Matrix differential calculus with applications to simple, hadamard, and kronecker products

A number of indices exist to calculate lifespan variation, each with different underlying properties. Here we present new formulae for the response of seven of these indices to changes in the underlying mortality schedule (life disparity, the Gini coefficient, the standard deviation, the variance, Theil's index, the mean logarithmic deviation, and the inter-quartile range). We derive each of these indices from an absorbing Markov chain formulation of the life table, and use matrix calculus to obtain the sensitivity and the elasticity (i.e., the proportional sensitivity) to changes in age-specific mortality. Using empirical French and Russian male data we compare the underlying sensitivities to mortality change under different mortality regimes in order to test under which conditions the indices might differ in their conclusions about the magnitude of lifespan variation. Finally, we demonstrate how the sensitivities can be used to decompose temporal changes in the indices into contributions of age-specific mortality changes. The result is an easily computable method for calculating the properties of this important class of longevity indices.

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“…then according to the "first identification theorem" of Magnus and Neudecker (Magnus and Neudecker 1985) …”

Section: Matrix Calculus Preliminariesmentioning

confidence: 99%

Perturbation Analysis of Indices of Lifespan Variability

2013

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“…Magnus and Neudecker (1985) proposed to use the following quantity that is thoroughly consistent with all the properties of the classical differentiation frameworks (in particular with the chain rule, the notion of the Jacobian, and Cauchy's rule of invariance) [52]:…”

Section: Appendix Matrix Differentiation Tools For Lyapunov's Indirecmentioning

confidence: 99%

On Weight-Prioritized Multitask Control of Humanoid Robots

Bouyarmane

Kheddar

2018

IEEE Trans. Automat. Contr.

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Abstract-We propose a formal analysis with some theoretical properties of weight-prioritized multi-task inverse-dynamics-like control of humanoid robots, being a case of redundant "manipulators" with a non-actuated free-floating base and multiple unilateral frictional contacts with the environment. The controller builds on a weighted sum scalarization of a multiobjective optimization problem under equality and inequality constraints, which appears as a straightforward solution to account for state and control input viability constraints characteristic of humanoid robots that were usually absent from early existing pseudo-inverse and null-space projection-based prioritized multitask approaches. We argue that our formulation is indeed well founded and justified from a theoretical standpoint, and we propose an analysis of some stability properties of the approach: Lyapunov stability is demonstrated for the closed-loop dynamical system that we analytically derive in the unconstrained multiobjective optimization case. Stability in terms of solution existence, uniqueness, continuity, and robustness to perturbations, is then formally demonstrated for the constrained quadratic program.

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“…These definitions (unlike some alternatives; see Magnus and Neudecker 1985) lead to the familiar chain rule. If Y is a function of X and X is a function of Z, then…”

Section: Matrix Calculusmentioning

confidence: 99%