Summary A variety of demographic statistical models exist for studying population dynamics when individuals can be tracked over time. In cases where data are missing due to imperfect detection of individuals, the associated measurement error can be accommodated under certain study designs (e.g. those that involve multiple surveys or replication). However, the interaction of the measurement error and the underlying dynamic process can complicate the implementation of statistical agent‐based models (ABMs) for population demography. In a Bayesian setting, traditional computational algorithms for fitting hierarchical demographic models can be prohibitively cumbersome to construct. Thus, we discuss a variety of approaches for fitting statistical ABMs to data and demonstrate how to use multi‐stage recursive Bayesian computing and statistical emulators to fit models in such a way that alleviates the need to have analytical knowledge of the ABM likelihood. Using two examples, a demographic model for survival and a compartment model for COVID‐19, we illustrate statistical procedures for implementing ABMs. The approaches we describe are intuitive and accessible for practitioners and can be parallelised easily for additional computational efficiency.

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Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.

The human brain is a complex system of neural tissue that varies significantly between individuals. Although the technology that delineates these neural pathways does not currently exist, medical imaging modalities, such as diffusion magnetic resonance imaging (dMRI), can be leveraged for mathematical identification. The purpose of this work is to develop a novel method employing machine learning techniques to determine intravoxel nerve number and direction from dMRI data. The method was tested on multiple synthetic datasets and showed promising estimation accuracy and robustness for multi-nerve systems under a variety of conditions, including highly noisy data and imprecision in parameter assumptions.

Many arrhythmia datasets are multimodal due to the simultaneous collection of physiological signals of a subject. These datasets frequently have missing modalities or missing block-wise data, a characteristic that various recent applications of neural networks fail to consider. Most arrhythmic detection models only use electrocardiogram and blood pressure recordings. Unconsidered physiological signals may be strongly correlated with other modalities despite having missing data. To improve robustness and accuracy of heartbeat detection, all available modalities should be considered in multimodal arrhythmia datasets. Several hybrid neural networks are proposed to robustly analyze heartbeats by considering every available physiological signal. These networks combine elements from convolutional neural networks, recurrent neural networks, and a deep learning architecture. This enables researchers to analyze every signal of subjects while the set of signals collected among subjects may differ. The proposed hybrid neural networks provide more robust results in heartbeat detection when utilizing missing data modalities. INDEX TERMS Multimodal, heartbeat detection, deep learning, neural networks.