Correlation matrix analysis is an important method to find dependence between variables. Computing correlation matrix and drawing correlogram is explained here. The aim of this article is to show you how to get the lower and the upper triangular part of a correlation matrix. We will also use the xtable R package to display a nice correlation table in html or latex formats. Note that online software is also available here to compute correlation matrix and to plot a correlogram without any installation.
Jing, Xiao-Yuan (Wuhan University) | Hu, Rui-Min (Wuhan University) | Zhu, Yang-Ping (Nanjing University of Posts and Telecommunications) | Wu, Shan-Shan (Nanjing University of Posts and Telecommunications) | Liang, Chao (Wuhan University) | Yang, Jing-Yu (Nanjing University of Science and Technology)
Multi-view feature learning is an attractive research topic with great practical success. Canonical correlation analysis (CCA) has become an important technique in multi-view learning, since it can fully utilize the inter-view correlation. In this paper, we mainly study the CCA based multi-view supervised feature learning technique where the labels of training samples are known. Several supervised CCA based multi-view methods have been presented, which focus on investigating the supervised correlation across different views. However, they take no account of the intra-view correlation between samples. Researchers have also introduced the discriminant analysis technique into multi-view feature learning, such as multi-view discriminant analysis (MvDA). But they ignore the canonical correlation within each view and between all views. In this paper, we propose a novel multi-view feature learning approach based on intra-view and inter-view supervised correlation analysis (I2SCA), which can explore the useful correlation information of samples within each view and between all views. The objective function of I2SCA is designed to simultaneously extract the discriminatingly correlated features from both inter-view and intra-view. It can obtain an analytical solution without iterative calculation. And we provide a kernelized extension of I2SCA to tackle the linearly inseparable problem in the original feature space. Four widely-used datasets are employed as test data. Experimental results demonstrate that our proposed approaches outperform several representative multi-view supervised feature learning methods.
One explanation for the divergence of many of the thermodynamic properties of water is that there is a critical point in deeply supercooled water at some positive pressure. For bulk water samples, these conditions are described as "no man's land," because ice nucleates before such temperatures can be reached. Kim et al. used femtosecond x-ray laser pulses to probe micrometer-sized water droplets cooled to 227 K (see the Perspective by Gallo and Stanley). The temperature dependence of the isothermal compressibility and correlation length extracted from x-ray scattering functions showed maxima at 229 K for H2O and 233 K for D2O, rather than diverging to infinity. These results point to the existence of the Widom line, a locus of maximum correlation lengths emanating from a critical point in the supercooled regime.
How much information does a neural population convey about a stimulus? Answers to this question are known to strongly depend on the correlation of response variability in neural populations. These noise correlations, however, are essentially immeasurable as the number of parameters in a noise correlation matrix grows quadratically with population size. Here, we suggest to bypass this problem by imposing a parametric model on a noise correlation matrix. Our basic assumption is that noise correlations arise due to common inputs between neurons. On average, noise correlations will therefore reflect signal correlations, which can be measured in neural populations. We suggest an explicit parametric dependency between signal and noise correlations. We show how this dependency can be used to fill the gaps" in noise correlations matrices using an iterative application of the Wishart distribution over positive definitive matrices. We apply our method to data from the primary somatosensory cortex of monkeys performing a two-alternative-forced choice task. We compare the discrimination thresholds read out from the population of recorded neurons with the discrimination threshold of the monkey and show that our method predicts different results than simpler, average schemes of noise correlations."
Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.