TY - JOUR

T1 - An Introduction to Kristof’s Theorem for Solving Least-Square Optimization Problems Without Calculus

AU - Waller, Niels

PY - 2018/3/4

Y1 - 2018/3/4

N2 - Kristof’s Theorem (Kristof, 1970) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof’s Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof’s (1964, 1970) writings. In this article, I describe the underlying logic of Kristof’s Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem’s proof. I then show how Kristof’s Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017) code to perform the calculations described in the text.

AB - Kristof’s Theorem (Kristof, 1970) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof’s Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof’s (1964, 1970) writings. In this article, I describe the underlying logic of Kristof’s Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem’s proof. I then show how Kristof’s Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017) code to perform the calculations described in the text.

KW - Optimization

KW - inequalities

KW - least squares

KW - multivariate statistics

UR - http://www.scopus.com/inward/record.url?scp=85041132073&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041132073&partnerID=8YFLogxK

U2 - 10.1080/00273171.2017.1412294

DO - 10.1080/00273171.2017.1412294

M3 - Article

C2 - 29323539

AN - SCOPUS:85041132073

VL - 53

SP - 190

EP - 198

JO - Multivariate Behavioral Research

JF - Multivariate Behavioral Research

SN - 0027-3171

IS - 2

ER -