Concept# Student's t-distribution

Summary

In probability and statistics, Student's t-distribution (or simply the t-distribution) t_\nu is
a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.
However, t_\nu has heavier tails and the amount of probability mass in the tails is controlled by the parameter \nu. For \nu = 1 the Student's t distribution t_\nu becomes the standard Cauchy distribution, whereas for \nu \rightarrow \infty it becomes the standard normal distribution N(0,1).
The Student's t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.
In the form of the location-scale

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Rahul Kumar Gupta, Mario Paolone

The objective of this activity is to validate an advanced real-time control framework for power distribution networks in order to control grid-connected battery energy storage system (BESSs) to satisfy the control’s objectives such as tracking a day-ahead dispatch plan of a distribution network hosting controllable and stochastic heterogenous resources. The control framework accounts for the grid constraints on the nodal voltages and lines and transformer capacities. The control algorithms rely on the availability of both short and day-ahead forecasting of the demand and PV generation developed in the context of feeder dispatching. In the scheduling phase on the day before operations, a stochastic optimization problem computes an aggregated dispatch plan at the grid connection point (GCP), accounting for the uncertainties of demand and PV generation via scenarios, and constraints of the grid and the controllable resource. For the day-ahead scenarios, we develop a Markov’s chain based forecasting method where we cluster the historical measurements for each day type and ﬁt multi-variate Gaussian distribution for each cluster. In the real-time phase, a grid-aware model predictive control (MPC) computes the active and reactive power set-points of the battery so that it tracks the dispatch plan at the GCP while obeying to its constraints and those of the grid. The MPC problem leverages short-term forecasting of the demand and PV generation. The proposed control framework is validated by dispatching the operation of a 12kV/20MVA MV distribution network in Aigle, Switzerland (i.e. the REeL demonstrator) using a 1.5 MW/2.5 MWh BESS, which is controlled in real-time given the online grid state estimation enabled by the deployed distributed PMU-based sensing infrastructure.

This paper examines a stochastic formulation of the generalized Nash equilibrium problem where agents are subject to randomness in the environment of unknown statistical distribution. We focus on fully distributed online learning by agents and employ penalized individual cost functions to deal with coupled constraints. Three stochastic gradient strategies are developed with constant step-sizes. We allow the agents to use heterogeneous step-sizes and show that the penalty solution is able to approach the Nash equilibrium in a stable manner within O(μmax), for small step-size value μmax and sufficiently large penalty parameters. The operation of the algorithm is illustrated by considering the network Cournot competition problem.

Daniel Gallichan, Frédéric Gretsch

Purpose To compare motion tracking by 2 modern methods (fat navigators [FatNavs] and Moir' phase tracking [MPT]) as well as their performance for retrospective correction of very high resolution acquisitions. Methods A direct comparison of FatNavs and MPT motion parameters was performed for several deliberate motion patterns to estimate the agreement between methods. In addition, 2 different navigator resolution were applied. 0.5 mm isotropic MP2RAGE images with simultaneous MPT and FatNavs tracking were acquired in 9 cooperative subjects with no intentional motion. Retrospective motion corrections based on both tracking modalities were compared qualitatively and quantitatively. The FatNavs impact on quantitative T-1 maps was also investigated. Results Both methods showed good agreement within a 0.3 mm/degrees margin in subjects that moved very little. Higher resolution FatNavs (2 mm) showed overall better agreement with MPT than 4 mm resolution ones, except for fast and large motion. The retrospective motion corrections based on MPT or FatNavs were at par in 33 cases out of 36, and visibly improved image quality compared to the uncorrected images. In separate fringe cases, both methods suffered from their respective potential shortcomings: unreliable marker attachment for MPT and poor temporal resolution for FatNavs. The magnetization transfer induced by the navigator RF pulses had a visible impact on the T-1 values distribution, with a shift of the gray and white matter peaks of 12 ms at most. Conclusion This work confirms both FatNavs and MPT as excellent retrospective motion correction methods for very high resolution imaging of cooperative subjects.

2020