For any class of binary functions on [n] = {1, . . . , n} a classical result by Sauer states a sufficient condition for its VC-dimension to be at least d: its cardinality should be at least O(n). A necessary condition is that its cardinality be at least 2 (which is O(1) with respect to n). How does the size of a ‘typical’ class of VC-dimension d compare to these two extreme thresholds ? To answ...

Background: Two main issues that challenge model building are number of Events Per Variable and multicollinearity among exploratory variables. Our aim is to review statistical methods that tackle these issues with emphasize on penalized Lasso regression model.  The present study aimed to explain problems of traditional regressions due to small sample size and m...

Modern data sets, though typically high dimensional, are often generated by processes possessing few essential degrees of freedom, as is the case with human speech. In recent years, such considerations have lead to the notion that high dimensional data may be modeled to lie on a submanifold of low intrinsic dimension. We derive bounds on the number of random samples needed before it is possible...

The goal of filtering theory is to compute the filter distribution, that is, the conditional distribution of a stochastic model given observed data. While exact computations are rarely possible, sequential Monte Carlo algorithms known as particle filters have been successfully applied to approximate the filter distribution, providing estimates whose error is uniform in time. However, the number...

Estimating causal interaction effects is essential for the exploration of heterogeneous treatment effects. In the presence of multiple treatment variables with each having several levels, researchers are often interested in identifying the combinations of treatments that induce large additional causal effects beyond the sum of separate effects attributable to each treatment. We show, however, t...

Dimension reduction is a useful tool for visualization of such high-dimensional data to make data analysis feasible for such vast volume and high-dimensional scientific data. Among the known dimension reduction algorithms, multidimensional scaling algorithm is investigated in this proposal due to its theoretical robustness and high applicability. Multidimensional scaling is known as a non-linea...