Jun Zhu

Topological data analysis (TDA) uses persistent homology to quantify loops and higher-dimensional holes in data, making it particularly relevant for examining the characteristics of images of cells in the field of cell biology. In the context of a cell injury, as time progresses, a wound in the form of a ring emerges in the cell image and then gradually vanishes. Performing statistical inference on this ring-like pattern in a single image is challenging due to the absence of repeated samples. In this paper, we develop a novel framework leveraging TDA to estimate underlying structures within an individual image and quantify associated uncertainties through confidence regions. Our proposed method partitions the image into the background and the damaged cell regions. Then, pixels within the affected cell region are used to establish confidence regions in the space of persistence diagrams (topological summary statistics). The proposed method establishes an estimate of a persistence diagram for an image that mitigates the bias of a traditional persistence diagram computation. A simulation study is conducted to evaluate the coverage probabilities of the proposed confidence regions in comparison to an alternative approach that is proposed in this paper. We also illustrate our methodology by a real-world example provided by biological cell repair.

Topological Data Analysis (TDA) can be used to detect and characterize holes in an image, such as zero-dimensional holes (connected components) or one-dimensional holes (loops). However, there is currently no widely accepted statistical framework for modeling spatiotemporal dependence in the evolution of topological features, such as holes, within a time series of images. We propose a hypothesis testing framework to identify statistically significant topological features of images in space and time, simultaneously. This addition of time may induce higher-dimensional topological features which can be used to establish temporal connections between the lower-dimensional features at each point in time. The temporal evolution of these lower-dimensional features is then represented on a zigzag persistence diagram, as a topological summary statistic focused on time dynamics. We demonstrate that the method effectively captures the emergence and progression of topological features in a study of a series of images of a wounded cell as it repairs. The proposed method outperforms a current approach in a simulation study that includes features of the wound healing process. Since, the wounded cell images exhibit nonlinear, dynamic, spatial, and temporal structures during single-cell repair, they provide a good application for this method.

Extreme weather events associated with climate change have caused significant damages. In particular, hail storms damage millions of properties in the U.S. and result in billion-dollar insured losses each year in the recent decade. To facilitate the insurance claims management operations in insurance companies, we construct a hierarchical dependence model, which accommodates the complex dependence within and between the outcomes of interests including the propensity of filing a claim, time to report a claim, and the claim amount. The storm-specific and property-specific characteristics are incorporated through marginal models, such as generalized linear models and survival analysis models. The dependence within the hail event is captured by spatial factor copula, while the dependence between different outcomes is captured by bivariate copula. For parameter estimation we develop a two-step procedure that first maximizes the marginal likelihood function and then maximizes the pairwise likelihood, which ensures computational feasibility for big data. We apply this modeling framework to analyze a large dataset involving hail storms in Colorado from 2011 to 2015 impacting hundreds of thousands of insured properties and demonstrate that the predictive performance can be improved by our proposed methodology.

Bretziella fagacearum, the ascomycete fungus oak wilt, is considered a virulent threat to North American oaks, but the influence of the physical environment on this pathosystem remains unclear, particularly at the forest scale. This study explored the influence of terrain and soil factors on B. fagacearum infections, applying discrete and continuous spatial models to investigate the question, besides proximity to other infections, which environmental factors influenced B. fagacearum incidence? Locations of infections were recorded from 586 confirmed B. fagacearum sites, identified from 2004 through 2021 in a 76 km² area of deep, sandy glacial outwash in Chequamegon-Nicolet National Forest, northern Wisconsin. Public datasets derived from remote sensing were incorporated as covariates, describing terrain elevation (USGS 10-m DEM), soil physical and chemical properties (POLARIS), and forest composition (WiscLand2). Spatial models included generalized additive models (GAMs) and Neyman-Scott cluster process models. The results indicated that spatial dependence and the distribution of oak forests were the most important drivers of B. fagacearum distribution in this area, with more minor influence from elevation, hill shade, and drainage patterns. Comparison between modeling approaches indicated that—at this scale and in this area—the most accurate models were those that included host distribution, spatial dependence, and quantitative terrain and soil descriptions. However, a close approximation could be attained using nonlinear models (GAMs) that incorporated only host distribution and spatial dependence.

In this work, we propose a deep learning-based method to perform semiparametric regression analysis for spatially dependent data. To be specific, we use a sparsely connected deep neural network with rectified linear unit (ReLU) activation function to estimate the unknown regression function that describes the relationship between response and covariates in the presence of spatial dependence. Under some mild conditions, the estimator is proven to be consistent, and the rate of convergence is determined by three factors: (1) the architecture of neural network class, (2) the smoothness and (intrinsic) dimension of true mean function, and (3) the magnitude of spatial dependence. Our method can handle well large data set owing to the stochastic gradient descent optimization algorithm. Simulation studies on synthetic data are conducted to assess the finite sample performance, the results of which indicate that the proposed method is capable of picking up the intricate relationship between response and covariates. Finally, a real data analysis is provided to demonstrate the validity and effectiveness of the proposed method.

The accumulation of soil organic matter (SOM) is vital to the agronomic and environmental functioning of agroecosystems, yet the relative influence of inherent soil properties and agricultural management practices on SOM dynamics are not often addressed in individual studies. Using a network of 218 operating farm fields across Wisconsin and southern Minnesota, USA, this research employs single variable analysis (ANOVA and regression) and regression tree analysis to assess the effects of soil properties (texture, drainage class, and pH) and management variables related to crop rotation, tillage, cover cropping, and manure application on SOM, as well as total organic carbon (TOC) and total nitrogen (TN) in the upper 15 cm. Single variable analysis revealed that greater SOM, TOC, and TN were associated with poorly drained soil, tile‐drained fields, high clay content soil, and high biomass crop rotations. SOM and TOC were strongly related (R² = 0.71), but different regression trees were produced; SOM was most influenced by clay content, while TOC was most influenced by drainage class. Future assessment for the building of SOM or TOC should be conducted with drainage and texture class categories and on a regional basis, given that these factors influence the practices that occur within landscapes. A rapid building of datasets through unstructured sampling, including an abundance of metadata, should be a research priority in agricultural science to identify practices to build SOM on a regional basis.

We explore a hierarchical generalized latent factor model for discrete and bounded response variables and in particular, binomial responses. Specifically, we develop a novel two-step estimation procedure and the corresponding statistical inference that is computationally efficient and scalable for the high dimension in terms of both the number of subjects and the number of features per subject. We also establish the validity of the estimation procedure, particularly the asymptotic properties of the estimated effect size and the latent structure, as well as the estimated number of latent factors. The results are corroborated by a simulation study and for illustration, the proposed methodology is applied to analyze a dataset in a gene–environment association study.