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Cross-validation (CV) is one of the most widely used techniques in statistical learning for estimating the test error of a model, but its behavior is not yet fully understood. It has been shown that standard confidence intervals for test error using estimates from CV may have coverage below nominal levels. This phenomenon occurs because each sample is used in both the training and testing procedures during CV and as a result, the CV estimates of the errors become correlated. Without accounting for this correlation, the estimate of the variance is smaller than it should be. One way to mitigate this issue is by estimating the mean squared error of the prediction error instead using nested CV. This approach has been shown to achieve superior coverage compared to intervals derived from standard CV. In this work, we generalize the nested CV idea to the Cox proportional hazards model and explore various choices of test error for this setting.

This paper formulates a general cross validation framework for signal denoising. The general framework is then applied to nonparametric regression methods such as Trend Filtering and Dyadic CART. The resulting cross validated versions are then shown to attain nearly the same rates of convergence as are known for the optimally tuned analogues. There did not exist any previous theoretical analyses of cross validated versions of Trend Filtering or Dyadic CART. To illustrate the generality of the framework we also propose and study cross validated versions of two fundamental estimators; lasso for high dimensional linear regression and singular value thresholding for matrix estimation. Our general framework is inspired by the ideas in Chatterjee and Jafarov (2015) and is potentially applicable to a wide range of estimation methods which use tuning parameters.

Cross-validation is the standard approach for tuning parameter selection in many non-parametric regression problems. However its use is less common in change-point regression, perhaps as its prediction error-based criterion may appear to permit small spurious changes and hence be less well-suited to estimation of the number and location of change-points. We show that in fact the problems of cross-validation with squared error loss are more severe and can lead to systematic under- or over-estimation of the number of change-points, and highly suboptimal estimation of the mean function in simple settings where changes are easily detectable. We propose two simple approaches to remedy these issues, the first involving the use of absolute error rather than squared error loss, and the second involving modifying the holdout sets used. For the latter, we provide conditions that permit consistent estimation of the number of change-points for a general change-point estimation procedure. We show these conditions are satisfied for optimal partitioning using new results on its performance when supplied with the incorrect number of change-points. Numerical experiments show that the absolute error approach in particular is competitive with common change-point methods using classical tuning parameter choices when error distributions are well-specified, but can substantially outperform these in misspecified models. An implementation of our methodology is available in the R package crossvalidationCP on CRAN.

A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a kernel. In particular, this strategy is highly efficient (both in terms of accuracy and complexity) when the kernel is data-adapted using Kernel Flows (KF)~\cite{Owhadi19} (which uses gradient-based optimization to learn a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used for interpolation). Despite its previous successes, this strategy (based on interpolating the vector field driving the dynamical system) breaks down when the observed time series is not regularly sampled in time. In this work, we propose to address this problem by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the (KF) data-adapted kernels. We compare our approach with the classical one over different benchmark dynamical systems and show that it significantly improves the forecasting accuracy while remaining simple, fast, and robust.

Not suitable for Time Series data: For Time Series data the order of the samples matter. But in Stratified Cross-Validation, samples are selected in random order. LeavePOut cross-validation is an exhaustive cross-validation technique, in which p-samples are used as the validation set and remaining n-p samples are used as the training set. Suppose we have 100 samples in the dataset. If we use p 10 then in each iteration 10 values will be used as a validation set and the remaining 90 samples as the training set. This process is repeated till the whole dataset gets divided on the validation set of p-samples and n-p training samples. All the data samples get used as both training and validation samples.

"Cross Validation" Anyone who has just started dabbling in the world of Data Science must have seen this term. In this article cross validation, the need for cross validation, and some types of cross validation will be explained in very simple words. Let's just start with a basic question before going into details. Well, it helps us to compare different machine learning algorithms. To evaluate how well a machine learning algorithm is working we need to train the algorithm on some part of data (known as "Training Data") and to test on unseen data (known as "Testing Data").

In many applications, we have access to the complete dataset but are only interested in the prediction of a particular region of predictor variables. A standard approach is to find the globally best modeling method from a set of candidate methods. However, it is perhaps rare in reality that one candidate method is uniformly better than the others. A natural approach for this scenario is to apply a weighted $L_2$ loss in performance assessment to reflect the region-specific interest. We propose a targeted cross-validation (TCV) to select models or procedures based on a general weighted $L_2$ loss. We show that the TCV is consistent in selecting the best performing candidate under the weighted $L_2$ loss. Experimental studies are used to demonstrate the use of TCV and its potential advantage over the global CV or the approach of using only local data for modeling a local region. Previous investigations on CV have relied on the condition that when the sample size is large enough, the ranking of two candidates stays the same. However, in many applications with the setup of changing data-generating processes or highly adaptive modeling methods, the relative performance of the methods is not static as the sample size varies. Even with a fixed data-generating process, it is possible that the ranking of two methods switches infinitely many times. In this work, we broaden the concept of the selection consistency by allowing the best candidate to switch as the sample size varies, and then establish the consistency of the TCV. This flexible framework can be applied to high-dimensional and complex machine learning scenarios where the relative performances of modeling procedures are dynamic.

Gradient Boosting Machines (GBM) are among the go-to algorithms on tabular data, which produce state of the art results in many prediction tasks. Despite its popularity, the GBM framework suffers from a fundamental flaw in its base learners. Specifically, most implementations utilize decision trees that are typically biased towards categorical variables with large cardinalities. The effect of this bias was extensively studied over the years, mostly in terms of predictive performance. In this work, we extend the scope and study the effect of biased base learners on GBM feature importance (FI) measures. We show that although these implementation demonstrate highly competitive predictive performance, they still, surprisingly, suffer from bias in FI. By utilizing cross-validated (CV) unbiased base learners, we fix this flaw at a relatively low computational cost. We demonstrate the suggested framework in a variety of synthetic and real-world setups, showing a significant improvement in all GBM FI measures while maintaining relatively the same level of prediction accuracy.

Time Series Cross-Validation Method 14. Blocked Cross-Validation Method 15.