Dimension reduction for regression is a prominent issue today because technological advances now allow scientists to routinely formulate regressions in which the number of predictors is considerably larger than in the past. While several methods have been proposed to deal with such regressions, principal components (PCs) still seem to be the most widely used across the applied sciences. We give a broad overview of ideas underlying a particular class of methods for dimension reduction that includes PCs, along with an introduction to the corresponding methodology. New methods are proposed for prediction in regressions with many predictors.
Multiple therapeutic opportunities have been suggested for compounds capable of selective activation of metabotropic glutamate 3 (mGlu) receptors, but small molecule tools are lacking. As part of our ongoing efforts to identify potent, selective, and systemically bioavailable agonists for mGlu and mGlu receptor subtypes, a series of C4-N-linked variants of (1 S,2 S,5 R,6 S)-2-amino-bicyclo[3.1.0]hexane-2,6-dicarboxylic acid 1 (LY354740) were prepared and evaluated for both mGlu and mGlu receptor binding affinity and functional cellular responses. From this investigation we identified (1 S,2 S,4 S,5 R,6 S)-2-amino-4-[(3-methoxybenzoyl)amino]bicyclo[3.1.0]hexane-2,6-dicarboxylic acid 8p (LY2794193), a molecule that demonstrates remarkable mGlu receptor selectivity. Crystallization of 8p with the amino terminal domain of hmGlu revealed critical binding interactions for this ligand with residues adjacent to the glutamate binding site, while pharmacokinetic assessment of 8p combined with its effect in an mGlu receptor-dependent behavioral model provides estimates for doses of this compound that would be expected to selectively engage and activate central mGlu receptors in vivo.
Due to current data collection technology, our ability to gather data has surpassed our ability to analyze it. In particular, k-means, one of the simplest and fastest clustering algorithms, is ill-equipped to handle extremely large datasets on even the most powerful machines. Our new algorithm uses a sample from a dataset to decrease runtime by reducing the amount of data analyzed. We perform a simulation study to compare our sampling based k-means to the standard k-means algorithm by analyzing both the speed and accuracy of the two methods. Results show that our algorithm is significantly more efficient than the existing algorithm with comparable accuracy.
Manifold optimization appears in a wide variety of computational problems in the applied sciences. In recent statistical methodologies such as sufficient dimension reduction and regression envelopes, estimation relies on the optimization of likelihood functions over spaces of matrices such as the Stiefel or Grassmann manifolds. Recently, Huang, Absil, Gallivan, and Hand (2016) have introduced the library ROPTLIB, which provides a framework and state of the art algorithms to optimize real-valued objective functions over commonly used matrix-valued Riemannian manifolds. This article presents ManifoldOptim, an R package that wraps the C++ library ROPTLIB. ManifoldOptim enables users to access functionality in ROPTLIB through R so that optimization problems can easily be constructed, solved, and integrated into larger R codes. Computationally intensive problems can be programmed with Rcpp and RcppArmadillo, and otherwise accessed through R. We illustrate the practical use of ManifoldOptim through several motivating examples involving dimension reduction and envelope methods in regression.
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