We will show how to perform multivariate random sampling using one of the Markov Chain Monte Carlo (MCMC) algorithms, called the Gibbs sampler. To start, what are MCMC algorithms and what are they based on? Suppose we are interested in generating a random variable with a distribution of, over . If we are not able to do this directly, we will be satisfied with generating a sequence of random variables, which in a sense tending to a distribution of . Build a Markov chain, for, whose stationary distribution is .

Then we use another map transformation, which will apply the ParseObs function to transform each Array of Double in the RDD into an Array of Cancer Observation objects. The toDF() method transforms the RDD of Array[[Cancer Observation]] into a Dataframe with the Cancer Observation class schema. Below the data is split into a training data set and a test data set, 70% of the data is used to train the model, and 30% will be used for testing. In this blog post, we showed you how to get started using Apache Spark's machine learning Logistic Regression for classification.

I am trying to create a network in tensor flow with multiple softmax outputs, each of a different size. The network architecture is: Input - LSTM - Dropout. Then I have 2 softmax layers: Softmax of 10 outputs and Softmax of 20 Outputs. I would like to multiply the result of one of the softmax with some data as well. I'm not sure how to do this in Tensorflow.

Many real-world networks are complex dynamical systems, where both local (e.g., changing node attributes) and global (e.g., changing network topology) processes unfold over time. Local dynamics may provoke global changes in the network, and the ability to detect such effects could have profound implications for a number of real-world problems. Most existing techniques focus individually on either local or global aspects of the problem or treat the two in isolation from each other. In this paper we propose a novel network model that simultaneously accounts for both local and global dynamics. To the best of our knowledge, this is the first attempt at modeling and detecting local and global change points on dynamic networks via a unified generative framework. Our model is built upon the popular mixed membership stochastic blockmodels (MMSB) with sparse co-evolving patterns. We derive an efficient stochastic gradient Langevin dynamics (SGLD) sampler for our proposed model, which allows it to scale to potentially very large networks. Finally, we validate our model on both synthetic and real-world data and demonstrate its superiority over several baselines.

Clustering samples according to an effective metric and/or vector space representation is a challenging unsupervised learning task with a wide spectrum of applications. Among several clustering algorithms, k-means and its kernelized version have still a wide audience because of their conceptual simplicity and efficacy. However, the systematic application of the kernelized version of k-means is hampered by its inherent square scaling in memory with the number of samples. In this contribution, we devise an approximate strategy to minimize the kernel k-means cost function in which the trade-off between accuracy and velocity is automatically ruled by the available system memory. Moreover, we define an ad-hoc parallelization scheme well suited for hybrid cpu-gpu state-of-the-art parallel architectures. We proved the effectiveness both of the approximation scheme and of the parallelization method on standard UCI datasets and on molecular dynamics (MD) data in the realm of computational chemistry. In this applicative domain, clustering can play a key role for both quantitively estimating kinetics rates via Markov State Models or to give qualitatively a human compatible summarization of the underlying chemical phenomenon under study. For these reasons, we selected it as a valuable real-world application scenario.

We propose new algorithms for topic modeling when the number of topics is unknown. Our approach relies on an analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Our algorithms are shown in practice to have accuracy comparable to a Gibbs sampler in terms of topic estimation, which requires the number of topics be given. Moreover, they are one of the fastest among several state of the art parametric techniques. Statistical consistency of our estimator is established under some conditions.

Maximum regularized likelihood estimators (MRLEs) are arguably the most established class of estimators in high-dimensional statistics. In this paper, we derive guarantees for MRLEs in Kullback-Leibler divergence, a general measure of prediction accuracy. We assume only that the densities have a convex parametrization and that the regularization is definite and positive homogenous. The results thus apply to a very large variety of models and estimators, such as tensor regression and graphical models with convex and non-convex regularized methods. A main conclusion is that MRLEs are broadly consistent in prediction - regardless of whether restricted eigenvalues or similar conditions hold.

How can we find patterns and anomalies in a tensor, or multi-dimensional array, in an efficient and directly interpretable way? How can we do this in an online environment, where a new tensor arrives each time step? Finding patterns and anomalies in a tensor is a crucial problem with many applications, including building safety monitoring, patient health monitoring, cyber security, terrorist detection, and fake user detection in social networks. Standard PARAFAC and Tucker decomposition results are not directly interpretable. Although a few sampling-based methods have previously been proposed towards better interpretability, they need to be made faster, more memory efficient, and more accurate. In this paper, we propose CTD, a fast, accurate, and directly interpretable tensor decomposition method based on sampling. CTD-S, the static version of CTD, provably guarantees a high accuracy that is 17 ~ 83x more accurate than that of the state-of-the-art method. Also, CTD-S is made 5 ~ 86x faster, and 7 ~ 12x more memory-efficient than the state-of-the-art method by removing redundancy. CTD-D, the dynamic version of CTD, is the first interpretable dynamic tensor decomposition method ever proposed. Also, it is made 2 ~ 3x faster than already fast CTD-S by exploiting factors at previous time step and by reordering operations. With CTD, we demonstrate how the results can be effectively interpreted in the online distributed denial of service (DDoS) attack detection.

With the maturation of metabolomics science and proliferation of biobanks, clinical metabolic profiling is an increasingly opportunistic frontier for advancing translational clinical research. Automated Machine Learning (AutoML) approaches provide exciting opportunity to guide feature selection in agnostic metabolic profiling endeavors, where potentially thousands of independent data points must be evaluated. In previous research, AutoML using high-dimensional data of varying types has been demonstrably robust, outperforming traditional approaches. However, considerations for application in clinical metabolic profiling remain to be evaluated. Particularly, regarding the robustness of AutoML to identify and adjust for common clinical confounders. In this study, we present a focused case study regarding AutoML considerations for using the Tree-Based Optimization Tool (TPOT) in metabolic profiling of exposure to metformin in a biobank cohort. First, we propose a tandem rank-accuracy measure to guide agnostic feature selection and corresponding threshold determination in clinical metabolic profiling endeavors. Second, while AutoML, using default parameters, demonstrated potential to lack sensitivity to low-effect confounding clinical covariates, we demonstrated residual training and adjustment of metabolite features as an easily applicable approach to ensure AutoML adjustment for potential confounding characteristics. Finally, we present increased homocysteine with long-term exposure to metformin as a potentially novel, non-replicated metabolite association suggested by TPOT; an association not identified in parallel clinical metabolic profiling endeavors. While considerations are recommended, including adjustment approaches for clinical confounders, AutoML presents an exciting tool to enhance clinical metabolic profiling and advance translational research endeavors.

We propose a top-down algorithm for multiple testing on directed acyclic graphs (DAGs), where nodes represent hypotheses and edges specify a partial ordering in which hypotheses must be tested. The procedure is guaranteed to reject a sub-DAG with bounded false discovery rate (FDR) while satisfying the logical constraint that a rejected node's parents must also be rejected. It is designed for sequential testing settings, when the DAG structure is known a priori, but the p-values are obtained selectively (such as sequential conduction of experiments), but the algorithm is also applicable in non-sequential settings when all p-values can be calculated in advance (such as variable/model selection). Our DAGGER algorithm, shorthand for Greedily Evolving Rejections on DAGs, allows for independence, positive or arbitrary dependence of the p-values, and is guaranteed to work on two different types of DAGs: (a) intersection DAGs in which all nodes are intersection hypotheses, with parents being supersets of children, or (b) general DAGs in which all nodes may be elementary hypotheses. The DAGGER procedure has the appealing property that it specializes to known algorithms in the special cases of trees and line graphs, and simplifies to the classic Benjamini-Hochberg procedure when the DAG has no edges. We explore the empirical performance of DAGGER using simulations, as well as a real dataset corresponding to a gene ontology DAG, showing that it performs favorably in terms of time and power.