Accounting for Autocorrelation in the Linear Regression Problem by an Example of Analysis of the Atmospheric Column NO2 Content

We formulate a generalized version of the classical linear regression problem on Riemannian manifolds and derive the counterpart to the normal equations for the manifold of symmetric and positive definite matrices, equipped with the only metric that is invariant under the natural action of the general linear group.

Download Full-text

Asymptotically optimal estimation in the linear regression problem in the case of violation of some classical assumptions

Siberian Mathematical Journal ◽

10.1007/s11202-009-0035-2 ◽

2009 ◽

Vol 50 (2) ◽

pp. 302-315 ◽

Cited By ~ 7

Author(s):

Yu. Yu. Linke ◽

A. I. Sakhanenko

Keyword(s):

Linear Regression ◽

Optimal Estimation ◽

Regression Problem ◽

Asymptotically Optimal ◽

Linear Regression Problem

Download Full-text

PAC-Bayes Unleashed: Generalisation Bounds with Unbounded Losses

Entropy ◽

10.3390/e23101330 ◽

2021 ◽

Vol 23 (10) ◽

pp. 1330

Author(s):

Maxime Haddouche ◽

Benjamin Guedj ◽

Omar Rivasplata ◽

John Shawe-Taylor

Keyword(s):

Linear Regression ◽

Supervised Learning ◽

Loss Function ◽

Loss Functions ◽

Learning Problems ◽

Regression Problem ◽

Learning Framework ◽

Actual Computation ◽

Linear Regression Problem

We present new PAC-Bayesian generalisation bounds for learning problems with unbounded loss functions. This extends the relevance and applicability of the PAC-Bayes learning framework, where most of the existing literature focuses on supervised learning problems with a bounded loss function (typically assumed to take values in the interval [0;1]). In order to relax this classical assumption, we propose to allow the range of the loss to depend on each predictor. This relaxation is captured by our new notion of HYPothesis-dependent rangE (HYPE). Based on this, we derive a novel PAC-Bayesian generalisation bound for unbounded loss functions, and we instantiate it on a linear regression problem. To make our theory usable by the largest audience possible, we include discussions on actual computation, practicality and limitations of our assumptions.

Download Full-text

An Aeromagnetic Compensation Method Based on a Multimodel for Mitigating Multicollinearity

Sensors ◽

10.3390/s19132931 ◽

2019 ◽

Vol 19 (13) ◽

pp. 2931 ◽

Cited By ~ 1

Author(s):

Guanyi Zhao ◽

Qi Han ◽

Xiang Peng ◽

Pengyi Zou ◽

Haidong Wang ◽

...

Keyword(s):

Magnetic Field ◽

Linear Regression ◽

Field Effect ◽

Compensation Method ◽

Interference Field ◽

Regression Problem ◽

Geophysical Exploration ◽

Linear Regression Problem ◽

The Impact

Aeromagnetic surveys play an important role in geophysical exploration and many other fields. In many applications, magnetometers are installed aboard an aircraft to survey large areas. Due to its composition, an aircraft has its own magnetic field, which degrades the reliability of the measurements, and thus a technique (named aeromagnetic compensation) that reduces the magnetic interference field effect is required. Commonly, based on the Tolles–Lawson model, this issue is solved as a linear regression problem. However, multicollinearity, which refers to the case when more than two model variables are highly linearly related, creates accuracy problems when estimating the model coefficients. The analysis in this study indicates that the variables that cause multicollinearity are related to the flight heading. To take this point into account, a multimodel compensation method is proposed. By selecting the variables that contribute less to the multicollinearity, different sub-models are built to describe the magnetic interference of the aircraft when flying in different orientations. This method restricts the impact of multicollinearity and improves the reliability of the measurements. Compared with the existing methods, the proposed method reduces the interference field more effectively, which is verified by a set of airborne tests.

Download Full-text

Multiple linear regression is a useful alternative to traditional analyses of variance

AJP Regulatory Integrative and Comparative Physiology ◽

10.1152/ajpregu.1988.255.3.r353 ◽

1988 ◽

Vol 255 (3) ◽

pp. R353-R367 ◽

Cited By ~ 19

Author(s):

B. K. Slinker ◽

S. A. Glantz

Keyword(s):

Missing Data ◽

Linear Regression ◽

Multiple Linear Regression ◽

Analysis Of Variance ◽

Continuous Variable ◽

Parameter Estimates ◽

Regression Problem ◽

Regression Approach ◽

Linear Regression Problem ◽

Analyze Data

Physiologists often wish to compare the effects of several different treatments on a continuous variable of interest, which requires an analysis of variance. Analysis of variance, as presented in most statistics texts, generally requires that there be no missing data and often that each sample group be the same size. Unfortunately, this requirement is rarely satisfied, and investigators are confronted with the problem of how to analyze data that do not strictly fit the traditional analysis of variance paradigm. One can avoid these pitfalls by recasting the analysis of variance as a multiple linear regression problem. When there are no missing data, the results of a traditional analysis of variance and the corresponding multiple regression problem are identical; when the sample sizes are unequal or there are missing data, one can use a regression formulation to analyze data that cannot be easily handled in a traditional analysis of variance paradigm and thus overcome a practical computational limitation of traditional analysis of variance. In addition to overcoming practical limitations of traditional analysis of variance, the multiple linear regression approach is more efficient because in one run of a statistics routine, not only is the analysis of variance done but also one obtains estimates of the size of the treatment effects (as opposed to just an indication of whether such effects are present or not), and many of the pairwise multiple comparisons are done (they are equivalent to t tests for significance of the regression parameter estimates). Finally, interaction between the different treatment factors is easier to interpret than it is in traditional analysis of variance.

Download Full-text