Quantile regression

Summary

Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met. Advantages and applications One advantage of quantile regression relative to ordinary least squares regression is that the quantile regression estimates are more robust against outliers in the response measurements. However, the main attraction of quantile regression goes beyond this and is advantageous when conditional quantile functions are of interest. Different measures of central tendency and statistical dispersion can be used to more comprehensively analyze the relationship between variables. In ecology, quantile regression has been

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

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Generalized Linear Models have become a commonly used tool of data analysis. Such models are used to fit regressions for univariate responses with normal, gamma, binomial or Poisson distribution. Maximum likelihood is generally applied as fitting method. In the usual regression setting the least absolute-deviations estimator (L1-norm) is a popular alternative to least squares (L2-norm) because of its simplicity and its robustness properties. In the first part of this thesis we examine the question of how much of these robustness features carry over to the setting of generalized linear models. We study a robust procedure based on the minimum absolute deviation estimator of Morgenthaler (1992), the Lq quasi-likelihood when q = 1. In particular, we investigate the influence function of these estimates and we compare their sensitivity to that of the maximum likelihood estimate. Furthermore we particularly explore the Lq quasi-likelihood estimates in binary regression. These estimates are difficult to compute. We derive a simpler estimator, which has a similar form as the Lq quasi-likelihood estimate. The resulting estimating equation consists in a simple modification of the familiar maximum likelihood equation with the weights wq(μ). This presents an improvement compared to other robust estimates discussed in the literature that typically have weights, which depend on the couple (xi, yi) rather than on μi = h(xiT β) alone. Finally, we generalize this estimator to Poisson regression. The resulting estimating equation is a weighted maximum likelihood with weights that depend on μ only.

EPFL2009

Classification of upper limb center-out reaching tasks by means of EEG-based continuous decoding techniques

Ricardo Andres Chavarriaga Lozano

Background: One of the current challenges in brain-machine interfacing is to characterize and decode upper limb kinematics from brain signals, e.g. to control a prosthetic device. Recent research work states that it is possible to do so based on low frequency EEG components. However, the validity of these results is still a matter of discussion. In this paper, we assess the feasibility of decoding upper limb kinematics from EEG signals in center-out reaching tasks during passive and active movements. Methods: The decoding of arm movement was performed using a multidimensional linear regression. Passive movements were analyzed using the same methodology to study the influence of proprioceptive sensory feedback in the decoding. Finally, we evaluated the possible advantages of classifying reaching targets, instead of continuous trajectories. Results: The results showed that arm movement decoding was significantly above chance levels. The results also indicated that EEG slow cortical potentials carry significant information to decode active center-out movements. The classification of reached targets allowed obtaining the same conclusions with a very high accuracy. Additionally, the low decoding performance obtained from passive movements suggests that discriminant modulations of low-frequency neural activity are mainly related to the execution of movement while proprioceptive feedback is not sufficient to decode upper limb kinematics. Conclusions: This paper contributes to the assessment of feasibility of using linear regression methods to decode upper limb kinematics from EEG signals. From our findings, it can be concluded that low frequency bands concentrate most of the information extracted from upper limb kinematics decoding and that decoding performance of active movements is above chance levels and mainly related to the activation of cortical motor areas. We also show that the classification of reached targets from decoding approaches may be a more suitable real-time methodology than a direct decoding of hand position.

BioMed Central2017

La détection de périodicités cachées

Jean-Marc Nicoletti

This thesis is a small part of the preparation of the launch of the Gaia mission, a satellite of the European Space Agency (ESA). One of the goals of the mission is to perform a classification among variable stars considering different attributes. Periodic behavior in the observed light curve is such an attribute. It is of importance to determine these hidden cycles with as high an accuracy as possible. A key difficulty is connected to the fact that observations are taken at irregularly distributed time points. Classical methods of frequency analysis do not work in this situation. In a first step, we made a catalogue of potential solutions and applied them to real and simulated data. The performances have been compared in order to select the best method. In a second step, we considered the asymptotic performance of estimators based on regression methods. We derived the asymptotic distribution under the hypothesis that the observation model contains a periodic signal with period P0 > 0, continuous and square integrable on [0; P0), with Gaussian correlated errors. The irregular sampling has to satisfy the asymptotic property that the whole interval [0, P0) can be observed. We have also considered the special case of a periodic signal with period P0 > 0, piecewise constant on [0; P0) and the asymptotic distribution of the estimator. If an irregular sampling scheme does not satisfy the asymptotic property that the signal can be fully observed, the period can still be estimated reliably. We determined the asymptotic distribution of an estimator in a particular situation and under the assumption that the observation model contains a periodic signal with period P0 > 0, continuous and square integrable on [0; P0), and is observed with additive errors that are independent and identically distributed.

EPFL2012

Classification of upper limb center-out reaching tasks by means of EEG-based continuous decoding techniques

Ricardo Andres Chavarriaga Lozano

BioMed Central2017