The Bayesian information criterion (BIC), defined as the observed data log likelihood minus a penalty term based on the sample size $N$, is a popular model selection criterion for factor analysis with complete data. This definition has also been suggested for incomplete data. However, the penalty term based on the `complete' sample size $N$ is the same no matter whether in a complete or incomplete data case. For incomplete data, there are often only $N_i

In unsupervised learning, Bayesian Nonparametric (BNP) mixture models, exemplified by the Dirichlet-Process Mixture Model (DPMM), provide a principled approach for Bayesian modeling while adapting the model complexity to the data. This contrasts with finite mixture models whose complexity is determined manually or via model-selection methods. To fix ideas, an important DPMM example is the Dirichlet-Process Gaussian Mixture Model (DPGMM), a Bayesian -dimensional extension of the classical Gaussian Mixture Model (GMM). Despite their potential, however, and although researchers have used them successfully in numerous applications during the last two decades, DPMMs still do not enjoy wide popularity among practitioners, largely due to computational bottlenecks that exist in current algorithms and/or implementations. In particular, one of the missing pieces is the availability of software tools that: 1) can efficiently handle DPMM inference in large datasets; 2) are user-friendly and can also be easily modified. We argue that in order for DPMMs to become a practical choice for large-scale data analysis, implementations of DPMM inference must leverage parallel-and distributed-computing resources (in an analogy, consider how advances in GPU computing and GPU software contributed to the success of deep learning). This is because of not only potential speedups but also memory and storage considerations. For example, this is especially true in distributed mobile robotic sensing applications where multiple autonomous agents working together have limited computational and communication resources. As another motivating example, consider unsupervised dataanalysis tasks in large and high-dimensional computer-vision datasets.

In the beginning, let's have some common terminologies overview, A cluster is a group of objects that lie under the same class, or in other words, objects with similar properties are grouped in one cluster, and dissimilar objects are collected in another cluster. And, clustering is the process of classifying objects into a number of groups wherein each group, objects are very similar to each other than those objects in other groups. Simply, segmenting groups with similar properties/behaviour and assign them into clusters. Being an important analysis method in machine learning, clustering is used for identifying patterns and structure in labelled and unlabelled datasets. Clustering is exploratory data analysis techniques that can identify subgroups in data such that data points in each same subgroup (cluster) are very similar to each other and data points in separate clusters have different characteristics.

To understand how organizations like Google, Amazon, and even Udemy use machine learning and artificial intelligence (AI) to extract meaning and insights from enormous data sets, this machine learning course will provide you with the essentials. According to Glassdoor and Indeed, data scientists earn an average income of $120,000, and that is just the norm! When it comes to being attractive, data scientists are already there. In a highly competitive job market, it is tough to keep them after they have been hired. People with a unique mix of scientific training, computer expertise, and analytical abilities are hard to find.

A model can be considered as transparent when it (or all parts of it) can be fully understood as such, or when the learning process is understandable. A natural and common candidate that fits, at first sight, these criteria is the linear regression algorithm, where coefficients are usually seen as the individual contributions of the input features. Another candidate is the decision tree approach where model predictions can be broken down into a series of understandable operations. One can reasonably consider these models as transparent: one can easily identify the features that were used to take the decision. However, one may need to be cautious not to push too far the medical interpretation.

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems, which we call dynamical dimension reduction (DDR). In the DDR model, each point is evolved via a nonlinear flow towards a lower-dimensional subspace; the projection onto the subspace gives the low-dimensional embedding. Training the model involves identifying the nonlinear flow and the subspace. Following the equation discovery method, we represent the vector field that defines the flow using a linear combination of dictionary elements, where each element is a pre-specified linear/nonlinear candidate function. A regularization term for the average total kinetic energy is also introduced and motivated by optimal transport theory. We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method. We also show how the DDR method can be trained using a gradient-based optimization method, where the gradients are computed using the adjoint method from optimal control theory. The DDR method is implemented and compared on synthetic and example datasets to other dimension reductions methods, including PCA, t-SNE, and Umap.

In this study, we examine a clustering problem in which the covariates of each individual element in a dataset are associated with an uncertainty specific to that element. More specifically, we consider a clustering approach in which a pre-processing applying a non-linear transformation to the covariates is used to capture the hidden data structure. To this end, we approximate the sets representing the propagated uncertainty for the pre-processed features empirically. To exploit the empirical uncertainty sets, we propose a greedy and optimistic clustering (GOC) algorithm that finds better feature candidates over such sets, yielding more condensed clusters. As an important application, we apply the GOC algorithm to synthetic datasets of the orbital properties of stars generated through our numerical simulation mimicking the formation process of the Milky Way. The GOC algorithm demonstrates an improved performance in finding sibling stars originating from the same dwarf galaxy. These realistic datasets have also been made publicly available.

We propose the AdaPtive Noise Augmentation (PANDA) procedure to regularize the estimation and inference of generalized linear models (GLMs). PANDA iteratively optimizes the objective function given noise augmented data until convergence to obtain the regularized model estimates. The augmented noises are designed to achieve various regularization effects, including $l_0$, bridge (lasso and ridge included), elastic net, adaptive lasso, and SCAD, as well as group lasso and fused ridge. We examine the tail bound of the noise-augmented loss function and establish the almost sure convergence of the noise-augmented loss function and its minimizer to the expected penalized loss function and its minimizer, respectively. We derive the asymptotic distributions for the regularized parameters, based on which, inferences can be obtained simultaneously with variable selection. PANDA exhibits ensemble learning behaviors that help further decrease the generalization error. Computationally, PANDA is easy to code, leveraging existing software for implementing GLMs, without resorting to complicated optimization techniques. We demonstrate the superior or similar performance of PANDA against the existing approaches of the same type of regularizers in simulated and real-life data. We show that the inferences through PANDA achieve nominal or near-nominal coverage and are far more efficient compared to a popular existing post-selection procedure.

Abstract: We propose a didactic approach to use the Machine Learning protocol in order to perform weather forecast. This study is motivated by the possibility to apply this method to predict weather conditions in proximity of the Etna and Stromboli volcanic areas, located in Sicily (south Italy). Here the complex orography may significantly influence the weather conditions due to Stau and Foehn effects, with possible impact on the air traffic of the nearby Catania and Reggio Calabria airports. We first introduce a simple thermodynamic approach, suited to provide information on temperature and pressure when the Stau and Foehn effect takes place. In order to gain information to the rainfall accumulation, the Machine Learning approach is presented: according to this protocol, the model is able to learn'' from a set of input data which are the meteorological conditions (in our case dry, light rain, moderate rain and heavy rain) associated to the rainfall, measured in mm.

Feature Scaling: The dataset may contain more than 1 column and in that case, if the range of one of the columns is 100–1000 and the other column is 0–1. Then linear regression may give more weightage to the first column than the other column because of its high value. That is why it is good to always scale your data before fitting the model. Linear Assumption: This is kinda self-explanatory. As the algorithm is literally named Linear Regression, it will not be able to handle data that does not fit into linear behavior.