Graduate Student Seminar

Speaker 1: Neha Agarwala

Title

On the relative efficiency of meta-analysis using literature summary data vs. individual patient data

Abstract

Meta-analysis is widely used to synthesize the results of multiple studies. The main statistical goal in meta-analysis is to estimate a common effect based on summary data from independent studies. If we have access to data on all individual patients, we might consider modeling the entire response vector to obtain an alternative estimator. An interesting question is whether these two approaches yield the same estimator in general, and if not, under which condition(s) the two estimators coincide. To be precise we want to study the relative efficiency of estimators obtained from the two approaches.

In this seminar, I am presenting results from some papers on estimating the linear function of mean vector and illustrate with an examplethe intuitive implication of the result.

Speaker 2: Katelynn Huneycutt

Title

Block-Symmetric Linearizations for Even Degree Matrix Polynomials with Good Numerical Properties

Abstract

Given a matrix polynomial, a polynomial with matrix coefficients, the Polynomial Eigenvalue Problem (PEP) involves finding scalars, delta, and nonzero vectors, x, such that P(delta)x=0.  The PEP arises in various fields, including engineering, control theory, and computer-aided graphic design. The PEP is usually solved using a linearization, which is a pencil (a matrix polynomial of degree one), with the same spectral structure. It is advantageous to choose linearizations that (1) are easy to construct; (2) share any algebraic structure; and (3) behave well numerically with respect to conditioning and backward error.

In applications, symmetric (or Hermitian) matrix polynomials appear frequently. For these polynomials, a block-symmetric linearization automatically satisfies (2). It was proven that within a vector space of block-symmetric pencils the joint use of two particular pencils were found to be numerically optimal. These pencils satisfy (1) above; however, they are not linearizations of every matrix polynomial.

Recently, a block-symmetric pencil was found which is a linearization of every matrix polynomial of odd degree and outperforms the pencils noted above. It has also been proven that there is no single block-symmetric linearization of every polynomial of even degree. In our work, we present a pair of block-symmetric pencils associated with even degree polynomials which satisfy (1) and show numerically and theoretically that their combined use outperforms more commonly used pencils, thus making them a better candidate for implementation in algorithms.