Actuarial Modelling with Mixtures of Markov Chains

Multi-state models are widely used in actuarial science because that they provide a convenient way of representing changes in people\u27s statuses. Calculations are easy if one assumes that the model is a Markov chain. However, the memoryless property of a Markov chain is rarely appropriate. This thesis considers several mixtures of Markov chains to capture the heterogeneity of people\u27s mortality rates, morbidity rates, recovery rates, and ageing speeds. This heterogeneity may be the result of unobservable factors that affect individuals\u27 health. The focus of this thesis is on investigating the behaviours of intensities of the observable transitions in the mixture models and assessing the applicability of the models. We first investigate the disability process. Using a mixture model allows the future of the process to be dependent on by its history. We use mixtures of Markov chains with appropriate assumptions to investigate how the intensities of these processes depend on their histories. We next explore an approach of using mixtures of Markov chains to model the dependence of two lifetimes. The mixture models allow the history of survivorship to affect future survival probabilities, which indicates a non-Markov behaviour. We discuss a simple mixture of two four-state Markov chains and a generalized mixture model. Finally, we model the physiological ageing process by using mixtures. The traditional physiological ageing process assumes a homogeneous ageing speed. In fact, the ageing speed of each individual is characterized by his/her own health status. Using mixture models allows the process to reflect health status differences

Visualization of Thawing and Desaturation in Frozen Gas Diffusion Layers of Proton Exchange Membrane Fuel Cells via Synchrotron X-ray Computed Tomography

Proton exchange membrane fuel cells (PEM fuel cells) are one of the promising clean energy solutions to replace fossil fuel in applications such as automobiles and stationary power systems. Though significant research progress has been made, there are still some key technical challenges to be solved including the water management and cold-start problems, hindering large scale commercialization of this technology. A successful water management requires the amount of water content in a PEM fuel cell system to be kept at an optimal level. A poor water management would lead to membrane dehydration or liquid water flooding, which would cause temporary or permanent losses in performance and durability. The water flooding problem would become more serious in the subzero temperature during the cold-start process, which could lead to irreversible damages on cell components, or even cell failure in some extreme cases. Due to opaque nature of PEM fuel cell components, visualization and understanding of water transport behavior remains a challenge. Therefore, thawing and desaturation processes of gas diffusion layers (GDLs) under cold-start operating conditions were studied in this research via synchrotron X-ray computed tomography (CT) imaging techniques. The high speed and high resolution CT scan made it possible to capture the dynamic water behavior during the thawing and desaturation process for both qualitative and quantitative analyses. The experiments were performed on a half cell (cathode side) with a 40 mm serpentine channel, where Sigracet® 35AA and 35BA graphite GDLs were selected in different trials, with the superficial gas velocity of the purging air set to 2.88 m/s, 4.26 m/s, 5.98 m/s and 9.02 m/s. A similar desaturation pattern was observed in both global and local GDL regions; however, heterogeneity in water transfer was found over the entire GDL domains, both in-plane and through-plane. It was also found that the air purging rate, purging distance, and flow field geometry would affect the desaturation pattern, while the GDL hydrophobicity would mainly affect the initial saturation level. These data provide valuable information for future experimental and modeling studies that involve the thawing process in the GDL, and could be used to optimize the cell design and develop the cold-start protocols

On the Meromorphic Integrability of the Critical Systems for Optimal Sums of Eigenvalues

The popularity of estimation to bounds for sums of eigenvalues started from P. Li and S. T. Yau for the study of the P\'{o}lya conjecture. This subject is extended to different types of differential operators. This paper explores for the sums of the first $m$ eigenvalues of Sturm-Liouville operators from two aspects. Firstly, by the complete continuity of eigenvalues, we propose a family of critical systems consisting of nonlinear ordinary differential equations, indexed by the exponent $p\in(1,\infty)$ of the Lebesgue spaces concerned. There have profound relations between the solvability of these systems and the optimal lower or upper bounds for the sums of the first $m$ eigenvalues of Sturm-Liouville operators, which provides a novel idea to study the optimal bounds. Secondly, we investigate the integrability or solvability of the critical systems. With suitable selection of exponents $p$, the critical systems are equivalent to the polynomial Hamiltonian systems of $m$ degrees of freedom. Using the differential Galois theory, we perform a complete classification for meromorphic integrability of these polynomial critical systems. As a by-product of this classification, it gives a positive answer to the conjecture raised by Tian, Wei and Zhang [J. Math. Phys. 64, 092701 (2023)] on the critical systems for optimal eigenvalue gaps. The numerical simulations of the Poincar\'{e} cross sections show that the critical systems for sums of eigenvalues can appear complex dynamical phenomena, such as periodic trajectories, quasi-periodic trajectories and chaos

Regularity theory for nonlocal equations with general growth in the Heisenberg group

We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate in the spirit of De Giorgi-Nash-Moser theory, some local properties of weak solutions to such kind of problems, involving boundedness, H\"{o}lder continuity and Harnack inequality. To this end, an improved nonlocal Caccioppoli-type estimate as the main auxiliary ingredient is exploited several times

Feature Generation by Convolutional Neural Network for Click-Through Rate Prediction

Click-Through Rate prediction is an important task in recommender systems, which aims to estimate the probability of a user to click on a given item. Recently, many deep models have been proposed to learn low-order and high-order feature interactions from original features. However, since useful interactions are always sparse, it is difficult for DNN to learn them effectively under a large number of parameters. In real scenarios, artificial features are able to improve the performance of deep models (such as Wide & Deep Learning), but feature engineering is expensive and requires domain knowledge, making it impractical in different scenarios. Therefore, it is necessary to augment feature space automatically. In this paper, We propose a novel Feature Generation by Convolutional Neural Network (FGCNN) model with two components: Feature Generation and Deep Classifier. Feature Generation leverages the strength of CNN to generate local patterns and recombine them to generate new features. Deep Classifier adopts the structure of IPNN to learn interactions from the augmented feature space. Experimental results on three large-scale datasets show that FGCNN significantly outperforms nine state-of-the-art models. Moreover, when applying some state-of-the-art models as Deep Classifier, better performance is always achieved, showing the great compatibility of our FGCNN model. This work explores a novel direction for CTR predictions: it is quite useful to reduce the learning difficulties of DNN by automatically identifying important features