We consider prediction in multiple studies with potential differences in the relationships between predictors and outcomes. Our objective is to integrate data from multiple studies to develop prediction models for unseen studies. We propose and investigate two cross-validation approaches applicable to multi-study stacking, an ensemble method that linearly combines study-specific ensemble members to produce generalizable predictions. Among our cross-validation approaches are some that avoid reuse of the same data in both the training and stacking steps, as done in earlier multi-study stacking. We prove that under mild regularity conditions the proposed cross-validation approaches produce stacked prediction functions with oracle properties. We also identify analytically in which scenarios the proposed cross-validation approaches increase prediction accuracy compared to stacking with data reuse. We perform a simulation study to illustrate these results. Finally, we apply our method to predicting mortality from long-term exposure to air pollutants, using collections of datasets.
Determining the number of factors is essential to factor analysis. In this paper, we propose {an efficient cross validation (CV)} method to determine the number of factors in approximate factor models. The method applies CV twice, first along the directions of observations and then variables, and hence is referred to hereafter as double cross-validation (DCV). Unlike most CV methods, which are prone to overfitting, the DCV is statistically consistent in determining the number of factors when both dimension of variables and sample size are sufficiently large. Simulation studies show that DCV ha…
This article is an invited discussion of the article by Gronau and Wagenmakers (2018) that can be found at https://dx.doi.org/10.1007/s42113-018-0011-7.
We extend the standard Bayesian multivariate Gaussian generative data classifier by considering a generalization of the conjugate, normal-Wishart prior distribution and by deriving the hyperparameters analytically via evidence maximization. The behaviour of the optimal hyperparameters is explored in the high-dimensional data regime. The classification accuracy of the resulting generalized model is competitive with state-of-the art Bayesian discriminant analysis methods, but without the usual computational burden of cross-validation.
This paper develops detailed mathematical statistical theory of a new class of cross-validation techniques of local linear kernel hazards and their multiplicative bias corrections. The new class of cross-validation combines principles of local information and recent advances in indirect cross-validation. A few applications of cross-validating multiplicative kernel hazard estimation do exist in the literature. However, detailed mathematical statistical theory and small sample performance are introduced via this paper and further upgraded to our new class of best one-sided cross-validation. Best…
Cross-validation is a well-known and widely used bandwidth selection method in nonparametric regression estimation. However, this technique has two remarkable drawbacks: (i) the large variability of the selected bandwidths, and (ii) the inability to provide results in a reasonable time for very large sample sizes. To overcome these problems, bagging cross-validation bandwidths are analyzed in this paper. This approach consists in computing the cross-validation bandwidths for a finite number of subsamples and then rescaling the averaged smoothing parameters to the original sample size. Under a …
Cross-validation is a popular non-parametric method for evaluating the accuracy of a predictive rule. The usefulness of cross-validation depends on the task we want to employ it for. In this note, I discuss a simple non-parametric setting, and find that cross-validation is asymptotically uninformative about the expected test error of any given predictive rule, but allows for asymptotically consistent model selection. The reason for this phenomenon is that the leading-order error term of cross-validation doesn't depend on the model being evaluated, and so cancels out when we compare two models.
This note corrects a mistake in the paper "consistent cross-validatory model-selection for dependent data: $hv$-block cross-validation" by Racine (2000). In his paper, he implied that the therein proposed $hv$-block cross-validation is consistent in the sense of Shao (1993). To get this intuition, he relied on the speculation that $hv$-block is a balanced incomplete block design (BIBD). This note demonstrates that this is not the case, and thus the theoretical consistency of $hv$-block remains an open question. In addition, I also provide a Python program counting the number of occurrences of …
It is shown that the log-likelihood of a hypothesis or model given some data is equivalent to an average of all leave-one-out cross-validation log-scores that can be calculated from all subsets of the data. This relation can be generalized to any $k$-fold cross-validation log-scores.
Cross validation residuals are well known for the ordinary least squares model. Here leave-M-out cross validation is extended to generalised least squares. The relationship between cross validation residuals and Cook's distance is demonstrated, in terms of an approximation to the difference in the generalised residual sum of squares for a model fit to all the data (training and test) and a model fit to a reduced dataset (training data only). For generalised least squares, as for ordinary least squares, there is no need to refit the model to reduced size datasets as all the values for K fold cr…
Cross-validation is one of the most popular model selection methods in statistics and machine learning. Despite its wide applicability, traditional cross validation methods tend to select overfitting models, due to the ignorance of the uncertainty in the testing sample. We develop a new, statistically principled inference tool based on cross-validation that takes into account the uncertainty in the testing sample. This new method outputs a set of highly competitive candidate models containing the best one with guaranteed probability. As a consequence, our method can achieve consistent variable…
One of the common goals of time series analysis is to use the observed series to inform predictions for future observations. In the absence of any actual new data to predict, cross-validation can be used to estimate a model's future predictive accuracy, for instance, for the purpose of model comparison or selection. Exact cross-validation for Bayesian models is often computationally expensive, but approximate cross-validation methods have been developed, most notably methods for leave-one-out cross-validation (LOO-CV). If the actual prediction task is to predict the future given the past, LOO-…
While many statistical models and methods are now available for network analysis, resampling network data remains a challenging problem. Cross-validation is a useful general tool for model selection and parameter tuning, but is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure. Here we propose a new network resampling strategy based on splitting node pairs rather than nodes applicable to cross-validation for a wide range of network model selection tasks. We provide a theoretical justification for our…
Cross-validation can be used to measure a model's predictive accuracy for the purpose of model comparison, averaging, or selection. Standard leave-one-out cross-validation (LOO-CV) requires that the observation model can be factorized into simple terms, but a lot of important models in temporal and spatial statistics do not have this property or are inefficient or unstable when forced into a factorized form. We derive how to efficiently compute and validate both exact and approximate LOO-CV for any Bayesian non-factorized model with a multivariate normal or Student-t distribution on the outcom…
In Bayesian statistics, the marginal likelihood, also known as the evidence, is used to evaluate model fit as it quantifies the joint probability of the data under the prior. In contrast, non-Bayesian models are typically compared using cross-validation on held-out data, either through $k$-fold partitioning or leave-$p$-out subsampling. We show that the marginal likelihood is formally equivalent to exhaustive leave-$p$-out cross-validation averaged over all values of $p$ and all held-out test sets when using the log posterior predictive probability as the scoring rule. Moreover, the log poster…
K-fold cross-validation (CV) with squared error loss is widely used for evaluating predictive models, especially when strong distributional assumptions cannot be taken. However, CV with squared error loss is not free from distributional assumptions, in particular in cases involving non-i.i.d. data. This paper analyzes CV for correlated data. We present a criterion for suitability of standard CV in presence of correlations. When this criterion does not hold, we introduce a bias corrected cross-validation estimator which we term $CV_c,$ that yields an unbiased estimate of prediction error in man…
Cross-validation (CV) methods are popular for selecting the tuning parameter in the high-dimensional variable selection problem. We show the mis-alignment of the CV is one possible reason of its over-selection behavior. To fix this issue, we propose a version of leave-$n_v$-out cross-validation (CV($n_v$)), for selecting the optimal model among the restricted candidate model set for high-dimensional generalized linear models. By using the same candidate model sequence and a proper order of construction sample size $n_c$ in each CV split, CV($n_v$) avoids the potential hurdles in developing the…
Efficient Cross-Validation for Sparse Linear Regression
Department of Analytics, Marketing and Operations, Imperial Business School, London, UK（分析、营销与运营系，帝国商务学院，伦敦，英国）；Department of Industrial and Systems Engineering, Viterbi School of Engineering, University of Southern California, CA（工业与系统工程系，维特比工程学院，美国南加州大学，CA）
The adaptive lasso refers to a class of methods that use weighted versions of the $L_1$-norm penalty, with weights derived from an initial estimate of the parameter vector to be estimated. Irrespective of the method chosen to compute this initial estimate, the performance of the adaptive lasso critically depends on the value of a hyperparameter, which controls the magnitude of the weighted $L_1$-norm in the penalized criterion. As for other machine learning methods, cross-validation is very popular for the calibration of the adaptive lasso, that this for the selection of a data-driven optimal …
Recently, new methods for model assessment, based on subsampling and posterior approximations, have been proposed for scaling leave-one-out cross-validation (LOO) to large datasets. Although these methods work well for estimating predictive performance for individual models, they are less powerful in model comparison. We propose an efficient method for estimating differences in predictive performance by combining fast approximate LOO surrogates with exact LOO subsampling using the difference estimator and supply proofs with regards to scaling characteristics. The resulting approach can be orde…
We develop an approximation formula for the cross-validation error (CVE) of a sparse linear regression penalized by $\ell_1$-norm and total variation terms, which is based on a perturbative expansion utilizing the largeness of both the data dimensionality and the model. The developed formula allows us to reduce the necessary computational cost of the CVE evaluation significantly. The practicality of the formula is tested through application to simulated black-hole image reconstruction on the event-horizon scale with super resolution. The results demonstrate that our approximation reproduces th…
Posterior distributions often feature intractable normalizing constants, called marginal likelihoods or evidence, that are useful for model comparison via Bayes factors. This has motivated a number of methods for estimating ratios of normalizing constants in statistics. In computational physics the logarithm of these ratios correspond to free energy differences. Combining unbiased Markov chain Monte Carlo estimators with path sampling, also called thermodynamic integration, we propose new unbiased estimators of the logarithm of ratios of normalizing constants. As a by-product, we propose unbia…
