Functional Regression Research Articles

Impact of COVID-19-Related Lockdown Measures on Economic and Social Outcomes in Lithuania

The current world crisis caused by the COVID-19 pandemic has transformed into an economic crisis, becoming a problem and a challenge not only for individual national economies but also for the world economy as a whole. The first global lockdown, which started in mid-March of 2020 and lasted for three months in Lithuania, affected the movement and behavior of the population, and had an impact on the economy. This research presents results on the impact of lockdown measures on the economy using nonparametric methods in combination with parametric ones. The impact on unemployment and salary inequality was estimated. To assess the impact of lockdown on the labor market, the analysis of the dynamics of the unemployment rate was performed using the results of the cluster analysis. The Lithuanian data were analyzed in the context of other countries, where the dynamics of the spread of the virus were similar. The salary inequality was measured by the Gini coefficient and analyzed using change point analysis, functional data analysis and linear regression. The study found that the greatest impact of the closure restrictions on socio-economic indicators was recorded in 2020, with a lower impact in 2021. The proposed multi-step approach could be applied to other countries and to various types of shocks and interventions, not only the COVID-19 crisis, in order to avoid adverse economic and social outcomes.

Functional Logistic Regression for Motor Fault Classification Using Acoustic Data in Frequency Domain

Motor diagnostics is an important subject for consideration. Electric motors of different types are present in a multitude of object, from consumer goods through everyday use devices to specialized equipment. Diagnostic assessment of motors using acoustic signals is an interesting field, as microphones are present everywhere and are relatively easy sensors to process. In this paper, we analyze acoustic signals for the purpose of motor diagnostics using functional data analysis. We represent the spectrum (FFT) of the acoustic signals on a B-Spline basis and construct a classifier based on that representation. The results are promising, especially for binary classifiers, while multiclass (softmax regression) shows more sensitivity to dataset size. In particular, we show that while we are able to obtain almost perfect classification for binary cases, multiclass classifiers can struggle depending on the training/testing split. This is especially visible for determining the number of broken teeth, which is a non-issue for binary classifiers.

Functional linear regression with Huber loss

In this paper, we consider the robust regression problem associated with Huber loss in the framework of functional linear model and reproducing kernel Hilbert spaces. We propose an Ivanov regularized empirical risk minimization estimation procedure to approximate the slope function of the linear model in the presence of outliers or heavy-tailed noises. By appropriately tuning the scale parameter of the Huber loss, we establish explicit rates of convergence for our estimates in terms of excess prediction risk under mild assumptions. Our study in the paper justifies the efficiency of Huber regression for functional data from a theoretical viewpoint.

Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design

In this work, we consider a nonparametric prediction of a spatiofunctional process observed under a non-random sampling design. The proposed predictor is based on functional regression and depends on two kernels, one of which controls the spatial structure and the other measures the proximity between the functional observations. It can be considered, in particular, as a supervised classification method when the variable of interest belongs to a predefined discrete finite set. The mean square error and almost complete (or sure) convergence are obtained when the sample considered is a locally stationary α-mixture sequence. Numerical studies were performed to illustrate the behavior of the proposed predictor. The finite sample properties based on simulated data show that the proposed prediction method outperformsthe classical predictor which not taking into account the spatial structure.

함수형 자료 기법을 활용한 로지스틱 회귀 모형과 응용

Stock prices around the world dramatically decreased in 2020 due to COVID-19. Although the general trend seemed to be rebounding in 2021, the degree of recovery might vary by industry. This paper intends to investigate the relationship between the type of the industry companies and the pattern of individual stock indexes for 2021. Specifically, we investigate whether the functional pattern of individual stock indexes can be an important factor that can distinguish the type of the industry company. Traditionally, the stock price data has been analyzed using a time series approach. However, with the recent development of automated devices, the interest on the functional data analysis has been increasing. If individual stock prices are assumed to be random functions defined above continuous time, functional data techniques can be applied. Following this approach, we propose to use a functional logistic regression model that can explain the type of the industry company with the pattern of individual stock prices especially for 2021.

The Functional Regression With Reconstructed Functions From Hybrid Principal Components Analysis: With EEG-fMRI Application

Objective: In this article, we reconstruct the hybrid data with hybrid principal component analysis (HPCA) as a feature extraction step and model them with functional regression as a modeling step, and comparing models and choose the best number of HPCA based on the prediction accuracy as an evaluation step. Method: We decompose the hybrid data to the eigencomponents with HPCA. The reconstructed data from HPCA were divided into the training and testing dataset. The function-on-function signal compression and scalar-on-function regressions were used. Three simulation scenarios and their applications in the neuroimaging datasets (EEG-fMRI) were studied. The number of HPCA was selected with the mean squared prediction error (MSPE). Result: The simulation shows that the raw data, reconstructed from the first and all HPCAs for the training dataset has median MSPE 0.1001, 0.0028, and 0.0174 respectively, and for the testing, the dataset has 0.3207, 0.1118, and 0.2484 respectively. The EEG-fMRI suggests that in both auditory and visual tasks and standard and target stimuli for different regions of the brain the first HPCA has the smallest MSPE. Conclusions: We conclude this method improves the prediction accuracy of the experiments with the EEG datasets. And we recommend that instead of using the functional PCA on the desired dimension, reconstruct the data with HPCA and average it on the other two dimensions for functional regression models.

Local influence of climate on grapevine: an analytical process involving a functional and Bayesian exploration of farm data time series synchronised with an eGDD thermal index

Climate influence on grapevine physiology is prevalent and this influence is expected to increase with climate change. Climate influence on grapevine physiology can vary depending on the terroir. A better understanding of these local terroir variations is likely to be achieved with analyses that use local data; i.e., farm/vineyard data. Thus, the challenge lies in exploiting farm data to enable grape growers to understand their own terroir and consequently adapt their practices to the local conditions. In such a context, this article proposes an analytical process to site-specifically study climate influence on grapevine physiology by focusing on time series of the weather data often contained in farm data sets. This article focuses on temperature and precipitation influence on yield in the form of a case study. The analytical process includes the Extended Growing Degree Days (eGDD) and the Bayesian functional Linear regression with Sparse Steps functions (BLiSS) methods in order to detect site-specific periods of strong climate influence on grapevine yield. It uses data from three commercial vineyards situated in the Bordeaux region (France), California (USA) and Israel. In general, the periods of climate influence on grapevine yield detected for the three vineyards identified the same stages of yield development, which have already been studied in the scientific literature. However, some vineyard differences were observed, including: i) different periods of influence associated with a given stage of yield development between the vineyards, ii) different influential weather variables between the three vineyards for a given period, and iii) differing duration of the period of influence associated with a given stage of yield development between the vineyards. These results show the potential of the proposed analytical process for analysing the time series of farm weather data in order to extract site-specific climate indicators of grapevine yield.

SPF: A spatial and functional data analytic approach to cell imaging data.

The tumor microenvironment (TME), which characterizes the tumor and its surroundings, plays a critical role in understanding cancer development and progression. Recent advances in imaging techniques enable researchers to study spatial structure of the TME at a single-cell level. Investigating spatial patterns and interactions of cell subtypes within the TME provides useful insights into how cells with different biological purposes behave, which may consequentially impact a subject's clinical outcomes. We utilize a class of well-known spatial summary statistics, the K-function and its variants, to explore inter-cell dependence as a function of distances between cells. Using techniques from functional data analysis, we introduce an approach to model the association between these summary spatial functions and subject-level outcomes, while controlling for other clinical scalar predictors such as age and disease stage. In particular, we leverage the additive functional Cox regression model (AFCM) to study the nonlinear impact of spatial interaction between tumor and stromal cells on overall survival in patients with non-small cell lung cancer, using multiplex immunohistochemistry (mIHC) data. The applicability of our approach is further validated using a publicly available multiplexed ion beam imaging (MIBI) triple-negative breast cancer dataset.

Modelling time-varying covariates effect on survival via functional data analysis: application to the MRC BO06 trial in osteosarcoma

Time-varying covariates are of great interest in clinical research since they represent dynamic patterns which reflect disease progression. In cancer studies biomarkers values change as functions of time and chemotherapy treatment is modified by delaying a course or reducing the dose intensity, according to patient’s toxicity levels. In this work, a Functional covariate Cox Model (FunCM) to study the association between time-varying processes and a time-to-event outcome is proposed. FunCM first exploits functional data analysis techniques to represent time-varying processes in terms of functional data. Then, information related to the evolution of the functions over time is incorporated into functional regression models for survival data through functional principal component analysis. FunCM is compared to a standard time-varying covariate Cox model, commonly used despite its limiting assumptions that covariate values are constant in time and measured without errors. Data from MRC BO06/EORTC 80931 randomised controlled trial for treatment of osteosarcoma are analysed. Time-varying covariates related to alkaline phosphatase levels, white blood cell counts and chemotherapy dose during treatment are investigated. The proposed method allows to detect differences between patients with different biomarkers and treatment evolutions, and to include this information in the survival model. These aspects are seldom addressed in the literature and could provide new insights into the clinical research.

Functional linear regression with truncated signatures

We place ourselves in a functional regression setting and propose a novel methodology for regressing a real output on vector-valued functional covariates. This methodology is based on the notion of signature, which is a representation of a function as an infinite series of its iterated integrals. The signature depends crucially on a truncation parameter for which an estimator is provided, together with theoretical guarantees. An empirical study on both simulated and real-world datasets shows that the resulting methodology is competitive with traditional functional linear models, in particular when the functional covariates take their values in a high dimensional space.

Model selection for functional linear regression with hierarchical structure

Scalar-on-function regression allows for a scalar response to be dependent on functional predictors; however, not much work has been done when interaction effects between the functional predictors are included. In this paper, we introduce a multiple functional linear regression model with interaction terms. Meanwhile, we enforce the hierarchical structure constraint on the model, that is, interaction terms can be selected into the model only if the associated main effects are in the model. Based on the functional principal component analysis and group smoothly clipped absolute deviation (SCAD) penalty, we propose a new penalized estimation procedure to select the important functional predictors and interactions while automatically obeying the hierarchical structure. With appropriate selection of the tuning parameters, the rates of convergence of the proposed estimators and the consistency of the model selection procedure are established under some regularity conditions. At last, we illustrate the finite sample performance of our proposed methods with some simulation studies and a real data application.

Estimation of sparse functional quantile regression with measurement error: a SIMEX approach.

Quantile regression is a semiparametric method for modeling associations between variables. It is most helpful when the covariates have complex relationships with the location, scale, and shape of the outcome distribution. Despite the method's robustness to distributional assumptions and outliers in the outcome, regression quantiles may be biased in the presence of measurement error in the covariates. The impact of function-valued covariates contaminated with heteroscedastic error has not yet been examined previously; although, studies have investigated the case of scalar-valued covariates. We present a two-stage strategy to consistently fit linear quantile regression models with a function-valued covariate that may be measured with error. In the first stage, an instrumental variable is used to estimate the covariance matrix associated with the measurement error. In the second stage, simulation extrapolation (SIMEX) is used to correct for measurement error in the function-valued covariate. Point-wise standard errors are estimated by means of nonparametric bootstrap. We present simulation studies to assess the robustness of the measurement error corrected for functional quantile regression. Our methods are applied to National Health and Examination Survey data to assess the relationship between physical activity and body mass index among adults in the United States.

Predicting Global Functional Outcomes Among Post-traumatic Epilepsy Patients After Moderate-to-Severe Traumatic Brain Injury: Development of a Prognostic Model.

ObjectiveThe development of post-traumatic epilepsy (PTE) following traumatic brain injury (TBI) is associated with unfavorable functional outcomes, and the global function of PTE patients might change dynamically overtime. Predicting the long-term functional outcomes of patients with PTE may help to develop accurate rehabilitation programs and improve their quality of life. Based on this, the objective of this study is to use clinical data to derive and validate a model for predicting the functional outcomes of patients with PTE after moderate-to-severe TBI.MethodsThis study retrospectively analyzed 721 patients with PTE after moderate-to-severe TBI in the Epilepsy Centre, Beijing Tiantan Hospital, from January 2013 to December 2018. All patients had favorable global function as indicated by the Glasgow Outcome Scale-Extended (GOSE) at the time of their first late post-traumatic seizure (PTS) onset, and the 5-year global function after the first late PTS onset was chosen as the principal outcome of interest. To identify possible predictors for the global functional outcomes, univariate and multivariate logistic regression techniques were used. A prognostic model was established using these identified predictors, the internal validation with the bootstrapping method was performed, and the model was then visualized as a graphical score chart.ResultsThe 5-year global functional outcome of 98 (13.59%) patients was unfavorable, and the temporal lobe lesion was found as the strongest predictor of unfavorable outcomes. The final prognostic model also included the following other predictors: gender, age at TBI, multiple injuries, the severity of TBI, and latency of PTE. Discrimination was satisfactory with C-statistic of 0.754 (0.707 – 0.800), the goodness-of-fit test indicated good calibration (P = 0.137), and the C-statistic was 0.726 for internal validation. A graphical score chart was also constructed to provide the probability of an unfavorable 5-year global functional outcomes more readily.ConclusionsClearer treatment strategies are essential to help ameliorate the global functional outcomes of patients with PTE. Our proposed prognostic model has significant potential to be used in the clinic for predicting global functional outcomes among patients with PTE after moderate-to-severe TBI.

Identification of a seven-gene signature predicting clinical outcome of liver cancer based on tumor mutational burden.

The total number of somatic mutations may affect the prognosis of cancer, so we applied bioinformatics methods to investigate the association between the TMB (tumor mutational burden)-related differentially expressed genes (DEGs) and the prognosis of hepatocellular carcinoma (HCC). We calculated the TMB value of the patients with HCC in TCGA database and identified the differentially expressed genes between the high-TMB and low-TMB patients. We performed functional enrichment analysis and LASSO Cox regression analysis of the DEGs, and seven genes were screened to establish a risk score model. A nomogram based on the risk scores was drawn to assess the predictive outcomes compared to the actual outcomes. The expression level of the seven genes was verified in cancer cell lines. Moreover, we explored the difference in immune cells infiltration and immune checkpoints between the high-risk and low-risk groups. The results showed that the DEGs between the high-TMB and low-TMB patients were enriched in extracellular matrix organization. A seven-gene risk score model (PAGE1, CHGA, OGN, MMP7, TRIM55, MAGEA6, and MAGEA12) was established for predicting HCC prognosis. Patients with lower risk scores had longer survival time and lower mortality rate. The nomogram based on risk scores and TNM staging showed good performance and reliability in predicting the clinical outcomes. Significant differences in cell infiltration and checkpoints were found between the high-risk and low-risk groups. Our study demonstrated a seven-gene signature and a nomogram based on the risk score model to predict the prognosis of HCC. Some of the newly identified DEGs may be potential biomarkers or therapeutic targets.

Some non‐parametric regression models for interval‐valued functional data

The interval‐valued functional data belong to the category of symbolic data in which the values of each observation consist of two functions, a lower limit function and an upper limit function. This study introduces some functional non‐parametric approaches to fit a functional regression model on interval‐valued functional data. First, using kernel functions, two non‐parametric regression models are presented to predict a scalar interval‐valued response variable based on interval‐valued functional predictors. For this purpose, a local linear regression and a kernel regression estimator of the Nadaraya–Watson type are proposed. These approaches are based on two independent non‐parametric models, which may not hold in practice. Therefore, some non‐parametric methods are introduced, including bivariate functional Nadaraya–Watson, bivariate functional locally linear smoothing and another method based on bootstrap, which maintain selecting points randomly in the intervals of the proposed data. The applicability and advantages of the proposed models are investigated through simulation experiments and a real data example.

Partially linear functional quantile regression in a reproducing kernel Hilbert space

We consider quantile functional regression with a functional part and a scalar linear part. We establish the optimal prediction rate for the model under mild assumptions in the reproducing kernel Hilbert space (RKHS) framework. Under stronger assumptions related to the capacity of the RKHS, the non-functional linear part is shown to have asymptotic normality. The estimators are illustrated in simulation studies.

Highly Irregular Functional Generalized Linear Regression with Electronic Health Records

Abstract This work presents a new approach, called Multiple Imputation of Sparsely-sampled Functions at Irregular Times (MISFIT), for fitting generalized functional linear regression models with sparsely and irregularly sampled data. Current methods do not allow for consistent estimation unless one assumes that the number of observed points per curve grows sufficiently quickly with the sample size. In contrast, MISFIT is based on a multiple imputation framework, which, as we demonstrate empirically, has the potential to produce consistent estimates without such an assumption. Just as importantly, it propagates the uncertainty of not having completely observed curves, allowing for a more accurate assessment of the uncertainty of parameter estimates, something that most methods currently cannot accomplish. This work is motivated by a longitudinal study on macrocephaly, or atypically large head size, in which electronic medical records allow for the collection of a great deal of data. However, the sampling is highly variable from child to child. Using MISFIT we are able to clearly demonstrate that the development of pathologic conditions related to macrocephaly is associated with both the overall head circumference of the children as well as the velocity of their head growth.

Functional data analysis for Brazilian term structure of interest rate curves

The term structure of the interest rate is a crucial tool to underline the decision-making process of several agents: investors, regulators, and risk managers. The data that form the basis of empirical/theoretical studies are particular observations of a function [representing the term structure of the interest rate for each instant of time]. Most of the studies in the literature do not take into account that the data are discretizations of functions. And, therefore, they do not incorporate information that reflects important aspects of the data, such as the smoothness characteristics of the curves. In this work, we propose the use of a set of techniques known by functional data analysis to describe the sources of variability of the interest curves. We also estimate functional linear regression models, where the covariates are some macroeconomic variables.

Smart Whether Prediction using Machine Learning Algorithm

Weather forecasting with traditional technique is mainly done by physical model, still in many parts of the world. Though not neglecting the importance of the model there is an alternative method where recorded data of past can be used for predicting the future weather data. The predicted data may not be exact, but less time consuming and more efficient. Minimum temperature, maximum temperature, average temperature, precipitation percentage these are the common parameters and predicting these with another less resource-based method with some precision will help us going. Machine learning can be used for processing the data based on models like linear regression, functional regression, circular, statistical which processes the data and reduces the error. On comparing the result of model based on different location one can used a model based on requirement. The result obtained then can be analyzed and further improved upon input variables and data size.

The European Lambert-Eaton Myasthenic Syndrome Registry: Long-Term Outcomes Following Symptomatic Treatment.

IntroductionLambert–Eaton myasthenic syndrome (LEMS) is characterized by autoantibodies against voltage-gated calcium channels (VGCC) at the neuromuscular junction causing proximal muscle weakness, decreased tendon reflexes, and autonomic changes. The European LEMS registry aimed to collate observational safety data for 3,4-diaminopyridine phosphate (3,4-DAPP) and examine long-term outcomes for patients with LEMS.MethodsThirty centers across four countries participated in the non-interventional European LEMS registry. Any patients diagnosed with LEMS by means of clinical assessment and abnormal neurophysiological testing, or clinical assessment and positive for VGCC antibodies were eligible to participate. Patients were monitored using standard assessments for LEMS-related clinical manifestations.ResultsAmong 96 evaluable participants, 50 (52.1%) were being treated with 3,4-DAPP, 21 (21.9%) with 3,4-diaminopyridine (3,4-DAP), and 25 (26.0%) with other treatments (e.g., pyridostigmine, corticosteroids, immunoglobulins, and azathioprine); 74 participants (77.1%) were exposed to 3,4-DAPP at any time. Quantitative myasthenia gravis scores were similar across treatment groups. Muscle strength was generally good and maintained during follow-up. Cerebellar ataxia, defined as a negative Romberg’s test and at least one other positive ataxia test, was observed in 30 (56.6%) patients. Most participants had reduced reflex tone and limited functioning. Sustained or improved functioning was observed in participants administered 3,4-DAPP. Inconsistent and sporadic functional improvement and regression was observed with 3,4-DAP and other treatments. Fifty-five treatment-related adverse events (AEs) were reported by 32 (33.3%) participants. Eight (8.3%) participants reported nine treatment-related serious AEs. No new safety signals were identified.ConclusionNo new safety signals were observed following long-term management of LEMS with 3,4-DAPP.Supplementary InformationThe online version contains supplementary material available at 10.1007/s40120-022-00354-8.