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 a group approach to the least squares regression. Using an all positive correlations arrangement of the strongly correlated variables, we first characterize group effects that are meaningful and can be accurately estimated. We then present the group approach with numerical examples and demonstrate its advantages over existing methods for handling multicollinearity. We also address a common misconception about prediction accuracy of the least squares estimated model.
相關內容
We provide finite sample bounds on the Normal approximation to the law of the least squares estimator of the projection parameters normalized by the sandwich-based standard errors. Our results hold in the increasing dimension setting and under minimal assumptions on the data generating distribution. In particular, we do not assume a linear regression function and only require the existence of finitely many moments for the response and the covariates. Furthermore, we construct confidence sets for the projection parameters in the form of hyper-rectangles and establish finite sample bounds on their coverage and accuracy. We derive analogous results for partial correlations among the entries of sub-Gaussian vectors. \end{abstract}
We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. In this distribution-free regression setting, we show that boundedness of the conditional second moment of the response given the covariates is a necessary and sufficient condition for achieving nontrivial guarantees. As a starting point, we prove an optimal version of the classical in-expectation bound for the truncated least squares estimator due to Gy\"{o}rfi, Kohler, Krzy\.{z}ak, and Walk. However, we show that this procedure fails with constant probability for some distributions despite its optimal in-expectation performance. Then, combining the ideas of truncated least squares, median-of-means procedures, and aggregation theory, we construct a non-linear estimator achieving excess risk of order $d/n$ with an optimal sub-exponential tail. While existing approaches to linear regression for heavy-tailed distributions focus on proper estimators that return linear functions, we highlight that the improperness of our procedure is necessary for attaining nontrivial guarantees in the distribution-free setting.
Requirements driven search-based testing (also known as falsification) has proven to be a practical and effective method for discovering erroneous behaviors in Cyber-Physical Systems. Despite the constant improvements on the performance and applicability of falsification methods, they all share a common characteristic. Namely, they are best-effort methods which do not provide any guarantees on the absence of erroneous behaviors (falsifiers) when the testing budget is exhausted. The absence of finite time guarantees is a major limitation which prevents falsification methods from being utilized in certification procedures. In this paper, we address the finite-time guarantees problem by developing a new stochastic algorithm. Our proposed algorithm not only estimates (bounds) the probability that falsifying behaviors exist, but also it identifies the regions where these falsifying behaviors may occur. We demonstrate the applicability of our approach on standard benchmark functions from the optimization literature and on the F16 benchmark problem.
Doubly truncated data arise in many areas such as astronomy, econometrics, and medical studies. For the regression analysis with doubly truncated response variables, the existence of double truncation may bring bias for estimation as well as affect variable selection. We propose a simultaneous estimation and variable selection procedure for the doubly truncated regression, allowing a diverging number of regression parameters. To remove the bias introduced by the double truncation, a Mann-Whitney-type loss function is used. The adaptive LASSO penalty is then added into the loss function to achieve simultaneous estimation and variable selection. An iterative algorithm is designed to optimize the resulting objective function. We establish the consistency and the asymptotic normality of the proposed estimator. The oracle property of the proposed selection procedure is also obtained. Some simulation studies are conducted to show the finite sample performance of the proposed approach. We also apply the method to analyze a real astronomical data.
We propose a method to reduce the complexity of Generalized Linear Models in the presence of categorical predictors. The traditional one-hot encoding, where each category is represented by a dummy variable, can be wasteful, difficult to interpret, and prone to overfitting, especially when dealing with high-cardinality categorical predictors. This paper addresses these challenges by finding a reduced representation of the categorical predictors by clustering their categories. This is done through a numerical method which aims to preserve (or even, improve) accuracy, while reducing the number of coefficients to be estimated for the categorical predictors. Thanks to its design, we are able to derive a proximity measure between categories of a categorical predictor that can be easily visualized. We illustrate the performance of our approach in real-world classification and count-data datasets where we see that clustering the categorical predictors reduces complexity substantially without harming accuracy.
We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the $\ell_0$-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with $5 \times 10^6$ features and $10^3$ observations in minutes to hours -- over $1000$ times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the $\ell_0$-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes. We provide an open-source implementation of our proposed framework.
We theoretically investigate the typical learning performance of $\ell_{1}$-regularized linear regression ($\ell_1$-LinR) for Ising model selection using the replica method from statistical mechanics. For typical random regular (RR) graphs in the paramagnetic phase, an accurate estimate of the typical sample complexity of $\ell_1$-LinR is obtained, demonstrating that, for an Ising model with $N$ variables, $\ell_1$-LinR is model selection consistent with $M=\mathcal{O}\left(\log N\right)$ samples. Moreover, we provide a computationally efficient method to accurately predict the non-asymptotic behavior of $\ell_1$-LinR for moderate $M$ and $N$, such as the precision and recall rates. Simulations show a fairly good agreement between the theoretical predictions and experimental results, even for graphs with many loops, which supports our findings. Although this paper focuses on $\ell_1$-LinR, our method is readily applicable for precisely investigating the typical learning performances of a wide class of $\ell_{1}$-regularized M-estimators including $\ell_{1}$-regularized logistic regression and interaction screening.
In this paper, we are interested in nonparametric kernel estimation of a generalized regression function, including conditional cumulative distribution and conditional quantile functions, based on an incomplete sample $(X_t, Y_t, \zeta_t)_{t\in \mathbb{ R}^+}$ copies of a continuous-time stationary ergodic process $(X, Y, \zeta)$. The predictor $X$ is valued in some infinite-dimensional space, whereas the real-valued process $Y$ is observed when $\zeta= 1$ and missing whenever $\zeta = 0$. Pointwise and uniform consistency (with rates) of these estimators as well as a central limit theorem are established. Conditional bias and asymptotic quadratic error are also provided. Asymptotic and bootstrap-based confidence intervals for the generalized regression function are also discussed. A first simulation study is performed to compare the discrete-time to the continuous-time estimations. A second simulation is also conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Finally, it is worth noting that our results are stated under ergodic assumption without assuming any classical mixing conditions.
We present a new clustering method in the form of a single clustering equation that is able to directly discover groupings in the data. The main proposition is that the first neighbor of each sample is all one needs to discover large chains and finding the groups in the data. In contrast to most existing clustering algorithms our method does not require any hyper-parameters, distance thresholds and/or the need to specify the number of clusters. The proposed algorithm belongs to the family of hierarchical agglomerative methods. The technique has a very low computational overhead, is easily scalable and applicable to large practical problems. Evaluation on well known datasets from different domains ranging between 1077 and 8.1 million samples shows substantial performance gains when compared to the existing clustering techniques.
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191