Solving multiple linear regression problem using artificial neural network

<span>Multiple linear regressions are an important tool used to find the relationship between a set of variables used in various scientific experiments. In this article we are going to introduce a simple method of solving a multiple rectilinear regressions (MLR) problem that uses an artificial neural network to find the accurate and expected output from MLR problem. Different artificial neural network (ANN) types with different architecture will be tested, the error between the target outputs and the calculated ANN outputs will be investigated. A recommendation of using a certain type of ANN based on the experimental results will be raised.</span>

Preserving gauge invariance in neural networks

In these proceedings we present lattice gauge equivariant convolutional neural networks (L-CNNs) which are able to process data from lattice gauge theory simulations while exactly preserving gauge symmetry. We review aspects of the architecture and show how L-CNNs can represent a large class of gauge invariant and equivariant functions on the lattice. We compare the performance of L-CNNs and non-equivariant networks using a non-linear regression problem and demonstrate how gauge invariance is broken for non-equivariant models.

PAC-Bayes Unleashed: Generalisation Bounds with Unbounded Losses

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.

An Aeromagnetic Compensation Method Based on a Multimodel for Mitigating Multicollinearity

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.

Аccounting for autocorrelation in a linear regression problem on an example of analysis of atmospheric column NO2 content

A method is proposed for taking into account a serial correlation (an autocorrelation) of data in a linear regression problem, which allows accounting for the autocorrelation on long scales. A residual series is presented as an autoregressive process of an order, k, that can be much larger than 1, and the autocorrelation function of the processes is calculated by solving the system of the Yule–Walker equations. Given the autocorrelation function, the autocorrelation matrix is constructed which enters the formulas for estimates of regression coefficients and their errors. The efficiency of the method is demonstrated on the base of the multiple regression analysis of data of 26-year measurements of the column NO2 contents at the Zvenigorod Research Station of the Institute of Atmospheric Physics. Estimates of regression coefficients and their errors depend on the autoregression order k. At first the error increases with increasing k. Then it approaches its maximum and thereafter begins to decrease. In the case of NO2 at the Zvenigorod Station the error more than doubled in its maximum compared to the beginning value. The decrease in the error after approaching the maximum stops if k approaches the value such that the autoregressive process of this order allows accounting for important features of the autocorrelation function of the residual series. Estimates have been obtained of seasonally dependent linear trends and effects on NO2 of nature factors such that the 11-year solar cycle, the quasi-biennial oscillation, the North Atlantic Oscillation and other.