Functional Linear Model Research Articles

Estimation of functional regression model via functional dimension reduction

The functional linear regression model corresponding to a scalar response and a functional predictor is becoming increasingly common. Since the predictor is infinite dimensional, some form of dimension reduction is essential. There are two popular dimension reduction methods such as expanding the functional predictor or regression parameter function on the functional principal component basis or on a fixed bases (such as B-spline, Wavelet). In the present paper, we estimate the functional linear model by using the functional sufficient dimension reduction (FSDR) basis. Compared to the existing methods, the proposed method is appealing because the FSDR basis is related to both the functional predictor and the response variable, whereas the functional principal component basis is only related to the functional predictor and the fixed basis (B-spline, Wavelet) is independent from both the functional predictor and the response variable. Our techniques involve methods for giving a new expansion for the predictor, for giving a specific expression for the regression parameter function, for estimating the FSDR space by a new method and some asymptotical properties about the regression parameter function and the prediction for the test samples. Numerical studies, including both simulation studies and applications on real-life data, are presented to demonstrate the accuracy of the proposed method.

Rank-Based Test for Partial Functional Linear Regression Models

This paper investigates the hypothesis test of the parametric component in partial functional linear regression models. Based on a rank score function, the authors develop a rank test using functional principal component analysis, and establish the asymptotic properties of the resulting test under null and local alternative hypotheses. A simulation study shows that the proposed test procedure has good size and power with finite sample sizes. The authors also present an illustration through fitting the Berkeley Growth Data and testing the effect of gender on the height of kids.

Robust Estimation for Partial Functional Linear Regression Model Based on Modal Regression

This paper presents a robust estimation procedure by using modal regression for the partial functional linear regression, which combines the common linear model with the functional linear regression model. The outstanding merit of the new method is that it is robust against outliers or heavy-tail error distributions while performs no worse than the least-square-based estimation method for normal error cases. The slope function is fitted by B-spline. Under suitable conditions, the authors obtain the convergence rates and asymptotic normality of the estimators. Finally, simulation studies and a real data example are conducted to examine the finite sample performance of the proposed method. Both the simulation results and the real data analysis confirm that the newly proposed method works very well.

Rank-based estimation in varying coefficient partially functional linear regression models

This paper focuses on robust estimation for varying coefficient partially functional linear regression model (VCPFLM) without specification of the error distributions. The proposed estimators of the nonparametric coefficients and slope function are based on B-spline approximation and rank regression, which demonstrates that the proposed estimation is robust for non-normal error distributions compared with that of least square estimate. In addition, the optimal convergence rates of the estimators are established under some mild regularity conditions. Finally, simulation study and a real data analysis are conducted to illustrate the finite sample performance of the proposed method.

Bayesian robust estimation of partially functional linear regression models using heavy-tailed distributions

Functional linear regression (FLR) is a popular method that studies the relationship between a scalar response and a functional predictor. A common estimation procedure for the FLR model is using maximum likelihood by assuming normal distributions for measurement errors; however this method may make inferences vulnerable to the presence of outliers. In this article, we introduce a robust estimation method of partially functional linear model by considering a class of scale mixtures of normal (SMN) distributions for measurement errors. Due to intractable closed form of likelihood function with the SMN distributions, a Bayesian framework is adopted and an MCMC algorithm is developed to carry out posterior inference on model parameters. The finite sample performance of our proposed method is evaluated by using some simulation studies and a real dataset.

Analyzing speech in both time and space: Generalized additive mixed models can uncover systematic patterns of variation in vocal tract shape in real-time MRI

We present a method of using generalized additive mixed models (GAMMs) to analyze midsagittal vocal tract data obtained from real-time magnetic resonance imaging (rt-MRI) video of speech production. Applied to rt-MRI data, GAMMs allow for observation of factor effects on vocal tract shape throughout two key dimensions: time (vocal tract change over the temporal course of a speech segment) and space (location of change within the vocal tract). Examples of this method are provided for rt-MRI data collected at a temporal resolution of 20 ms and a spatial resolution of 1.41 mm, for 36 native speakers of German. The rt-MRI data were quantified as 28-point semi-polar-grid aperture functions. Three test cases are provided as a way of observing vocal tract differences between: (1) /aː/ and /iː/, (2) /aː/ and /aɪ/, and (3) accentuated and unstressed /aː/. The results for each GAMM are independently validated using functional linear mixed models (FLMMs) constructed from data obtained at 20% and 80% of the vowel interval. In each case, the two methods yield similar results. In light of the method similarities, we propose that GAMMs are a robust, powerful, and interpretable method of simultaneously analyzing both temporal and spatial effects in rt-MRI video of speech.

Bayesian bandwidth estimation and semi-metric selection for a functional partial linear model with unknown error density

ABSTRACT This study examines the optimal selections of bandwidth and semi-metric for a functional partial linear model. Our proposed method begins by estimating the unknown error density using a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, can be estimated by functional principal component and functional Nadayara-Watson estimators. The estimation accuracy of the regression function and error density crucially depends on the optimal estimations of bandwidth and semi-metric. A Bayesian method is utilized to simultaneously estimate the bandwidths in the regression function and kernel error density by minimizing the Kullback-Leibler divergence. For estimating the regression function and error density, a series of simulation studies demonstrate that the functional partial linear model gives improved estimation and forecast accuracies compared with the functional principal component regression and functional nonparametric regression. Using a spectroscopy dataset, the functional partial linear model yields better forecast accuracy than some commonly used functional regression models. As a by-product of the Bayesian method, a pointwise prediction interval can be obtained, and marginal likelihood can be used to select the optimal semi-metric.

Forecasting value at risk with intra-day return curves

Methods for incorporating high resolution intra-day asset price data into risk forecasts are being developed at an increasing pace. Existing methods such as those based on realized volatility depend primarily on reducing the observed intra-day price fluctuations to simple scalar summaries. In this study, we propose several methods that incorporate full intra-day price information as functional data objects in order to forecast value at risk (VaR). Our methods are based on the recently proposed functional generalized autoregressive conditionally heteroscedastic (GARCH) models and a new functional linear quantile regression model. In addition to providing daily VaR forecasts, these methods can be used to forecast intra-day VaR curves, which we considered and studied with companion backtests to evaluate the quality of these intra-day risk measures. Using high-frequency trading data from equity and foreign exchange markets, we forecast the one-day-ahead daily and intra-day VaR with the proposed methods and various benchmark models. The empirical results suggested that the functional GARCH models estimated based on the overnight cumulative intra-day return curves exhibited competitive performance with benchmark models for daily risk management, and they produced valid intra-day VaR curves.

Analisis Risiko Produksi Bunga Krisan Potong (Chrysanthemum) di Desa Sidomulyo Kota Batu

This study aims to analyze the income of cut chrysanthemum farming and the effect of the use of chrysanthemum cut production inputs on the risk of chrysanthemum production in Sidomulyo Village, Batu City. This research was carried out during May and June for one time planting at the last harvest in 2018 year. Data collection was obtained through interviews to 34 farmers that taken as a whole (census method). Farm income is analyzed using Economic Analysis while the factors that influence the risk of chrysanthemum production are cut analyzed using multiple linear regression analysis Cobb-Douglass production function model and Just and Pope production function. The results of the analysis show that Chrysanthemum farming income cut profitable and efficientc, seen from the results of the R / C Ratio amounting to 1,90. Production factors significantly affect the productivity of cut chrysanthemums is seeds, chemical fertilizers, and labor while the factors that significantly affect the risk of production is chemical fertilizers.

Cross-Validation Model Averaging for Generalized Functional Linear Model

Functional data is a common and important type in econometrics and has been easier and easier to collect in the big data era. To improve estimation accuracy and reduce forecast risks with functional data, in this paper, we propose a novel cross-validation model averaging method for generalized functional linear model where the scalar response variable is related to a random function predictor by a link function. We establish asymptotic theoretical result on the optimality of the weights selected by our method when the true model is not in the candidate model set. Our simulations show that the proposed method often performs better than the commonly used model selection and averaging methods. We also apply the proposed method to Beijing second-hand house price data.

Estimating Truncated Functional Linear Models With a Nested Group Bridge Approach

We study a scalar-on-function truncated linear regression model which assumes that the functional predictor does not influence the response when the time passes a certain cutoff point. We approach this problem from the perspective of locally sparse modeling, where a function is locally sparse if it is zero on a substantial portion of its defining domain. In the truncated linear model, the slope function is exactly a locally sparse function that is zero beyond the cutoff time. A locally sparse estimate then gives rise to an estimate of the cutoff time. We propose a nested group bridge penalty that is able to specifically shrink the tail of a function. Combined with the B-spline basis expansion and penalized least squares, the nested group bridge approach can identify the cutoff time and produce a smooth estimate of the slope function simultaneously. The proposed nested group bridge estimator is shown to be consistent, while its numerical performance is illustrated by simulation studies. The proposed nested group bridge method is demonstrated with an application of determining the effect of the past engine acceleration on the current particulate matter emission. Supplementary materials for this article are available online.

Robust estimation with a modified Huber’s loss for partial functional linear models based on splines

In this article, we consider a new robust estimation procedure for the partial functional linear model (PFLM) with the slope function approximated by spline basis functions. This robust estimation procedure applies a modified Huber’s function with tail function replaced by the exponential squared loss (ESL) to achieve robustness against outliers. A data-driven procedure is presented for selecting the tuning parameters of the new estimation method, which enables us to reach better robustness and efficiency than other methods in the presence of outliers or non-normal errors. We construct robust estimators of both parametric coefficients and function coefficient in the PFLM. Moreover, some asymptotic properties of the resulting estimators are established. The finite sample performance of our proposed method is studied through simulations and illustrated with a data example.

Energy Disaggregation for Commercial Building with Unmonitored Facilities by Using Diurnal Linear Basis Function Model

Energy Disaggregation for Commercial Building with Unmonitored Facilities by Using Diurnal Linear Basis Function Model

Load-bearing characteristics of fibreglass uplift anchors in weathered rock

Pull-out destructive tests on six fibreglass anchors and four steel anchors for resisting the uplift of foundations in medium-weathered granite were conducted. The test results revealed two modes of shear failure for the anchors: at the interface between the anchor rod and the mortar interface (the first interface) and at the interface between the mortar and surrounding rock (the second interface). With M32·5 mortar and a 2 m anchorage length, the ultimate pull-out bearing capacity of the fibreglass anchor rods and steel rods of 28 mm diameter was 225 kN and that of fibreglass anchor rods of 32 mm diameter was 250 kN, which satisfied engineering requirements. An exponential–power–linear function model was established that better described the load–slip curves of the anchor rods. The average bond strengths at the first and second interfaces were found to be 1·50–1·54 MPa and 0·32–0·37 MPa, respectively, which were slightly lower than those for steel rods. The average bond strength at the second interface of the fibreglass anchor rods increased with the rod diameter.

A functional‐model‐adjusted spatial scan statistic

This paper introduces a new spatial scan statistic designed to adjust cluster detection for longitudinal confounding factors indexed in space. The functional-model-adjusted statistic was developed using generalized functional linear models in which longitudinal confounding factors were considered to be functional covariates. A general framework was developed for application to various probability models. Application to a Poisson model showed that the new method is equivalent to a conventional spatial scan statistic that adjusts the underlying population for covariates. In a simulation study with single and multiple covariate models, we found that our new method adjusts the cluster detection procedure more accurately than other methods. Use of the new spatial scan statistic was illustrated by analyzing data on premature mortality in France over the period from 1998 to 2013, with the quarterly unemployment rate as a longitudinal confounding factor.

Functional single-index quantile regression models

It is known that functional single-index regression models can achieve better prediction accuracy than functional linear models or fully nonparametric models, when the target is to predict a scalar response using a function-valued covariate. However, the performance of these models may be adversely affected by extremely large values or skewness in the response. In addition, they are not able to offer a full picture of the conditional distribution of the response. Motivated by using trajectories of $$\hbox {PM}_{{10}}$$ concentrations of last day to predict the maximum $$\hbox {PM}_{{10}}$$ concentration of the current day, a functional single-index quantile regression model is proposed to address those issues. A generalized profiling method is employed to estimate the model. Simulation studies are conducted to investigate the finite sample performance of the proposed estimator. We apply the proposed framework to predict the maximal value of $$\hbox {PM}_{{10}}$$ concentrations based on the intraday $$\hbox {PM}_{{10}}$$ concentrations of the previous day.

Functional measurement error in functional regression

AbstractMeasurement error is an important problem that has not been studied very well in the context of functional data analysis. To the best of our knowledge, there are no existing methods that address the presence of functional measurement errors in generalized functional linear models. In this article, a novel approach is proposed to estimate the slope function in the presence of measurement error in the generalized functional linear model with a scalar response. This work significantly advances the existing conditional score method to accommodate the case where both the measurement error and independent variables lie in infinite dimensional spaces. Asymptotic results are established for the proposed estimate, and its behaviour is studied via simulations, where the response is continuous or binary. Analysis of Canadian Weather data highlights the practical utility of our method. The Canadian Journal of Statistics 48: 238–258; 2020 © 2020 Statistical Society of Canada

The Application of Partially Functional Linear Regression Model in Health Science

With the rapid development of information technology, data information also presents the Characteristics of diversity. Meanwhile more and more datum are presented in the form of functions. Therefore, functional data has become the focus of researchers. Functional data analysis has also proved to be of great value in the fields of biology, medicine and metrology. A partially functional linear regression model is proposed for the regression cases in which the response variables are scalar types and the predictive variables are both variable types and functional types. For the functional predictive variables, we use the functional principal component analysis method to reduce the dimension of the functional data.The least square method is used to calculate the estimate of parameters.With the improvement of people's living standard, people pay more and more attention to health. And an increasing number of people are eager to live a healthy life and keep healthy. Healthy and comfortable sleep has become a topic of increasing concern to researchers. Using data from PhysioNet Databases on activity and sleep in healthy people for this study, we found that the predicted variables in the model could well explain the response variables. The application of partially functional linear model is further extended.

Study on the Influence of Air Pressure and Temperature on PM2.5 by Multivariate Functional Linear Regression Model

In recent years, the Internet has developed rapidly, and we have more and more ways to collect data. We find that many data have the characteristics of functions. We can use the important method of functional data analysis to analyze these data. The basic idea of functional data analysis is to treat data with functional properties as a whole for analysis and corresponding processing. In this paper, the daily air pressure, temperature and PM2.5 data of 49 cities with serious PM2.5 pollution in 2017 are sorted out. We use a multivariate functional linear regression model to discuss the influence of pressure and temperature on PM2.5 when the number of basis functions is different.

Estimation and inference for functional linear regression models with partially varying regression coefficients

In this paper, we present a class of functional linear regression models with varying coefficients of a functional response on one or multiple functional predictors and scalar predictors. In particular, the approach can accommodate densely or sparsely sampled functional responses as well as multiple scalar and functional predictors. It also allows for the combination of continuous or categorical covariates. Tensor product B‐spline basis is proposed for the estimation of the bivariate coefficient functions. We show that our estimators hold asymptotic consistency and normality. Several numerical examples demonstrate superior performance of the proposed methods against two existing approaches. The proposed method is also applied to a real data example.