In the AGEMAP genomics study, researchers were interested in detecting genes related to age in a variety of tissue types. After not finding many age-related genes in some of the analyzed tissue types, the study was criticized for having low power. It is possible that the low power is due to the presence of important unmeasured variables, and indeed we find that a latent factor model appears to explain substantial variability not captured by measured covariates. We propose including the estimated latent factors in a multiple regression model. The key difficulty in doing so is assigning appropriate degrees of freedom to the estimated factors to obtain unbiased error variance estimators and enable valid hypothesis testing. When the number of responses is large relative to the sample size, treating the estimated factors like observed covariates leads to a downward bias in the variance estimates. Many ad-hoc solutions to this problem have been proposed in the literature without the backup of a careful theoretical analysis. Using recent results from random matrix theory, we derive a simple, easy to use expression for degrees of freedom. Our estimate gives a principled alternative to ad-hoc approaches in common use. Extensive simulation results show excellent agreement between the proposed estimator and its theoretical value. Applying our methodology to the AGEMAP genomics study, we found an order of magnitude increase in the number of significant genes. Although we focus on the AGEMAP study, the methods developed in this paper are widely applicable to other multivariate models, and thus are of independent interest.
Overparametrized interpolating models have drawn increasing attention from machine learning. Some recent studies suggest that regularized interpolating models can generalize well. This phenomenon seemingly contradicts the conventional wisdom that interpolation tends to overfit the data and performs poorly on test data. Further, it appears to defy the bias-variance trade-off. As one of the shortcomings of the existing theory, the classical notion of model degrees of freedom fails to explain the intrinsic difference among the interpolating models since it focuses on estimation of in-sample predi…
Heteroscedasticity is common in real world applications and is often handled by incorporating case weights into a modeling procedure. Intuitively, models fitted with different weight schemes would have a different level of complexity depending on how well the weights match the inverse of error variances. However, existing statistical theories on model complexity, also known as model degrees of freedom, were primarily established under the assumption of equal error variances. In this work, we focus on linear regression procedures and seek to extend the existing measures to a heteroscedastic set…
In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest. The parameter function is estimated by combining ideas from f…
Conformal Prediction is a framework that produces prediction intervals based on the output from a machine learning algorithm. In this paper we explore the case when training data is made up of multiple parts available in different sources that cannot be pooled. We here consider the regression case and propose a method where a conformal predictor is trained on each data source independently, and where the prediction intervals are then combined into a single interval. We call the approach Non-Disclosed Conformal Prediction (NDCP), and we evaluate it on a regression dataset from the UCI machine l…
Data collected from wearable devices and smartphones can shed light on an individual's pattern of behavioral and circadian routine. Phone use can be modeled as alternating event process, between the state of active use and the state of being idle. Markov chains and alternating recurrent event models are commonly used to model state transitions in cases such as these, and the incorporation of random effects can be used to introduce diurnal effects. While state labels can be derived prior to modeling dynamics, this approach omits informative regression covariates that can influence state members…
The effectiveness of non-parametric, kernel-based methods for function estimation comes at the price of high computational complexity, which hinders their applicability in adaptive, model-based control. Motivated by approximation techniques based on sparse spectrum Gaussian processes, we focus on models given by regularized trigonometric linear regression. This paper provides an analysis of the performance of such an estimation set-up within the statistical learning framework. In particular, we derive a novel bound for the sample error in finite-dimensional spaces, accounting for noise with po…
The partial copula provides a method for describing the dependence between two random variables $X$ and $Y$ conditional on a third random vector $Z$ in terms of nonparametric residuals $U_1$ and $U_2$. This paper develops a nonparametric test for conditional independence by combining the partial copula with a quantile regression based method for estimating the nonparametric residuals. We consider a test statistic based on generalized correlation between $U_1$ and $U_2$ and derive its large sample properties under consistency assumptions on the quantile regression procedure. We demonstrate thro…
Compared to the conditional mean as a simple point estimator, the conditional density function is more informative to describe the distributions with multi-modality, asymmetry or heteroskedasticity. In this paper, we propose a novel parametric conditional density estimation method by showing the connection between the general density and the likelihood function of inhomogeneous Poisson process models. The maximum likelihood estimates can be obtained via weighted logistic regressions, and the computation can be significantly relaxed by combining a block-wise alternating maximization scheme and …
Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first consider the ridge regression estimator and review earlier results using classical notions from non-parametric statistics, namely degrees of freedom, also known as effective dimensionality. We then compute asymptotic equivalents of the generalization performance (in terms of squared bias and variance) of the minimum norm least-squares fit with random projectio…
Testing heteroscedasticity of the errors is a major challenge in high-dimensional regressions where the number of covariates is large compared to the sample size. Traditional procedures such as the White and the Breusch-Pagan tests typically suffer from low sizes and powers. This paper proposes two new test procedures based on standard OLS residuals. Using the theory of random Haar orthogonal matrices, the asymptotic normality of both test statistics is obtained under the null when the degree of freedom tends to infinity. This encompasses both the classical low-dimensional setting where the nu…
The statistical challenges in using big data for making valid statistical inference in the finite population have been well documented in literature. These challenges are due primarily to statistical bias arising from under-coverage in the big data source to represent the population of interest and measurement errors in the variables available in the data set. By stratifying the population into a big data stratum and a missing data stratum, we can estimate the missing data stratum by using a fully responding probability sample, and hence the population as a whole by using a data integration es…
We present a linear regression method for predictions on a small data set making use of a second possibly biased data set that may be much larger. Our method fits linear regressions to the two data sets while penalizing the difference between predictions made by those two models. The resulting algorithm is a shrinkage method similar to those used in small area estimation. We find a Stein-type finding for Gaussian responses: when the model has 5 or more coefficients and 10 or more error degrees of freedom, it becomes inadmissible to use only the small data set, no matter how large the bias is. …
We propose a novel resampling-based method to construct an asymptotically exact test for any subset of hypotheses on coefficients in high-dimensional linear regression. It can be embedded into any multiple testing procedure to make confidence statements on relevant predictor variables. The method constructs permutation test statistics for any individual hypothesis by means of repeated splits of the data and a variable selection technique; then it defines a test for any subset by suitably aggregating its variables' test statistics. The resulting procedure is extremely flexible, as it allows dif…
Policy gradient based reinforcement learning algorithms coupled with neural networks have shown success in learning complex policies in the model free continuous action space control setting. However, explicitly parameterized policies are limited by the scope of the chosen parametric probability distribution. We show that alternatively to the likelihood based policy gradient, a related objective can be optimized through advantage weighted quantile regression. Our approach models the policy implicitly in the network, which gives the agent the freedom to approximate any distribution in each acti…
We present a model for generating probabilistic forecasts by combining kernel density estimation (KDE) and quantile regression techniques, as part of the probabilistic load forecasting track of the Global Energy Forecasting Competition 2014. The KDE method is initially implemented with a time-decay parameter. We later improve this method by conditioning on the temperature or the period of the week variables to provide more accurate forecasts. Secondly, we develop a simple but effective quantile regression forecast. The novel aspects of our methodology are two-fold. First, we introduce symmetry…
TU Dortmund University（多特蒙德工业大学）；Dortmund Data Science Center（多特蒙德数据科学中心）
We study the $p$-generalized probit regression model, which is a generalized linear model for binary responses. It extends the standard probit model by replacing its link function, the standard normal cdf, by a $p$-generalized normal distribution for $p\in[1, \infty)$. The $p$-generalized normal distributions \citep{Sub23} are of special interest in statistical modeling because they fit much more flexibly to data. Their tail behavior can be controlled by choice of the parameter $p$, which influences the model's sensitivity to outliers. Special cases include the Laplace, the Gaussian, and the u…
Institute of Statistics and Data Science, Nanjing Audit University（南京审计大学统计与数据科学学院）
China has made great achievements in electric power industry during the long-term deepening of reform and opening up. However, the complex regional economic, social and natural conditions, electricity resources are not evenly distributed, which accounts for the electricity deficiency in some regions of China. It is desirable to develop a robust electricity forecasting model. Motivated by which, we propose a Panel Semiparametric Quantile Regression Neural Network (PSQRNN) by utilizing the artificial neural network and semiparametric quantile regression. The PSQRNN can explore a potential linear…
Combining matching and regression for causal inference provides double-robustness in removing treatment effect estimation bias due to confounding variables. In most real-world applications, however, treatment and control populations are not large enough for matching to achieve perfect or near-perfect balance on all confounding variables and their nonlinear/interaction functions, leading to trade-offs. [this fact is independent of regression, so a bit disjointed from first sentence.] Furthermore, variance is as important of a contributor as bias towards total error in small samples, and must th…
If error distribution has heteroscedasticity, it voliates the assumption of linear regression. Expectile regression is a powerful tool for estimating the conditional expectiles of a response variable in this setting. Since multiple levels of expectile regression modelhas been well studied, we propose composite expectile regression by combining different levels of expectile regression to improve the efficacy. In this paper, we study the sparse composite expectile regression under high dimensional setting. It is realized by implementing a coordinate descent algorithm. We also prove its selection…
Can two separate case-control studies, one about Hepatitis disease and the other about Fibrosis, for example, be combined together? It would be hugely beneficial if two or more separately conducted case-control studies, even for entirely irrelevant purposes, can be merged together with a unified analysis that produces better statistical properties, e.g., more accurate estimation of parameters. In this paper, we show that, when using the popular logistic regression model, the combined/integrative analysis produces a more accurate estimation of the slope parameters than the single case-control s…
Studies of the effects of medical interventions increasingly take place in distributed research settings using data from multiple clinical data sources including electronic health records and administrative claims. In such settings, privacy concerns typically prohibit sharing of individual patient data, and instead, analyses can only utilize summary statistics from the individual databases. In the specific but very common context of the Cox proportional hazards model, we show that standard meta analysis methods then lead to substantial bias when outcome counts are small. This bias derives prim…
Important objectives in cancer research are the prediction of a patient's risk based on molecular measurements such as gene expression data and the identification of new prognostic biomarkers (e.g. genes). In clinical practice, this is often challenging because patient cohorts are typically small and can be heterogeneous. In classical subgroup analysis, a separate prediction model is fitted using only the data of one specific cohort. However, this can lead to a loss of power when the sample size is small. Simple pooling of all cohorts, on the other hand, can lead to biased results, especially …
Multivariate regression techniques are commonly applied to explore the associations between large numbers of outcomes and predictors. In real-world applications, the outcomes are often of mixed types, including continuous measurements, binary indicators, and counts, and the observations may also be incomplete. Building upon the recent advances in mixed-outcome modeling and sparse matrix factorization, generalized co-sparse factor regression (GOFAR) is proposed, which utilizes the flexible vector generalized linear model framework and encodes the outcome dependency through a sparse singular val…
