Latent tree models

Latent tree models are graphical models defined on trees, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributions to generalise star-decomposable distributions such as the latent class model. Latent tree models, or their submodels, are widely used in: phylogenetic analysis, network tomography, computer vision, causal modeling, and data clustering. They also contain other well-known classes of models like hidden Markov models, Brownian motion tree model, the Ising model on a tree, and many popular models used in phylogenetics. This article offers a concise introduction to the theory of latent tree models. We emphasise the role of tree metrics in the structural description of this model class, in designing learning algorithms, and in understanding fundamental limits of what and when can be learned

Outlier detection algorithms over fuzzy data with weighted least squares

In the classical leave-one-out procedure for outlier detection in regression analysis, we exclude an observation and then construct a model on the remaining data. If the difference between predicted and observed value is high we declare this value an outlier. As a rule, those procedures utilize single comparison testing. The problem becomes much harder when the observations can be associated with a given degree of membership to an underlying population, and the outlier detection should be generalized to operate over fuzzy data. We present a new approach for outlier detection that operates over fuzzy data using two inter-related algorithms. Due to the way outliers enter the observation sample, they may be of various order of magnitude. To account for this, we divided the outlier detection procedure into cycles. Furthermore, each cycle consists of two phases. In Phase 1, we apply a leave-one-out procedure for each non-outlier in the dataset. In Phase 2, all previously declared outliers are subjected to Benjamini–Hochberg step-up multiple testing procedure controlling the false-discovery rate, and the non-confirmed outliers can return to the dataset. Finally, we construct a regression model over the resulting set of non-outliers. In that way, we ensure that a reliable and high-quality regression model is obtained in Phase 1 because the leave-one-out procedure comparatively easily purges the dubious observations due to the single comparison testing. In the same time, the confirmation of the outlier status in relation to the newly obtained high-quality regression model is much harder due to the multiple testing procedure applied hence only the true outliers remain outside the data sample. The two phases in each cycle are a good trade-off between the desire to construct a high-quality model (i.e., over informative data points) and the desire to use as much data points as possible (thus leaving as much observations as possible in the data sample). The number of cycles is user defined, but the procedures can finalize the analysis in case a cycle with no new outliers is detected. We offer one illustrative example and two other practical case studies (from real-life thrombosis studies) that demonstrate the application and strengths of our algorithms. In the concluding section, we discuss several limitations of our approach and also offer directions for future research

Outlier detection algorithms over fuzzy data with weighted least squares

In the classical leave-one-out procedure for outlier detection in regression analysis, we exclude an observation and then construct a model on the remaining data. If the difference between predicted and observed value is high we declare this value an outlier. As a rule, those procedures utilize single comparison testing. The problem becomes much harder when the observations can be associated with a given degree of membership to an underlying population, and the outlier detection should be generalized to operate over fuzzy data. We present a new approach for outlier detection that operates over fuzzy data using two inter-related algorithms. Due to the way outliers enter the observation sample, they may be of various order of magnitude. To account for this, we divided the outlier detection procedure into cycles. Furthermore, each cycle consists of two phases. In Phase 1, we apply a leave-one-out procedure for each non-outlier in the dataset. In Phase 2, all previously declared outliers are subjected to Benjamini–Hochberg step-up multiple testing procedure controlling the false-discovery rate, and the non-confirmed outliers can return to the dataset. Finally, we construct a regression model over the resulting set of non-outliers. In that way, we ensure that a reliable and high-quality regression model is obtained in Phase 1 because the leave-one-out procedure comparatively easily purges the dubious observations due to the single comparison testing. In the same time, the confirmation of the outlier status in relation to the newly obtained high-quality regression model is much harder due to the multiple testing procedure applied hence only the true outliers remain outside the data sample. The two phases in each cycle are a good trade-off between the desire to construct a high-quality model (i.e., over informative data points) and the desire to use as much data points as possible (thus leaving as much observations as possible in the data sample). The number of cycles is user defined, but the procedures can finalize the analysis in case a cycle with no new outliers is detected. We offer one illustrative example and two other practical case studies (from real-life thrombosis studies) that demonstrate the application and strengths of our algorithms. In the concluding section, we discuss several limitations of our approach and also offer directions for future research