arXiv subjects

Sagnik Halder

Publications and source records attributed to Sagnik Halder.

3 recordsLinked to original sources

arXiv2022

Robust Estimation of Sparse, High Dimensional Time Series with Polynomial Tails

High dimensional Vector Autoregressions (VAR) have received a lot of interest recently due to novel applications in health, engineering, finance and the social sciences. Three issues arise when analyzing VAR's: (a) The high dimensional nature of the model in the presence of many time series that poses challenges for consistent estimation of its parameters; (b) the presence of temporal dependence introduces additional challenges for theoretical analysis of various estimation procedures; (b) the presence of heavy tails in a number of applications. Recent work, e.g. [Basu and Michailidis, 2015],[Kock and Callot,2015], has addressed consistent estimation of sparse high dimensional, stable Gaussian VAR models based on an $\ell_1$ LASSO procedure. Further, the rates obtained are optimal, in the sense that they match those for iid data, plus a multiplicative factor (which is the "price" paid) for temporal dependence. However, the third issue remains unaddressed in extant literature. This paper extends existing results in the following important direction: it considers consistent estimation of the parameters of sparse high dimensional VAR models driven by heavy tailed homoscedastic or heteroskedastic noise processes (that do not possess all moments). A robust penalized approach (e.g., LASSO) is adopted for which optimal consistency rates and corresponding finite sample bounds for the underlying model parameters are obtain that match those for iid data, albeit paying a price for temporal dependence. The theoretical results are illustrated on VAR models and also on other popular time series models. Notably, the key technical tool used, is a single concentration bound for heavy tailed dependent processes.

math.ST↗

arXiv2022

Optimal Sparse Estimation of High Dimensional Heavy-tailed Time Series

Recently, high dimensional vector auto-regressive models (VAR), have attracted a lot of interest, due to novel applications in the health, engineering and social sciences. The presence of temporal dependence poses additional challenges to the theory of penalized estimation techniques widely used in the analysis of their iid counterparts. However, recent work (e.g., [Basu and Michailidis, 2015, Kock and Callot, 2015]) has established optimal consistency of $\ell_1$-LASSO regularized estimates applied to models involving high dimensional stable, Gaussian processes. The only price paid for temporal dependence is an extra multiplicative factor that equals 1 for independent and identically distributed (iid) data. Further, [Wong et al., 2020] extended these results to heavy tailed VARs that exhibit "$\beta$-mixing" dependence, but the rates rates are sub-optimal, while the extra factor is intractable. This paper improves these results in two important directions: (i) We establish optimal consistency rates and corresponding finite sample bounds for the underlying model parameters that match those for iid data, modulo a price for temporal dependence, that is easy to interpret and equals 1 for iid data. (ii) We incorporate more general penalties in estimation (which are not decomposable unlike the $\ell_1$ norm) to induce general sparsity patterns. The key technical tool employed is a novel, easy-to-use concentration bound for heavy tailed linear processes, that do not rely on "mixing" notions and give tighter bounds.

math.ST↗

arXiv2021

High Dimensional Logistic Regression Under Network Dependence

Logistic regression is key method for modeling the probability of a binary outcome based on a collection of covariates. However, the classical formulation of logistic regression relies on the independent sampling assumption, which is often violated when the outcomes interact through an underlying network structure, such as over a temporal/spatial domain or on a social network. This necessitates the development of models that can simultaneously handle both the network `peer-effect' and the effect of high-dimensional covariates. In this paper, we develop a framework for incorporating such dependencies in a high-dimensional logistic regression model by introducing a quadratic interaction term, as in the Ising model, designed to capture the pairwise interactions from the underlying network. The resulting model can also be viewed as an Ising model, where the node-dependent external fields linearly encode the high-dimensional covariates. We propose a penalized maximum pseudo-likelihood method for estimating the network peer-effect and the effect of the covariates (the regression coefficients), which, in addition to handling the high-dimensionality of the parameters, conveniently avoids the computational intractability of the maximum likelihood approach. Under various standard regularity conditions, we show that the corresponding estimate attains the classical high-dimensional rate of consistency. Our results imply that even under network dependence it is possible to consistently estimate the model parameters at the same rate as in classical (independent) logistic regression, when the true parameter is sparse and the underlying network is not too dense. We also develop an efficient algorithm for computing the estimates and validate our theoretical results in numerical experiments. An application to selecting genes in clustering spatial transcriptomics data is also discussed.

math.ST↗