Geometric standard deviation

Summary

In probability theory and statistics, the geometric standard deviation (GSD) describes how spread out are a set of numbers whose preferred average is the geometric mean. For such data, it may be preferred to the more usual standard deviation. Note that unlike the usual arithmetic standard deviation, the geometric standard deviation is a multiplicative factor, and thus is dimensionless, rather than having the same dimension as the input values. Thus, the geometric standard deviation may be more appropriately called geometric SD factor. When using geometric SD factor in conjunction with geometric mean, it should be described as "the range from (the geometric mean divided by the geometric SD factor) to (the geometric mean multiplied by the geometric SD factor), and one cannot add/subtract "geometric SD factor" to/from geometric mean. Definition If the geometric mean of a set of numbers {A1, A2, ..., An} is denoted as μg, then the geometric standard deviation is : \sig

Official source

https://en.wikipedia.org/wiki/Geometric_standard_deviation

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Objectives: Contralesional brain connectivity plasticity was previously reported after stroke. This study aims at disentangling the biological mechanisms underlying connectivity plasticity in the uninjured motor network after an ischemic lesion. In particular, we measured generalized fractional anisotropy (GFA) and magnetization transfer ratio (MTR) to assess whether post-stroke connectivity remodeling depend on axonal and/or myelin changes. Materials and Methods: Diffusion Spectrum Imaging (DSI) and Magnetization Transfer MRI at 3T were performed in 10 patients in acute phase, at one and six months after stroke, which was affecting motor cortical and/or subcortical areas. Ten age- and gender- matched healthy volunteers were scanned one month apart for longitudinal comparison. Clinical assessment was also performed in patients prior to MRI. In the contra-lesional hemisphere, average measures and tract-based quantitative analysis of GFA and MTR was performed to assess axonal integrity and myelination along motor connections as well as their variations in time. Results and Conclusions: Mean and tract-based measures of MTR and GFA showed significant changes in a number of contralesional motor connections, confirming both axonal and myelin plasticity in our cohort of patients. Moreover, density-derived features (peak height, standard deviation-SD and skewness) of GFA and MTR along the tracts showed additional correlation with clinical scores than mean values. These findings reveal the interplay between contralateral myelin and axonal remodeling after stroke.

2015

MATHICSE Technical Report : Analytic regularity and collocation approximation for PDEs with random domain deformations

Fabio Nobile

In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped on to a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in CN with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and standard deviation of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments.

MATHICSE2013