Functional Regression Approach Research Articles

Functional Continuum Regression Approach to Wavelet Transformation Data in a Non-Invasive Glucose Measurement Calibration Model

Functional Continuum Regression Approach to Wavelet Transformation Data in a Non-Invasive Glucose Measurement Calibration Model

The spatial decay of human capital externalities - A functional regression approach with precise geo-referenced data

This paper analyzes human capital externalities from high-skilled workers by applying functional regression to precise geocoded register data. Functional regression enables us to describe the concentration of high-skilled workers around workplaces as continuous curves and to efficiently estimate a spillover function determined by distance. Furthermore, our rich panel data allow us to address the sorting of workers and disentangle human capital externalities from supply effects by using an extensive set of time-varying fixed effects. Our estimates reveal that human capital externalities attenuate with increasing distance and disappear after 25 km. Externalities from the immediate neighborhood of an establishment are twice as large as externalities from surroundings 10 km away.

Using UAV-based hyperspectral imaging and functional regression to assist in predicting grain yield and related traits in wheat under heat-related stress environments for the purpose of stable yielding genotypes

Quantifying certain physiological traits under heat-stress is crucial for maximizing genetic gain for wheat yield and yield-related components. In-season estimation of different physiological traits related to heat stress tolerance can ensure the finding of germplasm, which could help in making effective genetic gains in yield. However, estimation of those complex traits is time- and labor-intensive. Unmanned aerial vehicle (UAV) based hyperspectral imaging could be a powerful tool to estimate indirectly in-season genetic variation for different complex physiological traits in plant breeding that could improve genetic gains for different important economic traits, like grain yield. This study aims to predict in-season genetic variations for cellular membrane thermostability (CMT), yield and yield related traits based on spectral data collected from UAVs; particularly, in cases where there is a small sample size to collect data from and a large range of features collected per sample. In these cases, traditional methods of yield-prediction modeling become less robust. To handle this, a functional regression approach was employed that addresses limitations of previous techniques to create a model for predicting CMT, grain yield and other traits in wheat under heat stress environmental conditions and when data availability is constrained. The results preliminarily indicate that the overall models of each trait studied presented a good accuracy compared to their data’s standard deviation. The yield prediction model presented an average error of 13.42%, showing the function-on-function algorithm chosen for the model as reliable for small datasets with high dimensionality.

Validating intra-day risk premium in cross-sectional return curves

This paper investigates the cross-sectional asset pricing for intra-day return curves. By introducing a functional Fama-MacBeth regression approach, the validation of the intra-day risk premium associated with the Fama-French Carhart factors is examined. The empirical evidence reveals that these common risk factors show weak explainability to the entire cross-sectional intra-day returns, despite significant risk premiums that are discovered in specific half-hour time-spans in bullish sentiment.

Landscape complexity is associated with crop yields across a large temperate grassland region

Establishing semi-natural areas within annual croplands can provide habitat for beneficial organisms and ecosystem services to crops through a spillover effect. However, this approach to increasing landscape complexity may have no effect on crops, or it may promote pests, weeds and other disservices that reduce productivity. An argument for changing landscape complexity may be more persuasive if it is associated with higher crop yields. Here, we examine regions that vary in their landscape complexity and, therefore, may also naturally differ in the potential for ecosystem services, disservices and crop yields. Specifically, we examine crop-growing districts in the Canadian province of Alberta to test whether the presence of more non-crop land covers has increased crop yields. Our dataset covered about one-quarter of the seeded area in Canada between 2012 and 2017 consisting of 10,069 records representing average field-level yields reported to a crop insurance provider. In total, we analyzed summary data for 250,000 km2 of seeded area for seven grain crops. Using a functional regression approach, we found evidence for a plausibly positive association between yield and the non-crop land covers found within and near fields in four of seven crops. Landscape complexity, therefore, represented a measurable yield benefit for farmers, although the variance in yield explained by the landscape was small. These findings suggest there may be a low risk of disservices to crops from non-crop land covers in this region. Our study adds support at a broad geographic extent for initiatives that restore perennial and other semi-natural vegetation in annual cropping systems and suggests that, in this temperate grassland region, their promotion (e.g., as carbon stores or as biodiversity refugia) may have no adverse effects for crop production.

Differential equation model of carbon dioxide emission using functional linear regression

ABSTRACTCarbon dioxide is one of the major contributors to Global Warming. In the present study, we develop a differential equation to model the carbon dioxide emission data in the atmosphere using functional linear regression approach. In the proposed method, a differential operator is defined as data smoother and we use the penalized least square fitting criteria to smooth the data. The profile error sum of squares is optimized to estimate the differential operators using functional regression. The solution of the developed differential equation estimates and predicts the rate of change of carbon dioxide in the atmosphere at a particular time. We apply the proposed model to fit the emission of carbon dioxide data in the continental United States. Numerical simulations of a number of test cases depict a satisfactory agreement with real data.

A Functional Regression Model of the Retinal Nerve Fiber Layer Thickness in Healthy Subjects.

PurposeA new functional regression model is presented to explain the intersubject variability of the circumpapillary retinal nerve fiber layer (RNFL) thickness in healthy subjects.MethodsTo evaluate the functional regression approach we used data from 202 healthy volunteers, divided equally into training samples (TS) and validation samples (VS). Covariates included RNFL, fovea distance, fovea angle, optic disk ratio, orientation and area provided by Fourier-domain–optical coherence tomography, age, and refractive error. Root mean square errors (RMSE) were calculated for each of the 256 sectors and for the 12 clock-hour sectors in the TS and VS and were compared to the RMSE of the previous model and the standard deviation of the raw data.ResultsWith the functional regression approach, we were able to explain on average 27.4% of the variation in the TS and 25.1% of the variation in the VS. The new model performed better compared to a multivariate linear regression model. It performed best in the superior-temporal and inferior-temporal clock-hour sectors where the percentage of RMSE reduction ranged between 26.3% and 44.1% for the TS and between 20.6% and 35.4% for the VS.ConclusionsThe new functional regression approach improves on the multivariate linear regression model and allows an even larger reduction of the amount of intersubject variability, while at the same time using a substantially smaller number of parameters to be estimated.Translational RelevanceThe demonstrated reduction of interindividual variation is expected to translate into an improved diagnostic separation between healthy and glaucomatous subjects, but this remains to be demonstrated in further studies.

A Functional Regression Approach to Facial Landmark Tracking.

Linear regression is a fundamental building block in many face detection and tracking algorithms, typically used to predict shape displacements from image features through a linear mapping. This paper presents a Functional Regression solution to the least squares problem, which we coin Continuous Regression, resulting in the first real-time incremental face tracker. Contrary to prior work in Functional Regression, in which B-splines or Fourier series were used, we propose to approximate the input space by its first-order Taylor expansion, yielding a closed-form solution for the continuous domain of displacements. We then extend the continuous least squares problem to correlated variables, and demonstrate the generalisation of our approach. We incorporate Continuous Regression into the cascaded regression framework, and show its computational benefits for both training and testing. We then present a fast approach for incremental learning within Cascaded Continuous Regression, coined iCCR, and show that its complexity allows real-time face tracking, being 20 times faster than the state of the art. To the best of our knowledge, this is the first incremental face tracker that is shown to operate in real-time. We show that iCCR achieves state-of-the-art performance on the 300-VW dataset, the most recent, large-scale benchmark for face tracking.

Evaluation of the cooling trend in the ionosphere using functional regression with incomplete curves

We develop a statistical framework to test the hypothesis of the existence of an ionospheric cooling trend related to the global warming hypothesis; both are attributed to the same driver, namely the increased concentration of greenhouse gases. However, the study of a temporal trend in the ionosphere is easier because there are fewer covariates to be taken into account. The hypothesis that a cooling trend in the ionosphere exists has been an important focus of space physics research for over two decades. A central difficulty in reaching broadly agreed—on conclusions has been the absence of data with sufficiently long temporal and sufficiently broad spatial coverage. Complete time series of data that cover several decades exist only in a few separated (industrialized) regions. The space physics community has struggled to combine the information contained in these data, and often contradictory conclusions have been reported based on the analyses relying on one or a few locations. We present a statistical analysis that uses all data, even those with incomplete temporal coverage. It is based on a new functional regression approach that can handle spatially indexed curves whose temporal domain depends on location and may contain gaps. The test statistic combines spatial and temporal dependence in the data and is approximately normally distributed. We conclude that a statistically significant cooling trend exists in the Northern Hemisphere. This confirms the hypothesis put forward in the space physics community over two decades ago.

Gaussian process emulators for quantifying uncertainty in [formula omitted] spreading predictions in heterogeneous media

We explore the use of Gaussian process emulators (GPE) in the numerical simulation of CO2 injection into a deep heterogeneous aquifer. The model domain is a two-dimensional, log-normally distributed stochastic permeability field. We first estimate the cumulative distribution functions (CDFs) of the CO2 breakthrough time and the total CO2 mass using a computationally expensive Monte Carlo (MC) simulation. We then show that we can accurately reproduce these CDF estimates with a GPE, using only a small fraction of the computational cost required by traditional MC simulation. In order to build a GPE that can predict the simulator output from a permeability field consisting of 1000s of values, we use a truncated Karhunen-Loève (K-L) expansion of the permeability field, which enables the application of the Bayesian functional regression approach. We perform a cross-validation exercise to give an insight of the optimization of the experiment design for selected scenarios: we find that it is sufficient to use 100s values for the size of training set and that it is adequate to use as few as 15 K-L components. Our work demonstrates that GPE with truncated K-L expansion can be effectively applied to uncertainty analysis associated with modelling of multiphase flow and transport processes in heterogeneous media.

Hydrological Controls on Ecosystem Dynamics in Lake Fryxell, Antarctica.

The McMurdo Dry Valleys constitute the largest ice free area of Antarctica. The area is a polar desert with an annual precipitation of ∼ 3 cm water equivalent, but contains several lakes fed by glacial melt water streams that flow from four to twelve weeks of the year. Over the past ∼20 years, data have been collected on the lakes located in Taylor Valley, Antarctica as part of the McMurdo Dry Valley Long-Term Ecological Research program (MCM-LTER). This work aims to understand the impact of climate variations on the biological processes in all the ecosystem types within Taylor Valley, including the lakes. These lakes are stratified, closed-basin systems and are perennially covered with ice. Each lake contains a variety of planktonic and benthic algae that require nutrients for photosynthesis and growth. The work presented here focuses on Lake Fryxell, one of the three main lakes of Taylor Valley; it is fed by thirteen melt-water streams. We use a functional regression approach to link the physical, chemical, and biological processes within the stream-lake system to evaluate the input of water and nutrients on the biological processes in the lakes. The technique has been shown previously to provide important insights into these Antarctic lacustrine systems where data acquisition is not temporally coherent. We use data on primary production (PPR) and chlorophyll-A (CHL)from Lake Fryxell as well as discharge observations from two streams flowing into the lake. Our findings show an association between both PPR, CHL and stream input.

Multivariate functional linear regression and prediction

We propose a multivariate functional linear regression (mFLR) approach to analysis and prediction of multivariate functional data in cases in which both the response and predictor variables contain multivariate random functions. The mFLR model, coupled with the multivariate functional principal component analysis approach, takes the advantage of cross-correlation between component functions within the multivariate response and predictor variables, respectively. The estimate of the matrix of bivariate regression functions is consistent in the sense of the multi-dimensional Gram–Schmidt norm and is asymptotically normally distributed. The prediction intervals of the multivariate random trajectories are available for predictive inference. We show the finite sample performance of mFLR by a simulation study and illustrate the method through predicting multivariate traffic flow trajectories for up-to-date and partially observed traffic streams.

Marginal Functional Regression Models for Analyzing the Feeding Behavior of Pigs

We observe a group of pigs over a period of about 100 days. Using high frequency radio frequency identification, it is recorded when each pig is feeding, leading to very dense binary functional data for each pig and day. One aim of the data analysis is to find pig-specific feeding profiles showing us the typical feeding pattern of each pig. For modeling the data, we use a marginal functional logistic regression approach, allowing us to model the densely observed binary measurements by assuming an underlying smooth subject-specific profile. The method also allows to incorporate additional covariates such as temperature and humidity that may influence the pigs’ behavior. To account for correlation of measurements, we use robust standard errors and corresponding pointwise confidence intervals. Before analyzing the feeding behavior of pigs, the method employed is evaluated in simulation studies. As our approach is rather general, it may also be applied to other types of generalized functional data with similar characteristics as the pig data.

Global Analysis of Methylation Profiles From High Resolution CpG Data

New high throughput technologies are now enabling simultaneous epigenetic profiling of DNA methylation at hundreds of thousands of CpGs across the genome. A problem of considerable practical interest is identification of large scale, global changes in methylation that are associated with environmental variables, clinical outcomes, or other experimental conditions. However, there has been little statistical research on methods for global methylation analysis using technologies with individual CpG resolution. To address this critical gap in the literature, we develop a new strategy for global analysis of methylation profiles using a functional regression approach wherein we approximate either the density or the cumulative distribution function (CDF) of the methylation values for each individual using B-spline basis functions. The spline coefficients for each individual are allowed to summarize the individual's overall methylation profile. We then test for association between the overall distribution and a continuous or dichotomous outcome variable using a variance component score test that naturally accommodates the correlation between spline coefficients. Simulations indicate that our proposed approach has desirable power while protecting type I error. The method was applied to detect methylation differences, both genome wide and at LINE1 elements, between the blood samples from rheumatoid arthritis patients and healthy controls and to detect the epigenetic changes of human hepatocarcinogenesis in the context of alcohol abuse and hepatitis C virus infection. A free implementation of our methods in the R language is available in the Global Analysis of Methylation Profiles (GAMP) package at http://research.fhcrc.org/wu/en.html.

Functional logistic regression approach to detecting gene by longitudinal environmental exposure interaction in a case-control study.

Most complex human diseases are likely the consequence of the joint actions of genetic and environmental factors. Identification of gene-environment (G × E) interactions not only contributes to a better understanding of the disease mechanisms, but also improves disease risk prediction and targeted intervention. In contrast to the large number of genetic susceptibility loci discovered by genome-wide association studies, there have been very few successes in identifying G × E interactions, which may be partly due to limited statistical power and inaccurately measured exposures. Although existing statistical methods only consider interactions between genes and static environmental exposures, many environmental/lifestyle factors, such as air pollution and diet, change over time, and cannot be accurately captured at one measurement time point or by simply categorizing into static exposure categories. There is a dearth of statistical methods for detecting gene by time-varying environmental exposure interactions. Here, we propose a powerful functional logistic regression (FLR) approach to model the time-varying effect of longitudinal environmental exposure and its interaction with genetic factors on disease risk. Capitalizing on the powerful functional data analysis framework, our proposed FLR model is capable of accommodating longitudinal exposures measured at irregular time points and contaminated by measurement errors, commonly encountered in observational studies. We use extensive simulations to show that the proposed method can control the Type I error and is more powerful than alternative ad hoc methods. We demonstrate the utility of this new method using data from a case-control study of pancreatic cancer to identify the windows of vulnerability of lifetime body mass index on the risk of pancreatic cancer as well as genes that may modify this association.

Modeling Conditional Distributions for Functional Responses, With Application to Traffic Monitoring via GPS-Enabled Mobile Phones

Motivated by problems involving a traffic monitoring system in which trajectory data are obtained from Global Positioning System-enabled mobile phones, we propose a novel approach to functional regression modeling, where instead of the usual mean regression the entire distribution of functional responses is modeled conditionally on predictors. An approach that sensibly balances flexibility and stability is obtained by assuming that the response functions are drawn from a Gaussian process, the mean and covariance function of which depend on predictors. The dependence of the mean function and covariance function of the response on the predictors is modeled additively. We demonstrate the proposed methods by constructing predicted curves and corresponding prediction regions for traffic velocity trajectories for a future time period, using current traffic velocity fields as predictor functions. The proposed functional regression and conditional distribution approach is of general interest for functional response settings, where in addition to predicting the conditional mean response function one is also interested in predicting the covariance surface of the random response functions, conditional on predictor curves.

Statistical modelling of collocation uncertainty in atmospheric thermodynamic profiles

Abstract. The quantification of measurement uncertainty of atmospheric parameters is a key factor in assessing the uncertainty of global change estimates given by numerical prediction models. One of the critical contributions to the uncertainty budget is related to the collocation mismatch in space and time among observations made at different locations. This is particularly important for vertical atmospheric profiles obtained by radiosondes or lidar. In this paper we propose a statistical modelling approach capable of explaining the relationship between collocation uncertainty and a set of environmental factors, height and distance between imperfectly collocated trajectories. The new statistical approach is based on the heteroskedastic functional regression (HFR) model which extends the standard functional regression approach and allows a natural definition of uncertainty profiles. Along this line, a five-fold decomposition of the total collocation uncertainty is proposed, giving both a profile budget and an integrated column budget. HFR is a data-driven approach valid for any atmospheric parameter, which can be assumed smooth. It is illustrated here by means of the collocation uncertainty analysis of relative humidity from two stations involved in the GCOS reference upper-air network (GRUAN). In this case, 85% of the total collocation uncertainty is ascribed to reducible environmental error, 11% to irreducible environmental error, 3.4% to adjustable bias, 0.1% to sampling error and 0.2% to measurement error.

A semiparametric model for electricity spot prices

This article proposes a semiparametric single-index model for short-term forecasting day-ahead electricity prices. The approach captures the dependency of electricity prices on covariates, such as demand for electricity, amount of energy produced by intermittent sources, and weather-dependent variables. To obtain parsimonious models, principal component analysis is used for dimension reduction. The approach is tested on two data sets from different markets and its performance is analyzed in terms of fit, forecast quality, and computational efficiency. The results are encouraging, in that the proposed method leads to a good in-sample fit and performs well out-of-sample compared with four benchmark models, including a SARIMA model as well as a functional nonparametric regression approach recently proposed in the literature.

Local wavelet‐vaguelette‐based functional classification of gene expression data

This paper focuses on the problem of functional statistical classification of gene expression curves. A local-wavelet-vaguelette-based functional logistic regression approach is presented. This approach is specially suitable for the classification of non-stationary singular (non-differentiable) curves. The performance of the methodology proposed is illustrated by implementing it for the classification of yeast cell-cycle temporal gene expression profiles. A simulation study is also carried out for comparison with other functional classification methodologies.

A Flexible Estimating Equations Approach for Mapping Function-Valued Traits

In genetic studies, many interesting traits, including growth curves and skeletal shape, have temporal or spatial structure. They are better treated as curves or function-valued traits. Identification of genetic loci contributing to such traits is facilitated by specialized methods that explicitly address the function-valued nature of the data. Current methods for mapping function-valued traits are mostly likelihood-based, requiring specification of the distribution and error structure. However, such specification is difficult or impractical in many scenarios. We propose a general functional regression approach based on estimating equations that is robust to misspecification of the covariance structure. Estimation is based on a two-step least-squares algorithm, which is fast and applicable even when the number of time points exceeds the number of samples. It is also flexible due to a general linear functional model; changing the number of covariates does not necessitate a new set of formulas and programs. In addition, many meaningful extensions are straightforward. For example, we can accommodate incomplete genotype data, and the algorithm can be trivially parallelized. The framework is an attractive alternative to likelihood-based methods when the covariance structure of the data is not known. It provides a good compromise between model simplicity, statistical efficiency, and computational speed. We illustrate our method and its advantages using circadian mouse behavioral data.