Penalized principal machines (P2M) are proposed for sparse sufficient dimension reduction. While the earlier penalized principal logistic regression (P2LR) method enjoys desirable theoretical properties and promising finite-sample performance, its difference-of-convex (DC) solver is computationally intensive. To address this computational limitation, a group coordinate descent (GCD) algorithm is employed to efficiently solve P2M with various loss functions. The GCD algorithm was originally developed for group-penalized least squares problems and is thus directly applicable to P2M with the squared loss. This algorithm is further extended to accommodate various loss functions, both differentiable and non-differentiable, via a quadratic approximation, resulting in a unified computational framework. Simulation studies and real data analyses demonstrate that the proposed methods are highly efficient in both computation and estimation of the central subspace, while producing interpretable, sparse solutions. The R package ppmSDR implements the proposed GCD algorithm for P2Ms.
The hemodynamic response (HR) in event-related functional magnetic resonance imaging is typically assumed to be stationary. While there are some approaches in the literature to model nonstationary HRs, few focus on change points. Two novel procedures to investigate change points in the HR are proposed. Both procedures make inference on the existence of sudden changes for multi-subject data. The change point locations may vary between subjects, conditions, and brain regions. The first procedure utilizes available information about the change point locations to compare multiple shape characteristics of the HR over time. In the second procedure, the change point locations are determined for each subject separately. To account for the estimation of the change point locations, the notion of post selection variance is proposed. The power of the proposed procedures is assessed in simulation studies. The procedure for pre-specified change point locations is applied to data from a category learning experiment.
A new model is introduced, the Gaussian Factorial Explicit Duration Hidden Markov Model (G-F-ED-HMM), that overcomes limiting assumptions of hidden Markov models by allowing flexible choice of distributions for the sojourn time, handling semi-Markov dynamics, and explicitly modelling a multidimensional hidden state with factorial structure. The combination of these extensions is relevant for the representation of complex dynamics that can occur in ecology as well as in signal analysis. A novel Variational EM (VEM) algorithm is provided for parameter estimation of the G-F-ED-HMM. The algorithm is computationally efficient due to the selection of a variational conditional distribution within the family of independent hidden chains. The consequence is that the VM step relies only on quantities related to a monochain hidden semi-Markov model, which are straightforward to compute. The behavior of the VEM algorithm is illustrated through its application to the problem of inferring migratory bird routes, using both simulated and real datasets. These experiments show that the proposed VEM algorithm recovers parameters with low error and performs significantly faster than a Monte Carlo EM algorithm.
Quantifying the effect of covariates on reciprocity, or the tendency for actors to form mutual relationships, and homophily, or the tendency for actors to form ties with similar actors, is a fundamental task in analyzing directed networks. However, in many dyad-dependent network models, the parameters governing a covariate’s effect on these two mechanisms are confounded, which makes their interpretation difficult. To address this issue, a new latent space model is introduced that disentangles the effect of covariates on homophily and reciprocity by separating the marginal probability model for tie formation from the dependence structure between ties. Two Bayesian inference procedures are proposed to ensure feasible estimation across network sizes. For small to medium-sized networks, a Hamiltonian Monte Carlo algorithm is introduced to sample from the parameters’ posterior distribution. For large networks, a structured mean-field variational approximation to this posterior and a fast stochastic variational inference algorithm is developed. The method’s ability to determine how covariates influence reciprocity and homophily is demonstrated through simulation studies and applications to friendship networks from the National Longitudinal Survey of Adolescent Health.
A limit order book (LOB) queueing system is considered in which limit orders are generated by Markov modulated Poisson processes (MMPP) to capture the clustered nature of order arrivals. The queueing dynamics are represented by a multidimensional birth-death type Markov chain, and the probability distributions of the state variables are obtained through matrix computing procedures for Markov chains. By modifying the Markov chain structure and assigning selected states as absorbing states, two conditional probabilities relevant to high-frequency trading are computed: the probability of a midprice increase and the probability of order execution before a midprice change. Numerical results show that clustered MMPP arrivals substantially influence both probabilities, indicating that clustering in the limit order arrival process is an important factor in limit order book queueing models.
