New Parsimonious Multivariate Spatial Model: Spatial Envelope

Abstract

Dimension reduction provides a useful tool for analyzing high dimensional data. The recently developed Envelope method is a parsimonious version of the classical multivariate regression model through identifying a minimal reducing subspace of the responses. However, existing envelope methods assume an independent error structure in the model. While the assumption of independence is convenient, it does not address the additional complications associated with spatial or temporal correlations in the data. In this article, we introduce a Spatial Envelope method for dimension reduction in the presence of dependencies across space. We study the asymptotic properties of the proposed estimators and show that the asymptotic variance of the estimated regression coefficients under the spatial envelope model is smaller than that from the traditional maximum likelihood estimation. Furthermore, we present a computationally efficient approach for inference. The efficacy of the new approach is investigated through simulation studies and an analysis of an Air Quality Standard (AQS) dataset from the Environmental Protection Agency (EPA).

Envelopes: A new chapter in partial least squares regression

“…There are now many articles extending and refining various aspects of envelope methodology, including envelopes for quantile regression, 30 spatial modeling, 31 and matrix-variate and tensor-variate regressions. 32,33 Chemometricians may find results on tensor envelope partial least squares regression 34 to be of particular interest.…”

A Review of Envelope Models

“…In such cases, it is better to develop an envelope estimator based on the specific parametric structure of the model, which is normally more efficient than the estimator produced from the general procedure. Besides the GLM example, other examples on using the specific parametric structure to develop an envelope estimator can be found in Cook et al (2015), Rekabdarkolaee et al (2019) and Forzani and Su (2019).…”

“…Up to now, all envelope models require that observations are independent to each other. Under the multivariate spatial regression model, Rekabdarkolaee et al (2019) derived a spatial envelope model that allows for dependent observations. Wang and Ding (2018) developed the envelope models for time series data in the context of vector autoregression model.…”

Envelope methods

“…Envelope methodology for sparse regressions was developed by Su, Zhu, Chen, and Yang (), and Khare, Pal, and Su () constructed a Bayesian version of envelopes, both based on Model 1 as the starting point. L. Li and Zhang () proposed tensor envelopes for analysis of neuroimaging applications with tensor‐valued responses, Ding and Cook () extended response envelopes to regressions with matrix‐valued responses and Rekabdarkolaee, Wang, Naji, and Fluentes () reported good efficiency gains in their adaptation of envelopes to spatial data. Ding, Su, Zhu, and Wang () adapted envelopes for use in quantile regression, and Su and Cook () adapted envelopes for estimation of the means μ k of several normal population N r ( μ k , ∑ k ), k = 1,…, K , with different variance covariance matrices.…”