Semi- and non-parametric mixture of normal regression models are a flexible class of mixture of regression models. These models assume that the component mixing proportions, regression functions and/or variances are non-parametric functions of the covariates. Among this class of models, the semi-parametric mixture of partially linear models (SPMPLMs) combine the desirable interpretability of a parametric model and the flexibility of a non-parametric model. However, local-likelihood estimation of the non-parametric term poses a computational challenge. Traditional EM optimisation of the local-likelihood functions is not appropriate due to the label-switching problem. Separately applying the EM algorithm on each local-likelihood function will likely result in non-smooth function estimates. This is because the local responsibilities calculated at the E-step of each local EM are not guaranteed to be aligned. To prevent this, the EM algorithm must be modified so that the same (global) responsibilities are used at each local M-step. In this paper, we propose a one-step backfitting EM-type algorithm to estimate the SPMPLMs and effectively address the label-switching problem. The proposed algorithm estimates the non-parametric term using each set of local responsibilities in turn and then incorporates a smoothing step to obtain the smoothest estimate. In addition, to reduce the computational burden imposed by the use of the partial-residuals estimator of the parametric term, we propose a plug-in estimator. The performance and practical usefulness of the proposed methods was tested using a simulated dataset and two real datasets, respectively. Our finite sample analysis revealed that the proposed methods are effective at solving the label-switching problem and producing reasonable and interpretable results in a reasonable amount of time.

The non-parametric Gaussian mixture of regressions (NPGMRs) model serves as a flexible approach for the determination of latent heterogeneous regression relationships. This model assumes that the component means, variances and mixing proportions are smooth unknown functions of the covariates where the error distribution of each component is assumed to be Gaussian and hence symmetric. These functions are estimated over a set of grid points using the Expectation-Maximization (EM) algorithm to maximise the local-likelihood functions. However, maximizing each local-likelihood function separately does not guarantee that the local responsibilities and corresponding labels, obtained at the E-step of the EM algorithm, align at each grid point leading to a label-switching problem. This results in non-smooth estimated component regression functions. In this paper, we propose an estimation procedure to account for label switching by tracking the roughness of the estimated component regression functions. We use the local responsibilities to obtain a global estimate of the responsibilities which are then used to maximize each local-likelihood function. The performance of the proposed procedure is demonstrated using a simulation study and through an application using real world data. In the case of well-separated mixture regression components, the procedure gives similar results to competitive methods. However, in the case of poorly separated mixture regression components, the procedure outperforms competitive methods.

Semi-parametric Gaussian mixtures of non-parametric regressions (SPGMNRs) are a flexible extension of Gaussian mixtures of linear regressions (GMLRs). The model assumes that the component regression functions (CRFs) are non-parametric functions of the covariate(s) whereas the component mixing proportions and variances are constants. Unfortunately, the model cannot be reliably estimated using traditional methods. A local-likelihood approach for estimating the CRFs requires that we maximize a set of local-likelihood functions. Using the Expectation-Maximization (EM) algorithm to separately maximize each local-likelihood function may lead to label-switching. This is because the posterior probabilities calculated at the local E-step are not guaranteed to be aligned. The consequence of this label-switching is wiggly and non-smooth estimates of the CRFs. In this paper, we propose a unified approach to address label-switching and obtain sensible estimates. The proposed approach has two stages. In the first stage, we propose a model-based approach to address the label-switching problem. We first note that each local-likelihood function is a likelihood function of a Gaussian mixture model (GMM). Next, we reformulate the SPGMNRs model as a mixture of these GMMs. Lastly, using a modified version of the Expectation Conditional Maximization (ECM) algorithm, we estimate the mixture of GMMs. In addition, using the mixing weights of the local GMMs, we can automatically choose the local points where local-likelihood estimation takes place. In the second stage, we propose one-step backfitting estimates of the parametric and non-parametric terms. The effectiveness of the proposed approach is demonstrated on simulated data and real data analysis.

Joint modelling of longitudinal and time-to-event data is a powerful tool for analysing complex medical data. Joint latent class models (JLCMs), in particular, provide additional prognostic insight by clustering subjects into latent subgroups within a mixture of experts framework. Traditionally, the mixing proportions are modelled using a multinomial logistic regression. Recent work in mixture regression has proposed a neural network alternative, implemented through the neural network expectation–maximization algorithm, which offers greater flexibility by capturing nonlinearities that may be missed by logistic regression. In this article, we extend the JLCM by applying the neural network to the modelling the mixing proportions captured in the class-membership submodel. The model estimated using an expectation–maximization-type algorithm. We call this the neural-network JLCM (NN-JLCM). We formally define the method, evaluate its performance through extensive simulations, and apply it to real data. The results show that NN-JLCM improves predictive accuracy and class separation compared with traditional JLCMs, while maintaining competitiveness in terms of overall model fit. The code for the article is available at https://github.com/tharris0924/nn-jlcm_p1 .

Semi- and non-parametric mixture of regressions are a very useful flexible class of mixture of regressions in which some or all of the parameters are non-parametric functions of the covariates. These models are, however, based on the Gaussian assumption of the component error distributions. Thus, their estimation is sensitive to outliers and heavy-tailed error distributions. In this paper, we propose semi- and non-parametric contaminated Gaussian mixture of regressions to robustly estimate the parametric and/or non-parametric terms of the models in the presence of mild outliers. The virtue of using a contaminated Gaussian error distribution is that we can simultaneously perform model-based clustering of observations and model-based outlier detection. We propose two algorithms, an expectation-maximization (EM)-type algorithm and an expectation-conditional-maximization (ECM)-type algorithm, to perform maximum likelihood and local-likelihood kernel estimation of the parametric and non-parametric of the proposed models, respectively. The robustness of the proposed models is examined using an extensive simulation study. The practical utility of the proposed models is demonstrated using real data.

Regression models, where the response variable is circular, are common in areas such as biology, geology and meteorology. A typical model assumes that the conditional distribution of the response follows a von-Mises distribution. However, this assumption is inadequate when the response variable is multimodal. For this reason, in this paper, a finite mixture of regressions model is proposed for the case of a circular response variable and a set of circular and/or linear covariates. Mixture models are very useful when the underlying population is multimodal. Despite the prevalence of multimodality in regression modelling of circular data, the use of mixtures of regressions has received no attention in the literature. This paper aims to close this knowledge gap. To estimate the proposed model, we develop a maximum likelihood estimation procedure via the Expectation-Maximization algorithm. An extensive simulation study is used to demonstrate the practical use and performance of the proposed model and estimation procedure. In addition, the model is shown to be useful as a model-based clustering tool. Lastly, the model is applied to a real dataset from a wind farm in South Africa which highlights its advantage compared to a competitive approach for model-based clustering of circular data.