If only love were so simple – How to graph a heart using R from a fun site called Date By Number. I highly recommend checking it out. If only love were so simple – How to graph a heart using R from a fun site called Date By Number. I highly recommend checking it out.

In this continuation of "Hybrid content-based and collaborative filtering recommendations wi..." I will describe the application of the {ordinal} clm() function to test a new, hybrid content-based, collaborative filtering approach to recommender engines by fitting a class of ordinal logistic (aka ordered logit) models to ratings data from the MovieLens 100K dataset. All R code used in this project can be obtained from the respective GitHub repository; the chunks of code present in the body of the post illustrate the essential steps only. The MovieLens 100K dataset can be obtained from the GroupLens research laboratory of the Department of Computer Science and Engineering at the University of Minnesota. This second part of the study relies on the R code in OrdinalRecommenders_3.R and presents the model training, cross-validation, and the analyses. But before we proceed to approach the recommendation problem from a viewpoint of discrete choice modeling, let me briefly remind you of the results of the feature engineering phase and explain what happens in OrdinalRecommenders_2.R which prepares the data frames that are passed to clm().

In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems. These are regimes in which the underlying statistical problem is information-theoretically possible although no efficient algorithm exists, rendering the problem essentially unsolvable for large instances. The methods we describe here are based on mature, albeit non-rigorous, tools from statistical physics. These notes are based on a lecture series given by the authors at the Courant Institute of Mathematical Sciences in New York City, on May 16th, 2017.

Machine-learning algorithms have gained popularity in recent years in the field of ecological modeling due to their promising results in predictive performance of classification problems. While the application of such algorithms has been highly simplified in the last years due to their well-documented integration in commonly used statistical programming languages such as R, there are several practical challenges in the field of ecological modeling related to unbiased performance estimation, optimization of algorithms using hyperparameter tuning and spatial autocorrelation. We address these issues in the comparison of several widely used machine-learning algorithms such as Boosted Regression Trees (BRT), k-Nearest Neighbor (WKNN), Random Forest (RF) and Support Vector Machine (SVM) to traditional parametric algorithms such as logistic regression (GLM) and semi-parametric ones like generalized additive models (GAM). Different nested cross-validation methods including hyperparameter tuning methods are used to evaluate model performances with the aim to receive bias-reduced performance estimates. As a case study the spatial distribution of forest disease Diplodia sapinea in the Basque Country in Spain is investigated using common environmental variables such as temperature, precipitation, soil or lithology as predictors. Results show that GAM and RF (mean AUROC estimates 0.708 and 0.699) outperform all other methods in predictive accuracy. The effect of hyperparameter tuning saturates at around 50 iterations for this data set. The AUROC differences between the bias-reduced (spatial cross-validation) and overoptimistic (non-spatial cross-validation) performance estimates of the GAM and RF are 0.167 (24%) and 0.213 (30%), respectively. It is recommended to also use spatial partitioning for cross-validation hyperparameter tuning of spatial data.

Variational Bayes (VB), also known as independent mean-field approximation, has become a popular method for Bayesian network inference in recent years. Its application is vast, e.g. in neural network, compressed sensing, clustering, etc. to name just a few. In this paper, the independence constraint in VB will be relaxed to a conditional constraint class, called copula in statistics. Since a joint probability distribution always belongs to a copula class, the novel copula VB (CVB) approximation is a generalized form of VB. Via information geometry, we will see that CVB algorithm iteratively projects the original joint distribution to a copula constraint space until it reaches a local minimum Kullback-Leibler (KL) divergence. By this way, all mean-field approximations, e.g. iterative VB, Expectation-Maximization (EM), Iterated Conditional Mode (ICM) and k-means algorithms, are special cases of CVB approximation. For a generic Bayesian network, an augmented hierarchy form of CVB will also be designed. While mean-field algorithms can only return a locally optimal approximation for a correlated network, the augmented CVB network, which is an optimally weighted average of a mixture of simpler network structures, can potentially achieve the globally optimal approximation for the first time. Via simulations of Gaussian mixture clustering, the classification's accuracy of CVB will be shown to be far superior to that of state-of-the-art VB, EM and k-means algorithms.

As the size of modern data sets exceeds the disk and memory capacities of a single computer, machine learning practitioners have resorted to parallel and distributed computing. Given that optimization is one of the pillars of machine learning and predictive modeling, distributed optimization methods have recently garnered ample attention, in particular when either observations or features are distributed, but not both. We propose a general stochastic algorithm where observations, features, and gradient components can be sampled in a double distributed setting, i.e., with both features and observations distributed. Very technical analyses establish convergence properties of the algorithm under different conditions on the learning rate (diminishing to zero or constant). Computational experiments in Spark demonstrate a superior performance of our algorithm versus a benchmark in early iterations of the algorithm, which is due to the stochastic components of the algorithm.

Gradient-based Monte Carlo algorithms are useful tools for sampling posterior distributions. Similar to gradient descent algorithms, gradient-based Monte Carlo generates posterior samples iteratively using the gradient of loglikelihood. Langevin dynamics (LD) and Hamiltonian Monte Carlo (HMC) [Duane et al., 1987, Neal et al., 2011] are two important examples of gradient-based Monte Carlo sampling algorithms that are widely used in Bayesian inference. Since calculating likelihood on large datasets is expensive, people use stochastic gradients [Robbins and Monro, 1951] in place of full gradient, and have, for both Langevin dynamics and Hamiltonian Monte Carlo, developed their stochastic gradient counterparts[Welling and Teh, 2011, Chen et al., 2014]. Stochastic gradient Hamiltonian Monte Carlo (SGHMC) usually converges faster than stochastic gradient Langevin dynamics (SGLD) in practical machine learning tasks like covariance estimation of bivariate Gaussian and Bayesian neural networks for classification on MNIST dataset, as demonstrated in [Welling and Teh, 2011]. Similar phenomenon was also observed in [Chen et al., 2015] where SGHMC and SGLD were compared on both synthetic and real-world datasets. Intuitively speaking, comparing against SGLD, SGHMC has a momentum term that enables it to explore the parameter space of posterior distribution much faster when the gradient of log-likelihood becomes smaller. Very recently, [Dubey et al., 2016] borrowed the standard variance reduction techniques from the stochastic optimization literature [Johnson and Zhang, 2013, Defazio et al., 2014] and applied them on SGLD to obtain two variancereduced SGLD algorithms (called SAGA-LD and SVRG-LD) with improved theoretical results and practical performance. Because of the superiority of SGHMC over SGLD in terms of convergence rate in a wide range of machine learning tasks, it would be a natural question whether such variance reduction techniques can be applied on SGHMC to achieve better results than variance-reduced SGLD.

SMOTE is one of the oversampling techniques for balancing the datasets and it is considered as a pre-processing step in learning algorithms. In this paper, four new enhanced SMOTE are proposed that include an improved version of KNN in which the attribute weights are defined by mutual information firstly and then they are replaced by maximum entropy, Renyi entropy and Tsallis entropy. These four pre-processing methods are combined with 1NN and J48 classifiers and their performance are compared with the previous methods on 11 imbalanced datasets from KEEL repository. The results show that these pre-processing methods improves the accuracy compared with the previous stablished works. In addition, as a case study, the first pre-processing method is applied on transportation data of Tehran-Bazargan Highway in Iran with IR equal to 36.

This paper explores conformal prediction in the learning under privileged information (LUPI) paradigm. We use the SVM+ realization of LUPI in an inductive conformal predictor, and apply it to the MNIST benchmark dataset and three datasets in drug discovery. The results show that using privileged information produces valid models and improves efficiency compared to standard SVM, however the improvement varies between the tested datasets and is not substantial in the drug discovery applications. More importantly, using SVM+ in a conformal prediction framework enables valid prediction intervals at specified significance levels.

Quality assessments of models in unsupervised learning and clustering verification in particular have been a long-standing problem in the machine learning research. The lack of robust and universally applicable cluster validity scores often makes the algorithm selection and hyperparameter evaluation a tough guess. In this paper, we show that cluster ensemble aggregation techniques such as consensus clustering may be used to evaluate clusterings and their hyperparameter configurations. We use normalized mutual information to compare individual objects of a clustering ensemble to the constructed consensus of the whole ensemble and show, that the resulting score can serve as an overall quality measure for clustering problems. This method is capable of highlighting the standout clustering and hyperparameter configuration in the ensemble even in the case of a distorted consensus. We apply this very general framework to various data sets and give possible directions for future research.