Rank Restrictions Research Articles

Robust fault reconstruction using sliding mode observers with nonlinear nominal systems

This paper is concerned with fault reconstruction based on sliding mode observers; its contribution is twofold: on the one hand, it develops a novel sliding mode observer whose nominal system is nonlinear, thus relaxing former limitations on the subject, namely, rank restrictions, linear outputs, and conservative bounds; on the other hand, based on the referred sliding mode observer, a novel robust fault reconstruction scheme is constructed which can solve a variety of problems that former methodologies cannot treat. Nonlinearities are cast as convex expressions, which enables obtaining design conditions in the form of linear matrix inequalities. Examples are provided to compare the proposal with former methodologies, thus highlighting the contributions.

Small area estimation of trends in household living standards in Uganda using a GMANOVA-MANOVA model and repeated surveys

Panel survey data from Uganda, as well as data from the 2014 Uganda Population and Housing Census have been analyzed. The Growth Curve model with rank restrictions on parameters was used to estimate the small area means. The aim of the analysis was to assess change over time in household living standards (welfare), i.e., to investigate whether households display growth in living standards? whether households grow at the same rates? and whether households in different geographical areas of the country grow at the same rates? Using a GMANOVA-MANOVA model with rank restrictions on parameters, it was established that growth in household standards of living in Uganda varied across small areas. Sub-regions (small areas) with the highest standards of living in Uganda at the endline were Central Urban region, Kampala Urban region and South Western Urban region, while the sub-regions with the lowest standards of living at the endline were North East Rural region, North East Urban region and Eastern Rural region. The sub-regions with the highest growth rates in standards of living were Mid West Urban region, Mid North Rural region, and South Western Urban region. The sub-regions with the highest decline in standards of living were East Central Rural region, East Rural region and West Nile Urban region.

Matrix pencils completions under double rank restrictions

In this paper we study the General Matrix Pencil Completion Problem under double rank restrictions. Considered rank restrictions are not structural, hence the obtained results deal with full sets of Kronecker invariants of the involved matrix pencils, and they generalize many of the existing results in the literature, for example [1, 5, 7, 9, 11, 12, 21, 24, 45, 46]. Main methods consist of combining the celebrated S?-Thompson?s result [1, 39, 44] with novel results on rank restrictions in completions of matrix pencils. All of the obtained results are explicit and constructive.

Completion of matrix pencils with a single rank restriction

In this paper we consider and solve the General Matrix Pencil Completion Problem (GMPCP) under single rank restrictions. These constraints are less limiting than any other restrictions on the GMPCP studied in the past. Thus, the obtained results directly generalize the existing results in the literature on the GMPCP. Finally, as one direct corollary of the main results, we obtain a novel result on determining the possible Kronecker invariants of a quasi-regular matrix pencil with a prescribed subpencil.

Robust Stabilization and Weighted Damping of Bounded Disturbances in Descriptor Control Systems

We establish necessary and sufficient conditions for the existence of dynamic regulators guaranteeing the prescribed estimation of the weighted level of damping of bounded disturbances and the asymptotic stability of linear descriptor systems. An algorithm for the construction of these regulators in the problems of robust stabilization and generalized H∞-optimization is proposed for the descriptor systems with controlled and observed outputs. The main computational procedures of the algorithm are reduced to the solution of linear matrix inequalities with additional rank restrictions. The efficiency of the algorithm is demonstrated by an illustrative example of descriptor stabilization system with bounded disturbances.

Identification of the linear factor model

This paper provides several new results on identification of the linear factor model. The model allows for correlated latent factors and dependence among the idiosyncratic errors. I also illustrate identification under a dedicated measurement structure and other reduced rank restrictions. I use these results to study identification in a model with both observed covariates and latent factors. The analysis emphasizes the different roles played by restrictions on the error covariance matrix, restrictions on the factor loadings and the factor covariance matrix, and restrictions on the coefficients on covariates. The identification results are simple, intuitive, and directly applicable to many settings.

Bilinear regression with random effects and reduced rank restrictions

Bilinear models with three types of effects are considered: fixed effects, random effects and latent variable effects. In the literature, bilinear models with random effects and bilinear models with latent variables have been discussed but there are no results available when combining random effects and latent variables. It is shown, via appropriate vector space decompositions, how to remove the random effects so that a well-known model comprising only fixed effects and latent variables is obtained. The spaces are chosen so that the likelihood function can be factored in a convenient and interpretable way. To obtain explicit estimators, an important standardization constraint on the random effects is assumed to hold. A theorem is presented where a complete solution to the estimation problem is given.

On the Distance to Singular Descriptor Dynamical Systems With Impulsive Initial Conditions

In this paper, we study the problem of computing the distance between a given singular descriptor system $ (E,A)$ , and a nearest descriptor system that has impulsive initial conditions. The link between existence of impulsive initial conditions and zeros at infinity for the associated matrix pencil $sE - A$ is well-known. Much of the literature focusses on the case when only one of $E$ and $A$ is perturbed. We give a closed form expression of the distance to a nearest descriptor system having impulsive solutions via rank-1 perturbations when both $E$ and $A$ are perturbed. Next, for the case of perturbations without rank restrictions, we propose and evaluate the bounds for the distance. In the context of structured perturbations, we formulate and obtain an explicit expression for the distance, when $E$ and $A$ are Hermitian and are perturbed by Hermitian matrices. For a suitable class of systems, we also show that upper and lower bounds are within a factor of $\sqrt{2}$ . We finally construct examples and compare the bounds obtained from our results with those from the literature as well as with computed values of the distance obtained via three numerical optimization techniques such as the structured low rank approximation, the Broyden–Fletcher–Goldfarb–Shanno algorithm, and direct optimization tools like globalsearch .

Principal minor ideals and rank restrictions on their vanishing sets

All matrices we consider have entries in a fixed algebraically closed field K. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size t principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of n×n rank n−2 matrices whose size n−2 principal minors vanish; this set has dimension n2−n−4.

Testing for Granger causality in large mixed-frequency VARs

We analyze Granger causality testing in a mixed-frequency VAR, where the difference in sampling frequencies of the variables is large, implying parameter proliferation problems in case we attempt to estimate the model unrestrictedly. We propose several tests based on reduced rank restrictions, including bootstrap versions thereof to account for factor estimation uncertainty and improve the finite sample properties of the tests, and a Bayesian VAR extended to mixed frequencies. We compare these methods to a test based on an aggregated model, the max-test (Ghysels et al., 2016a) and an unrestricted VAR-based test (Ghysels et al., 2016b) using Monte Carlo simulations. An empirical application illustrates the techniques.

Structural analysis with Multivariate Autoregressive Index models

We address the issue of parameter dimensionality reduction in Vector Autoregressive models (VARs) for many variables by imposing specific reduced rank restrictions on the coefficient matrices that simplify the VARs into Multivariate Autoregressive Index (MAI) models. We derive the Wold representation implied by the MAIs and show that it is closely related to that associated with dynamic factor models. Then, the theoretical analysis is extended to the case of general rank restrictions on the VAR coefficients. Next, we describe classical and Bayesian estimation of large MAIs, and discuss methods for rank determination. Finally, the performance of the MAIs is compared with that of large Bayesian VARs in the context of Monte Carlo simulations and two empirical applications, on the transmission mechanism of monetary policy and on the propagation of demand and supply shocks.

Testing for Granger Causality in Large Mixed-Frequency VARs

We analyze Granger causality testing in a mixed-frequency VAR, where the difference in sampling frequencies of the variables is large. Given a realistic sample size, the number of high-frequency observations per low-frequency period leads to parameter proliferation problems in case we attempt to estimate the model unrestrictedly. We propose several tests based on reduced rank restrictions, and implement bootstrap versions to account for the uncertainty when estimating factors and to improve the finite sample properties of these tests. We also consider a Bayesian VAR that we carefully extend to the presence of mixed frequencies. We compare these methods to an aggregated model, the max-test approach introduced by Ghysels et al. (2015a) as well as to the unrestricted VAR using Monte Carlo simulations. The techniques are illustrated in an empirical application involving daily realized volatility and monthly business cycle fluctuations.

Infinite Groups with Rank Restrictions on Subgroups

Classical results by Mal’cev and Sunkov show that locally nilpotent groups and locally finite groups of infinite rank must contain some Abelian subgroups of infinite rank. In recent years, many authors have studied groups in which all subgroups of infinite rank have a given property (which can be either absolute or of embedding type). Results from these researches and some new contributions to this topic are described in this paper.

Groups Admitting an Automorphism of Prime Order with Finite Centralizer

We consider a group G with an automorphism of finite, usually prime, order. If G has finite Hirsch number, and also if G satisfies various stronger rank restrictions, we study the consequences and equivalent hypotheses of having only finitely many fixed-points. In particular we prove that if a group G with finite Hirsch number $${\mathfrak{h}}$$ admits an automorphism $${\varphi}$$ of prime order p such that $${\vert C_{G}(\varphi) \vert = n < \infty,}$$ then G has a subgroup of finite index bounded in terms of p, n and $${\mathfrak{h}}$$ that is nilpotent of p-bounded class.

Periodic subsemigroups of endomorphism monoids

If S is a periodic subsemigroup of the endomorphism monoid of a polycyclic group, then Endimioni (Mediterr J Math 8:307–313, 2011) proved that S is locally finite. Here we present an alternative proof that also extends the result to groups with suitable rank restrictions. Further we give an alternative proof of McNaughton and Zalcstein’s (J Algebra 34:292–299, 1975) theorem that periodic multiplicative subsemigroups of a matrix ring over a field are also locally finite. Finally we extend the latter to periodic subsemigroups of the endomorphism ring of a finitely generated module over a commutative ring.

Modules over group rings of solvable groups with rank restrictions on subgroups

Let A be an RG-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), CG(A) = 1, A is not a Noetherian R-module, and the quotient A/CA(H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.

On the Fixed-Point Set and Commutator Subgroup of an Authomorphism of a Group of Finite Rank

Let \phi be an automorphism of a group G . For G polycyclic, Endimioni and Moravec in [1] discuss the relationship between the fixed-point set C_G(\phi) and the commutator subgroup [G,\phi] of \phi in G . Here we extend these results to soluble groups satisfying various rank restrictions.

Joint variable and rank selection for parsimonious estimation of high-dimensional matrices

We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor selection and rank reduction are the most popular strategies for obtaining lower-dimensional approximations of the parameter matrix in such models. We show in this article that important gains in prediction accuracy can be obtained by considering them jointly. We motivate a new class of sparse multivariate regression models, in which the coefficient matrix has low rank and zero rows or can be well approximated by such a matrix. Next, we introduce estimators that are based on penalized least squares, with novel penalties that impose simultaneous row and rank restrictions on the coefficient matrix. We prove that these estimators indeed adapt to the unknown matrix sparsity and have fast rates of convergence. We support our theoretical results with an extensive simulation study and two data analyses.

On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

We study a $ \mathbb{Z}G $ -module A such that $ \mathbb{Z} $ is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C G (A) = 1, A is not a minimax $ \mathbb{Z} $ -module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/C A (H) is a minimax $ \mathbb{Z} $ -module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed.

Forecasting large datasets with Bayesian reduced rank multivariate models

The paper addresses the issue of forecasting a large set of variables using multivariate models. In particular, we propose three alternative reduced rank forecasting models and compare their predictive performance for US time series with the most promising existing alternatives, namely, factor models, large scale Bayesian VARs, and multivariate boosting. Speci…cally, we focus on classical reduced rank regression, a two-step procedure that applies, in turn, shrinkage and reduced rank restrictions, and the reduced rank Bayesian VAR of Geweke (1996). We …nd that using shrinkage and rank reduction in combination rather than separately improves substantially the accuracy of forecasts, both when the whole set of variables is to be forecast, and for key variables such as industrial production growth, in‡ation, and the federal funds rate. The robustness of this …nding is con…rmed by a Monte Carlo experiment based on bootstrapped data. We also provide a consistency result for the reduced rank regression valid when the dimension of the system tends to in…nity, which opens the