Functional Regression Research Articles

Functional coefficient quantile regression model with time-varying loadings

ABSTRACT This paper proposes a functional coefficient quantile regression model with heterogeneous and time-varying regression coefficients and factor loadings. Estimation of the model coefficients is done in two stages. First, we estimate the unobserved common factors from a linear factor model with exogenous covariates. Second, we plug-in an affine transformation of the estimated common factors to obtain the functional coefficient quantile regression model. The quantile parameter estimators are consistent and asymptotically normal. The application of this model to the quantile process of a cross-section of U.S. firms’ excess returns confirms the predictive ability of firm-specific covariates and the good performance of the local estimator of the heterogeneous and time-varying quantile coefficients.

Generalizable Descriptors of Highly Sensitive Detection of As(III) over Transition-Metal Single Atoms: A Combined Density Function Theory and Gradient Boosting Regression Approach.

Traditional nanomodified electrodes have made great achievements in electrochemical stripping voltammetry of sensing materials for As(III) detection. Moreover, the intermediate states are complicated to probe because of the ultrashort lifetime and complex reaction conditions of the electron transfer process in electroanalysis, which seriously hinder the identification of the actual active site. Herein, the intrinsic interaction of highly sensitive analytical behavior of nanomaterials is elucidated from the perspective of electronic structure through density functional theory (DFT) and gradient boosting regression (GBR). It is revealed that the atomic radius, d-band center (εd), and the largest coordinative TM-N bond length play a crucial role in regulating the arsenic reduction reaction (ARR) performance by the established ARR process for 27 sets of transition-metal single atoms supported on N-doped graphene. Furthermore, the database composed of filtered intrinsic electronic structural properties and the calculated descriptors of the central metal atom in TM-N4-Gra were also successfully extended to oxygen evolution reaction (OER) systems, which effectively verified the reliability of the whole approach. Generally, a multistep workflow is developed through GBR models combined with DFT for valid screening of sensing materials, which will effectively upgrade the traditional trial-and-error mode for electrochemical interface designing.

Modeling the evolution of deaths from infectious diseases with functional data models: The case of COVID-19 in Brazil.

In this paper, we apply statistical methods for functional data to explore the heterogeneity in the registered number of deaths of COVID-19, over time. The cumulative daily number of deaths in regions across Brazil is treated as continuous curves (functional data). The first stage of the analysis applies clustering methods for functional data to identify and describe potential heterogeneity in the curves and their functional derivatives. The estimated clusters are labeled with different "levels of alert" to identify cities in a possible critical situation. In the second stage of the analysis, we apply a functional quantile regression model for the death curves to explore the associations with functional rates of vaccination and stringency and also with several scalar geographical, socioeconomic and demographic covariates. The proposed model gave a better curve fit at different levels of the cumulative number of deaths when compared to a functional regression model based on ordinary least squares. Our results add to the understanding of the development of COVID-19 death counts.

Theoretical error performance analysis for variational quantum circuit based functional regression

The noisy intermediate-scale quantum devices enable the implementation of the variational quantum circuit (VQC) for quantum neural networks (QNN). Although the VQC-based QNN has succeeded in many machine learning tasks, the representation and generalization powers of VQC still require further investigation, particularly when the dimensionality of classical inputs is concerned. In this work, we first put forth an end-to-end QNN, TTN-VQC, which consists of a quantum tensor network based on a tensor-train network (TTN) for dimensionality reduction and a VQC for functional regression. Then, we aim at the error performance analysis for the TTN-VQC in terms of representation and generalization powers. We also characterize the optimization properties of TTN-VQC by leveraging the Polyak-Lojasiewicz condition. Moreover, we conduct the experiments of functional regression on a handwritten digit classification dataset to justify our theoretical analysis.

Diagnostic identification of chronic insomnia using ALFF and FC features of resting-state functional MRI and logistic regression approach

This study investigated whether the amplitude of low-frequency fluctuation (ALFF) and functional connectivity (FC) features could be used as potentially neurological markers to identify chronic insomnia (CI) using resting-state functional MRI and machine learning method logistic regression (LR). This study included 49 CI patients and 47 healthy controls (HC). Voxel-wise features, including the amplitude of low-frequency fluctuations (ALFF) and functional connectivity (FC), were extracted from resting-state functional magnetic resonance brain images. Then, we divided the data into two independent cohorts for training (44 CI patients and 42 HC patients), and independent validation (5 CI patients and 5 HC patients) by using logistic regression. The model was evaluated using 20 rounds of fivefold cross‑validation for training. In particular, a two-sample t-test (GRF corrected, p-voxel < 0.001, p-cluster < 0.05) was used for feature selection during the model training. Finally, single‑shot testing of the final model was performed on the independent validation cohort. A correlation analysis (Bonferroni correction, p < 0.05/4) was also conducted to determine whether the features contributing to the prediction were correlated with clinical characteristics, including the Insomnia Severity Index (ISI), Pittsburgh sleep quality index (PSQI), self-rating anxiety scale (SAS), and self-rating depression scale (SDS). Results showed that resting-state features had a discrimination accuracy of 86.40%, with a sensitivity of 93.00% and specificity of 79.80%. The area under the curve (AUC) was 0.89 (all {P}_{mathrm{permutation}}< 0.001). The ALFF and FC features showed significant differences between the CI patients and HC. The regions contributing to the prediction mainly included the anterior cingulate, prefrontal cortex, orbital part of the frontal lobe, angular gyrus, cingulate gyrus, praecuneus, parietal lobe, temporal gyrus, superior temporal gyrus, and middle temporal gyrus. Furthermore, some specific functional connectivity among related regions was positively correlated with the ISI, and also negatively related to the SDS in correlation analysis. Our current study suggested that ALFF and FC in the regions contributing to diagnostic identification might serve as potential neuromarkers for CI.

Robust Estimation for Semi-Functional Linear Model with Autoregressive Errors

It is well-known that the traditional functional regression model is mainly based on the least square or likelihood method. These methods usually rely on some strong assumptions, such as error independence and normality, that are not always satisfied. For example, the response variable may contain outliers, and the error term is serially correlated. Violation of assumptions can result in unfavorable influences on model estimation. Therefore, a robust estimation procedure of a semi-functional linear model with autoregressive error is developed to solve this problem. We compare the efficiency of our procedure to the least square method through a simulation study and two real data analyses. The conclusion illustrates that the proposed method outperforms the least square method, providing random errors follow the heavy-tail distribution.

On estimation and prediction in spatial functional linear regression model

We consider a spatial functional linear regression, where a scalar response is related to a square integrable spatial functional process. We use a smoothing spline estimator for the functional slope parameter and establish a finite sample bound for variance of this estimator under mixing spatial dependence. Then, we give a bound of the prediction error. Finally, we illustrate our results by simulations

Testing Linear Operator Constraints in Functional Response Regression with Incomplete Response Functions.

Hypothesis testing procedures are developed to assess linear operator constraints in function-on-scalar regression when incomplete functional responses are observed. The approach enables statistical inferences about the shape and other aspects of the functional regression coefficients within a unified framework encompassing three incomplete sampling scenarios: (i) partially observed response functions as curve segments over random sub-intervals of the domain; (ii) discretely observed functional responses with additive measurement errors; and (iii) the composition of former two scenarios, where partially observed response segments are observed discretely with measurement error. The latter scenario has been little explored to date, although such structured data is increasingly common in applications. For statistical inference, deviations from the constraint space are measured via integrated -distance between the model estimates from the constrained and unconstrained model spaces. Large sample properties of the proposed test procedure are established, including the consistency, asymptotic distribution and local power of the test statistic. Finite sample power and level of the proposed test are investigated in a simulation study covering a variety of scenarios. The proposed methodologies are illustrated by applications to U.S. obesity prevalence data, analyzing the functional shape of its trends over time, and motion analysis in a study of automotive ergonomics.

Lasso in infinite dimension: application to variable selection in functional multivariate linear regression

Lasso in infinite dimension: application to variable selection in functional multivariate linear regression

Single Functional Index Quantile Regression for Functional Data with Missing Data at Random

The primary goal of this research was to estimate the quantile of a conditional distribution using a semi-parametric approach in the presence of randomly missing data, where the predictor variable belongs to a semi-metric space. The authors assumed a single index structure to link the explanatory and response variable. First, a kernel estimator was proposed for the conditional distribution function, assuming that the data were selected from a stationary process with missing data at random (MAR). By imposing certain general conditions, the study established the model’s uniform almost complete consistencies with convergence rates.

Predicting fatigue from heart rate signatures using functional logistic regression

Physical fatigue can have adverse effects on humans in extreme environments. Therefore, being able to predict fatigue using easy to measure metrics such as heart rate (HR) signatures has potential to have an impact in real‐life scenarios. We apply a functional logistic regression model that uses HR signatures to predict physical fatigue, where physical fatigue is defined in a data‐driven manner. Data were collected using commercially available wearable devices on 47 participants hiking the 20.7‐mile Grand Canyon rim‐to‐rim trail in a single day. Fitted model provides good predictions and interpretable parameters for real‐life application.

Heterogeneous Beliefs and the Phillips Curve

We establish a set of novel empirical facts concerning cross-section distributions of inflation expectations reported in surveys. Almost all the variation in expectations about their mean may be summarized via three factors we call disagreement, skew, and shape. We adopt a functional principal component regression approach to estimating forward-looking models of inflation that exploits the heterogeneity present in individual-level data. By using survey information more effectively, our approach reveals an enhanced role for expectations in inflation dynamics that is robust to lagged inflation, trend inflation, and supply factors. Our findings hold in similar form across two major economies.

Partially Linear Additive Functional Regression

Partially Linear Additive Functional Regression

Developing multivariable models for predicting headache improvement in patients with acute post-traumatic headache attributed to mild traumatic brain injury: A preliminary study.

Post-traumatic headache (PTH) is a common symptom after mild traumatic brain injury (mTBI). Although there have been several studies that have used clinical features of PTH to attempt to predict headache recovery, currently no accurate methods exist for predicting individuals' improvement from acute PTH. This study investigated the utility of clinical questionnaires for predicting (i) headache improvement at 3 and 6 months, and (ii) headache trajectories over the first 3 months. We conducted a clinic-based observational longitudinal study of patients with acute PTH who completed a battery of clinical questionnaires within 0-59 days post-mTBI. The battery included headache history, symptom evaluation, cognitive tests, psychological tests, and scales assessing photosensitivity, hyperacusis, insomnia, cutaneous allodynia, and substance use. Each participant completed a web-based headache diary, which was used to determine headache improvement. Thirty-seven participants with acute PTH (mean age=42.7, standard deviation [SD]=12.0; 25 females/12 males) completed questionnaires at an average of 21.7 (SD=13.1) days post-mTBI. The classification of headache improvement or non-improvement at 3 and 6 months achieved cross-validation area under the curve (AUC) of 0.72 (95% confidence interval [CI] 0.55 to 0.89) and 0.84 (95% CI 0.66 to 1.00). Sub-models trained using only the top five features still achieved 0.72 (95% CI 0.55 to 0.90) and 0.77 (95% CI 0.52 to 1.00) AUC. The top five contributing features were from three questionnaires: Pain Catastrophizing Scale total score and helplessness sub-domain score; Sports Concussion Assessment Tool Symptom Evaluation total score and number of symptoms; and the State-Trait Anxiety Inventory score. The functional regression model achieved for modeling headache trajectory over the first 3 months. Questionnaires completed following mTBI have good utility for predicting headache improvement at 3 and 6 months in the future as well as the evolving headache trajectory. Reducing the battery to only three questionnaires, which assess post-concussive symptom load and biopsychosocialecologic factors, was helpful to determine a reasonable prediction accuracy for headache improvement.

LogitFD: an R package for functional principal component logit regression

The functional logit regression model was proposed by [@Escabias04] with the objective of modeling a scalar binary response variable from a functional predictor. The model estimation proposed in that case was performed in a subspace of $L^2(T)$ of squared integrable functions of finite dimension, generated by a finite set of basis functions. For that estimation it was assumed that the curves of the functional predictor and the functional parameter of the model belong to the same finite subspace. The estimation so obtained was affected by high multicollinearity problems and the solution given to these problems was based on different functional principal component analysis. The [logitFD](https://CRAN.R-project.org/package=logitFD) package introduced here provides a toolbox for the fit of these models by implementing the different proposed solutions and by generalizing the model proposed in 2004 to the case of several functional and non-functional predictors. The performance of the functions is illustrated by using data sets of functional data included in the [fda.usc](https://CRAN.R-project.org/package=fda.usc) package from R-CRAN.

Women-only versus mixed-gender groups in multimodal, day clinic treatment of trauma-related disorders.

Background: In the German healthcare system benefits include inpatient and day clinic psychotherapy of trauma-related disorders, which are often provided in gender specific groups. Despite this widespread practice, the actual impact and potential outcome have been studied relatively scarcely and are still open to debate. The present study therefore examines whether the therapeutic relationships in the group vary depending on the group composition and whether this has an effect on therapy outcome. Methods: 66 patients (N = 55 women) were treated in our multimodal, day clinic groupbased treatment program and filled out symptom questionnaires (ETI, BDI-II, PHQ-15, Questionnaire on social support) at admission and discharge and the Group Questionnaire (GQ; including subscales positive bonding, positive working and negative relationships) at the end of each of the eight weeks of treatment. Patients took part in five women-only and six mixed-gender groups and underwent therapy in a closed group format, so that they began and ended therapy within the same group. All patients were diagnosed with trauma-related disorders, which included PTSD, cPTSD, depression, somatoform disorders and further mental diseases. Outcome data were examined using functional regression analysis and linear mixed models. Results: In women-only groups, positive bonding and positive working relationship were significantly higher than in mixed-gender groups. Negative relationship ratings were significantly lower in women-only groups, especially in the middle phase of treatment. However, no effect of an exclusively female group composition on treatment outcome was found, except for the higher increase of perceived social support for patients treated in women-only groups. Conclusions: Group relationships were considered more pleasant in women-only groups than in mixed-gender groups, but this had no effect on the symptom-based therapy outcome. To enhance feelings of safety and comfort, traumatized women should have the possibility to choose whether to be treated in a women-only or a mixed-gender group.

Revealing the influence of oxygen-containing functional groups on mercury adsorption via density functional theory and multiple linear regression analysis

Oxygen-containing functional groups (OFGs) are important species on activated carbons, which have received extensive attention due to the predominant chemisorption activity towards elemental mercury. Based on density functional theory (DFT) and multiple linear regression (MLR) studies, this work is focused on the following five aspects: (1) The Gibbs free energy-temperature dependences of OFGs are compared, and CO2 -1, CO3 -7, C2O -2, C2O2 -9, C2O3 -24 are proved as the most dominant candidates among their respective species. (2) The adsorption of OFGs on carbonaceous surface is studied, among which C, O and C2 species belong to chemisorption, on the contrary, the adsorption of CO and CO2 is weak. (3) The mechanism of mercury adsorption by OFGs is sorted into three categories, and the strongest adsorption can be obtained when mercury is co-adsorbed by OFGs and substrate. (4) Six descriptors are extracted from the binary relationships to provide data source for MLR analysis. (5) A regression equation is successfully established, which can effectively predict mercury adsorption energy. This theoretical work estimating mercury adsorption will provide new insights for the development of mercury adsorbents with OFG synergy.

Psychosis and autism without functional regression in a patient with Kleefstra syndrome.

Kleefstra syndrome is a rare genetic disorder caused by haploinsufficiency of the euchromatic histone lysine methyltransferase 1 (EHMT1) gene. It is characterized by a variety of dysmorphic features, comorbid medical issues, and developmental delays/intellectual disability. Neuropsychiatric symptoms may also occur, including autistic features and psychosis, and are often accompanied by functional regression. However, the phenomenology of psychotic symptoms in this syndrome has not been well described in the literature. As such, in this brief report, we review the literature with respect to the occurrence of psychosis in Kleefstra syndrome and describe the symptom profile of a 35-year-old affected male with an intellectual disability, autism spectrum disorder, and schizophrenia (in association with manic features). This is the first report of psychotic symptoms fully remitting in response to zuclopenthixol therapy in an individual with Kleefstra syndrome. This case is also unique as it demonstrates that functional regression does not necessarily coincide with the development of schizophrenia-like presentations in affected individuals.

Deep transfer operator learning for partial differential equations under conditional shift

Transfer learning enables the transfer of knowledge gained while learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labelling, potential computational power limitations and dataset distribution mismatches. We propose a new transfer learning framework for task-specific learning (functional regression in partial differential equations) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of the target data. Inspired by conditional embedding operator theory, we minimize the statistical distance between labelled target data and the surrogate prediction on unlabelled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various transfer learning scenarios involving nonlinear partial differential equations under diverse conditions due to shifts in the geometric domain and model dynamics. Our transfer learning framework enables fast and efficient learning of heterogeneous tasks despite considerable differences between the source and target domains.

Joint classification and prediction of random curves using heavy‐tailed process functional regression

We propose a heavy-tailed process functional regression to jointly perform classification and prediction of time-varying functional data. We use two independent scale mixtures of Gaussian Processes to respectively model random effects and random errors, yielding robust inferences against both magnitude and shape outliers. We classify random curves by posterior predictive probabilities of class labels and offer a weighted prediction of future curve trends. A Bayesian estimation procedure is implemented through an MCMC sampling algorithm. The performance of classification and prediction of the proposed model is evaluated using simulated studies and some real data sets.