Statistics

2406 Submissions

[3] viXra:2406.0160 [pdf] submitted on 2024-06-27 17:11:50

Likelihood Measures for Classifying Frequency Response Functions from Posture Control Experiments

Authors: Vittorio Lippi
Comments: 4 Pages. presented at International Conference on Mathematical Analysis and Applications in Science and Engineering 20 - 22 June 2024, Porto, Portugal

The frequency response function (FRF) is an established way to describe the outcome of experiments in posture control literature. The FRF is an empirical transfer function between an input stimulus and the induced body segment sway profile, represented as a vector of complex values associated with a vector of frequencies. Having obtained an FRF from a trial with a subject, it can be useful to quantify the likelihood it belongs to a certain population, e.g., to diagnose a condition or to evaluate the human likeliness of a humanoid robot or a wearable device. In this work, a recently proposed method for FRF statistics based on confidence bands computed with Bootstrap will be summarized, and, on its basis, possible ways to quantify the likelihood of FRFs belonging to a given set will be proposed
Category: Statistics

[2] viXra:2406.0159 [pdf] submitted on 2024-06-27 20:15:06

Random Field Theory for Testing Differences Between Frequency Response Functions in Posturography

Authors: Vittorio Lippi
Comments: 2 Pages. Presented at 9th International Posture Symposium, Smolenice 2023 (Note by viXra Admin: An abstract on the article is required)

The frequency response function (FRF) is an established way to describe the outcome of experiments in posture control literature. Specifically, the FRF is an empirical transfer function between an input stimulus and the induced body movement. By definition, the FRF is a complex function of frequency. When statistical analysis is performed to assess differences between groups of FRFs (e.g., obtained under different conditions or from a group of patients and a control group), the FRF's structure should be considered. Usually, the statistics are performed defined a scalar variable to be studied, such as the norm of the difference between FRFs, or considering the components independently (that can be applied to real and complex components separately), in some cases both approaches are integrated, e.g., the comparison frequency-by frequency is used as a post hoc test when the null hypothesis is rejected on the scalar value. The two components of the complex values can be tested with multivariate methods such as Hotelling’s T2 as done in on the averages of the FRF over all the frequencies, where a further post hoc test is performed applying bootstrap on magnitude and phase separately. The problem with the definition of a scalar variable as the norm of the differences or the difference of the averages in the previous examples is that it introduces an arbitrary metric that, although reasonable, has no substantial connection with the experiment unless the scalar value is assumed a priori as the object of the study as in where a human-likeness score for humanoid robots is defined on the basis of FRFs difference. On the other hand, testing frequencies (and components) separately does not consider that the FRF's values are not independent, and applying corrections for multiple comparisons, e.g., Bonferroni can result in a too conservative approach destroying the power of the experiment. In order to properly consider the nature of the FRF, a method based on random field theory is presented. A case study with data from posture control experiments is presented. To take into account the two components (imaginary and real) as two independent variables, the fact that the same subject repeated the test in the two conditions, a 1-D implementation of the Hotelling T2 is used as presented in but applied in the frequency domain instead of the time domain.
Category: Statistics

[1] viXra:2406.0055 [pdf] replaced on 2025-04-08 18:36:58

A Note on the Area Under the Likelihood and the Fake Evidence for Model Selection

Authors: L. Martino, F. Llorente
Comments: 29 Pages.

Improper priors are not allowed for the computation of the Bayesian evidence Z = p(y) (a.k.a., marginal likelihood), since in this case Z is not completely specified due to an arbitrary constant involved in the computation. However, in this work, we remark that they can be employed in a specific type of model selection problem: when we have several (possibly infinite) models belonging to the same parametric family (i.e., for tuning parameters of a parametric model). However, the quantities involved in this type of selection cannot be considered as Bayesian evidences: we suggest to use the name "fake evidences" (or "areas under the likelihood" in the case of uniform improper priors). We also show that, in this model selection scenario, using a use prior and increasing its scale parameter asymptotically to infinity, we cannot recover the value of the area under the likelihood, obtained with a uniform improper prior. We first discuss it from a general point of view. Then we provide, as an applicative example, all the details for Bayesian regression models with nonlinear bases, considering two cases: the use of a uniform improper prior and the use of a Gaussian prior, respectively. A numerical experiment is also provided confirming and checking all the previous statements.
Category: Statistics