Fang Han

JMLR Journal 2015 Journal Article

A Direct Estimation of High Dimensional Stationary Vector Autoregressions

• Fang Han
• Huanran Lu
• Han Liu

The vector autoregressive (VAR) model is a powerful tool in learning complex time series and has been exploited in many fields. The VAR model poses some unique challenges to researchers: On one hand, the dimensionality, introduced by incorporating multiple numbers of time series and adding the order of the vector autoregression, is usually much higher than the time series length; On the other hand, the temporal dependence structure naturally present in the VAR model gives rise to extra difficulties in data analysis. The regular way in cracking the VAR model is via "least squares" and usually involves adding different penalty terms (e.g., ridge or lasso penalty) in handling high dimensionality. In this manuscript, we propose an alternative way in estimating the VAR model. The main idea is, via exploiting the temporal dependence structure, formulating the estimating problem to a linear program. There is instant advantage of the proposed approach over the lasso-type estimators: The estimation equation can be decomposed to multiple sub-equations and accordingly can be solved efficiently using parallel computing. Besides that, we also bring 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, 2015) are based on stringent assumptions that are not transparent. Our results, on the other hand, show that the spectral norms of the transition matrices play an important role in estimation accuracy and build estimation and prediction consistency accordingly. Moreover, we provide some experiments on both synthetic and real-world equity data. We show that there are empirical advantages of our method over the lasso-type estimators in parameter estimation and forecasting. [abs] [ pdf ][ bib ] &copy JMLR 2015. ( edit, beta )

ICML Conference 2015 Conference Paper

Robust Estimation of Transition Matrices in High Dimensional Heavy-tailed Vector Autoregressive Processes

• Huitong Qiu
• Sheng Xu
• Fang Han
• Han Liu 0001
• Brian Caffo

Gaussian vector autoregressive (VAR) processes have been extensively studied in the literature. However, Gaussian assumptions are stringent for heavy-tailed time series that frequently arises in finance and economics. In this paper, we develop a unified framework for modeling and estimating heavy-tailed VAR processes. In particular, we generalize the Gaussian VAR model by an elliptical VAR model that naturally accommodates heavy-tailed time series. Under this model, we develop a quantile-based robust estimator for the transition matrix of the VAR process. We show that the proposed estimator achieves parametric rates of convergence in high dimensions. This is the first work in analyzing heavy-tailed high dimensional VAR processes. As an application of the proposed framework, we investigate Granger causality in the elliptical VAR process, and show that the robust transition matrix estimator induces sign-consistent estimators of Granger causality. The empirical performance of the proposed methodology is demonstrated by both synthetic and real data. We show that the proposed estimator is robust to heavy tails, and exhibit superior performance in stock price prediction.

NeurIPS Conference 2015 Conference Paper

Robust Portfolio Optimization

• Huitong Qiu
• Fang Han
• Han Liu
• Brian Caffo

We propose a robust portfolio optimization approach based on quantile statistics. The proposed method is robust to extreme events in asset returns, and accommodates large portfolios under limited historical data. Specifically, we show that the risk of the estimated portfolio converges to the oracle optimal risk with parametric rate under weakly dependent asset returns. The theory does not rely on higher order moment assumptions, thus allowing for heavy-tailed asset returns. Moreover, the rate of convergence quantifies that the size of the portfolio under management is allowed to scale exponentially with the sample size of the historical data. The empirical effectiveness of the proposed method is demonstrated under both synthetic and real stock data. Our work extends existing ones by achieving robustness in high dimensions, and by allowing serial dependence.

JMLR Journal 2013 Journal Article

CODA: High Dimensional Copula Discriminant Analysis

• Fang Han
• Tuo Zhao
• Han Liu

We propose a high dimensional classification method, named the Copula Discriminant Analysis (CODA). The CODA generalizes the normal-based linear discriminant analysis to the larger Gaussian Copula models (or the nonparanormal) as proposed by Liu et al. (2009). To simultaneously achieve estimation efficiency and robustness, the nonparametric rank-based methods including the Spearman's rho and Kendall's tau are exploited in estimating the covariance matrix. In high dimensional settings, we prove that the sparsity pattern of the discriminant features can be consistently recovered with the parametric rate, and the expected misclassification error is consistent to the Bayes risk. Our theory is backed up by careful numerical experiments, which show that the extra flexibility gained by the CODA method incurs little efficiency loss even when the data are truly Gaussian. These results suggest that the CODA method can be an alternative choice besides the normal-based high dimensional linear discriminant analysis. [abs] [ pdf ][ bib ] &copy JMLR 2013. ( edit, beta )

ICML Conference 2013 Conference Paper

Principal Component Analysis on non-Gaussian Dependent Data

• Fang Han
• Han Liu 0001

In this paper, we analyze the performance of a semiparametric principal component analysis named Copula Component Analysis (COCA) (Han & Liu, 2012) when the data are dependent. The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. We study the scenario where the observations are drawn from non-i. i. d. processes ($m$-dependency or a more general $\phi$-mixing case). We show that COCA can allow weak dependence. In particular, we provide the generalization bounds of convergence for both support recovery and parameter estimation of COCA for the dependent data. We provide explicit sufficient conditions on the degree of dependence, under which the parametric rate can be maintained. To our knowledge, this is the first work analyzing the theoretical performance of PCA for the dependent data in high dimensional settings. Our results strictly generalize the analysis in Han & Liu (2012) and the techniques we used have the separate interest for analyzing a variety of other multivariate statistical methods.

NeurIPS Conference 2013 Conference Paper

Robust Sparse Principal Component Regression under the High Dimensional Elliptical Model

• Fang Han
• Han Liu

In this paper we focus on the principal component regression and its application to high dimension non-Gaussian data. The major contributions are in two folds. First, in low dimensions and under a double asymptotic framework where both the dimension $d$ and sample size $n$ can increase, by borrowing the strength from recent development in minimax optimal principal component estimation, we first time sharply characterize the potential advantage of classical principal component regression over least square estimation under the Gaussian model. Secondly, we propose and analyze a new robust sparse principal component regression on high dimensional elliptically distributed data. The elliptical distribution is a semiparametric generalization of the Gaussian, including many well known distributions such as multivariate Gaussian, rank-deficient Gaussian, $t$, Cauchy, and logistic. It allows the random vector to be heavy tailed and have tail dependence. These extra flexibilities make it very suitable for modeling finance and biomedical imaging data. Under the elliptical model, we prove that our method can estimate the regression coefficients in the optimal parametric rate and therefore is a good alternative to the Gaussian based methods. Experiments on synthetic and real world data are conducted to illustrate the empirical usefulness of the proposed method.

ICML Conference 2013 Conference Paper

Transition Matrix Estimation in High Dimensional Time Series

• Fang Han
• Han Liu 0001

In this paper, we propose a new method in estimating transition matrices of high dimensional vector autoregressive (VAR) models. Here the data are assumed to come from a stationary Gaussian VAR time series. By formulating the problem as a linear program, we provide a new approach to conduct inference on such models. In theory, under a doubly asymptotic framework in which both the sample size T and dimensionality d of the time series can increase, we provide explicit rates of convergence between the estimator and the population transition matrix under different matrix norms. Our results show that the spectral norm of the transition matrix plays a pivotal role in determining the final rates of convergence. This is the first work analyzing the estimation of transition matrices under a high dimensional doubly asymptotic framework. Experiments are conducted on both synthetic and real-world stock data to demonstrate the effectiveness of the proposed method compared with the existing methods. The results of this paper have broad impact on different applications, including finance, genomics, and brain imaging.

ICML Conference 2012 Conference Paper

High Dimensional Semiparametric Gaussian Copula Graphical Models

• Han Liu 0001
• Fang Han
• Ming Yuan 0001
• John D. Lafferty
• Larry A. Wasserman

NeurIPS Conference 2012 Conference Paper

Semiparametric Principal Component Analysis

• Fang Han
• Han Liu

We propose two new principal component analysis methods in this paper utilizing a semiparametric model. The according methods are named Copula Component Analysis (COCA) and Copula PCA. The semiparametric model assumes that, af- ter unspecified marginally monotone transformations, the distributions are multi- variate Gaussian. The COCA and Copula PCA accordingly estimate the leading eigenvectors of the correlation and covariance matrices of the latent Gaussian dis- tribution. The robust nonparametric rank-based correlation coefficient estimator, Spearman’s rho, is exploited in estimation. We prove that, under suitable condi- tions, although the marginal distributions can be arbitrarily continuous, the COCA and Copula PCA estimators obtain fast estimation rates and are feature selection consistent in the setting where the dimension is nearly exponentially large relative to the sample size. Careful numerical experiments on the synthetic and real data are conducted to back up the theoretical results. We also discuss the relationship with the transelliptical component analysis proposed by Han and Liu (2012).

NeurIPS Conference 2012 Conference Paper

Transelliptical Component Analysis

• Fang Han
• Han Liu

We propose a high dimensional semiparametric scale-invariant principle compo- nent analysis, named TCA, by utilize the natural connection between the ellipti- cal distribution family and the principal component analysis. Elliptical distribu- tion family includes many well-known multivariate distributions like multivari- ate Gaussian, t and logistic and it is extended to the meta-elliptical by Fang et. al (2002) using the copula techniques. In this paper we extend the meta-elliptical distribution family to a even larger family, called transelliptical. We prove that TCA can obtain a near-optimal splog d/n estimation consistency rate in recover- ing the leading eigenvector of the latent generalized correlation matrix under the transelliptical distribution family, even if the distributions are very heavy-tailed, have infinite second moments, do not have densities and possess arbitrarily con- tinuous marginal distributions. A feature selection result with explicit rate is also provided. TCA is further implemented in both numerical simulations and large- scale stock data to illustrate its empirical usefulness. Both theories and experi- ments confirm that TCA can achieve model flexibility, estimation accuracy and robustness at almost no cost.

NeurIPS Conference 2012 Conference Paper

Transelliptical Graphical Models

• Han Liu
• Fang Han
• Cun-Hui Zhang

We advocate the use of a new distribution family—the transelliptical—for robust inference of high dimensional graphical models. The transelliptical family is an extension of the nonparanormal family proposed by Liu et al. (2009). Just as the nonparanormal extends the normal by transforming the variables using univariate functions, the transelliptical extends the elliptical family in the same way. We propose a nonparametric rank-based regularization estimator which achieves the parametric rates of convergence for both graph recovery and parameter estima- tion. Such a result suggests that the extra robustness and flexibility obtained by the semiparametric transelliptical modeling incurs almost no efficiency loss. We also discuss the relationship between this work with the transelliptical component analysis proposed by Han and Liu (2012).