Statistical Regularization Method Research Articles

Extract the information via multiple repeated observations under randomly distributed noise

Abstract Extracting the useful information has been used almost everywhere in many fields of mathematics and applied mathematics. It is a classical ill-posed problem due to the unstable dependence of approximations on small perturbation of the data. The traditional regularization methods depend on the choice of the regularization parameter, which are closely related to an available accurate upper bound of noise level; thus it is not appropriate for the randomly distributed noise with big or unknown variance. In this paper, a purely data driven statistical regularization method is proposed, effectively extracting the information from randomly noisy observations. The rigorous upper bound estimation of confidence interval of the error in L 2 L^{2} norm is established, and some numerical examples are provided to illustrate the effectiveness and computational performance of the method.

Extract the information from big data with randomly distributed noise

Abstract In this manuscript, a purely data-driven statistical regularization method is proposed for extracting the information from big data with randomly distributed noise. Since the variance of the noise may be large, the method can be regarded as a general data preprocessing method in ill-posed problems, which is able to overcome the difficulty that the traditional regularization method is unable to solve, and has superior advantage in computing efficiency. The unique solvability of the method is proved, and a number of conditions are given to characterize the solution. The regularization parameter strategy is discussed, and the rigorous upper bound estimation of the confidence interval of the error in the L 2 L^{2} norm is established. Some numerical examples are provided to illustrate the appropriateness and effectiveness of the method.

Reconstruction of Spontaneous Fission Neutron Multiplicity Distribution Spectra by the Statistical Regularization Method

Reconstruction of Spontaneous Fission Neutron Multiplicity Distribution Spectra by the Statistical Regularization Method

Risk based arsenic rational sampling design for public and environmental health management

Groundwater contaminated with arsenic has been recognized as a global threat, which negatively impacts human health. Populations that rely on private wells for their drinking water are vulnerable to the potential arsenic-related health risks such as cancer and birth defects. Arsenic exposure through drinking water is among one of the primary arsenic exposure routes that can be effectively managed by active testing and water treatment. From the public and environmental health management perspective, it is critical to allocate the limited resources to establish an effective arsenic sampling and testing plan for health risk mitigation. We present a spatially adaptive sampling design approach based on an estimation of the spatially varying underlying contamination distribution. The method is different from traditional sampling design methods that often rely on a spatially constant or smoothly varying contamination distribution. In contrast, we propose a statistical regularization method to automatically detect spatial clusters of the underlying contamination risk from the currently available private well arsenic testing data in the USA, Iowa. This approach allows us to develop a sampling design method that is adaptive to the changes in the contamination risk across the identified clusters. We provide the spatially adaptive sample size calculation and sampling location determination at different acceptance precision and confidence levels for each cluster. The spatially adaptive sampling approach may effectively mitigate the arsenic risk from the resource management perspectives. The model presents a framework that can be widely used for other environmental contaminant monitoring and sampling for public and environmental health.

Multiplication Method for Fine-Tuning Regularization Parameter of a Sparse Modeling Technique Tentatively Optimized via Cross Validation

Sparse modeling is generally applied to data analysis in the field of experimental physics, using statistical regularization methods such as the least absolute shrinkage and selection operator (LAS...

Enhancing the Efficiency of the Reconstruction of the Temperature and Humidity Profiles of the Cloud Atmosphere by the Data of Satellite Microwave Spectrometers

It is shown that the a priori data on the atmospheric conditions, in addition to the climate’s characteristics, can enhance the efficiency of the algorithms for reconstructing the atmospheric profiles using satellite microwave radiometric observations. Such additional data are sought and their efficiency in remote sensing is estimated. The possibility of expanding the statistical approach by including new types of a priori data on the temperature and humidity of atmospheric conditions is discussed. It is demonstrated that the developed technique can be used to estimate the efficiency of using the following types of additional a priori data in the statistical regularization method: (i) the covariance matrix of the full vector of temperature and humidity variations within a vertical atmospheric column, (ii) the covariance matrix of the temperature and humidity variations within horizontal atmospheric strata, (iii) physical limits of the humidity variation amplitude, and (iv) statistically average model representations of the microwave radiation transfer parameters in the cloud layer.

Bayesian Maximum-A-Posteriori Approach with Global and Local Regularization to Image Reconstruction Problem in Medical Emission Tomography

The Bayesian approach Maximum a Posteriori (MAP) provides a common basis for developing statistical methods for solving ill-posed image reconstruction problems. MAP solutions are dependent on a priori model. Approaches developed in literature are based on prior models that describe the properties of the expected image rather than the properties of the studied object. In this paper, such models have been analyzed and it is shown that they lead to global regularization of the solution. Prior models that are based on the properties of the object under study are developed and conditions for local and global regularization are obtained. A new reconstruction algorithm has been developed based on the method of local statistical regularization. Algorithms with global and local regularization were compared in numerical simulations. The simulations were performed close to the real oncologic single photon emission computer tomography (SPECT) study. It is shown that the approach with local regularization produces more accurate images of ‘hot spots’, which is especially important to tumor diagnostics in nuclear oncology.

Simultaneous Reconstruction of the Complex Refractive Index and the Particle Size Distribution Function from Lidar Measurements: Testing the Developed Algorithms

A method for the joint determination of microphysical aerosol characteristics, namely, the complex refractive index $$m = m_{\text{real}}^{{}} + im_{\text{image}}^{{}}$$ and spherical-particle size distribution function U(r), from the data of nighttime lidar sensing at wavelengths of 355–1064 nm is proposed. During their simultaneous estimations, it is useful to directly minimize the discrepancy functional Φ(m) in the range of the physically justified m. The principal limitations due to a wider region of the global minima of Φ(m) appear at $$m_{{{\text{image}}}}^{{{\text{true}}}} \in $$ [0.01, 0.04] and give rise to a potential shift of the resulting values of $$m_{\text{real}}^{\text{est}}$$ and $$m_{\text{image}}^{\text{est}}$$. A simultaneous use of several functionals gives a better estimate of m due to different sets of the respective optical characteristics. The problem in retrieving the size distribution function is caused by the information content of the coarse particle measurements. The statistical regularization method offers an unambiguous estimation of U(r) for the mean radius up to 3 µm and gives an admissible estimate for larger radii. The algorithms are tested on eight values of absorption, when one value corresponding to one $$m_{\text{image}}^{\text{true}}$$ is associated with 50 empirical models of the distribution function.

Simultaneous reconstruction of two microphysical aerosol characteristics from the lidar data

A method is proposed for joint determination of two aerosol microphysical characteristics: the complex refractive index m = mreal + i*mimage and the spherical-particle size distribution function U(r) from the data of the nighttime vertical lidar sensing at the wavelengths 355–1064 nm. The values of the refractive index are nonlinearly related to the size distribution function. During their simultaneous estimation it is useful to minimize the discrepancy functional Φ(m) in the range of the physically justified m. The variation in the functional behavior with absorption is investigated. The principal limitations due to a wider region of the global minima of Φ(m) appear at mimage ∈ [0.01, 0.04] and give rise to a potential shift of the resulting values of mrealest and mimageest. A simultaneous use of several functionals gives a better estimate of mdue to different sets of the respective optical characteristics. The problem in retrieving the distribution function is caused by the information content of the coarse particle measurements. The statistical regularization method offers an unambiguous estimation of U(r)for the mean radius up to 3 µm, and gives an admissible estimate for larger radii. The algorithms have been tested using 50 empirical models obtained from the Zvenigorod AERONET site. In addition, the vertical profiles of aerosol parameters reconstructed from the data of the LOZA-S lidar measurements during the polar air mass transfer are presented.

Application of Turchin's method of statistical regularization

During analysis of experimental data, one usually needs to restore a signal after it has been convoluted with some kind of apparatus function. According to Hadamard's definition this problem is ill-posed and requires regularization to provide sensible results. In this article we describe an implementation of the Turchin's method of statistical regularization based on the Bayesian approach to the regularization strategy.

Model-driven regularization approach to straight line program genetic programming

This paper presents a regularization method for program complexity control of linear genetic programming tuned for transcendental elementary functions. Our goal is to improve the performance of evolutionary methods when solving symbolic regression tasks involving Pfaffian functions such as polynomials, analytic algebraic and transcendental operations like sigmoid, inverse trigonometric and radial basis functions. We propose the use of straight line programs as the underlying structure for representing symbolic expressions. Our main result is a sharp upper bound for the Vapnik Chervonenkis dimension of families of straight line programs containing transcendental elementary functions. This bound leads to a penalization criterion for the mean square error based fitness function often used in genetic programming for solving inductive learning problems. Our experiments show that the new fitness function gives very good results when compared with classical statistical regularization methods (such as Akaike and Bayesian Information Criteria) in almost all studied situations, including some benchmark real-world regression problems.

SAXSEV 2.1 cross-platform application for data analysis of small-angle X-ray scattering from polydisperse systems

The present paper discusses development and implementation of the cross-platform application with a graphical user interface for estimation of the particle volume fraction distribution function and fitting specific surface area to this distribution pattern. SAXSEV implements the method of statistical regularization for ill-posed mathematical tasks being solved with the use of Numpy, Scipy and MathPlotlib libraries. The main features of this software application are the ability to adjust the arguments grid of the desired function and the ability to select the optimal value of the regularization parameter. This parameter is selected by several specific and one common criteria. The software application consists of modules written in Python3. The modules are combined by common interface based on Tkinter library. Current version SAXSEV 2.1 was tested on the basis of Windows XP / Vista / 7/8, Ubuntu 14.1. SAXSEV 2.1 was used successfully at effectiveness study of statistical regularization method for analyzing dispersed system by SAXS, at research of the powder consisting from nanoparticles and composite materials with nanoparticles inclusion.

The problem of concentration of masses

The problem of the concentration of mass for two-dimensional (2D) and three-dimensional (3D) cases is considered. The problem is stated for a square and a cube and is solved by the method of statistical regularization, including the minimization of the discrepancy and Monte Carlo testing for constructing the sought densities.

Retrieving cloudy atmosphere parameters from RPG-HATPRO radiometer data

An algorithm for simultaneously determining both tropospheric temperature and humidity profiles and cloud liquid water content from ground-based measurements of microwave radiation is presented. A special feature of this algorithm is that it combines different types of measurements and different a priori information on the sought parameters. The features of its use in processing RPG-HATPRO radiometer data obtained in the course of atmospheric remote sensing experiments carried out by specialists from the Faculty of Physics of St. Petersburg State University are discussed. The results of a comparison of both temperature and humidity profiles obtained using a ground-based microwave remote sensing method with those obtained from radiosonde data are analyzed. It is shown that this combined algorithm is comparable (in accuracy) to the classical method of statistical regularization in determining temperature profiles; however, this algorithm demonstrates better accuracy (when compared to the method of statistical regularization) in determining humidity profiles.

Lidar positioning of higher clear-air turbulence regions

The possibility of lidar positioning of regions with higher clear-air turbulence (CAT) is shown. The turbulence is indicated by air density fluctuations generated by it. A scheme with a lidar based on using the backscattering enhecement (BSE) effect in a turbulent medium is considered. A stable solution of the positioning problem is obtained using the statistical regularization method. As is shown on models, CAT regions that are dangerous for civil aviation flights can be detected using such a lidar.

Decomposition of Complex Signal on Lorenz Components

Algorithm of complex signal decomposition on elementary components having Lorenz form is proposed.The non-linear minimization problem to the problem of linear equation solving. The number of components is the necessary aprioir information. The algorithm can be combined with the method of statistical regularization. The results of numerical experiments are represented.

Signal Processing with Regularized Multistep Support Vector Method

A new method is proposed for processing the signal distorted by random noise. The processing model is based on a statistical regularization method, and the obtained system of linear equations and inequalities is solved using a multistep support vector method. An advantage of this approach is that the iterative nature of the algorithm makes it possible to take into account the a priori information on the solution represented by the inequalities. The results of numerical experiments showing the efficiency of the algorithm are given.

An inverse random source problem in quantifying the elastic modulus of nanomaterials

Nanotechnology deals with structures sized between 1 and 100 nanometer and involves manipulating and controlling materials or devices within that scale. Due to its small size, it is difficult to quantify the mechanical properties of nanomaterials which significantly rely on measurement techniques, conditions and environment. In this paper, to describe the elastic deformation of nanobelts obtained from an atomic force microscope (AFM) under contact mode, we propose a novel model based on a Euler–Bernoulli equation with a stochastic source term accounting for effects of initial bending, surface roughness and white noise during measurements. With the random source, the forward problem is demonstrated to have a unique and explicit path-wise solution. Furthermore, the inverse problem consists of identifying the elastic modulus of nanobelts and reconstructing the random source structure, i.e. the mean and the variance. Based upon explicit formulae of direct problems, the corresponding inverse problem can be reduced into a first kind of Fredholm-type integral equation where two kinds of regularization techniques are applied to obtain stable solutions, i.e. Tikhonov regularization (TR) for smooth solutions and a statistical regularization method from Bayes' formula and maximum likelihood estimation for discontinuous solutions, respectively. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods.

One method for restoration of blurred images

A new method is proposed to reduce image distortions caused by random noise and rectilinear uniform motion of the object or recording system. The of a restoration model is based on a statistical regularization method, and the obtained system of linear equations and inequalities is solved using a multistep support vector method. An advantage of this approach is that the iterative nature of the algorithm makes it possible to take into account the a priori information on the solution represented by the inequalities. The results of numerical experiments showing the efficiency of the algorithm are given.

From Matched Spatial Filtering towards the Fused Statistical Descriptive Regularization Method for Enhanced Radar Imaging

We address a new approach to solve the ill-posed nonlinear inverse problem of high-resolution numerical reconstruction of the spatial spectrum pattern (SSP) of the backscattered wavefield sources distributed over the remotely sensed scene. An array or synthesized array radar (SAR) that employs digital data signal processing is considered. By exploiting the idea of combining the statistical minimum risk estimation paradigm with numerical descriptive regularization techniques, we address a new fused statistical descriptive regularization (SDR) strategy for enhanced radar imaging. Pursuing such an approach, we establish a family of the SDR-related SSP estimators, that encompass a manifold of existing beamforming techniques ranging from traditional matched filter to robust and adaptive spatial filtering, and minimum variance methods.