C
Filomena Pereira-Maxwell in Medical Statistics, 2018
The joint variance of two variables (or random variables). It is estimated as the average cross-product between deviations from the mean, where for each observation, the difference between the observation value for variable x and the mean of x is calculated, and likewise for variable y. The covariances between pairs of variables may be displayed in a covariance or variance-covariance matrix, a symmetrical matrix in which the elements in the main diagonal represent the variances of the variables, and the off-diagonal elements represent the covariance between pairs of variables. With standardized variables (z-scores), the covariance matrix is given by the correlation matrix. Correlation is a standardized covariance (HAMILTON, 1992). The covariance matrix often holds the necessary and sufficient building blocks for statistical analysis.
Exploratory Data Analysis with Unsupervised Machine Learning
Altuna Akalin in Computational Genomics with R, 2020
One thing that is new in Figure 4.11 is the concept of eigenarrays. The eigenarrays, sometimes called eigenassays, represent the sample space and can be used to plot the relationship between samples rather than genes. In this way, SVD offers additional information than the PCA using the covariance matrix. It offers us a way to summarize both genes and samples. As we can project the gene expression profiles over the top two eigengenes and get a 2D representation of genes, but with the SVD, we can also project the samples over the top two eigenarrays and get a representation of samples in 2D scatter plot. The eigenvector could represent independent expression programs across samples, such as cell-cycle, if we had time-based expression profiles. However, there is no guarantee that each eigenvector will be biologically meaningful. Similarly each eigenarray represents samples with specific expression characteristics. For example, the samples that have a particular pathway activated might be correlated to an eigenarray returned by SVD.
Methodological Issues in Health Economic Analysis
Demissie Alemayehu, Joseph C. Cappelleri, Birol Emir, Kelly H. Zou in Statistical Topics in Health Economics and Outcomes Research, 2017
An attractive feature of the GPQ methodology is that it can address the interval estimation of an arbitrary function of the mean vectors and covariance matrices in the model (Equation 5.4). For example, suppose that, in addition to the costs, the effectiveness measures are also log-normally distributed. That is, we have the model where is the population mean of the log-transformed effectiveness measures, j = 1, 2. Now, the ICER and INB () have the expressions where is the second diagonal element of the covariance matrix , . The parameter can also be similarly modified. It should be clear that the GPQ methodology can be easily adopted in this scenario in which the costs and effectiveness are both lognormally distributed.
Robust principal component analysis for compositional tables
Published in Journal of Applied Statistics, 2021
J. de Sousa, K. Hron, K. Fačevicová, P. Filzmoser
The covariance matrix n. The columns of the matrix scores and the columns of loadings. Typically, only the first few principal components are considered for further analysis. Taking into account only two PCs, a graphical outcome called biplot can depict both loadings as arrows and scores as points in one plot, where associations can be revealed.
Sparse graphical models via calibrated concave convex procedure with application to fMRI data
Published in Journal of Applied Statistics, 2020
Sungtaek Son, Cheolwoo Park, Yongho Jeon
The regularization process illustrated above uses a single λ value for the estimation of all columns of the inverse covariance matrix. In what follows, we introduce a column-by-column method, which searches for an optimal tuning parameter for each column. With this approach, the estimated Step 2 of the calibrated CCCP algorithm is given as ith column of the inverse covariance matrix. For parameter tuning, if a separate testing set is unavailable, we can divide the data into training and testing sets. Then, the optimal ith column can be obtained from
Clustering of longitudinal interval-valued data via mixture distribution under covariance separability
Published in Journal of Applied Statistics, 2020
Seongoh Park, Johan Lim, Hyejeong Choi, Minjung Kwak
To this end, Fraley et al. [17] provide a series of covariance matrices based on the eigenvalue decomposition within their R package mclust. The decomposition is represented as kth cluster, respectively. If we assume that volumes are different, orientations are the same across K components, and shape is an identity matrix, the covariance matrix is expressed by et al. [17], where the letters V, E, and I, stand for variable, equal, and identity, respectively. In addition, VII and EII options stand for et al. [17] for more available combinations.
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