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### Mass-Univariate Hypothesis Testing on MEEG Data using Cross-Validation

Recent advances in statistical theory, together with advances in the computational power of computers, provide alternative methods to do mass-univariate hypothesis testing in which a large number of univariate tests, can be properly used to compare MEEG data at a large number of time-frequency points and scalp locations. One of the major problematic aspects of this kind of mass-univariate analysis is due to high number of accomplished hypothesis tests. Hence procedures that remove or alleviate the increased probability of false discoveries are crucial for this type of analysis. Here, I propose a new method for mass-univariate analysis of MEEG data based on cross-validation scheme. In this method, I suggest a hierarchical classification procedure under k-fold cross-validation to detect which sensors at which time-bin and which frequency-bin contributes in discriminating between two different stimuli or tasks. To achieve this goal, a new feature extraction method based on the discrete cosine transform (DCT) employed to get maximum advantage of all three data dimensions. Employing cross-validation and hierarchy architecture alongside the DCT feature space makes this method more reliable and at the same time enough sensitive to detect the narrow effects in brain activities.

### Cross validation in sparse linear regression with piecewise continuous nonconvex penalties and its acceleration

We investigate the signal reconstruction performance of sparse linear regression in the presence of noise when piecewise continuous nonconvex penalties are used. Among such penalties, we focus on the smoothly clipped absolute deviation (SCAD) penalty. The contributions of this study are three-fold: We first present a theoretical analysis of a typical reconstruction performance, using the replica method, under the assumption that each component of the design matrix is given as an independent and identically distributed (i.i.d.) Gaussian variable. This clarifies the superiority of the SCAD estimator compared with $\ell_1$ in a wide parameter range, although the nonconvex nature of the penalty tends to lead to solution multiplicity in certain regions. This multiplicity is shown to be connected to replica symmetry breaking in the spin-glass theory, and associated phase diagrams are given. We also show that the global minimum of the mean square error between the estimator and the true signal is located in the replica symmetric phase. Second, we develop an approximate formula efficiently computing the cross-validation error without actually conducting the cross-validation, which is also applicable to the non-i.i.d. design matrices. It is shown that this formula is only applicable to the unique solution region and tends to be unstable in the multiple solution region. We implement instability detection procedures, which allows the approximate formula to stand alone and resultantly enables us to draw phase diagrams for any specific dataset. Third, we propose an annealing procedure, called nonconvexity annealing, to obtain the solution path efficiently. Numerical simulations are conducted on simulated datasets to examine these results to verify the consistency of the theoretical results and the efficiency of the approximate formula and nonconvexity annealing.

### A Leisurely Look at Versions and Variants of the Cross Validation Estimator

Many versions of cross-validation (CV) exist in the literature; and each version though has different variants. All are used interchangeably by many practitioners; yet, without explanation to the connection or difference among them. This article has three contributions. First, it starts by mathematical formalization of these different versions and variants that estimate the error rate and the Area Under the ROC Curve (AUC) of a classification rule, to show the connection and difference among them. Second, we prove some of their properties and prove that many variants are either redundant or "not smooth". Hence, we suggest to abandon all redundant versions and variants and only keep the leave-one-out, the $K$-fold, and the repeated $K$-fold. We show that the latter is the only among the three versions that is "smooth" and hence looks mathematically like estimating the mean performance of the classification rules. However, empirically, for the known phenomenon of "weak correlation", which we explain mathematically and experimentally, it estimates both conditional and mean performance almost with the same accuracy. Third, we conclude the article with suggesting two research points that may answer the remaining question of whether we can come up with a finalist among the three estimators: (1) a comparative study, that is much more comprehensive than those available in literature and conclude no overall winner, is needed to consider a wide range of distributions, datasets, and classifiers including complex ones obtained via the recent deep learning approach. (2) we sketch the path of deriving a rigorous method for estimating the variance of the only "smooth" version, repeated $K$-fold CV, rather than those ad-hoc methods available in the literature that ignore the covariance structure among the folds of CV.

### Network cross-validation by edge sampling

Many models and methods are now available for network analysis, but model selection and tuning remain challenging. Cross-validation is a useful general tool for these tasks in many settings, 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 cross-validation strategy based on splitting edges rather than nodes, which avoids losing information and is applicable to a wide range of network problems. We provide a theoretical justification for our method in a general setting, and in particular show that the method has good asymptotic properties under the stochastic block model. Numerical results on simulated networks show that our approach performs well for a number of model selection and parameter tuning tasks. We also analyze a citation network of statisticians, with meaningful research communities emerging from the analysis.

### Futility Analysis in the Cross-Validation of Machine Learning Models

Many machine learning models have important structural tuning parameters that cannot be directly estimated from the data. The common tactic for setting these parameters is to use resampling methods, such as cross--validation or the bootstrap, to evaluate a candidate set of values and choose the best based on some pre--defined criterion. Unfortunately, this process can be time consuming. However, the model tuning process can be streamlined by adaptively resampling candidate values so that settings that are clearly sub-optimal can be discarded. The notion of futility analysis is introduced in this context. An example is shown that illustrates how adaptive resampling can be used to reduce training time. Simulation studies are used to understand how the potential speed--up is affected by parallel processing techniques.