So I understand that variable selection is a part of model selection. But what exactly does model selection consist of? I ask this because I am reading an article Burnham & Anderson: AIC vs BIC where they talk about AIC and BIC in model selection. Reading this article I realize I have been thinking of'model selection' as'variable selection' (ref. An excerpt from the article where they talk about 12 models with increasing degrees of "generality" and these models show "tapering effects" (Figure 1) when KL-Information is plotted against the 12 models: DIFFERENT PHILOSOPHIES AND TARGET MODELS ... Despite that the target of BIC is a more general model than the target model for AIC, the model most often selected here by BIC will be less general than Model 7 unless n is very large.

The question of which model type to apply to a Machine Learning task can be a daunting one given the immense number of algorithms available in the literature. It can be difficult to compare the relative merits of two methods, as one can outperform the other in a certain class of problems while consistently coming in behind for another class. In this post, the last one of our series of posts about Logistic Regression, we'll explore the differences between Decision Trees and Logistic Regression for classification problems, and try to highlight scenarios where one might be recommended over the other. Logistic Regression and trees differ in the way that they generate decision boundaries i.e. the lines that are drawn to separate different classes. To illustrate this difference, let's look at the results of the two model types on the following 2-class problem: Decision Trees bisect the space into smaller and smaller regions, whereas Logistic Regression fits a single line to divide the space exactly into two.

I have read a book or some posts on machine learning. I have watched some of the Coursera machine learning course. I still don't know how to get started… How do you get started in machine learning? The most common question I'm asked by developers on my newsletter is: I honestly cannot remember how many times I have answered it. In this post, I lay out all of my very best thinking on this topic. You are a developer and you're interested in getting into machine learning. You read some blog posts. You tried to go deeper but the books are dreadful.

Did you have a good, relaxing break over the summer? Are you refreshed and re-energised, looking forward to a new start, a new you and brushing up on your data analysis skills? If so, I've thrown together a collection of a few excellent (and free!) statistics eBooks for your Kindle to sharpen up your stats while you're on the long commute to work. Just try not to read them while driving! These books require different levels of existing knowledge, and while some are for early-stage data scientists others are for more hard-core physicists and mathematicians.

Computed tomography (CT) equivalent information is needed for attenuation correction in PET imaging and for dose planning in radiotherapy. Prior work has shown that Gaussian mixture models can be used to generate a substitute CT (s-CT) image from a specific set of MRI modalities. This work introduces a more flexible class of mixture models for s-CT generation, that incorporates spatial dependency in the data through a Markov random field prior on the latent field of class memberships associated with a mixture model. Furthermore, the mixture distributions are extended from Gaussian to normal inverse Gaussian (NIG), allowing heavier tails and skewness. The amount of data needed to train a model for s-CT generation is of the order of 100 million voxels. The computational efficiency of the parameter estimation and prediction methods are hence paramount, especially when spatial dependency is included in the models. A stochastic Expectation Maximization (EM) gradient algorithm is proposed in order to tackle this challenge. The advantages of the spatial model and NIG distributions are evaluated with a cross-validation study based on data from 14 patients. The study show that the proposed model enhances the predictive quality of the s-CT images by reducing the mean absolute error with 17.9%. Also, the distribution of CT values conditioned on the MR images are better explained by the proposed model as evaluated using continuous ranked probability scores.

Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classification which is motivated by this objective. The proposed minimum density hyperplane minimises the integral of the empirical probability density function along it, thereby avoiding intersection with high density clusters. We show that the minimum density and the maximum margin hyperplanes are asymptotically equivalent, thus linking this approach to maximum margin clustering and semi-supervised support vector classifiers. We propose a projection pursuit formulation of the associated optimisation problem which allows us to find minimum density hyperplanes efficiently in practice, and evaluate its performance on a range of benchmark data sets. The proposed approach is found to be very competitive with state of the art methods for clustering and semi-supervised classification.

Correlation matrices play a key role in many multivariate methods (e.g., graphical model estimation and factor analysis). The current state-of-the-art in estimating large correlation matrices focuses on the use of Pearson's sample correlation matrix. Although Pearson's sample correlation matrix enjoys various good properties under Gaussian models, it is not an effective estimator when facing heavy-tailed distributions. As a robust alternative, Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] advocated the use of a transformed version of the Kendall's tau sample correlation matrix in estimating high dimensional latent generalized correlation matrix under the transelliptical distribution family (or elliptical copula). The transelliptical family assumes that after unspecified marginal monotone transformations, the data follow an elliptical distribution. In this paper, we study the theoretical properties of the Kendall's tau sample correlation matrix and its transformed version proposed in Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] for estimating the population Kendall's tau correlation matrix and the latent Pearson's correlation matrix under both spectral and restricted spectral norms. With regard to the spectral norm, we highlight the role of "effective rank" in quantifying the rate of convergence. With regard to the restricted spectral norm, we for the first time present a "sign sub-Gaussian condition" which is sufficient to guarantee that the rank-based correlation matrix estimator attains the fast rate of convergence. In both cases, we do not need any moment condition.

We propose a data-driven, coarse-graining formulation in the context of equilibrium statistical mechanics. In contrast to existing techniques which are based on a fine-to-coarse map, we adopt the opposite strategy by prescribing a probabilistic coarse-to-fine map. This corresponds to a directed probabilistic model where the coarse variables play the role of latent generators of the fine scale (all-atom) data. From an information-theoretic perspective, the framework proposed provides an improvement upon the relative entropy method and is capable of quantifying the uncertainty due to the information loss that unavoidably takes place during the CG process. Furthermore, it can be readily extended to a fully Bayesian model where various sources of uncertainties are reflected in the posterior of the model parameters. The latter can be used to produce not only point estimates of fine-scale reconstructions or macroscopic observables, but more importantly, predictive posterior distributions on these quantities. Predictive posterior distributions reflect the confidence of the model as a function of the amount of data and the level of coarse-graining. The issues of model complexity and model selection are seamlessly addressed by employing a hierarchical prior that favors the discovery of sparse solutions, revealing the most prominent features in the coarse-grained model. A flexible and parallelizable Monte Carlo - Expectation-Maximization (MC-EM) scheme is proposed for carrying out inference and learning tasks. A comparative assessment of the proposed methodology is presented for a lattice spin system and the SPC/E water model.

Non-negative matrix factorization (NMF) is the problem of determining two non-negative low rank factors $W$ and $H$, for the given input matrix $A$, such that $A \approx W H$. NMF is a useful tool for many applications in different domains such as topic modeling in text mining, background separation in video analysis, and community detection in social networks. Despite its popularity in the data mining community, there is a lack of efficient parallel algorithms to solve the problem for big data sets. The main contribution of this work is a new, high-performance parallel computational framework for a broad class of NMF algorithms that iteratively solves alternating non-negative least squares (NLS) subproblems for $W$ and $H$. It maintains the data and factor matrices in memory (distributed across processors), uses MPI for interprocessor communication, and, in the dense case, provably minimizes communication costs (under mild assumptions). The framework is flexible and able to leverage a variety of NMF and NLS algorithms, including Multiplicative Update, Hierarchical Alternating Least Squares, and Block Principal Pivoting. Our implementation allows us to benchmark and compare different algorithms on massive dense and sparse data matrices of size that spans for few hundreds of millions to billions. We demonstrate the scalability of our algorithm and compare it with baseline implementations, showing significant performance improvements. The code and the datasets used for conducting the experiments are available online.

We present a structural clustering algorithm for large-scale datasets of small labeled graphs, utilizing a frequent subgraph sampling strategy. A set of representatives provides an intuitive description of each cluster, supports the clustering process, and helps to interpret the clustering results. The projection-based nature of the clustering approach allows us to bypass dimensionality and feature extraction problems that arise in the context of graph datasets reduced to pairwise distances or feature vectors. While achieving high quality and (human) interpretable clusterings, the runtime of the algorithm only grows linearly with the number of graphs. Furthermore, the approach is easy to parallelize and therefore suitable for very large datasets. Our extensive experimental evaluation on synthetic and real world datasets demonstrates the superiority of our approach over existing structural and subspace clustering algorithms, both, from a runtime and quality point of view.