popular model for analyzing the relationship between a covariate and a survival endpoint. The standard Cox model assumes a constant covariate effect across the entire covariate domain. However, in many epidemiological and other applications, the covariate of main interest is subject to a threshold effect: a change in the slope at a certain point within the covariate domain. Often, the covariate of interest is subject to some degree of measurement error. In this paper, we study measurement error correction in the case where the threshold is known. Several bias correction methods are examined: two versions of regression calibration (RC1 and RC2, the latter of which is new), two methods based on the induced relative risk under a rare event assumption (RR1 and RR2, the latter of which is new), a maximum pseudo-partial likelihood estimator (MPPLE), and simulationextrapolation (SIMEX). We develop the theory, present simulations comparing the methods, and illustrate their use on data concerning the relationship between chronic air pollution exposure to particulate matter PM10 and fatal myocardial infarction (Nurses Health Study (NHS)), and on data concerning the effect of a subject's long-term underlying systolic blood pressure level on the risk of cardiovascular disease death (Framingham Heart Study (FHS)). The simulations indicate that the best methods are RR2 and MPPLE.

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