A Direct Estimation of High Dimensional Stationary Vector Autoregressions

The vector autoregressive (VAR) model is a powerful tool in modeling complex time series and has been exploited in many fields. However, fitting high dimensional VAR model poses some unique challenges: On one hand, the dimensionality, caused by modeling a large number of time series and higher order autoregressive processes, is usually much higher than the time series length; On the other hand, the temporal dependence structure in the VAR model gives rise to extra theoretical challenges. In high dimensions, one popular approach is to assume the transition matrix is sparse and fit the VAR model using the "least squares" method with a lasso-type penalty. In this manuscript, we propose an alternative way in estimating the VAR model. The main idea is, via exploiting the temporal dependence structure, to formulate the estimating problem into a linear program. There is instant advantage for the proposed approach over the lasso-type estimators: The estimation equation can be decomposed into multiple sub-equations and accordingly can be efficiently solved in a parallel fashion. In addition, our method brings new theoretical insights into the VAR model analysis. So far the theoretical results developed in high dimensions (e.g., Song and Bickel (2011) and Kock and Callot (2012)) mainly pose assumptions on the design matrix of the formulated regression problems. Such conditions are indirect about the transition matrices and not transparent. In contrast, our results show that the operator norm of the transition matrices plays an important role in estimation accuracy. We provide explicit rates of convergence for both estimation and prediction. In addition, we provide thorough experiments on both synthetic and real-world equity data to show that there are empirical advantages of our method over the lasso-type estimators in both parameter estimation and forecasting.Comment: 36 pages, 3 figur

Characterisation of microRNAs in the heart

MicroRNAs (miRNAs) are endogenous, non-coding RNA species that regulate gene expression at the post-transcriptional level. Recent studies have shown that miRNAs are important for cardiac hypertrophy and heart failure, and are critical determinants of tissue metabolism. To investigate the role(s) of miRNAs in the insulin resistant heart, left ventricular biopsies were collected from patients with normal ventricular function with or without type 2 diabetes, and patients with left ventricular dysfunction (LVD). Using TaqMan based reverse transcriptase PCR, quantitative expression levels of 155 mature miRNAs in normal and diabetic hearts were determined. Five miRNAs were significantly upregulated in the diabetic human heart. Among these, miR-223 was upregulated in both diabetic heart and patients with LVD. Adenoviral-mediated overexpression of miR-223 increased baseline glucose uptake in cardiac myocytes in vitro with an effect size similar to that observed for insulin stimulation. This increase was associated with increase in Glut4 protein expression but independent of PI3K/Akt signalling and AMPK activity. In contrast to findings in other cells, in cardiac myocytes miR-223 did not downregulate protein levels of Mef2c or Igf1r, and an unexpected increase in NfIa protein was observed, where all three genes are miR-223 targets in immune cells. Systemic inhibition of miR-223 in vivo decreased blood glucose level 48 hours after administration and increased Glut4 protein level in the skeletal muscle, however Glut4 levels were decreased in the heart. Cardiac-specific transgenic mice overexpressing miR-223 showed no detectable changes in Glut4 protein level and cardiac insulin signalling at baseline. Collectively, these data characterise the expression of miRNAs in the human heart,demonstrate that miRNAs regulate gene targets in a cell/tissue type specific manner, they can unexpectedly increase protein expression in cardiac myocytes, and miR-223 regulates cardiac glucose metabolism through a non-canonical pathway, which may have implications for future investigations and treatment of insulin resistance

Fine gradings of complex simple Lie algebras and Finite Root Systems

A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems, we define a finite root system $R$ to be some subset of a finite symplectic abelian group satisfying certain axioms. There always corresponds to $R$ a semisimple Lie algebra $L(R)$ together with a quasi-good grading on it. Thus one can construct nice basis of $L(R)$ by means of finite root systems. We classify finite maximal abelian subgroups $T$ in \Aut(L) for complex simple Lie algebras $L$ such that the grading induced by the action of $T$ on $L$ is quasi-good, and show that the set of roots of $T$ in $L$ is always a finite root system. There are five series of such finite maximal abelian subgroups, which occur only if $L$ is a classical simple Lie algebra