Regularized Principal Component Analysis for Spatial Data

In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at $p$ locations and $n$ time points with the possibility that $p>n$. While principal component analysis (PCA) is commonly applied to find the dominant patterns, the eigenimages produced from PCA may exhibit patterns that are too noisy to be physically meaningful when $p$ is large relative to $n$. To obtain more precise estimates of eigenimages, we propose a regularization approach incorporating smoothness and sparseness of eigenimages, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial random-effects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical example

Learning an L2 and L3 at the same time: Help or hinder?

The aim of this doctoral study is to investigate whether learning a third language (L3) alongside with a second language (L2) helps or hinders the L2 development. Assuming a Complex Dynamic Systems Theory perspective, the current study views language development as a complex dynamic system within which different subsystems are interconnected and in constant change. Such subsystems can be the linguistic subsystems, such as the learners’ L2 and L3, but also the language learning resources, such as the learners’ individual difference (ID) factors. The current study traced 160 participants for 9 months. Pre-test and post-test on the ID factors and L2 speaking proficiency were administrated. L2 writing texts were collected on a tri-weekly basis. The results show that overall, the L2+L3 learners developed their L2 writing to the same degree as the L2 learners did, but the beginning L2+L3 learners showed higher variability in writing fluency. In addition, the beginning L2+L3 learners started with a higher English motivation and gained more working memory than L2 learners during the observation. Moreover, the variability in L2 writing scores was the best and only predictor for L2 writing proficiency when compared to the ID factors at the pre-test. In sum, the current study suggested that 1) the addition of an L3 does not hinder L2 writing development within the 9 months of observation; 2) ID factors are constantly changing and may be trained by language learning; 3) the degree of variability indicates the dynamics in L2 writing development and predicts L2 writing gains