A partial least squares regression is proposed for estimating the function-on-function regression model where a functional response and multiple functional predictors consist of random curves with quadratic and interaction effects. The direct estimation of a function-on-function regression model is usually an ill-posed problem. To overcome this difficulty, in practice, the functional data that belong to the infinite-dimensional space are generally projected into a finite-dimensional space of basis functions. The function-on-function regression model is converted to a multivariate regression model of the basis expansion coefficients. In the estimation phase of the proposed method, the functional variables are approximated by a finite-dimensional basis function expansion method. We show that the partial least squares regression constructed via a functional response, multiple functional predictors, and quadratic/interaction terms of the functional predictors is equivalent to the partial least squares regression constructed using basis expansions of functional variables. From the partial least squares regression of the basis expansions of functional variables, we provide an explicit formula for the partial least squares estimate of the coefficient function of the function-on-function regression model. Because the true forms of the models are generally unspecified, we propose a forward procedure for model selection. The finite sample performance of the proposed method is examined using several Monte Carlo experiments and two empirical data analyses, and the results were found to compare favorably with an existing method.
Functional data analysis tools, such as function-on-function regression models, have received considerable attention in various scientific fields because of their observed high-dimensional and complex data structures. Several statistical procedures, including least squares, maximum likelihood, and maximum penalized likelihood, have been proposed to estimate such function-on-function regression models. However, these estimation techniques produce unstable estimates in the case of degenerate functional data or are computationally intensive. To overcome these issues, we proposed a partial least s…
Scalar-on-function logistic regression, where the response is a binary outcome and the predictor consists of random curves, has become a general framework to explore a linear relationship between the binary outcome and functional predictor. Most of the methods used to estimate this model are based on the least-squares type estimators. However, the least-squares estimator is seriously hindered by outliers, leading to biased parameter estimates and an increased probability of misclassification. This paper proposes a robust partial least squares method to estimate the regression coefficient funct…
The asymptotic optimality (a.o.) of various hyper-parameter estimators with different optimality criteria has been studied in the literature for regularized least squares regression problems. The estimators include e.g., the maximum (marginal) likelihood method, $C_p$ statistics, and generalized cross validation method, and the optimality criteria are based on e.g., the inefficiency, the expectation inefficiency and the risk. In this paper, we consider the regularized least squares regression problems with fixed number of regression parameters, choose the optimality criterion based on the risk…
The scalar-on-function regression model has become a popular analysis tool to explore the relationship between a scalar response and multiple functional predictors. Most of the existing approaches to estimate this model are based on the least-squares estimator, which can be seriously affected by outliers in empirical datasets. When outliers are present in the data, it is known that the least-squares-based estimates may not be reliable. This paper proposes a robust functional partial least squares method, allowing a robust estimate of the regression coefficients in a scalar-on-multiple-function…
In other FICO Technical Papers, I have shown how to fit Generalized Additive Models (GAM) with shape constraints using quadratic programming applied to B-Spline component functions. In this paper, I extend the method to Robust Least Squares Regression.
A general method of minimization using correlation coefficients and order statistics is evaluated relative to least squares procedures in the estimation of parameters for normal data in simple linear regression.
Galton's family heights data has been a preeminent historical dataset in regression analysis, on which the original model and basic results have survived the close scrutiny of statisticians for 125 years. However by revisiting Galton's family data, we challenge whether Galton's classic model and his regression towards mean interpretation are proper. Using Galton's data as a benchmark for different regression methods, such as least squares, orthogonal regression, geometric mean regression, and least sine squares regression - a newly developed nonparametric robust regression approach, we elucida…
In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via …
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 est…
We propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. We further extend our partial quantile covariance techniques to functional composite quantile regression (CQR) defining partial composite quantile covariance. There are three major contributions. (1) We define partial quantile covariance between two scalar variables through linear quantile regr…
Traditionally, the least squares regression is mainly concerned with studying the effects of individual predictor variables, but strongly correlated variables generate multicollinearity which makes it difficult to study their effects. Existing methods for handling multicollinearity such as ridge regression are complicated. To resolve the multicollinearity issue without abandoning the simple least squares regression, for situations where predictor variables are in groups with strong within-group correlations but weak between-group correlations, we propose to study the effects of the groups with…
In this paper, we apply empirical likelihood method to inference for the regression parameters in the partial functional linear regression models based on B spline. We prove that the empirical log likelihood ratio for the regression parameters converges in law to a weighted sum of independent chi square distributions and run simulations to assess the finite sample performance of our method.
We consider a resampling scheme for parameters estimates in nonlinear regression models. We provide an estimation procedure which recycles, via random weighting, the relevant parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic normality of the resampled estimates and demonstrate the applicability of the recycling approach in a small simulation study and via example.
To conduct regression analysis for data contaminated with outliers, many approaches have been proposed for simultaneous outlier detection and robust regression, so is the approach proposed in this manuscript. This new approach is called "penalized weighted least squares" (PWLS). By assigning each observation an individual weight and incorporating a lasso-type penalty on the log-transformation of the weight vector, the PWLS is able to perform outlier detection and robust regression simultaneously. A Bayesian point-of-view of the PWLS is provided, and it is showed that the PWLS can be seen as an…
In this article, the parameters of a hybrid log-linear model (log-Poisson) are estimated using the fuzzy least-squares (FLS) procedures (Celmiņš, 987a,b, D'Urso and Gastaldi, 2000, DUrso and Gastaldi, 2001). A goodness of fit have been derived in order to assess and compare this new model and the classical log-Poisson regression in loss reserving framework (Mack, 1991). Both the hybrid model and its goodness of fit are performed on a loss reserving data.
The function-on-function linear regression model in which the response and predictors consist of random curves has become a general framework to investigate the relationship between the functional response and functional predictors. Existing methods to estimate the model parameters may be sensitive to outlying observations, common in empirical applications. In addition, these methods may be severely affected by such observations, leading to undesirable estimation and prediction results. A robust estimation method, based on iteratively reweighted simple partial least squares, is introduced to i…
Sign-constrained least squares estimation for high-dimensional regression
高维回归问题的正则化方案很多。大多数需要使用模型选择标准或交叉验证方案来选择调优参数。我们表明，一个简单的非负或符号约束最小二乘是一个非常简单和有效的正则化技术，为某一类高维回归问题。符号约束必须通过先验知识或初始估计器导出，但不需要进一步的调整或交叉验证。成功取决于在实践中容易检验的条件。我们的结果的一个充分条件是，大多数变量具有相同的符号约束是正相关的。对于稀疏最优预报，证明了回归系数的L1-误差的非渐近界。不使用任何进一步的正则化，只要\log（p）s/n -> 0 for n -> \infty，就可以一致地估计回归向量，其中s是最佳回归向量的稀疏度，p是变量的数量，n是样本量。网络断层扫描被证明是一个应用程序，其中成功的非负最小二乘的必要条件自然满足和经验结果证实了稀疏恢复的符号约束的有效性。
People employ the function-on-function regression to model the relationship between two random curves. Fitting this model, widely used strategies include algorithms falling into the framework of functional partial least squares (typically requiring iterative eigen-decomposition). Here we introduce a route of functional partial least squares based upon Krylov subspaces. It can be expressed in two forms equivalent to each other (in exact arithmetic): one is non-iterative with explicit forms of estimators and predictions, facilitating the theoretical derivation and potential extensions (to more c…
We implement extensions of the partial least squares generalized linear regression (PLSGLR) due to Bastien et al. (2005) through its combination with logistic regression and linear discriminant analysis, to get a partial least squares generalized linear regression-logistic regression model (PLSGLR-log), and a partial least squares generalized linear regression-linear discriminant analysis model (PLSGLRDA). These two classification methods are then compared with classical methodologies like the k-nearest neighbours (KNN), linear discriminant analysis (LDA), partial least squares discriminant an…
Nonparametric partitioning-based least squares regression is an important tool in empirical work. Common examples include regressions based on splines, wavelets, and piecewise polynomials. This article discusses the main methodological and numerical features of the R software package lspartition, which implements modern estimation and inference results for partitioning-based least squares (series) regression estimation. This article discusses the main methodological and numerical features of the R software package lspartition, which implements results for partitioning-based least squares (seri…
The linear regression models are widely used statistical techniques in numerous practical applications. The standard regression model requires several assumptions about the regres- sors and the error term. The regression parameters are estimated using the least-squares method. In this paper, we consider the regression model with arbitrary regressors and with- out the error term. An explicit expression for the regression parameters vector is obtained. The unbiasedness approach is used to estimate the regression parameters and its various properties are investigated. It is shown that the resulti…
Functional regression is very crucial in functional data analysis and a linear relationship between scalar response and functional predictor is often assumed. However, the linear assumption may not hold in practice, which makes the methods for linear models invalid. To gain more flexibility, we focus on functional nonlinear models and aim to develop new method that requires no strict constraint on the nonlinear structure. Inspired by the idea of the kernel method in machine learning, we propose a kernel functional partial least squares (KFPLS) method for the functional nonlinear models. The in…
Subsampling methods aim to select a subsample as a surrogate for the observed sample. As a powerful technique for large-scale data analysis, various subsampling methods are developed for more effective coefficient estimation and model prediction. This review presents some cutting-edge subsampling methods based on the large-scale least squares estimation. Two major families of subsampling methods are introduced, respectively, the randomized subsampling approach and the optimal subsampling approach. The former aims to develop a more effective data-dependent sampling probability, while the latter…
In this paper, we consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation st…
In this paper, a subgroup least squares and a convex clustering are introduced for inferring a partially heterogenous linear regression that has potential application in the areas of precision marketing and precision medicine. The homogenous parameter and the subgroup-average of the heterogenous parameters can be consistently estimated by the subgroup least squares, without need of the sparsity assumption on the heterogenous parameters. The heterogenous parameters can be consistently clustered via the convex clustering. Unlike the existing methods for regression clustering, our clustering proc…
In the functional linear regression model, many methods have been proposed and studied to estimate the slope function while the functional predictor was observed in the entire domain. However, works on functional linear regression models with partially observed trajectories have received less attention. In this paper, to fill the literature gap we consider the scenario where individual functional predictor may be observed only on part of the domain. Depending on whether measurement error is presented in functional predictors, two methods are developed, one is based on linear functionals of the…
A new partial functional linear regression model for panel data with time varying parameters is introduced. The parameter vector of the multivariate model component is allowed to be completely time varying while the function-valued parameter of the functional model component is assumed to change over K unknown parameter regimes. Consistency is derived for the suggested estimators and for the classification procedure used to detect the K unknown parameter regimes. Additionally, the convergence rates of the estimators are derived under a double asymptotic differentiating between asymptotic scena…
Additive regression models are actively researched in the statistical field because of their usefulness in the analysis of responses determined by non-linear relationships with multivariate predictors. In this kind of statistical models, the response depends linearly on unknown functions of predictor variables and typically, the goal of the analysis is to make inference about these functions. In this paper, we study the problem of additive regression using a least squares approach based on periodic orthogonal wavelets on the interval [0,1]. For this estimator, we obtain strong consistency (wit…
Generalized estimating equations (GEE) are widely used to analyze longitudinal data; however, they are not appropriate for heteroscedastic data, because they only estimate regressor effects on the mean response{\textemdash}and therefore do not account for data heterogeneity. Here, we combine the GEE with the asymmetric least squares (expectile) regression to derive a new class of estimators, which we call generalized expectile estimating equations (GEEE). The GEEE model estimates regressor effects on the expectiles of the response distribution, which provides a detailed view of regressor effec…
In the famous least sum of trimmed squares (LTS) of residuals estimator (Rousseeuw (1984)), residuals are first squared and then trimmed. In this article, we first trim residuals - using a depth trimming scheme - and then square the rest of residuals. The estimator that can minimize the sum of squares of the trimmed residuals, is called an LST estimator. It turns out that LST is a robust alternative to the classic least sum of squares (LS) estimator. Indeed, it has a very high finite sample breakdown point, and can resist, asymptotically, up to $50\%$ contamination without breakdown - in sharp…
In statistics, the Durbin-Watson test is always employed to detect the presence of serial correlation of residuals from a least squares regression analysis. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the Durbin-Watson will be ineffectual because the value of Durbin-Watson's statistic depends on the sequences of data point arrangement. Based on the ideas from spatial autocorrelation, this paper presents two new statistics for testing serial correlation of residuals …
We review sufficient dimension reduction (SDR) estimators with multivariate response in this paper. A wide range of SDR methods are characterized as inverse regression SDR estimators or forward regression SDR estimators. The inverse regression family include pooled marginal estimators, projective resampling estimators, and distance-based estimators. Ordinary least squares, partial least squares, and semiparametric SDR estimators, on the other hand, are discussed as estimators from the forward regression family.
Columbia University（哥伦比亚大学）；Pontifical Catholic University of Rio de Janeiro (PUC-Rio)（里约热内卢天主教大学）；University of California, Irvine（加州大学尔湾分校）
In this paper, we introduce a new machine learning (ML) model for nonlinear regression called the Boosted Smooth Transition Regression Trees (BooST), which is a combination of boosting algorithms with smooth transition regression trees. The main advantage of the BooST model is the estimation of the derivatives (partial effects) of very general nonlinear models. Therefore, the model can provide more interpretation about the mapping between the covariates and the dependent variable than other tree-based models, such as Random Forests. We present several examples with both simulated and real data…
In linear models, the generalized least squares (GLS) estimator is applicable when the structure of the error dependence is known. When it is unknown, such structure must be approximated and estimated in a manner that may lead to misspecification. The large-sample analysis of incorrectly-specified GLS (IGLS) estimators requires careful asymptotic manipulations. When performing estimation in the frequency domain, the asymptotic normality of the IGLS estimator, under the so-called Grenander assumptions, has been proved for a broad class of error dependence models. Under the same assumptions, asy…
IIT Kanpur（印度理工学院坎普尔分校）；Microsoft Research India（微软印度研究院）
We provide the first global model recovery results for the IRLS (iteratively reweighted least squares) heuristic for robust regression problems. IRLS is known to offer excellent performance, despite bad initializations and data corruption, for several parameter estimation problems. Existing analyses of IRLS frequently require careful initialization, thus offering only local convergence guarantees. We remedy this by proposing augmentations to the basic IRLS routine that not only offer guaranteed global recovery, but in practice also outperform state-of-the-art algorithms for robust regression. …
kNN estimation in semi-functional partial linear regression with missing responses at random
Universidade da Coruña（拉科鲁尼亚大学）；Centro de Investigación en Tecnologías de la Información y las Comunicaciones (CITIC)（信息与通信技术研究中心）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
