652 research outputs found
Targeted Fused Ridge Estimation of Inverse Covariance Matrices from Multiple High-Dimensional Data Classes
We consider the problem of jointly estimating multiple inverse covariance
matrices from high-dimensional data consisting of distinct classes. An
-penalized maximum likelihood approach is employed. The suggested
approach is flexible and generic, incorporating several other
-penalized estimators as special cases. In addition, the approach
allows specification of target matrices through which prior knowledge may be
incorporated and which can stabilize the estimation procedure in
high-dimensional settings. The result is a targeted fused ridge estimator that
is of use when the precision matrices of the constituent classes are believed
to chiefly share the same structure while potentially differing in a number of
locations of interest. It has many applications in (multi)factorial study
designs. We focus on the graphical interpretation of precision matrices with
the proposed estimator then serving as a basis for integrative or meta-analytic
Gaussian graphical modeling. Situations are considered in which the classes are
defined by data sets and subtypes of diseases. The performance of the proposed
estimator in the graphical modeling setting is assessed through extensive
simulation experiments. Its practical usability is illustrated by the
differential network modeling of 12 large-scale gene expression data sets of
diffuse large B-cell lymphoma subtypes. The estimator and its related
procedures are incorporated into the R-package rags2ridges.Comment: 52 pages, 11 figure
rags2ridges:A One-Stop-â„“<sub>2</sub>-Shop for Graphical Modeling of High-Dimensional Precision Matrices
A graphical model is an undirected network representing the conditional independence properties between random variables. Graphical modeling has become part and parcel of systems or network approaches to multivariate data, in particular when the variable dimension exceeds the observation dimension. rags2ridges is an R package for graphical modeling of high-dimensional precision matrices through ridge (ℓ2) penalties. It provides a modular framework for the extraction, visualization, and analysis of Gaussian graphical models from high-dimensional data. Moreover, it can handle the incorporation of prior information as well as multiple heterogeneous data classes. As such, it provides a one-stop-ℓ2-shop for graphical modeling of high-dimensional precision matrices. The functionality of the package is illustrated with an example dataset pertaining to blood-based metabolite measurements in persons suffering from Alzheimer’s disease.</p
Structured penalties for functional linear models---partially empirical eigenvectors for regression
One of the challenges with functional data is incorporating spatial
structure, or local correlation, into the analysis. This structure is inherent
in the output from an increasing number of biomedical technologies, and a
functional linear model is often used to estimate the relationship between the
predictor functions and scalar responses. Common approaches to the ill-posed
problem of estimating a coefficient function typically involve two stages:
regularization and estimation. Regularization is usually done via dimension
reduction, projecting onto a predefined span of basis functions or a reduced
set of eigenvectors (principal components). In contrast, we present a unified
approach that directly incorporates spatial structure into the estimation
process by exploiting the joint eigenproperties of the predictors and a linear
penalty operator. In this sense, the components in the regression are
`partially empirical' and the framework is provided by the generalized singular
value decomposition (GSVD). The GSVD clarifies the penalized estimation process
and informs the choice of penalty by making explicit the joint influence of the
penalty and predictors on the bias, variance, and performance of the estimated
coefficient function. Laboratory spectroscopy data and simulations are used to
illustrate the concepts.Comment: 29 pages, 3 figures, 5 tables; typo/notational errors edited and
intro revised per journal review proces
Regularized Linear Discriminant Analysis Using a Nonlinear Covariance Matrix Estimator
Linear discriminant analysis (LDA) is a widely used technique for data
classification. The method offers adequate performance in many classification
problems, but it becomes inefficient when the data covariance matrix is
ill-conditioned. This often occurs when the feature space's dimensionality is
higher than or comparable to the training data size. Regularized LDA (RLDA)
methods based on regularized linear estimators of the data covariance matrix
have been proposed to cope with such a situation. The performance of RLDA
methods is well studied, with optimal regularization schemes already proposed.
In this paper, we investigate the capability of a positive semidefinite
ridge-type estimator of the inverse covariance matrix that coincides with a
nonlinear (NL) covariance matrix estimator. The estimator is derived by
reformulating the score function of the optimal classifier utilizing linear
estimation methods, which eventually results in the proposed NL-RLDA
classifier. We derive asymptotic and consistent estimators of the proposed
technique's misclassification rate under the assumptions of a double-asymptotic
regime and multivariate Gaussian model for the classes. The consistent
estimator, coupled with a one-dimensional grid search, is used to set the value
of the regularization parameter required for the proposed NL-RLDA classifier.
Performance evaluations based on both synthetic and real data demonstrate the
effectiveness of the proposed classifier. The proposed technique outperforms
state-of-art methods over multiple datasets. When compared to state-of-the-art
methods across various datasets, the proposed technique exhibits superior
performance.Comment: \c{opyright} 2024 IEEE. Personal use of this material is permitted.
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Sparse summaries of complex covariance structures : a thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics, School of Natural & Computational Sciences, Massey University, Auckland, New Zealand
A matrix that has most of its elements equal to zero is called a sparse matrix. The zero elements in a sparse matrix reduce the number of parameters for its potential interpretability. Bayesians desiring a sparse model frequently formulate priors that enhance sparsity. However, in most settings, this leads to sparse posterior samples, not to a sparse posterior mean. A decoupled shrinkage and selection posterior - variable selection approach was proposed by (Hahn & Carvalho, 2015) to address this problem in a regression setting to set some of the elements of the regression coefficients matrix to exact zeros. Hahn & Carvallho (2015) suggested to work on a decoupled shrinkage and selection approach in a Gaussian graphical models setting to set some of the elements of a precision matrix (graph) to exact zeros. In this thesis, I have filled this gap and proposed decoupled shrinkage and selection approaches to sparsify the precision matrix and the factor loading matrix that is an extension of Hahn & Carvallho’s (2015) decoupled shrinkage and selection approach. The decoupled shrinkage and selection approach proposed by me uses samples from the posterior over the parameter, sets a penalization criteria to produce progressively sparser estimates of the desired parameter, and then sets a rule to pick the final desired parameter from the generated parameters, based on the posterior distribution of fit. My proposed decoupled approach generally produced sparser graphs than a range of existing sparsification strategies such as thresholding the partial correlations, credible interval, adaptive graphical Lasso, and ratio selection, while maintaining a good fit based on the log-likelihood. In simulation studies, my decoupled shrinkage and selection approach had better sensitivity and specificity than the other strategies as the dimension p and sample size n grew. For low-dimensional data, my decoupled shrinkage and selection approach was comparable with the other strategies.
Further, I have extended my proposed decoupled shrinkage and selection approach for one population to two populations by modifying the ADMM (alternating directions method of multipliers) algorithm in the JGL (joint graphical Lasso) R – package (Danaher et al, 2013) to find sparse sets of differences between two inverse covariance matrices. The simulation studies showed that my decoupled shrinkage and selection approach for two populations for the sparse case had better sensitivity and specificity than the sensitivity and specificity using JGL. However, sparse sets of differences were challenging for the dense case and moderate sample sizes. My decoupled shrinkage and selection approach for two populations was also applied to find sparse sets of differences between the precision matrices for cases and controls in a metabolomics dataset.
Finally, decoupled shrinkage and selection is used to post-process the posterior mean covariance matrix to produce a factor model with a sparse factor loading matrix whose expected fit lies within the upper 95% of the posterior over fits. In the Gaussian setting, simulation studies showed that my proposed DSS sparse factor model approach performed better than fanc (factor analysis using non-convex penalties) (Hirose and Yamamoto, 2015) in terms of sensitivity, specificity, and picking the correct number of factors. Decoupled shrinkage and selection is also easily applied to models where a latent multivariate normal underlies non-Gaussian marginals, e.g., multivariate probit models. I illustrate my findings with moderate dimensional data examples from modelling of food frequency questionnaires and fish abundance
Regularised inference for changepoint and dependency analysis in non-stationary processes
Multivariate correlated time series are found in many modern socio-scientific domains such as neurology, cyber-security, genetics and economics. The focus of this thesis is on efficiently modelling and inferring dependency structure both between data-streams and across points in time. In particular, it is considered that generating processes may vary over time, and are thus non-stationary. For example, patterns of brain activity are expected to change when performing different tasks or thought processes. Models that can describe such behaviour must be adaptable over time. However, such adaptability creates challenges for model identification. In order to perform learning or estimation one must control how model complexity grows in relation to the volume of data. To this extent, one of the main themes of this work is to investigate both the implementation and effect of assumptions on sparsity; relating to model parsimony at an individual time- point, and smoothness; how quickly a model may change over time. Throughout this thesis two basic classes of non-stationary model are stud- ied. Firstly, a class of piecewise constant Gaussian Graphical models (GGM) is introduced that can encode graphical dependencies between data-streams. In particular, a group-fused regulariser is examined that allows for the estima- tion of changepoints across graphical models. The second part of the thesis focuses on extending a class of locally-stationary wavelet (LSW) models. Un- like the raw GGM this enables one to encode dependencies not only between data-streams, but also across time. A set of sparsity aware estimators are developed for estimation of the spectral parameters of such models which are then compared to previous works in the domain
Contributions to High-Dimensional Pattern Recognition
This thesis gathers some contributions to statistical pattern recognition particularly targeted
at problems in which the feature vectors are high-dimensional. Three pattern recognition
scenarios are addressed, namely pattern classification, regression analysis and score fusion.
For each of these, an algorithm for learning a statistical model is presented. In order to
address the difficulty that is encountered when the feature vectors are high-dimensional,
adequate models and objective functions are defined. The strategy of learning simultaneously
a dimensionality reduction function and the pattern recognition model parameters is shown to
be quite effective, making it possible to learn the model without discarding any discriminative
information. Another topic that is addressed in the thesis is the use of tangent vectors as
a way to take better advantage of the available training data. Using this idea, two popular
discriminative dimensionality reduction techniques are shown to be effectively improved. For
each of the algorithms proposed throughout the thesis, several data sets are used to illustrate
the properties and the performance of the approaches. The empirical results show that the
proposed techniques perform considerably well, and furthermore the models learned tend to
be very computationally efficient.Villegas SantamarÃa, M. (2011). Contributions to High-Dimensional Pattern Recognition [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/10939Palanci
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