Functional Linear Model Research Articles

Testing for changes in the error distribution in functional linear models

We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those estimated functions are more challenging in models with infinite-dimensional covariates than in regression models with scalar or vector-valued covariates due to a slower rate of convergence of the parameter estimators. Yet the suggested change point test is asymptotically distribution-free and consistent for one-change point alternatives. In the latter case we also show consistency of a change point estimator.

High-dimensional partially linear functional Cox models.

As a commonly employed method for analyzing time-to-event data involving functional predictors, the functional Cox model assumes a linear relationship between the functional principal component (FPC) scores of the functional predictors and the hazard rates. However, in practical scenarios, such as our study on the survival time of kidney transplant recipients, this assumption often fails to hold. To address this limitation, we introduce a class of high-dimensional partially linear functional Cox models, which accommodates the non-linear effects of functional predictors on the response and allows for diverging numbers of scalar predictors and FPCs as the sample size increases. We employ the group smoothly clipped absolute deviation method to select relevant scalar predictors and FPCs, and use B-splines to obtain a smoothed estimate of the non-linear effect. The finite sample performance of the estimates is evaluated through simulation studies. The model is also applied to a kidney transplant dataset, allowing us to make inferences about the non-linear effects of functional predictors on patients' hazard rates, as well as to identify significant scalar predictors for long-term survival time.

Analysis of Time–Frequency Characteristics and Influencing Factors of Air Quality Based on Functional Data in Fujian

Increased air pollution is driven by anthropogenic pollution emissions and climate change, which pose great challenges to environmental governance. Strengthening the monitoring of regional air quality levels and analyzing the causes of regional pollution is conducive to the management and sustainable development of the regional atmosphere. Functional data obtained on a wavelet basis were used in the fitting of air quality data of Fujian Province, and wavelet decomposition was performed to obtain low-frequency and high-frequency information. While the Fourier basis cannot adaptively adjust the time–frequency window, resulting in the loss of location information of special frequencies, the wavelet basis solves this problem. Functional analysis of variance was utilized for analyzing spatial differences in air pollution characteristics. Furthermore, the study established a multivariate functional linear regression model to explore the impact of meteorological factors and ozone precursor factors. The results indicated that the overall air quality was gradually improving in Fujian Province, but the concentration of ozone was progressively increasing. Air pollution in coastal areas was higher than that in inland areas. The p-values of the functional analysis of variance for energy values and crest values were less than 0.05. Moreover, the energy entropy and kurtosis values were greater than 0.05. There were significant differences of AQI in the fluctuation amplitude and variation characteristics of different cities. The total squared multiple correlation of regression model was above 50% on average. Ozone is currently the most serious pollution factor, mainly affected by wind speed, temperature, NO2, and CO. In summer, it was principally influenced by VOCs. The findings of this study could act as a reference in exploring the time–frequency characteristics of air quality data and support of air pollution control.

Functional partial linear regression with quadratic regression for the multivariate predictor

. In the last decades, functional semi-parametric models are widely studied such as functional additive model, functional partial linear model, functional index model, and so on. But some of the semi-parametric models suffer from the lower convergence rate for the non parametric parts (especially for the non parametric functional regression) and the assumption for the linear parts is too strict in practical. In this article, we propose a functional partial linear model corresponding to a scalar response, while the predictors contain both of function and multivariate predictor. We use a functional linear model to avoid the low convergence rate of the non parametric functional estimation, and for the multivariate predictor, we use a quadratic regression model to deal with the strict assumption for the linear model. To estimate this model, we first expand the functional predictor and functional regression parametric on the functional principal component basis, then, we estimate the functional coefficient by the least square method. For the theoretical studying, we will study the asymptotical normal distribution of the multivariate regression parameter and the specific convergence rate of the functional regression parameter for the dense functional observations. Furthermore, we illustrate the performance of the proposed method by simulation studies and one real data analysis.

Sustainable Development Goals (SDGs): The nexus of fintech and water productivity in 11 BRICS countries

In the context of a global water crisis, enhancing water productivity is becoming increasingly crucial. While previous research has predominantly addressed technical and policy aspects of water management, the role of fintech in improving water productivity has not been sufficiently explored. This research investigates the impact of fintech on water productivity, considering the moderating effect of education level. Using panel data from new BRICS countries spanning 2011 to 2021, we employ a partially linear functional model to analyze how fintech influences water productivity and assess how education levels moderate this relationship. Our findings reveal that: (i) Fintech holds significant potential for improving water productivity; (ii) The effect of fintech on water production varies with the education level; (iii) There is considerable spatial variation in how education level affects the impact of fintech, with a more pronounced effect observed in countries with higher education levels. Specifically, the impact of fintech on water productivity becomes substantially more significant when the education level index exceeds 2.3. These results remain robust across various tests. Based on these insights, the paper proposes policy recommendations to enhance water productivity through the integration of fintech and education improvements.

A Functional Linear Regression for High‐Resolution 3D Faces

ABSTRACTMany scientific disciplines are faced with the challenge of extracting meaningful information from large, complex and highly structured datasets. A significant portion of contemporary statistical research is dedicated to developing tools for handling such data. This paper introduces a functional linear regression model specifically designed for 3D facial shapes, which are viewed as manifolds. We propose a comprehensive framework that includes converting 3D facial data into functional objects, employing a functional principal component analysis method and utilising a function‐on‐scalar regression model. This framework facilitates computation for high‐dimensional data and is employed to investigate how individual traits, such as age and genetic ancestry, impact the diversity of human facial features.

Variable selection of partially functional linear spatial autoregressive model with a diverging number of parameters

In this article, we consider variable selection of partially functional linear spatial autoregressive model with a diverging number of parameters, in which the explanatory variables include infinite dimensional predictor procedure, treated as functional data, and multiple real valued scalar variate. By combining series approximation method, two-stage least squares method and a class of non convex penalty function, we propose a variable selection method to simultaneously select significant explanatory variables in the parametric component and estimate the corresponding parameter related to spatial lag of the response variable. Under appropriate conditions, we derive the rate of convergence of the series estimator of the functional and parametric component, and show that the proposed variable selection method processes the oracle property. That is, it can estimate the zero components as exact zero with high probability, and estimate the non zero components as efficiently as if the true model was known beforehand. Simulation result show that our proposed variable selection method has better finite sample property.Notably, in the case where the correlation among the explanatory variables in the parametric component is low, the proposed variable selection method performs well. An application of the proposed variable selection method serves as a practical illustration.

Design and economic analysis of energy storage optimization allocation for campus microgrid

In the twenty-first century, the "double carbon" plan is advancing steadily. Energy saving and emission reduction is one of the key points of the plan. A power grid independent park, if energy storage equipment can be configured, it can greatly improve the park microgrid power utilization, in order to achieve the purpose of energy saving. Therefore, research on the optimization of the proportion of power grid and microgrid in the park and the optimization of the configuration of energy storage devices can greatly improve the energy utilization rate, so as to achieve the significance of environmental protection. After collecting and sorting the daily load, daily wind power generation and 12-month daily wind power generation data of the three parks A, B and C, and processing the data, a linear function model is established. After using linear programming, operations research and genetic algorithm, the research results are obtained when the three parks are combined and equipped with 50kW/100kWh energy storage devices.

Locally sparse estimator for functional linear panel models with fixed effects

Locally sparse estimator for functional linear panel models with fixed effects

Exploration of bulk viscous Bianchi type cosmological model in f(T) theory of gravity

In the present paper, the expansion of Locally Rotational Symmetric (LRS) Bianchi type - I cosmological model have been investigated with Bulk Viscous matter in the context of f(T) theory of gravity, where T signifies the torsion scalar. The power model, exponential model and linear functional model of the universe have been discussed for choices of f(T)with utilization of the special form of time dependent varying deceleration parameter. The discussion involves the examination of the dynamical behaviour of these models using some dynamical parameters and its graphical representation.

Assessment of hybrid machine learning models for non-linear system identification of fatigue test rigs

The prediction of system responses for a given fatigue test bench drive signal is a challenging problem, since highly dynamic loads from measurement campaigns must be reproduced accurately. Linear frequency response function models are commonly used for this system identification task, but energy intensive experimental iterations are required to account for system non-linearities. Two novel hybrid modeling strategies are suggested, which augment existing approaches using non-linear Long Short-Term Memory networks. These are trained and deployed on short subsequences of measurement data and recombined using a windowing technique, which enables their application to measurement data with high sampling rates. In addition to fatigue test bench commissioning, these hybrid models can also be employed in the field of virtual sensing. The approach is tested using non-linear experimental data from a servo-hydraulic test rig and this dataset is made publicly available. A variety of metrics in time and frequency domains, as well as fatigue strength under variable amplitudes, are employed in the evaluation. It is shown, that hybrid models can successfully use frequency response function models as a linear baseline estimate, which is further improved by Long Short-Term Memory networks to enable non-linear predictions.

Generalized Partially Functional Linear Model with Interaction between Functional Predictors

This paper proposes a generalized partially functional linear model with interaction terms. It is suitable for cases where the response variable is scalar, and the predictor variables include a mix of functional and scalar types, while considering the correlations among functional predictor variables. The model uses principal component analysis for dimensionality reduction, employs maximum likelihood estimation to obtain parameter values, proves the asymptotic properties of the estimates, and validates the model’s accuracy through data simulation experiments. Finally, the proposed model was applied to investigate the influence of air quality, climate factors, and medical and social indicators, along with their interactions, on cancer incidence, which is a binary response.

Composite quantile estimation in partially functional linear regression model with randomly censored responses

Composite quantile estimation in partially functional linear regression model with randomly censored responses

Suitability Evaluation of Urban Underground Space Development: A Case Study of Qingdao City

Urban underground space, as an underutilized land resource, holds tremendous potential and value. Efficient and rational development and utilization of this resource are key to addressing current urban challenges. This study takes the main urban area of Qingdao City as an example and establishes a comprehensive evaluation system for the suitability of urban underground space development at different depth levels through the integration of geological, hydrological, and urban planning factors. By utilizing the Analytic Hierarchy Process to assign weights to evaluation criteria within the system, both a multi-objective linear weighting function model and a fuzzy comprehensive evaluation model are employed to assess the suitability of underground space development. The results delineate the distribution of underground space development suitability within the study area. Comparative analysis of the two models reveals that the fuzzy comprehensive evaluation model offers a more detailed and comprehensive reflection of the complexity and diversity of underground space development, providing forward-looking and insightful evaluation results for urban planning and development. The evaluation indicates that certain streets within the main urban area of Qingdao exhibit excellent prospects for underground space development.

On the functional regression model and its finite-dimensional approximations

AbstractThe problem of linearly predicting a scalar response Y from a functional (random) explanatory variable $$X=X(t),\ t\in I$$ X = X ( t ) , t ∈ I is considered. It is argued that the term “linearly” can be interpreted in several meaningful ways. Thus, one could interpret that (up to a random noise) Y could be expressed as a linear combination of a finite family of marginals $$X(t_i)$$ X ( t i ) of the process X, or a limit of a sequence of such linear combinations. This simple point of view (which has some precedents in the literature) leads to a formulation of the linear model in terms of the RKHS space generated by the covariance function of the process X(t). It turns out that such RKHS-based formulation includes the standard functional linear model, based on the inner product in the space $$L^2[0,1]$$ L 2 [ 0 , 1 ] , as a particular case. It includes as well all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite number of linear projections of X. Some consistency results are proved which, in particular, lead to an asymptotic approximation of the predictions derived from the general (functional) linear model in terms of finite-dimensional models based on a finite family of marginals $$X(t_i)$$ X ( t i ) , for an increasing grid of points $$t_j$$ t j in I. We also include a discussion on the crucial notion of coefficient of determination (aimed at assessing the fit of the model) in this setting. A few experimental results are given.

Interpreting and generalizing deep learning in physics-based problems with functional linear models

Interpreting and generalizing deep learning in physics-based problems with functional linear models

Study on repeated discharge frequency characteristics of plasma pulse equipment

In order to increase the production of unconventional oil and gas, it is necessary to improve its reservoir. Plasma pulse technology is one of a safe and environmentally friendly reservoir reconstruction technology. In this paper, the discharge test of plasma pulse equipment was carried out in the laboratory. The different discharge stages (pre-discharge, main-discharge, sinusoidal attenuation oscillation and post-discharge stage) of plasma pulse equipment in a single discharge were discussed, and the frequency characteristics in the repeated discharge were studied in detail. The discharge test shows that the energy and average power of main-discharge stage are about 109 J and 15.6 MW respectively, and the discharge frequency range is about 0–25 Hz. Furthermore, the linear attenuation function model of breakdown voltage change under the initial discharge frequency of 4 Hz, 9 Hz, 14 Hz and 19 Hz is established, and the change law of breakdown voltage decreases linearly with the increase of discharge frequency is revealed. The results show that the research content of this paper has an important guiding significance for revealing the variation law of repeated discharge frequency characteristics of plasma pulse equipment, which is helpful to further optimize the design of plasma pulse equipment and improve its working performance.

Zeroing Neural Network With Coefficient Functions and Adjustable Parameters for Solving Time-Variant Sylvester Equation.

To solve the time-variant Sylvester equation, in 2013, Li et al. proposed the zeroing neural network with sign-bi-power function (ZNN-SBPF) model via constructing a nonlinear activation function. In this article, to further improve the convergence rate, the zeroing neural network with coefficient functions and adjustable parameters (ZNN-CFAP) model as a variation in zeroing neural network (ZNN) model is proposed. On the basis of the introduced coefficient functions, an appropriate ZNN-CFAP model can be chosen according to the error function. The high convergence rate of the ZNN-CFAP model can be achieved by choosing appropriate adjustable parameters. Moreover, the finite-time convergence property and convergence time upper bound of the ZNN-CFAP model are proved in theory. Computer simulations and numerical experiments are performed to illustrate the efficacy and validity of the ZNN-CFAP model in time-variant Sylvester equation solving. Comparative experiments among the ZNN-CFAP, ZNN-SBPF, and ZNN with linear function (ZNN-LF) models further substantiate the superiority of the ZNN-CFAP model in view of the convergence rate. Finally, the proposed ZNN-CFAP model is successfully applied to the tracking control of robot manipulator to verify its practicability.

On the convergence of gradient descent for robust functional linear regression

Functional data analysis offers a set of statistical methods concerned with extracting insights from intrinsically infinite-dimensional data and has attracted considerable amount of attentions in the past few decades. In this paper, we study robust functional linear regression model with a scalar response and a functional predictor in the framework of reproducing kernel Hilbert spaces. A gradient descent algorithm with early stopping is introduced to solve the corresponding empirical risk minimization problem associated with robust loss functions. By appropriately selecting the early stopping rule and the scaling parameter of the robust losses, the convergence of the proposed algorithm is established when the response variable is bounded or satisfies a moment condition. Explicit learning rates with respect to both estimation and prediction error are provided in terms of regularity of the regression function and eigenvalue decay rate of the integral operator induced by the reproducing kernel and covariance function.

Exploring ionospheric dynamics: a comprehensive analysis of GNSS TEC estimations during the solar phases using linear function model

Abstract The modeling and forecasting of Total Electron Content (TEC) play a major role in influencing signals from satellite-based navigation systems and impact the performance of diverse satellite-dependent technologies. The intensity of solar ionizing radiation and the state of geomagnetic field activity influence the Global Navigation Satellite System (GNSS)-TEC. This paper uses a Linear TEC Function (LTF) climatology model to understand ionospheric behavior under solar and geomagnetic activities that cause variations in the electron distribution of the ionosphere medium. The LTF model integrates representations of solar EUV photon (MgII) and geomagnetic (SYMH) activities, incorporating solar-modulated oscillations (periodic variations) at four seasonal cycles and a linear trend. The LTF model examined the time series of GPS-TEC at a location (geographic 34.95° N, 134.05° E) with a time resolution of 1 h, from 1997 to 2016, covering solar cycles 23 and 24. The Root Mean Square Deviation (RMSD) and correlation coefficient between the GNSS-TEC and model TEC (LTF) was 5.30 TECU and 95 %. The results indicate that solar components, as well as annual and semi-annual variations, have a significant impact on the daily average TEC. Solar activity appears to be the predominant determining factor of TEC during the solar phases of cycles 23 and 24. In contrast, periodic influences primarily outline TEC during periods characterized by minimal solar activity. The geomagnetic component presents an increased influence, particularly during storm periods. The model demonstrates superior performance in Total TEC modeling compared to other state-of-the-art approaches.