Estimation Of Principal Components Research Articles

Penalized spline estimation of principal components for sparse functional data: Rates of convergence

Penalized spline estimation of principal components for sparse functional data: Rates of convergence

Improvement of simultaneous prediction using principal components approach

Prediction has a fundamental role both in linear regression analysis and in many applied sciences. The predictions of actual values or mean values of the response variable were obtained separately, earlier than defined the target function. It has been attracting a lot of interest because obtaining simultaneous predictions both of the mean values and actual values is more reasonable in many applied studies since the target function was reported. This paper purposes to improve the simultaneous prediction with the target function when the problem of multicollinearity exists in the linear regression model by defining new predictors based on the principal components estimator, which is more resistant to multicollinearity than the ordinary least squares estimator. The comparisons of proposed predictors are done theoretically by using the mean squared error of predictions. Then theoretical results are supported by an exhaustive simulation study. Also, performance comparisons of the newly suggested predictors are made for two real-life data sets, real estate valuation and cruise ship info. The findings show that the use of the proposed predictors will make a crucial contribution to applied studies in obtaining better predictions.

Penalized estimation in parametric frailty model

Frailty model examines the effect of observable and non-observable factors on time to event data. Presence of collinearity produces unstable estimates of parameters. Therefore, this research focus on the penalized estimation of frailty model and proposed the new estimator which is the extension of ridge and principal component estimators. Simulation is run to reveal the performance of proposed estimator. Moreover, the technique is applied on NFHS (National Family Health Survey) data to examine the infant mortality in India.

Minimax estimation of functional principal components from noisy discretized functional data

AbstractFunctional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.

Combination of the modified Kibria–Lukman and the principal component regression estimators

Statistical inference with the ordinary least squares (OLS) estimator is frequently influenced when there is a multicollinearity in the linear regression model. In this article, to reduce these effects of multicollinearity, we generalize the modified Kibria–Lukman principal component (MKLPC) estimator in the linear regression model by combining the principal component regression (PCR) estimator and the modified Kibria–Lukman (MKL) estimator. Meanwhile, the necessary and sufficient conditions for the superiority of the MKLPC estimator over OLS, PCR, Ridge, r-k, Liu, r-d, k-d, KL, and MKL estimators in the mean squared error (MSE) criterion are derived. Furthermore, we conduct Monte Carlo simulation and empirical analysis to compare these estimators under the MSE criterion.

Performance of Principal Components Estimation Method on the Quality of Factor Analysis without and with Varimax Rotation

The objective of this work was to evaluate the performance of the principal components estimation method on the quality of estimates of some parameters of exploratory factor analysis (EFA), without and with varimax rotation. To this end, 18 parametric correlation matrices (r) were imposed. These matrices were obtained from combinations between six different values of parametric communalities of four normally distributed random variables with three different proportions of distances between the parametric factorial loadings of the first two orthogonal factors. For each matrix r, the following parameters were defined: the first two eigenvalues (l1 and l2), the matrix of factor loadings (G), the four communalities () and the matrix of the specific factors (Y). After the 36 factor analyses, the respective estimates of these parameters were obtained, and their respective absolute deviations between the estimates obtained a posteriori and the parameters known a priori were evaluated. With the results of the Student's t test at 5% significance applied to the response surface analysis, it was concluded that the principal components estimation method for estimating orthogonal factors was not adequate and the varimax rotation improved relatively little the quality of the SFA estimates.

PFA-Nipals: An Unsupervised Principal Feature Selection Based on Nonlinear Estimation by Iterative Partial Least Squares

Unsupervised feature selection (UFS) has received great interest in various areas of research that require dimensionality reduction, including machine learning, data mining, and statistical analysis. However, UFS algorithms are known to perform poorly on datasets with missing data, exhibiting a significant computational load and learning bias. In this work, we propose a novel and robust UFS method, designated PFA-Nipals, that works with missing data without the need for deletion or imputation. This is achieved by considering an iterative nonlinear estimation of principal components by partial least squares, while the relevant features are selected through minibatch K-means clustering. The proposed method is successfully applied to select the relevant features of a robust health dataset with missing data, outperforming other UFS methods in terms of computational load and learning bias. Furthermore, the proposed method is capable of finding a consistent set of relevant features without biasing the explained variability, even under increasing missing data. Finally, it is expected that the proposed method could be used in several areas, such as machine learning and big data with applications in different areas of the medical and engineering sciences.

Sparse principal component analysis for high‐dimensional stationary time series

AbstractWe consider the sparse principal component analysis for high‐dimensional stationary processes. The standard principal component analysis performs poorly when the dimension of the process is large. We establish oracle inequalities for penalized principal component estimators for the large class of processes including heavy‐tailed time series. The rate of convergence of the estimators is established. We also elucidate the theoretical rate for choosing the tuning parameter in penalized estimators. The performance of the sparse principal component analysis is demonstrated by numerical simulations. The utility of the sparse principal component analysis for time series data is exemplified by the application to average temperature data.

Combining Kibria-Lukman and principal component estimators for the distributed lag models

Combining Kibria-Lukman and principal component estimators for the distributed lag models

Functional principal component analysis for partially observed elliptical process

The robust principal component estimators for partially observed functional data with heavy-tail behaviors are presented, where sample trajectories are collected over individual-specific subintervals. The method considers partially sampled trajectories as the elliptical process filtered by the missing indicator process and implements robust functional principal component analysis under this framework. The proposed method is computationally efficient and straightforward by estimating the robust correlation function through pairwise covariance computation combined with M-estimation. The asymptotic consistency of the estimators is established under general conditions. The simulation studies demonstrate the superior performance of the method in the approximation of the subspace of data and reconstruction of full trajectories. The proposed method is then applied to hourly monitored air pollutant data containing anomaly trajectories with random missing segments.

Approximate factor models with weaker loadings

Pervasive cross-section dependence is increasingly recognized as a characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure where Λ0′Λ0/Nα is positive definite in the limit when α=1, early work established convergence of the principal component estimates of the factors and loadings up to a rotation matrix. This paper shows that the estimates are still consistent and asymptotically normal when α∈(0,1] albeit at slower rates and under additional assumptions on the sample size. The results hold whether α is constant or varies across factor loadings. The framework developed for heterogeneous loadings and the simplified proofs that can be also used in strong factor analysis are of independent interest.

Milk-Compositional Study of Metabolites and Pathogens in the Milk of Bovine Animals Affected with Subclinical Mastitis.

Bovine milk is an important food component in the human diet due to its nutrient-rich metabolites. However, bovine subclinical mastitis alters the composition and quality of milk. In present study, California mastitis testing, somatic cell count, pH, and electrical conductivity were used as confirmatory tests to detect subclinical mastitis. The primary goal was to study metabolome and identify major pathogens in cows with subclinical mastitis. In this study, 29 metabolites were detected in milk using gas chromatography−mass spectrometry. Volatile acidic compounds, such as hexanoic acid, hexadecanoic acid, lauric acid, octanoic acid, n-decanoic acid, tricosanoic acid, tetradecanoic acid, and hypogeic acid were found in milk samples, and these impart good flavor to the milk. Metaboanalyst tool was used for metabolic pathway analysis and principal component estimation. In this study, EC and pH values in milk were significantly increased (p < 0.0001), whereas fat (p < 0.04) and protein (p < 0.0002) significantly decreased in animals with subclinical mastitis in comparison to healthy animals. Staphylococcus aureus was the predominant pathogen found (n = 54), followed by Escherichia coli (n = 30). Furthermore, antibiotic sensitivity revealed that Staphylococcus aureus was more sensitive to gentamicin (79.6%), whereas Escherichia coli showed more sensitivity to doxycycline hydrochloride (80%).

Simultaneous prediction using target function based on principal components estimator with correlated errors

Simultaneous prediction using target function based on principal components estimator with correlated errors

Adherence to Medication in Neurogeriatric Patients: Insights from the NeuroGerAd Study.

Nonadherence to medication is associated with increased morbidity, mortality, and healthcare costs, especially in older adults with higher chances of multimorbidity. However, comprehensive data on factors influencing adherence in this patient group are rare. Thus, data for 910 patients were acquired, including demographic data, nonadherence (Stendal Adherence to Medication), depression (Beck Depression Inventory), cognition (Montreal Cognitive Assessment), personality (Big Five Inventory), satisfaction with healthcare (Health Care Climate Questionnaire), quality of life (36-item Short Form Survey), mobility, diagnoses, and medication. Elastic net regularization was used to analyze the predictors of adherence. Principal component and general estimation equations were calculated to analyze the underlying patterns of adherence. Only 21.1% of patients were fully adherent. Nonadherence was associated with male gender, higher number of medications, diagnosis, depression, poor patient–physician relationship, personality, impaired cognition, and impaired mobility. Nonadherence was classified into three sub-factors: forgetting (46.2%), missing knowledge about medication (29%), and intentional modification of medication (24.8%). While depression exerted the strongest influence on modification, a high number of medications was associated with missing knowledge. The different patterns of nonadherence (i.e., modification, missing knowledge, and forgetting) are influenced differently by clinical factors, indicating that specific approaches are needed for interventions targeting adherence.

Analysis And Prediction of Influencing Factors of Total Retail Sales of Consumer Goods in China——Based on Multiple Linear Regression Models and ARIMA Models

This paper takes the annual data of total retail sales of consumer goods in China from 2001 to 2020 as the research object. Using methods such as principal component estimation, ridge regression and Lasso regression, a multivariate linear regression model is established, and the regression results of the three models are compared and analyzed, and the significant factors affecting the total retail sales of consumer goods in China are the income level of residents, the level of consumption of residents, and the development of the tertiary industry, and the less significant factors are demographic factors. In addition, this paper takes the annual data of china's total retail sales of social consumer goods from 1973 to 2020 as the research object, and uses the ALAMA (0,1,1) model to predict and analyze the total retail sales of social consumer goods in China in the next ten years, and concludes that the total retail sales of social consumer goods in China will gradually rise from 2021 to 2030, which is consistent with the actual situation. Thus, it puts forward rationalization suggestions for increasing the total retail sales of consumer goods in China and promoting economic development.

Further research on the modified ridge principal component estimator in linear model

Yang and Huang have introduced a generalized Mahalanobis loss function which can be applied to the estimator for regression coefficients whether its covariance matrix is singular or non singular. In this paper, we give the detailed comparisons among those estimators that can be derived from the modified ridge principal component estimator under the generalized Mahalanobis loss function by the average loss criterion. Also, we obtain conditions for the superiority of one estimator over the others. Furthermore, two numerical examples are given to illustrate the theoretical results.

Deep learning optical image denoising research based on principal component estimation.

High-quality denoising of optical interference images usually requires preliminary prediction of the noise level. Although blind denoising can filter the image at the pixel level without noise prediction, it inevitably loses a significant amount of phase information. This paper proposes a fast and high-quality denoising algorithm for optical interference images that combines the merits of a principal component analysis (PCA) and residual neural networks. The PCA is used to analyze the image noise and, in turn, establishes an accurate mapping between the estimated and true noise levels. The mapping helps to select a suitable residual neural network model for image processing, which maximizes the retention of image information and reduces the effect of noise. In addition, a comprehensive evaluation factor to account for the time complexity and denoising effect of the algorithm is proposed, since time complexity can be a dominant concern in some cases of actual measurement. The performance of the denoising algorithm and the effectiveness of the evaluation criterion are demonstrated to be high by processing a set of optical interference images and benchmarking with other denoising algorithms. The proposed algorithm outperforms the previously reported counterparts in a specific area of optical interference image preprocessing and provides an alternative paradigm for other denoising problems of optics, such as holograms and structured light measurements.

A robust approach to heteroskedasticity, error serial correlation and slope heterogeneity in linear models with interactive effects for large panel data

In this paper, we propose a robust approach against heteroskedasticity, error serial correlation and slope heterogeneity in linear models with interactive effects for large panel data. First, consistency and asymptotic normality of the pooled iterated principal component (IPC) estimator for random coefficient and homogeneous slope models are established. Then, we prove the asymptotic validity of the associated Wald test for slope parameter restrictions based on the panel heteroskedasticity and autocorrelation consistent (PHAC) variance matrix estimator for both random coefficient and homogeneous slope models, which does not require the Newey-West type time-series parameter truncation. These results asymptotically justify the use of the same pooled IPC estimator and the PHAC standard error for both homogeneous-slope and heterogeneous-slope models. This robust approach can significantly reduce the model selection uncertainty for applied researchers. In addition, we propose a Lagrange Multiplier (LM) test for correlated random coefficients with covariates. This test has non-trivial power against correlated random coefficients, but not for random coefficients and homogeneous slopes. The LM test is important because the IPC estimator becomes inconsistent with correlated random coefficients. The finite sample evidence and an empirical application support the reliability and the usefulness of our robust approach.

Two-Dimensional Functional Principal Component Analysis for Image Feature Extraction

Methodologies for functional principal component analysis are well established in the one-dimensional setting. However, for two-dimensional surfaces, for example, images, conducting functional principal component analysis is complicated and challenging, because the conventional eigendecomposition approach would require the estimation of a four-dimensional covariance function, which may incur high cost in terms of time and machine memory. To circumvent such computational difficulties, we propose a novel two-dimensional functional principal component analysis for extracting functional principal components and achieving dimensionality reduction for images. Different from the conventional eigendecomposition approach, our proposed method is based on the direct estimation of the optimal two-dimensional functional principal components via tensor product B-spline, which opens up a new avenue for estimating functional principal components. We present theoretical results that prove the consistency of the proposed approach. Our method is illustrated by analyzing brain images of subjects with the Alzheimer’s Disease and the handwritten digits images. The finite sample performance of our method is further assessed with some simulation studies. Supplementary materials for this article are available online.

Combining principal component and robust ridge estimators in linear regression model with multicollinearity and outlier

AbstractThe method of least squared suffers a setback when there is multicollinearity and outliers in the linear regression model. In this article, we developed a new estimator to jointly handle multicollinearity and outliers by pooling the following estimators together: the M‐estimator, the principal component and the ridge estimator. The new estimator is called the robust r‐k estimator and is employed. We established theoretically that the new estimator is better than some of the existing ones. The simulation studies and real‐life application supports the efficiency of the new method.