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>

Robust phase unwrapping algorithm based on Zernike polynomial fitting and Swin-Transformer network

Phase unwrapping plays an important role in optical phase measurements. In particular, phase unwrapping under heavy noise conditions remains an open issue. In this paper, a deep learning-based method is proposed to conduct the phase unwrapping task by combining Zernike polynomial fitting and a Swin-Transformer network. In this proposed method, phase unwrapping is regarded as a regression problem, and the Swin-Transformer network is used to map the relationship between the wrapped phase data and the Zernike polynomial coefficients. Because of the self-attention mechanism of the transformer network, the fitting coefficients can be estimated accurately even under extremely harsh noise conditions. Simulation and experimental results are presented to demonstrate the outperformance of the proposed method over the other two polynomial fitting-based methods. This is a promising phase unwrapping method in optical metrology, especially in electronic speckle pattern interferometry.

Leveraging pleiotropic association using sparse group variable selection in genomics data

Background Genome-wide association studies (GWAS) have identified genetic variants associated with multiple complex diseases. We can leverage this phenomenon, known as pleiotropy, to integrate multiple data sources in a joint analysis. Often integrating additional information such as gene pathway knowledge can improve statistical efficiency and biological interpretation. In this article, we propose statistical methods which incorporate both gene pathway and pleiotropy knowledge to increase statistical power and identify important risk variants affecting multiple traits. Methods We propose novel feature selection methods for the group variable selection in multi-task regression problem. We develop penalised likelihood methods exploiting different penalties to induce structured sparsity at a gene (or pathway) and SNP level across all studies. We implement an alternating direction method of multipliers (ADMM) algorithm for our penalised regression methods. The performance of our approaches are compared to a subset based meta analysis approach on simulated data sets. A bootstrap sampling strategy is provided to explore the stability of the penalised methods. Results Our methods are applied to identify potential pleiotropy in an application considering the joint analysis of thyroid and breast cancers. The methods were able to detect eleven potential pleiotropic SNPs and six pathways. A simulation study found that our method was able to detect more true signals than a popular competing method while retaining a similar false discovery rate. Conclusion We developed feature selection methods for jointly analysing multiple logistic regression tasks where prior grouping knowledge is available. Our method performed well on both simulation studies and when applied to a real data analysis of multiple cancers.

Multiscale regression on unknown manifolds

<abstract><p>We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.</p></abstract>

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.

ESTIMATING WITH A GIVEN ACCURACY OF THE COEFFICIENTS AT NONLINEAR TERMS OF UNIVARIATE POLYNOMIAL REGRESSION USING A SMALL NUMBER OF TESTS IN AN ARBITRARY LIMITED ACTIVE EXPERIMENT

We substantiate the structure of the efficient numerical axis segment an active experiment on which allows finding estimates of the coefficients fornonlinear terms of univariate polynomial regression with high accuracy using normalized orthogonal Forsyth polynomials with a sufficiently smallnumber of experiments. For the case when an active experiment can be executed on a numerical axis segment that does not satisfy these conditions, wesubstantiate the possibility of conducting a virtual active experiment on an efficient interval of the numerical axis. According to the results of the experiment, we find estimates for nonlinear terms of the univariate polynomial regression under research as a solution of a linear equalities system withan upper non-degenerate triangular matrix of constraints. Thus, to solve the problem of estimating the coefficients for nonlinear terms of univariatepolynomial regression, it is necessary to choose an efficient interval of the numerical axis, set the minimum required number of values of the scalarvariable which belong to this segment and guarantee a given value of the variance of estimates for nonlinear terms of univariate polynomial regressionusing normalized orthogonal polynomials of Forsythe. Next, it is necessary to find with sufficient accuracy all the coefficients of the normalized orthogonal polynomials of Forsythe for the given values of the scalar variable. The resulting set of normalized orthogonal polynomials of Forsythe allows us to estimate with a given accuracy the coefficients of nonlinear terms of univariate polynomial regression in an arbitrary limited active experiment: the range of the scalar variable values can be an arbitrary segment of the numerical axis. We propose to find an estimate of the constant and ofthe coefficient at the linear term of univariate polynomial regression by solving the linear univariate regression problem using ordinary least squaresmethod in active experiment conditions. Author and his students shown in previous publications that the estimation of the coefficients for nonlinearterms of multivariate polynomial regression is reduced to the sequential construction of univariate regressions and the solution of the correspondingsystems of linear equalities. Thus, the results of the paper qualitatively increase the efficiency of finding estimates of the coefficients for nonlinearterms of multivariate polynomial regression given by a redundant representation.

Expert-enhanced machine learning for cardiac arrhythmia classification

We propose a new method for the classification task of distinguishing atrial fibrillation (AFib) from regular atrial tachycardias including atrial flutter (AFlu) based on a surface electrocardiogram (ECG). Recently, many approaches for an automatic classification of cardiac arrhythmia were proposed and to our knowledge none of them can distinguish between these two. We discuss reasons why deep learning may not yield satisfactory results for this task. We generate new and clinically interpretable features using mathematical optimization for subsequent use within a machine learning (ML) model. These features are generated from the same input data by solving an additional regression problem with complicated combinatorial substructures. The resultant can be seen as a novel machine learning model that incorporates expert knowledge on the pathophysiology of atrial flutter. Our approach achieves an unprecedented accuracy of 82.84% and an area under the receiver operating characteristic (ROC) curve of 0.9, which classifies as “excellent” according to the classification indicator of diagnostic tests. One additional advantage of our approach is the inherent interpretability of the classification results. Our features give insight into a possibly occurring multilevel atrioventricular blocking mechanism, which may improve treatment decisions beyond the classification itself. Our research ideally complements existing textbook cardiac arrhythmia classification methods, which cannot provide a classification for the important case of AFib↔AFlu. The main contribution is the successful use of a novel mathematical model for multilevel atrioventricular block and optimization-driven inverse simulation to enhance machine learning for classification of the arguably most difficult cases in cardiac arrhythmia. A tailored Branch-and-Bound algorithm was implemented for the domain knowledge part, while standard algorithms such as Adam could be used for training.

Quantum Support Vector Regression for Disability Insurance

We propose a hybrid classical-quantum approach for modeling transition probabilities in health and disability insurance. The modeling of logistic disability inception probabilities is formulated as a support vector regression problem. Using a quantum feature map, the data are mapped to quantum states belonging to a quantum feature space, where the associated kernel is determined by the inner product between the quantum states. This quantum kernel can be efficiently estimated on a quantum computer. We conduct experiments on the IBM Yorktown quantum computer, fitting the model to disability inception data from a Swedish insurance company.

Reverse engineering of a mechanistic model of gene expression using metastability and temporal dynamics

Differentiation can be modeled at the single cell level as a stochastic process resulting from the dynamical functioning of an underlying Gene Regulatory Network (GRN), driving stem or progenitor cells to one or many differentiated cell types. Metastability seems inherent to differentiation process as a consequence of the limited number of cell types. Moreover, mRNA is known to be generally produced by bursts, which can give rise to highly variable non-Gaussian behavior, making the estimation of a GRN from transcriptional profiles challenging. In this article, we present CARDAMOM (Cell type Analysis from scRna-seq Data achieved from a Mixture MOdel), a new algorithm for inferring a GRN from timestamped scRNA-seq data, which crucially exploits these notions of metastability and transcriptional bursting. We show that such inference can be seen as the successive resolution of as many regression problem as timepoints, after a preliminary clustering of the whole set of cells with regards to their associated bursts frequency. We demonstrate the ability of CARDAMOM to infer a reliable GRN from in silico expression datasets, with good computational speed. To the best of our knowledge, this is the first description of a method which uses the concept of metastability for performing GRN inference.