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Quasi-likelihood-based EM algorithm for regime-switching SDE

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arXiv 2024年

作者： Cheng, Yuzhong Masuda, Hiroki Graduate School of Mathematics Kyushu University 744 Motooka Nishi-ku Fukuoka819-0395 Japan Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo153-8914 Japan

This paper considers estimating the parameters in a regime-switching stochastic differential equation(SDE) driven by Normal Inverse Gaussian(NIG) noise. The model under consideration incorporates a continuous-time finite state Markov chain to capture regime changes, enabling a more realistic representation of evolving market conditions or environmental factors. Although the continuous dynamics are typically observable, the hidden nature of the Markov chain introduces significant complexity, rendering standard likelihood-based methods less effective. To address these challenges, we propose an estimation algorithm designed for discrete, high-frequency observations, even when the Markov chain is not directly observed. Our approach integrates the expectation-maximization (EM) algorithm, which iteratively refines parameter estimates in the presence of latent variables, with a quasi-likelihood method adapted to NIG noise. Notably, this method can simultaneously estimate parameters within both the SDE coefficients and the driving noise. Simulation results are provided to evaluate the performance of the algorithm. These experiments demonstrate that the proposed method provides reasonable estimation under challenging conditions. © 2024, CC BY.

关键词： expectation maximization algorithm

A Large Confirmatory Dynamic Factor Model for Stock Market Returns in Different Time Zones

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arXiv 2022年

作者： Linton, Oliver B. Tang, Haihan Wu, Jianbin Faculty of Economics University of Cambridge Austin Robinson Building Sidgwick Avenue CambridgeCB3 9DD United Kingdom International School of Finance Fudan University 220 Handan Road Yangpu District Shanghai200433 China School of Business Nanjing University 22 Hankou Road Gulou District Nanjing210093 China

We propose a confirmatory dynamic factor model for a large number of stocks whose returns are observed daily across multiple time zones. The model has a global factor and a continental factor that both drive the individual stock return series. We propose two estimators of the model: a quasi-maximum likelihood estimator (QML-just-identified), and an improved estimator based on an expectation maximization (EM) algorithm (QML-all-res). Our estimators are consistent and asymptotically normal under the large approximate factor model setting. In particular, the asymptotic distributions of QML-all-res are the same as those of the infeasible OLS estimators that treat factors as known and utilize all the restrictions on the parameters of the model. We apply the model to MSCI equity indices of 42 developed and emerging markets, and find that most markets are more integrated when the CBOE Volatility Index (VIX) is high. © 2022, CC BY-NC-ND.

关键词： expectation maximization algorithm

In-context Learning for Mixture of Linear Regressions: Existence, Generalization and Training Dynamics

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arXiv 2024年

作者： Jin, Yanhao Balasubramanian, Krishnakumar Lai, Lifeng Department of Statistics University of California Davis United States Department of Electrical and Computer Engineering University of California Davis United States

We investigate the in-context learning capabilities of transformers for the d-dimensional mixture of linear regression model, providing theoretical insights into their existence, generalization bounds, and training dynamics. Specifically, we prove that there exists a transformer capable of achieving a prediction error of order O(√d/n) with high probability, where n represents the training prompt size in the high signal-to-noise ratio (SNR) regime. Moreover, we derive in-context excess risk bounds of order O(L/√B) for the case of two mixtures, where B denotes the number of training prompts, and L represents the number of attention layers. The dependence of L on the SNR is explicitly characterized, differing between low and high SNR settings. We further analyze the training dynamics of transformers with single linear self-attention layers, demonstrating that, with appropriately initialized parameters, gradient flow optimization over the population mean square loss converges to a global optimum. Extensive simulations suggest that transformers perform well on this task, potentially outperforming other baselines, such as the expectation-maximization algorithm. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

A Thorough Comparison Between Independent Cascade and Susceptible-Infected-Recovered Models

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arXiv 2024年

作者： Liu, Panfeng Qiu, Guoliang Tao, Biaoshuai Yang, Kuan Shanghai Jiao Tong University China

We study cascades in social networks with the independent cascade (IC) model and the Susceptible-Infected-recovered (SIR) model. The well-studied IC model fails to capture the feature of node recovery, and the SIR model is a variant of the IC model with the node recovery feature. In the SIR model, by computing the probability that a node successfully infects another before its recovery and viewing this probability as the corresponding IC parameter, the SIR model becomes an "out-going-edge-correlated" version of the IC model: the events of the infections along different out-going edges of a node become dependent in the SIR model, whereas these events are independent in the IC model. In this paper, we thoroughly compare the two models and examine the effect of this extra dependency in the SIR model. By a carefully designed coupling argument, we show that the seeds in the IC model have a stronger influence spread than their counterparts in the SIR model, and sometimes it can be significantly stronger. Specifically, we prove that, given the same network, the same seed sets, and the parameters of the two models being set based on the above-mentioned equivalence, the expected number of infected nodes at the end of the cascade for the IC model is weakly larger than that for the SIR model, and there are instances where this dominance is significant. We also study the influence maximization problem (the optimization problem of selecting a set of nodes as initial seeds in a social network to maximize their influence) with the SIR model. We show that the above-mentioned difference in the two models yields different seed-selection strategies, which motivates the design of influence maximization algorithms specifically for the SIR model. We design efficient approximation algorithms with theoretical guarantees by adapting the reverse-reachable-set-based algorithms, commonly used for the IC model, to the SIR model. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

Learning Submodular Sequencing from Samples

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arXiv 2024年

作者： Yuan, Jing Tang, Shaojie Department of Computer Science and Engineering The University of North Texas United States Department of Management Science and Systems University at Buffalo United States

This paper addresses the problem of sequential submodular maximization: selecting and ranking items in a sequence to optimize some composite submodular function. In contrast to most of the previous works, which assume access to the utility function, we assume that we are given only a set of samples. Each sample includes a random sequence of items and its associated utility. We present an algorithm that, given polynomially many samples drawn from a two-stage uniform distribution, achieves an approximation ratio dependent on the curvature of individual submodular functions. Our results apply in a wide variety of real-world scenarios, such as ranking products in online retail platforms, where complete knowledge of the utility function is often impossible to obtain. Our algorithm gives an empirically useful solution in such contexts, thus proving that limited data can be of great use in sequencing tasks. From a technical perspective, our results extend prior work on "optimization from samples" by generalizing from optimizing a set function to a sequence-dependent function. © 2024, CC BY.

关键词： expectation maximization algorithm

Fitting Multiple Machine Learning Models with Performance Based Clustering

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arXiv 2024年

作者： Lorasdagi, Mehmet E. Koc, Ahmet B. Koc, Ali T. Kozat, Suleyman S. The Department of Electrical and Electronics Engineering Bilkent University Turkey

Traditional machine learning approaches assume that data comes from a single generating mechanism, which may not hold for most real life data. In these cases, the single mechanism assumption can result in suboptimal performance. We introduce a clustering framework that eliminates this assumption by grouping the data according to the relations between the features and the target values, and we obtain multiple separate models to learn different parts of the data. We further extend our framework to applications having streaming data where we produce outcomes using an ensemble of models. For this, the ensemble weights are updated based on the incoming data batches. We demonstrate the performance of our approach over the widely-studied real life datasets, showing significant improvements over the traditional single-model approaches. © 2024, CC BY-NC-ND.

关键词： expectation maximization algorithm

Community Detection with Heterogeneous Block Covariance Model

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arXiv 2024年

作者： Li, Xiang Zhao, Yunpeng Pan, Qing Hao, Ning Department of Statistics George Washington University United States Department of Statistics Colorado State University United States Department of Biostatistics and Bioinformatics George Washington University United States Department of Mathematics University of Arizona United States

Community detection is the task of clustering objects based on their pairwise relationships. Most of the model-based community detection methods, such as the stochastic block model and its variants, are designed for networks with binary (yes/no) edges. In many practical scenarios, edges often possess continuous weights, spanning positive and negative values, which reflect varying levels of connectivity. To address this challenge, we introduce the heterogeneous block covariance model (HBCM) that defines a community structure within the covariance matrix, where edges have signed and continuous weights. Furthermore, it takes into account the heterogeneity of objects when forming connections with other objects within a community. A novel variational expectation-maximization algorithm is proposed to estimate the group membership. The HBCM provides provable consistent estimates of memberships, and its promising performance is observed in numerical simulations with different setups. The model is applied to a single-cell RNA-seq dataset of a mouse embryo and a stock price dataset. Supplementary materials for this article are available online. Copyright © 2024, The Authors. All rights reserved.

关键词： expectation maximization algorithm

High-Dimensional Markov-switching Ordinary Differential Processes

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arXiv 2024年

作者： Tsai, Katherine Kolar, Mladen Koyejo, Sanmi Department of Electrical and Computer Engineering University of Illinois Urbana-Champaign United States Marshall School of Business University of Southern California United States Department of Computer Science Stanford University United States

We investigate the parameter recovery of Markov-switching ordinary differential processes from discrete observations, where the differential equations are nonlinear additive models. This framework has been widely applied in biological systems, control systems, and other domains;however, limited research has been conducted on reconstructing the generating processes from observations. In contrast, many physical systems, such as human brains, cannot be directly experimented upon and rely on observations to infer the underlying systems. To address this gap, this manuscript presents a comprehensive study of the model, encompassing algorithm design, optimization guarantees, and quantification of statistical errors. Specifically, we develop a two-stage algorithm that first recovers the continuous sample path from discrete samples and then estimates the parameters of the processes. We provide novel theoretical insights into the statistical error and linear convergence guarantee when the processes are β-mixing. Our analysis is based on the truncation of the latent posterior processes and demonstrates that the truncated processes approximate the true processes under mixing conditions. We apply this model to investigate the differences in resting-state brain networks between the ADHD group and normal controls, revealing differences in the transition rate matrices of the two groups. © 2024, CC BY.

关键词： expectation maximization algorithm

A Model-Based Clustering Approach for Bounded Data Using Transformation-Based Gaussian Mixture Models

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arXiv 2024年

作者： Scrucca, Luca Department of Statistical Sciences University of Bologna Italy

The clustering of bounded data presents unique challenges in statistical analysis due to the constraints imposed on the data values. This paper introduces a novel method for model-based clustering specifically designed for bounded data. Building on the transformation-based approach to Gaussian mixture density estimation introduced by Scrucca (2019), we extend this framework to develop a probabilistic clustering algorithm for data with bounded support that allows for accurate clustering while respecting the natural bounds of the variables. In our proposal, a flexible range-power transformation is employed to map the data from its bounded domain to the unrestricted real space, hence enabling the estimation of Gaussian mixture models in the transformed space. This approach leads to improved cluster recovery and interpretation, especially for complex distributions within bounded domains. The performance of the proposed method is evaluated through real-world data applications involving both fully and partially bounded data, in both univariate and multivariate settings. The results demonstrate the effectiveness and advantages of our approach over traditional and advanced model-based clustering techniques that employ distributions with bounded support. © 2024, CC BY.

关键词： expectation maximization algorithm