Physical Sciences and Mathematics Commons^™

Lecture 14: Randomized Algorithms For Least Squares Problems, Ilse C.F. Ipsen

Mathematical Sciences Spring Lecture Series

The emergence of massive data sets, over the past twenty or so years, has lead to the development of Randomized Numerical Linear Algebra. Randomized matrix algorithms perform random sketching and sampling of rows or columns, in order to reduce the problem dimension or compute low-rank approximations. We review randomized algorithms for the solution of least squares/regression problems, based on row sketching from the left, or column sketching from the right. These algorithms tend to be efficient and accurate on matrices that have many more rows than columns. We present probabilistic bounds for the amount of sampling required to achieve a …

Commuting Perturbations Of Operator Equations, Xue Xu, Jiu Ding

Faculty Publications

Let X be a Banach space and let T: X →X be a bounded linear operator with closed range. We study a class of commuting perturbations of the corresponding operator equation, using the concept of the spectral radius of a bounded linear operator. Our results extend the classic perturbation theorem for invertible operators and its generalization for arbitrary operators under the commutability assumption.

Predicting The Winning Percentage Of Limited-Overs Cricket Using The Pythagorean Formula, Hasika K. W. Senevirathne, Ananda B.W. Manage

Mathematics & Statistics Faculty Publications

The Pythagorean Win-Loss formula can be effectively used to estimate winning percentages for sporting events. This formula was initially developed by baseball statistician Bill James and later was extended by other researchers to sports such as football, basketball, and ice hockey. Although one can calculate actual winning percentages based on the outcomes of played games, that approach does not take into account the margin of victory. The key benefit of the Pythagorean formula is its utilization of actual average runs scored and actual average runs allowed. This article presents the application of the Pythagorean Win-Loss formula to two different types …

Predictive Modeling Of Iphone 7 Charge Rates Using Least Squares Curve Fitting, Grace Cahill

Honors Theses

In a time where individuals depend on their cell phones, the need for a long lasting and quick charging battery life is imperative. As information regarding how long a battery can remained charged is highly advertised, there is no information regarding how long it would take for a dead phone battery to completely charge. This study determined the amount of time it will take an iPhone 7 to charge from 0% to 100% using the standard charging cable under four different charging conditions. The charge percentage was recorded every two minutes until it was fully charged with this process being …

Fitting Parameter Uncertainties In Least Squares Fitting, R. Steven Turley

Faculty Publications

This article review the theory and practice of computing uncertainties in the fit parameters in least squares fits. It shows how to estimate the uncertainties and gives some numerical examples in Julia of their use. Examples are given and validated for both linear and nonlinear fits.

Polynomial Fitting, R. Steven Turley

Faculty Publications

This article reviews the theory and some good practice for fitting polynomials to data. I show by theory and example why fitting using a basis of orthogonal polynomials rather than monomials is desirable. I also show how to scale the independent variable for a more stable fit. I also demonstrate how to compute the uncertainty in the fit parameters. Finally, I discuss regression analysis: how to determine whether adding an additional term to the fit is justified.

Enhancing Building Footprints With Squaring Operations, Imran Lokhat, Guillaume Touya

Journal of Spatial Information Science

Whatever the data source, or the capture process, the creation of a building footprint in a geographical dataset is error prone. Building footprints are designed with square angles, but once in a geographical dataset, the angles may not be exactly square. The almost-square angles blur the legibility of the footprints when displayed on maps, but might also be propagated in further applications based on the footprints, e.g., 3D city model construction. This paper proposes two new methods to square such buildings: a simple one, and a more complex one based on nonlinear least squares. The latter squares right and flat …

A 10-Point Approximating Subdivision Scheme Based On Least Squares Technique, Ghulam Mustafa, Muhammad T. Iqbal

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a 10-point approximating subdivision scheme is presented. Least squares technique for fitting the polynomial of degree 9 to data is used to develop this scheme. The proposed strategy can be used to generate a family of schemes. The important characteristics of the scheme are also discussed. Graphical efficiency of the scheme is shown by applying it on different types of data.

Stabilized Least Squares Migration, Graham Ganssle

University of New Orleans Theses and Dissertations

Before raw seismic data records are interpretable by geologists, geophysicists must process these data using a technique called migration. Migration spatially repositions the acoustic energy in a seismic record to its correct location in the subsurface. Traditional migration techniques used a transpose approximation to a true acoustic propagation operator. Conventional least squares migration uses a true inverse operator, but is limited in functionality by the large size of modern seismic datasets. This research uses a new technique, called stabilized least squares migration, to correctly migrate seismic data records using a true inverse operator. Contrary to conventional least squares migration, this …

On The Coupling Of Dpg And Bem, Thomas Führer, Norbert Heuer, Michael Karkulik

Mathematics and Statistics Faculty Publications and Presentations

We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with optimal test functions to (i) least-squares boundary elements and (ii) various variants of standard Galerkin boundary elements. An essential feature of our method is that, despite the use of boundary integral equations, optimal test functions have to be computed only locally. We apply our findings to a standard transmission problem in full space and present numerical experiments to validate our theory.

Parameter Identification And Fault Detection For Reliable Control Of Permanent Magnet Motors, Dusan Vukosav Progovac

Wayne State University Dissertations

The objective of this dissertation is to develop new fault detection, identification, estimation and control algorithms that will be used to detect winding stator fault, identify the motor parameters and optimally control machine during faulty condition. Quality or proposed algorithms for Fault detection, parameter identification and control under faulty condition will validated through analytical study (Cramer-Rao bound) and simulation. Simulation will be performed for three most applied control schemes: Proportional-Integral-Derivative (PID), Direct Torque Control (DTC) and Field Oriented Control (FOC) for Permanent Magnet Machines. New detection schemes forfault detection, isolation and machine parameter identification are presented and analyzed. Different control …

X2 Tests For The Choice Of The Regularization Parameter In Nonlinear Inverse Problems, J. L. Mead, C. C. Hammerquist

Jodi Mead

We address discrete nonlinear inverse problems with weighted least squares and Tikhonov regularization. Regularization is a way to add more information to the problem when it is ill-posed or ill-conditioned. However, it is still an open question as to how to weight this information. The discrepancy principle considers the residual norm to determine the regularization weight or parameter, while the χ2 method [J. Mead, J. Inverse Ill-Posed Probl., 16 (2008), pp. 175–194; J. Mead and R. A. Renaut, Inverse Problems, 25 (2009), 025002; J. Mead, Appl. Math. Comput., 219 (2013), pp. 5210–5223; R. A. Renaut, I. Hnetynkova, and J. L. …

Discontinuous Parameter Estimates With Least Squares Estimators, J. L. Mead

Jodi Mead

We discuss weighted least squares estimates of ill-conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices. Least squares estimators are the maximum likelihood estimate for normally distributed data and parameters, but here we do not assume particular probability distributions. Weights for the estimator are found by ensuring its minimum follows a χ2 distribution. Previous work with this approach has shown that it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [20]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization parameter. …

Discontinuous Parameter Estimates With Least Squares Estimators, J. L. Mead

Mathematics Faculty Publications and Presentations

We discuss weighted least squares estimates of ill-conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices. Least squares estimators are the maximum likelihood estimate for normally distributed data and parameters, but here we do not assume particular probability distributions. Weights for the estimator are found by ensuring its minimum follows a χ² distribution. Previous work with this approach has shown that it is competitive with regularization methods such as the L-curve and Generalized Cross Validation (GCV) [20]. In this work we extend the method to find diagonal weighting matrices, rather than a scalar regularization …

3d Velocity Retrieval And Storm Tracking Using Multiple Radars, Yong Zhang

Electronic Thesis and Dissertation Repository

Severe weather forecasting is one of the most important and urgent tasks in the meteorology field. This thesis builds on previous work by Barron and Mercer and their graduate students, concerning the use of 3D optical flow to retrieve 3D wind velocity from 3D Doppler radial velocity datasets and tracking 3D severe weather storms using fuzzy points realized as ellipsoids to represent storms and a fuzzy algebra machinery in a relaxation labeling framework to track storms in Doppler precipitation datasets.

We first extend the original 3D optical flow (both least squares and regularization methods) for recovering 3D wind velocity from …

Phase Retrieval For Characteristic Functions Of Convex Bodies And Reconstruction From Covariograms, Gabriele Bianchi, Richard J. Gardner, Markus Kiederlen

Mathematics Faculty Publications

The Phase Retrieval Problem of Fourier analysis involves determining a function f on Rn from the modulus |f�| of its Fourier transform f�. This problem arises naturally and frequently in various areas of science, such as X-ray crystallography, electron microscopy, optics, astronomy, and remote sensing, in which only the magnitude of the Fourier transform can be measured and the phase is lost.

A Least Squares Closure Approximation For Liquid Crystalline Polymers, Traci Ann Sievenpiper

Mathematics & Statistics Theses & Dissertations

An introduction to existing closure schemes for the Doi-Hess kinetic theory of liquid crystalline polymers is provided. A new closure scheme is devised based on a least squares fit of a linear combination of the Doi, Tsuji-Rey, Hinch-Leal I, and Hinch-Leal II closure schemes. The orientation tensor and rate-of-strain tensor are fit separately using data generated from the kinetic solution of the Smoluchowski equation. The known behavior of the kinetic solution and existing closure schemes at equilibrium is compared with that of the new closure scheme. The performance of the proposed closure scheme in simple shear flow for a variety …

Iteratively Reweighted Least Squares Minimization With Prior Information A New Approach, Dmitriy Popov

Electronic Theses and Dissertations

Iteratively reweighted least squares (IRLS) algorithms provide an alternative to the more standard 1 l -minimization approach in compressive sensing. Daubechies et al. introduced a particularly stable version of an IRLS algorithm and rigorously proved its convergence in 2010. They did not, however, consider the case in which prior information on the support of the sparse domain of the solution is available. In 2009, Miosso et al. proposed an IRLS algorithm that makes use of this information to further reduce the number of measurements required to recover the solution with specified accuracy. Although Miosso et al. obtained a number of …

A Comparison Between Unbiased Ridge And Least Squares Regression Methods Using Simulation Technique, Mowafaq M. Al-Kassab, Omar Q. Qwaider

Journal of Modern Applied Statistical Methods

The parameters of the multiple linear regression are estimated using least squares ( B̂_LS ) and unbiased ridge regression methods (B̂(KI,J)). Data was created for fourteen independent variables with four different values of correlation between these variables using Monte Carlo techniques. The above methods were compared using the mean squares error criterion. Results show that the unbiased ridge method is preferable to the least squares method.

A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut

Jodi Mead

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of …

Development Of An Adaptive Equalization Algorithm Using Acoustic Energy Density, Panu Tapani Puikkonen

Theses and Dissertations

Sound pressure equalization of audio signals using digital signal processors has been a subject of ongoing study for many years. The traditional approach is to equalize sound at a point in a listening environment, but because of its specific dependence on the room frequency response between a source and receiver position, this equalization generally causes the spectral response to worsen significantly at other locations in the room. This work presents both a time-invariant and a time-varying implementation of an adaptive acoustic energy density equalization filter for a one-dimensional sound field. Energy density equalization addresses the aforementioned challenge and others that …

A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut

Mathematics Faculty Publications and Presentations

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X² distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value …

Robust General Linear Models And Graphics Via A User Interface (Web Rglm), Kimberly Crimin, Asheber Abebe, Joseph W. Mckean

Journal of Modern Applied Statistical Methods

Rank-based procedures provide superior estimation and testing techniques when the data deviate from normality or contain gross outliers. However, these robust techniques are rarely incorporated in a nonparametric statistics or methods courses due to the lack of computational tools. One reason for this is the existence of certain unavoidable complexities in the numerical methods due to the absence of a closedform solution for the rank estimation problem. This article introduces a user interface, Web RGLM, which may be used to perform rank-based analyses of linear models across the World Wide Web. These models include simple location problems to complicated ANOVA …

An Evaluation Of Standard, Alternative, And Robust Slope Test Strategies, Tim Moses, Alan Klockars

Journal of Modern Applied Statistical Methods

The robustness and power of nine strategies for testing the differences between two groups’ regression slopes under nonnormality and residual variance heterogeneity are compared. The results showed that three most robust slope test strategies were the combination of the trimmed and Winsorized slopes with the James second order test, the combination of Theil-Sen with James, and Theil-Sen with percentile bootstrapping. The slope tests based on Theil-Sen slopes were more powerful than those based on trimmed and Winsorized slopes.

Spacecraft Proximity Operations Used To Estimate The Dynamical & Physical Properties Of A Resident Space Object, Abraham F. Brunner

Theses and Dissertations

When conducting a space proximity operation, developing high-fidelity estimates of the dynamical and physical properties of a Resident Space Object (RSO) based on post-rendezvous observational data acquired, is imperative for the understanding of the RSO itself and the operating environment. This research investigates the estimation of relative motion dynamics, rotational dynamics, and the feasibility of estimating the moments of inertia of a RSO. Using the Hill-Clohessy-Wiltshire equations, rigid-body dynamics, and estimation theory, a nonlinear least squares estimation algorithm is implemented in the processing of range data from tracked observation points on the RSO body. Through simulation, it was determined that …

Convergence Of Algorithms For Reconstructing Convex Bodies And Directional Measures, Richard J. Gardner, Markus Kiderlen, Peyman Milanfar

Mathematics Faculty Publications

We investigate algorithms for reconstructing a convex body K in Rⁿ from noisy measurements of its support function or its brightness function in k directions u₁, . . . , u_k. The key idea of these algorithms is to construct a convex polytope P_k whose support function (or brightness function) best approximates the given measurements in the directions u₁, . . . , u_k (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian. It is shown that this procedure is (strongly) consistent, meaning that, …

Modeling Inter-Plant Interactions, Jessica Larson

Electronic Theses and Dissertations

The purpose of this paper is to examine the interactions between two plant species endemic to Florida and develop a model for the growth of one of the plant species. An equation for the growth of Hypericum cumulicola is developed through analyzing how the distance to and the height of the nearest Ceratiola ericoides (Florida rosemary) affects the growth of Hypericum cumulicola. The hypericums were separated into five separate regions according to the distance to the nearest rosemary plant. The parameters for a basic growth equation were obtained in each of the five regions and compared to each other along …

Estimating The Slope Of Simple Linear Regression In The Presence Of Outliers, Mohammed Al-Haj Ebrahem, Amjad D. Al-Nasser

Journal of Modern Applied Statistical Methods

In this article, an estimation procedure to simple linear regression in the presence of outliers is proposed. The performance of the proposed estimator, the AM estimator, is compared with other traditional estimators: least squares, Theil type repeated median, and geometric mean. A numerical example is given to illustrate the proposed estimator. Simulation results indicate that the proposed estimator is accurate and has a high precision in the presence of outliers.

Characterization And Properties Of $(R,S)$-Symmetric, $(R,S)$-Skew Symmetric, And $(R,S)$-Conjugate Matrices, William F. Trench

William F. Trench

No abstract provided.

Histospline Method In Nonparametric Regression Models With Application To Clustered/Longitudinal Data, Raymond J. Carroll, Peter Hall, Tatiyana V. Apanasovich, Xihong Lin

The University of Michigan Department of Biostatistics Working Paper Series

Kernel and smoothing methods for nonparametric function and curve estimation have been particularly successful in "standard" settings, where function values are observed subject to independent errors. However, when aspects of the function are known parametrically, or where the sampling scheme has significant structure, it can be quite difficult to adapt standard methods in such a way that they retain good statistical performance and continue to enjoy easy computability and good numerical properties. In particular, when using local linear modeling it is often awkward to both respect the sampling scheme and produce an estimator with good variance properties, without resorting to …