Properties Of Principal Component Analysis Research Articles

Universal prediction of cell-cycle position using transfer learning

BackgroundThe cell cycle is a highly conserved, continuous process which controls faithful replication and division of cells. Single-cell technologies have enabled increasingly precise measurements of the cell cycle both as a biological process of interest and as a possible confounding factor. Despite its importance and conservation, there is no universally applicable approach to infer position in the cell cycle with high-resolution from single-cell RNA-seq data.ResultsHere, we present tricycle, an R/Bioconductor package, to address this challenge by leveraging key features of the biology of the cell cycle, the mathematical properties of principal component analysis of periodic functions, and the use of transfer learning. We estimate a cell-cycle embedding using a fixed reference dataset and project new data into this reference embedding, an approach that overcomes key limitations of learning a dataset-dependent embedding. Tricycle then predicts a cell-specific position in the cell cycle based on the data projection. The accuracy of tricycle compares favorably to gold-standard experimental assays, which generally require specialized measurements in specifically constructed in vitro systems. Using internal controls which are available for any dataset, we show that tricycle predictions generalize to datasets with multiple cell types, across tissues, species, and even sequencing assays.ConclusionsTricycle generalizes across datasets and is highly scalable and applicable to atlas-level single-cell RNA-seq data.

Quasar main sequence: A line or a plane

Context. A quasar main sequence is widely believed to reveal itself through objects represented in a plane spanned by two parameters: the full width at half maximum (FWHM) of Hβ and the ratio of Fe II to Hβ equivalent width. This sequence is related to the application to quasar properties of principal component analysis (PCA), which reveals that the main axis of variance (eigenvector 1) is codirectional with a strong anticorrelation between these two measurements. Aims. We aim to determine whether the dominance of two eigenvectors, originally discovered over two decades ago, is replicated in newer high-quality quasar samples. If so, we aim to test whether a nonlinear approach is an improvement on the linear PCA method by finding two new parameters that represent a more accurate projection of the variances than the eigenvectors recovered from PCA. Methods. We selected quasars from the X-shooter archive and a major quasar catalog to build high-quality samples. These samples were tested with PCA. Results. We find that the new high-quality samples indeed have two dominant eigenvectors as originally discovered. Subsequently, we find that fitting a nonlinear decay curve to the main sequence allows a new plane spanned by linearly independent axes to be defined; this is based on the distance along the decay curve as the main axis and the distance of each quasar data point from the curve as the secondary axis, respectively. Conclusions. The results show that it is possible to define a new plane based on the quasar main sequence, which accounts for the majority of the variance. The most likely candidate for the new main axis is an anticorrelation with a black hole mass. In this case the secondary axis likely represents luminosity. However, given the results of previous studies, the inclination angle likely plays a role in the Hβ width.

Asymptotic properties of principal component analysis and shrinkage-bias adjustment under the generalized spiked population model

With the development of high-throughput technologies, principal component analysis (PCA) in the high-dimensional regime is of great interest. Most of the existing theoretical and methodological results for high-dimensional PCA are based on the spiked population model in which all the population eigenvalues are equal except for a few large ones. Due to the presence of local correlation among features, however, this assumption may not be satisfied in many real-world datasets. To address this issue, we investigate the asymptotic behavior of PCA under the generalized spiked population model. Based on our theoretical results, we propose a series of methods for the consistent estimation of population eigenvalues, angles between the sample and population eigenvectors, correlation coefficients between the sample and population principal component (PC) scores, and the shrinkage bias adjustment for the predicted PC scores. Using numerical experiments and real data examples from the genetics literature, we show that our methods can greatly reduce bias and improve prediction accuracy.

The Statistics and Mathematics of High Dimension Low Sample Size Asymptotics.

The aim of this paper is to establish several deep theoretical properties of principal component analysis for multiple-component spike covariance models. Our new results reveal an asymptotic conical structure in critical sample eigendirections under the spike models with distinguishable (or indistinguishable) eigenvalues, when the sample size and/or the number of variables (or dimension) tend to infinity. The consistency of the sample eigenvectors relative to their population counterparts is determined by the ratio between the dimension and the product of the sample size with the spike size. When this ratio converges to a nonzero constant, the sample eigenvector converges to a cone, with a certain angle to its corresponding population eigenvector. In the High Dimension, Low Sample Size case, the angle between the sample eigenvector and its population counterpart converges to a limiting distribution. Several generalizations of the multi-spike covariance models are also explored, and additional theoretical results are presented.

A general framework for consistency of principal component analysis

A general asymptotic framework is developed for studying consistency properties of principal component analysis (PCA). Our framework includes several previously studied domains of asymptotics as special cases and allows one to investigate interesting connections and transitions among the various domains. More importantly, it enables us to investigate asymptotic scenarios that have not been considered before, and gain new insights into the consistency, subspace consistency and strong inconsistency regions of PCA and the boundaries among them. We also establish the corresponding convergence rate within each region. Under general spike covariance models, the dimension (or number of variables) discourages the consistency of PCA, while the sample size and spike information (the relative size of the population eigenvalues) encourage PCA consistency. Our framework nicely illustrates the relationship among these three types of information in terms of dimension, sample size and spike size, and rigorously characterizes how their relationships aect PCA consistency.

Acceleration of dynamic fluorescence molecular tomography with principal component analysis.

Dynamic fluorescence molecular tomography (FMT) is an attractive imaging technique for three-dimensionally resolving the metabolic process of fluorescent biomarkers in small animal. When combined with compartmental modeling, dynamic FMT can be used to obtain parametric images which can provide quantitative pharmacokinetic information for drug development and metabolic research. However, the computational burden of dynamic FMT is extremely huge due to its large data sets arising from the long measurement process and the densely sampling device. In this work, we propose to accelerate the reconstruction process of dynamic FMT based on principal component analysis (PCA). Taking advantage of the compression property of PCA, the dimension of the sub weight matrix used for solving the inverse problem is reduced by retaining only a few principal components which can retain most of the effective information of the sub weight matrix. Therefore, the reconstruction process of dynamic FMT can be accelerated by solving the smaller scale inverse problem. Numerical simulation and mouse experiment are performed to validate the performance of the proposed method. Results show that the proposed method can greatly accelerate the reconstruction of parametric images in dynamic FMT almost without degradation in image quality.

Minimizer of the Reconstruction Error for multi-class document categorization

In the present article we introduce and validate an approach for single-label multi-class document categorization based on text content features. The introduced approach uses the statistical property of Principal Component Analysis, which minimizes the reconstruction error of the training documents used to compute a low-rank category transformation matrix. Such matrix transforms the original set of training documents from a given category to a new low-rank space and then optimally reconstructs them to the original space with a minimum reconstruction error. The proposed method, called Minimizer of the Reconstruction Error (mRE) classifier, uses this property, and extends and applies it to new unseen test documents. Several experiments on four multi-class datasets for text categorization are conducted in order to test the stable and generally better performance of the proposed approach in comparison with other popular classification methods.

Subspace Learning from Image Gradient Orientations

We introduce the notion of subspace learning from image gradient orientations for appearance-based object recognition. As image data are typically noisy and noise is substantially different from Gaussian, traditional subspace learning from pixel intensities very often fails to estimate reliably the low-dimensional subspace of a given data population. We show that replacing pixel intensities with gradient orientations and the ℓ₂ norm with a cosine-based distance measure offers, to some extend, a remedy to this problem. Within this framework, which we coin Image Gradient Orientations (IGO) subspace learning, we first formulate and study the properties of Principal Component Analysis of image gradient orientations (IGO-PCA). We then show its connection to previously proposed robust PCA techniques both theoretically and experimentally. Finally, we derive a number of other popular subspace learning techniques, namely, Linear Discriminant Analysis (LDA), Locally Linear Embedding (LLE), and Laplacian Eigenmaps (LE). Experimental results show that our algorithms significantly outperform popular methods such as Gabor features and Local Binary Patterns and achieve state-of-the-art performance for difficult problems such as illumination and occlusion-robust face recognition. In addition to this, the proposed IGO-methods require the eigendecomposition of simple covariance matrices and are as computationally efficient as their corresponding ℓ₂ norm intensity-based counterparts. Matlab code for the methods presented in this paper can be found at http://ibug.doc.ic.ac.uk/resources.

A General Framework for Consistency of Principal Component Analysis

A general asymptotic framework is developed for studying consis- tency properties of principal component analysis (PCA). Our frame- work includes several previously studied domains of asymptotics as special cases and allows one to investigate interesting connections and transitions among the various domains. More importantly, it enables us to investigate asymptotic scenarios that have not been considered before, and gain new insights into the consistency, subspace consistency and strong inconsistency regions of PCA and the boundaries among them. We also establish the corresponding convergence rate within each region. Under general spike covariance models, the dimension (or the number of variables) discourages the consistency of PCA, while the sample size and spike information (the relative size of the population eigenvalues) encourages PCA consistency. Our framework nicely illustrates the relationship among these three types of information in terms of dimension, sample size and spike size, and rigorously characterizes how their relationships affect PCA consistency.

ChemInform Abstract: Structure-Protection Mechanism of Dipalmitoylphosphatidylcholine Liposomes by Adsorption of Sucrose Molecules in the Liposome Surface

Molecular mechanism of structure-protection of dipalmitoylphosphatidylcholine (DPPC) liposomes by sucrose adsorption on/in the liposome surface was studied by performing fluorescence decay, quartz-crystal microbalance (QCM), and infrared reflection-absorption (IRRA) spectroscopic measurements. Thermally hydrated DPPC monolayers were newly introduced in place of the conventional DPPC liposomes, since the hydrated monolayers were expected to have similar surface property to that of liposomes. Studies of the fluidity and permeability of membrane in the monolayers and liposomes suggested that the thermally hydrated DPPC monolayers could be used as a model membrane in place of DPPC liposomes. With the hydrated DPPC monolayers, the adsorption rate of sucrose molecules in the DPPC monolayer was investigated by a combined technique of QCM and IRRA by the use of Langmuir-Blodgett (LB) technique. It was found that the incorporation rate of sucrose into the monolayer in the liquid-expanded state was proportional to the bulk concentration of sucrose in the Langmuir trough. The linear relationship depicts the incorporation mechanism that sucrose molecules were deeply incorporated in the DPPC molecules, and the layer is formed as a single mingled layer. This molecular scheme readily supports the conventional hypotheses based upon water-replacement and vitrification mechanisms. The IRRA spectra were further studied using newly found property of principal component analysis (PCA). PCA was found to have great potential to detect minute bands hidden in dominantly large bands, after theoretical speculation and simulations. The new analytical technique was employed to search for an unknown minute spectrum in the collection of raw spectra. The calculated abstract spectra yielded by PCA strongly suggested that there remained minute quantity of water in the dried LB films. The water molecules were also found to bind to PO2 group in the head group moiety of DPPC. The mass change of adsorbed and incorporated sucrose molecules in the DPPC monolayer were readily distinguished by PCA without any probe molecule, and they were followed semi-quantitatively by calculation of scores.

Dimension Reduction of Optical Remote Sensing Images via Minimum Change Rate Deviation Method

This paper introduces a new dimension reduction (DR) method, called minimum change rate deviation (MCRD), which is applicable to the DR of remote sensing images. As the main shortcoming of the well-known principal component analysis (PCA) method is that it does not consider the spatial relation among image points, our proposed approach takes into account the spatial relation among neighboring image pixels while preserving all useful properties of PCA. These include uncorrelatedness property in resulted components and the decrease of error with the increasing of the number of selected components. Our proposed method can be considered as a generalization of PCA and, under certain conditions, reduces to it. The proposed MCRD method employs linear spatial operators to consider the spatiality of images. The superiority of MCRD over conventional PCA is demonstrated both mathematically and experimentally. It is shown that MCRD, with an acceptable speed, outperforms PCA in retaining the required information for classification purposes. Moreover, as the locally linear embedding (LLE) method also employs the spatial relations in its DR process, the performances of MCRD and LLE are compared, and the superiority of the proposed method in both classification accuracy and computational cost is shown.

Composition estimation of multicomponent reactive batch distillation with optimal sensor configuration

An inferential state estimation scheme based on extended Kalman filter (EKF) with optimal selection of sensor locations using principal component analysis (PCA) is presented for composition estimation in multicomponent reactive batch distillation. The properties of PCA are exploited to provide the most sensitive dynamic temperature measurement information of the process to the estimator for accurate estimation of compositions. The state estimator is supported by a simplified dynamic model of reactive batch distillation that includes component balance equations together with thermodynamic relations and reaction kinetics. The performance of the proposed scheme is evaluated by applying it for composition estimation on all trays, reboiler, reflux drum and products of a reactive batch distillation column, in which ethyl acetate is produced through an esterification reaction between acetic acid and ethanol. This quaternary system with azeotropism is highly nonlinear and typically suited for implementation of the proposed scheme. The results demonstrate that the proposed EKF estimation scheme with optimal temperature sensor configuration is effective for inferential estimation of compositions in multicomponent reactive batch distillation.

Structure-protection mechanism of dipalmitoylphosphatidylcholine liposomes by adsorption of sucrose molecules in the liposome surface

Molecular mechanism of structure-protection of dipalmitoylphosphatidylcholine (DPPC) liposomes by sucrose adsorption on/in the liposome surface was studied by performing fluorescence decay, quartz-crystal microbalance (QCM), and infrared reflection-absorption (IRRA) spectroscopic measurements. Thermally hydrated DPPC monolayers were newly introduced in place of the conventional DPPC liposomes, since the hydrated monolayers were expected to have similar surface property to that of liposomes. Studies of the fluidity and permeability of membrane in the monolayers and liposomes suggested that the thermally hydrated DPPC monolayers could be used as a model membrane in place of DPPC liposomes. With the hydrated DPPC monolayers, the adsorption rate of sucrose molecules in the DPPC monolayer was investigated by a combined technique of QCM and IRRA by the use of Langmuir-Blodgett (LB) technique. It was found that the incorporation rate of sucrose into the monolayer in the liquid-expanded state was proportional to the bulk concentration of sucrose in the Langmuir trough. The linear relationship depicts the incorporation mechanism that sucrose molecules were deeply incorporated in the DPPC molecules, and the layer is formed as a single mingled layer. This molecular scheme readily supports the conventional hypotheses based upon water-replacement and vitrification mechanisms. The IRRA spectra were further studied using newly found property of principal component analysis (PCA). PCA was found to have great potential to detect minute bands hidden in dominantly large bands, after theoretical speculation and simulations. The new analytical technique was employed to search for an unknown minute spectrum in the collection of raw spectra. The calculated abstract spectra yielded by PCA strongly suggested that there remained minute quantity of water in the dried LB films. The water molecules were also found to bind to PO2 group in the head group moiety of DPPC. The mass change of adsorbed and incorporated sucrose molecules in the DPPC monolayer were readily distinguished by PCA without any probe molecule, and they were followed semi-quantitatively by calculation of scores.

Principal Components Biplots and Alpha and Beta Diversity

Attention is drawn to some useful but not generally known properties of principal components analysis (PCA). Noncentered PCA of proportion data gives site ordinations that display approximate alpha diversities of sites and beta diversities of groups of sites, as measured by the Simpson index and mean squared Euclidean distance, respectively. Species centering allows a better approximation to beta diversities. Alpha diversities can still be visualized after centering if the true origin is projected into the plane of the ordination. The approximate species composition of each site can also be visualized if the site ordination is combined with a species ordination. The resulting plot of site scores and species loadings is called a PCA biplot. Finally, in a PCA biplot that displays both species composition and diversity, diversity values can be explained in terms of the main species contributing to diversity. In such a biplot the sum of squares of the species loadings must be scaled to unity, while the site scores must be scaled to a sum of squares equal to the corresponding eigenvalue. This type of biplot is termed a "distance biplot." For a simple illustration noncentered and species—centered distance biplots were produced for some diatom samples taken from Dutch moorland pools in the 1920s and 1978. The distance biplot is concluded to be among the most powerful analytical tools for species—composition data and derives some of its power from properties not possessed by, for example, reciprocal averaging. One problem is that it attaches little weight to rare species, but this problem can be solved by various possible data transformations based on the theory of diversity indices.

Understanding Canonical Correlation through the General Linear Model and Principal Components

Abstract Canonical correlation has been little used and little understood, even by otherwise sophisticated analysts. An alternative approach to canonical correlation, based on a general linear multivariate model, is presented. Properties of principal component analysis are used to help explain the method. Standard computational methods for full rank canonical correlation, techniques for canonical correlation on component scores, and canonical correlation with less than full rank are discussed. They are seen to be essentially equivalent when the model equation for canonical correlation on component scores is presented. The two approaches to less than full rank situations are equivalent in some senses, but quite different in usefulness, depending on the application. An example dataset is analyzed in detail to help demonstrate the conclusions.

Principal components and varimax-rotated components in event-related potential research: Some remarks on their interpretation

Some properties of principal components analysis (PCA) and simple structure rotation, which are relevant to the study of event-related potentials (ERPs), were examined both in theory and with simulated data. The analysis casts some doubt on whether it is useful and desirable to rotate the loading pattern of a PCA of ERPs to simple structure. The evidence presented is more in favour of the unrotated solution. In particular we were able to demonstrate with simulated data that a solution rotated to simple structure may lead to false conclusions about the functional independence of ERP peaks. Beyond this. the more general question is asked of whether principal components should be accepted as basic waveforms, i.e. as physiologically meaningful entities which, for example, represent different ERP generators.