Reconciling functional data regression with excess bases

Functional regression has been widely used on longitudinal data, but it is not clear how to apply functional regression to microbiome sequencing data. We propose a novel functional response regression model analyzing correlated longitudinal microbiome sequencing data, which extends the classic functional response regression model only working for independent functional responses. We derive the theory of generalized least squares estimators for predictors’ effects when functional responses are correlated, and develop a data transformation technique to solve the computational challenge for analyzing correlated functional response data using existing functional regression method. We show by extensive simulations that our proposed method provides unbiased estimations for predictors’ effect, and our model has accurate type I error and power performance for correlated functional response data, compared with classic functional response regression model. Finally we implement our method to a real infant gut microbiome study to evaluate the relationship of clinical factors to predominant taxa along time.

The calibration method is an alternative method that can be used to analyze the relationship between invasive and non-invasive blood glucose levels. Calibration modeling generally has a large dimension and contains multicolinearities because usually in functional data the number of independent variables (p) is greater than the number of observations (p>n). Both problems can be overcome using Functional Regression (FR) and Functional Principal Component Regression (FPCR). FPCR is based on Principal Component Analysis (PCA). In FPCR, the data is transformed using a polynomial basis before data reduction. This research tried to model the equations of spectral calibration of voltage value excreted by non-invasive blood glucose level monitoring devices to predict blood glucose using FR and FPCR. This study aimed to determine the best calibration model for measuring non-invasive blood glucose levels with the FR and FPCR. The results of this research showed that the FR model had a bigger coefficient determination (R2) value and lower Root Mean Square Error (RMSE) and Root Mean Square Error Prediction (RMSEP) value than the FPCR model, which was 12.9%, 5.417, and 5.727 respectively. Overall, the calibration modeling with the FR model is the best model for estimate blood glucose level compared to the FPCR model.

This article provides an overview of recent nonparametric and semiparametric advances in kernel regression estimation for functional data. In particular, it considers the various statistical techniques based on kernel smoothing ideas that have recently been developed for functional regression estimation problems. The article first examines nonparametric functional regression modelling before discussing three popular functional regression estimates constructed by means of kernel ideas, namely: the Nadaraya-Watson convolution kernel estimate, the kNN functional estimate, and the local linear functional estimate. Uniform asymptotic results are then presented. The article proceeds by reviewing kernel methods in semiparametric functional regression such as single functional index regression and partial linear functional regression. It also looks at the use of kernels for additive functional regression and concludes by assessing the impact of kernel methods on practical real-data analysis involving functional (curves) datasets.

Functional data are increasingly encountered in scientific studies, and their high dimensionality and complexity lead to many analytical challenges. Various methods for functional data analysis have been developed, including functional response regression methods that involve regression of a functional response on univariate/multivariate predictors with nonparametrically represented functional coefficients. In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions of curves, so is not robust. In this article, we introduce a new Bayesian method, robust functional mixed models (R-FMM), for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation induced by the experimental design. The underlying model involves a hierarchical scale mixture model for the fixed effects, random effect, and residual error functions. These modeling assumptions across curves result in robust nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with great flexibility for the data to determine the sparsity and the heaviness of the tails. Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large datasets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be applied more generally using other heavy-tailed distributions, higher dimensional functions (e.g., images), and using other invertible transformations as alternatives to wavelets. Supplementary materials for this article are available online.

This article studies a general regression model for a scalar quality response with mixed types of process predictors including process images, functional sensing signals, and scalar process setup attributes. To represent a set of time-dependent process images, a third-order tensor is employed for preserving not only the spatial correlation of pixels within one image but also the temporal dependency among a sequence of images. Although there exist some papers dealing with either tensorial or functional regression, there is little research to thoroughly study a regression model consisting of both tensorial and functional predictors. For simplicity, the presented regression model is called functional linear regression with tensorial and functional predictor (FLR-TFP). The advantage of the presented FLR-TFP model, which is compared to the classical stack-up strategy, is that FLR-TFP can handle both tensorial and functional predictors without destroying the data correlation structure. To estimate an FLR-TFP model, this article presents a new alternating Elastic Net (AEN) estimation algorithm, in which the problem is reformed as three sub-problems by iteratively estimating each group of tensorial, functional, and scalar parameters. To execute the proposed AEN algorithm, a systematic approach is developed to effectively determine the initial running sequence among three sub-problems. The performance of the FLR-TFP model is evaluated using simulations and a real-world case study of friction stir blind riveting process.

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay & Silverman's (1997) textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article focuses on functional regression, the area of FDA that has received the most attention in applications and methodological development. First, there is an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: (a) functional predictor regression (scalar-on-function), (b) functional response regression (function-on-scalar), and (c) function-on-function regression. For each, the role of replication and regularization is discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. The review concludes with a brief discussion describing potential areas of future development in this field.

In recent decades, functional data have become a commonly encountered type of data. Its ideal units of observation are functions defined on some continuous domain and the observed data are sampled on a discrete grid. An important problem in functional data analysis is how to fit regression models with scalar responses and functional predictors (scalar-on-function regression). This paper focuses on the nonparametric approaches to this problem. First there is a review of the classical k-nearest neighbors (kNN) method for functional regression. Then the mutual nearest neighbors (MNN) method, which is a variant of kNN method, is applied to functional regression. Compared with the classical kNN approach, the MNN method takes use of the concept of mutual nearest neighbors to construct regression model and the pseudo nearest neighbors will not be taken into account during the prediction process. In addition, any nonparametric method in the functional data cases is affected by the curse of infinite dimensionality. To prevent this curse, it is legitimate to measure the proximity between two curves via a semi-metric. The effectiveness of MNN method is illustrated by comparing the predictive power of MNN method with kNN method first on the simulated datasets and then on a real chemometrical example. The comparative experimental analyses show that MNN method preserves the main merits inherent in kNN method and achieves better performances with proper proximity measures.

BackgroundDSC is used to determine thermally-induced conformational changes of biomolecules within a blood plasma sample. Recent research has indicated that DSC curves (or thermograms) may have different characteristics based on disease status and, thus, may be useful as a monitoring and diagnostic tool for some diseases. Since thermograms are curves measured over a range of temperature values, they are considered functional data. In this paper we apply functional data analysis techniques to analyze differential scanning calorimetry (DSC) data from individuals from the Lupus Family Registry and Repository (LFRR). The aim was to assess the effect of lupus disease status as well as additional covariates on the thermogram profiles, and use FD analysis methods to create models for classifying lupus vs. control patients on the basis of the thermogram curves.MethodsThermograms were collected for 300 lupus patients and 300 controls without lupus who were matched with diseased individuals based on sex, race, and age. First, functional regression with a functional response (DSC) and categorical predictor (disease status) was used to determine how thermogram curve structure varied according to disease status and other covariates including sex, race, and year of birth. Next, functional logistic regression with disease status as the response and functional principal component analysis (FPCA) scores as the predictors was used to model the effect of thermogram structure on disease status prediction. The prediction accuracy for patients with Osteoarthritis and Rheumatoid Arthritis but without Lupus was also calculated to determine the ability of the classifier to differentiate between Lupus and other diseases. Data were divided 1000 times into separate 2/3 training and 1/3 test data for evaluation of predictions. Finally, derivatives of thermogram curves were included in the models to determine whether they aided in prediction of disease status.ResultsFunctional regression with thermogram as a functional response and disease status as predictor showed a clear separation in thermogram curve structure between cases and controls. The logistic regression model with FPCA scores as the predictors gave the most accurate results with a mean 79.22% correct classification rate with a mean sensitivity = 79.70%, and specificity = 81.48%. The model correctly classified OA and RA patients without Lupus as controls at a rate of 75.92% on average with a mean sensitivity = 79.70% and specificity = 77.6%. Regression models including FPCA scores for derivative curves did not perform as well, nor did regression models including covariates.ConclusionChanges in thermograms observed in the disease state likely reflect covalent modifications of plasma proteins or changes in large protein-protein interacting networks resulting in the stabilization of plasma proteins towards thermal denaturation. By relating functional principal components from thermograms to disease status, our Functional Principal Component Analysis model provides results that are more easily interpretable compared to prior studies. Further, the model could also potentially be coupled with other biomarkers to improve diagnostic classification for lupus.

With the continuous improvement of computer storage capabilities and online observation technology, functional data emerged as the times required. Among the various research methods, regression analysis of functional data stands out as one of the most commonly employed techniques. This study employed Fourier spline basis functions to smooth the gathered data and preprocess it to establish a functional linear regression model with scalar response. Specifically, we investigated temperature regression on Ozone concentration across 17 cities in Henan Province in 2022. In our functional regression prediction, we compared the results obtained with and without the addition of a roughness penalty. Through this comparison, we observed that the absence of a roughness penalty led to overfitting, while including the penalty yielded more detailed results. Moreover, the coefficient of determination R 2 approached one more closely when using the roughness penalty, indicating a better-fitting regression effect. Lastly, for those interested in monitoring the regression residuals of the estimated Ozone concentration, we provide a detailed flowchart outlining the process for monitoring these residuals.

. We review and extend some statistical tools that have proved useful for analysing functional data. Functional data analysis primarily is designed for the analysis of random trajectories and infinite-dimensional data, and there exists a need for the development of adequate statistical estimation and inference techniques. While this field is in flux, some methods have proven useful. These include warping methods, functional principal component analysis, and conditioning under Gaussian assumptions for the case of sparse data. The latter is a recent development that may provide a bridge between functional and more classical longitudinal data analysis. Besides presenting a brief review of functional principal components and functional regression, we develop some concepts for estimating functional principal component scores in the sparse situation. An extension of the so-called generalized functional linear model to the case of sparse longitudinal predictors is proposed. This extension includes functional binary regression models for longitudinal data and is illustrated with data on primary biliary cirrhosis.

BackgroundModern agriculture uses hyperspectral cameras with hundreds of reflectance data at discrete narrow bands measured in several environments. Recently, Montesinos-López et al. (Plant Methods 13(4):1–23, 2017a. https://doi.org/10.1186/s13007-016-0154-2; Plant Methods 13(62):1–29, 2017b. https://doi.org/10.1186/s13007-017-0212-4) proposed using functional regression analysis (as functional data analyses) to help reduce the dimensionality of the bands and thus decrease the computational cost. The purpose of this paper is to discuss the advantages and disadvantages that functional regression analysis offers when analyzing hyperspectral image data. We provide a brief review of functional regression analysis and examples that illustrate the methodology. We highlight critical elements of model specification: (i) type and number of basis functions, (ii) the degree of the polynomial, and (iii) the methods used to estimate regression coefficients. We also show how functional data analyses can be integrated into Bayesian models. Finally, we include an in-depth discussion of the challenges and opportunities presented by functional regression analysis.ResultsWe used seven model-methods, one with the conventional model (M1), three methods using the B-splines model (M2, M4, and M6) and three methods using the Fourier basis model (M3, M5, and M7). The data set we used comprises 976 wheat lines under irrigated environments with 250 wavelengths. Under a Bayesian Ridge Regression (BRR), we compared the prediction accuracy of the model-methods proposed under different numbers of basis functions, and compared the implementation time (in seconds) of the seven proposed model-methods for different numbers of basis. Our results as well as previously analyzed data (Montesinos-López et al. 2017a, 2017b) support that around 23 basis functions are enough. Concerning the degree of the polynomial in the context of B-splines, degree 3 approximates most of the curves very well. Two satisfactory types of basis are the Fourier basis for period curves and the B-splines model for non-periodic curves. Under nine different basis, the seven method-models showed similar prediction accuracy. Regarding implementation time, results show that the lower the number of basis, the lower the implementation time required. Methods M2, M3, M6 and M7 were around 3.4 times faster than methods M1, M4 and M5.ConclusionsIn this study, we promote the use of functional regression modeling for analyzing high-throughput phenotypic data and indicate the advantages and disadvantages of its implementation. In addition, many key elements that are needed to understand and implement this statistical technique appropriately are provided using a real data set. We provide details for implementing Bayesian functional regression using the developed genomic functional regression (GFR) package. In summary, we believe this paper is a good guide for breeders and scientists interested in using functional regression models for implementing prediction models when their data are curves.

Functional linear regression analysis aims to model regression relations which include a functional predictor. The analog of the regression parameter vector or matrix in conventional multivariate or multiple-response linear regression models is a regression parameter function in one or two arguments. If, in addition, one has scalar predictors, as is often the case in applications to longitudinal studies, the question arises how to incorporate these into a functional regression model. We study a varying-coefficient approach where the scalar covariates are modeled as additional arguments of the regression parameter function. This extension of the functional linear regression model is analogous to the extension of conventional linear regression models to varying-coefficient models and shares its advantages, such as increased flexibility; however, the details of this extension are more challenging in the functional case. Our methodology combines smoothing methods with regularization by truncation at a finite number of functional principal components. A practical version is developed and is shown to perform better than functional linear regression for longitudinal data. We investigate the asymptotic properties of varying-coefficient functional linear regression and establish consistency properties.

The aim of this study is to propose the use of a functional data analysis approach as an alternative to the classical statistical methods most commonly used in oceanography and water quality management. In particular we consider the prediction of total suspended solids (TSS) based on remote sensing (RS) data. For this purpose several functional linear regression models and classical non-functional regression models are applied to 10 years of RS data obtained from medium resolution imaging spectrometer sensor to predict the TSS concentration in the coastal zone of the Guadalquivir estuary. The results of functional and classical approaches are compared in terms of their mean square prediction error values and the superiority of the functional models is established. A simulation study has been designed in order to support these findings and to determine the best prediction model for the TSS parameter in more general contexts.

In genetic studies, many phenotypes have multiple naturally ordered discrete values. The phenotypes can be correlated with each other. If multiple correlated ordinal traits are analyzed simultaneously, the power of analysis may increase significantly while the false positives can be controlled well. In this study, we propose bivariate functional ordinal linear regression (BFOLR) models using latent regressions with cumulative logit link or probit link to perform a gene-based analysis for bivariate ordinal traits and sequencing data. In the proposed BFOLR models, genetic variant data are viewed as stochastic functions of physical positions, and the genetic effects are treated as a function of physical positions. The BFOLR models take the correlation of the two ordinal traits into account via latent variables. The BFOLR models are built upon functional data analysis which can be revised to analyze the bivariate ordinal traits and high-dimension genetic data. The methods are flexible and can analyze three types of genetic data: (1) rare variants only, (2) common variants only, and (3) a combination of rare and common variants. Extensive simulation studies show that the likelihood ratio tests of the BFOLR models control type I errors well and have good power performance. The BFOLR models are applied to analyze Age-Related Eye Disease Studydata, in which two genes, CFH and ARMS2, are found to strongly associate with eye drusen size, drusen area, age-related macular degeneration (AMD) categories, and AMD severity scale.

In functional data analysis, functional linear regression has attracted significant attention recently. Herein, we consider the case where both the response and covariates are functions. There are two available approaches for addressing such a situation: concurrent and nonconcurrent functional models. In the former, the value of the functional response at a given domain point depends only on the value of the functional regressors evaluated at the same domain point, whereas, in the latter, the functional covariates evaluated at each point of their domain have a non null effect on the response at any point of its domain. To balance these two extremes, we propose a locally sparse functional regression model in which the functional regression coefficient is allowed (but not forced) to be exactly zero for a subset of its domain. This is achieved using a suitable basis representation of the functional regression coefficient and exploiting an overlapping group-Lasso penalty for its estimation. We introduce efficient computational strategies based on majorization-minimization algorithms and discuss appealing theoretical properties regarding the model support and consistency of the proposed estimator. We further illustrate the empirical performance of the method through simulations and two applications related to human mortality and bidding in energy markets. Supplementary materials for this article are available online.