Generative artificial intelligence (GenAI) has been developing with many incredible achievements like ChatGPT and Bard. Deep generative model (DGM) is a branch of GenAI, which is preeminent in generating raster data such as image and sound due to strong points of deep neural network (DNN) in inference and recognition. The built-in inference mechanism of DNN, which simulates and aims to synaptic plasticity of human neuron network, fosters generation ability of DGM which produces surprised results with support of statistical flexibility. Two popular approaches in DGM are Variational Autoencoders (VAE) and Generative Adversarial Network (GAN). Both VAE and GAN have their own strong points although they share and imply underline theory of statistics as well as incredible complex via hidden layers of DNN when DNN becomes effective encoding/decoding functions without concrete specifications. In this research, VAE and GAN is unified into a consistent and consolidated model called Adversarial Variational Autoencoders (AVA) in which VAE and GAN complement each other, for instance, VAE is good at generator by encoding data via excellent ideology of Kullback-Leibler divergence and GAN is a significantly important method to assess reliability of data which is realistic or fake. In other words, AVA aims to improve accuracy of generative models, besides AVA extends function of simple generative models. In methodology this research focuses on combination of applied mathematical concepts and skillful techniques of computer programming in order to implement and solve complicated problems as simply as possible.

Machine learning forks into three main branches such as supervised learning, unsupervised learning, and reinforcement learning where reinforcement learning is much potential to artificial intelligence (AI) applications because it solves real problems by progressive process in which possible solutions are improved and finetuned continuously. The progressive approach, which reflects ability of adaptation, is appropriate to the real world where most events occur and change continuously and unexpectedly. Moreover, data is getting too huge for supervised learning and unsupervised learning to draw valuable knowledge from such huge data at one time. Bayesian optimization (BO) models an optimization problem as a probabilistic form called surrogate model and then directly maximizes an acquisition function created from such surrogate model in order to maximize implicitly and indirectly the target function for finding out solution of the optimization problem. A popular surrogate model is Gaussian process regression model. The process of maximizing acquisition function is based on updating posterior probability of surrogate model repeatedly, which is improved after every iteration. Taking advantages of acquisition function or utility function is also common in decision theory but the semantic meaning behind BO is that BO solves problems by progressive and adaptive approach via updating surrogate model from a small piece of data at each time, according to ideology of reinforcement learning. Undoubtedly, BO is a reinforcement learning algorithm with many potential applications and thus it is surveyed in this research with attention to its mathematical ideas. Moreover, the solution of optimization problem is important to not only applied mathematics but also AI.

Global optimization is an imperative development of local optimization because there are many problems in artificial intelligence and machine learning requires highly acute solutions over entire domain. There are many methods to resolve the global optimization, which can be classified into three groups such as analytic methods (purely mathematical methods), probabilistic methods, and heuristic methods. Especially, heuristic methods like particle swarm optimization and ant bee colony attract researchers because their effective and practical techniques which are easy to be implemented by computer programming languages. However, these heuristic methods are lacking in theoretical mathematical fundamental. Fortunately, minima distribution establishes a strict mathematical relationship between optimized target function and its global minima. In this research, I try to study minima distribution and apply it into explaining convergence and convergence speed of optimization algorithms. Especially, weak conditions of convergence and monotonicity within minima distribution are drawn so as to be appropriate to practical optimization methods.

Particle swarm optimization (PSO) is an effective algorithm to solve the optimization problem in case that derivative of target function is inexistent or difficult to be determined. Because PSO has many parameters and variants, I propose a general framework of PSO called GPSO which aggregates important parameters and generalizes important variants so that researchers can customize PSO easily. Moreover, two main properties of PSO are exploration and exploitation. The exploration property aims to avoid premature converging so as to reach global optimal solution whereas the exploitation property aims to motivate PSO to converge as fast as possible. These two aspects are equally important. Therefore, GPSO also aims to balance the exploration and the exploitation. It is expected that GPSO supports users to tune parameters for not only solving premature problem but also fast convergence.

Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data. In other words, the relationship helps us know hidden data by surveying observed data. The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data. Pioneers in EM algorithm proved its convergence. As a result, EM algorithm produces parameter estimators as well as MLE does. This tutorial aims to provide explanations of EM algorithm in order to help researchers comprehend it. Moreover some improvements of EM algorithm are also proposed in the tutorial such as combination of EM and third-order convergence Newton-Raphson process, combination of EM and gradient descent method, and combination of EM and particle swarm optimization (PSO) algorithm.

Dyadic data contains co-occurrences of objects, which is often modeled by finite mixture model which in turn is learned by expectation maximization (EM) algorithm. Objects in traditional dyadic data are identified by names, causing the drawback which is that it is impossible to extract implicit valuable knowledge under objects. In this research, I propose the so-called attributed dyadic data (ADD) in which each object has an informative attribute and each co-occurrence of two objects is associated with a value. ADD is flexible and covers most of structures / forms of dyadic data. Conditional mixture model (CMM), which is a variant of finite mixture model, is applied into learning ADD. Moreover, a significant feature of CMM is that any co-occurrence of two objects is based on some conditional variable. As a result, CMM can predict or estimate co-occurrent values based on regression model, which extends applications of ADD and CMM.

Dyadic data which is also called co-occurrence data (COD) contains co-occurrences of objects. Searching for statistical models to represent dyadic data is necessary. Fortunately, finite mixture model is a solid statistical model to learn and make inference on dyadic data because mixture model is built smoothly and reliably by expectation maximization (EM) algorithm which is suitable to inherent spareness of dyadic data. This research summarizes mixture models for dyadic data. When each co-occurrence in dyadic data is associated with a value, there are many unaccomplished values because a lot of co-occurrences are inexistent. In this research, these unaccomplished values are estimated as mean (expectation) of random variable given partial probabilistic distributions inside dyadic mixture model.

Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating statistical parameter when data sample contains hidden part and observed part. EM is applied to learn finite mixture model in which the whole distribution of observed variable is average sum of partial distributions. Coverage ratio of every partial distribution is specified by the probability of hidden variable. An application of mixture model is soft clustering in which cluster is modeled by hidden variable whereas each data point can be assigned to more than one cluster and degree of such assignment is represented by the probability of hidden variable. However, such probability in traditional mixture model is simplified as a parameter, which can cause loss of valuable information. Therefore, in this research I propose a so-called conditional mixture model (CMM) in which the probability of hidden variable is modeled as a full probabilistic density function (PDF) that owns individual parameter. CMM aims to extend mixture model. I also propose an application of CMM which is called adaptive regressive model (ARM). Traditional regression model is effective when data sample is scattered equally. If data points are grouped into clusters, regression model tries to learn a unified regression function which goes through all data points. Obviously, such unified function is not effective to evaluate response variable based on grouped data points. The concept “adaptive” of ARM means that ARM solves the ineffectiveness problem by selecting the best cluster of data points firstly and then evaluating response variable within such best cluster. In order words, ARM reduces estimation space of regression model so as to gain high accuracy in calculation.

Fetal weight estimation before delivery is important in obstetrics, which assists doctors diagnose abnormal or diseased cases. Linear regression based on ultrasound measures such as bi-parietal diameter (bpd), head circumference (hc), abdominal circumference (ac), and fetal length (fl) is common statistical method for weight estimation but the regression model requires that time points of collecting such measures must not be too far from last ultrasound scans. Therefore this research proposes a method of early weight estimation based on expectation maximization (EM) algorithm so that ultrasound measures can be taken at any time points in gestational period. In other words, gestational sample can lack some or many fetus weights, which gives facilities to practitioners because practitioners need not concern fetus weights when taking ultrasound examinations. The proposed method is called dual regression expectation maximization (DREM) algorithm. Experimental results indicate that accuracy of DREM decreases insignificantly when completion of ultrasound sample decreases significantly. So it is proved that DREM withstands missing values in incomplete sample or sparse sample.

Land use/land cover (LULC) and climate changes are two main factors directly affecting hydrologic conditions. However, very few studies in Vietnam have investigated changes in hydrological process under the impact of climate and land use changes on a basin scale. The objective of this study is to assess the individual and combined impacts of land use and climate changes on hydrological processes for the Nam Rom river basin, Northwestern Viet Nam using Remote Sensing (RS) and Soil and Water Assessment Tools (SWAT) model. SWAT model was used for hydrological process simulation. Results indicated that SWAT proved to be a powerful tool in simulating the impacts of land use and climate change on catchment hydrology. The change in historical land use between 1992 and 2015 strongly contributed to increasing hydrological processes (ET, percolation, ground water, and water yield), whereas, climate change led to significant decrease of all hydrological components. The combination of land use and climate changes significantly reduced surface runoff (-16.9%), ground water (-5.7%), water yield (-9.2%), and sediment load (-4.9%). Overall climatic changes had more significant effect on hydrological components than land use changes in the Nam Rom river basin during the 1992–2015. Under impacts of projected land use and climate change scenarios in 2030 on hydrological process of the upper Nam Rom river basin indicate that ET and surface flow are more sensitive to the changes in land use and climate in the future. In conclusion, the findings of this study will basic knowledge of the effects of climate and land-use changes on the hydrology for future development of integrated land use and water management practices in Nam Rom river basin.

Nonwoven fabrics have been widely used in textile manufacturing industry as a sheet or web structure because of soft, water-repellent, recycle, ecological and resilient functions. Ultrasonic welding method has been applied for bonding nonwoven fabrics due to clean, fast and reliable approach. In this work, the ultrasonic stepped horn is designed to generate uniform amplitudes on the working surface by using finite element analysis (FEA) simulation. Chromium carbon steels are utilized to produce ultrasonic horns due to high wear resistant and hardness. Isotactic polypropylene nonwoven fabrics fabricated by spunbond process were bonded by continuous ultrasonic sewing machine. Ultrasonic horn with 70 mm in diameter working at 20 kHz, polypropylene (PP) nonwoven density of 80 gsm and various design of welding joints were applied. A typical image in the case of number one was investigated by the scanning electron microscope (SEM) images of inter-facial micro-structure. However, welding joints of totally eight roller patterns was test the tensile strength of the ultrasonic welding joints on PP nonwoven fabrics. The tensile strength of the welding joints is proportional to the area ratio between the welding area and cycling area. The results showed that the melted zone without welding defects such as crack or blowhole can be seen. Furthermore, the tensile strength of welding joints in eight cases of roller patterns (No.1-No.8) was described in details. The ultrasonic welding joints with brick structures give higher tensile strength while the solid line in the pattern gave less strength.

The material flow dynamic and velocity distribution on the melted domain surface play a crucial role on the joint quality and formation of welding defects. In this study, authors investigated the effects of the low and high currents of plasma arc welding on the material flow and thermodynamics of molten pool and its relationship to the welding defects. The high-speed video camera (HSVC) was used to observe the convection of the melted domain and welded-joint appearance. Furthermore, to consider the Marangoni force activation, the temperature on the melted domain was measured by a thermal HSVC. The results revealed that the velocity distribution on the weld surface was higher than that inside the molten weld pool due to the difference of the massive density between the air and the steel. Moreover, in the case of low welding current (80A) the convection speed of molten was faster than that of the high welding current case (160A) owing to the difference of main driving forces direction and strength, which leading to undercut and humping defects on the weld surface and excessive convex (burn-through) defect at the bottom weld side, respectively. The medium welding current (120A) had two convection patterns with the main flow in backward direction, which resulted in better welding quality without defect. The interaction between the shear force and Marangoni force played a solid state on the convection and heat transportation processes in the plasma arc welding process.