An existence level for the residual sum of squares of the power-law regression with an unknown location parameter

In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate.

Double Differencing by Demeaning: Applications to Hypocenter Location and Wavespeed Tomography

ABSTRACT Double differencing of body-wave arrival times has proved to be a useful technique for increasing the resolution of earthquake locations and elastic wavespeed images, primarily because (1) differences in arrival times often can be determined with much greater precision than absolute onset times and (2) differencing reduces the effects of unknown, unmodeled, or otherwise unconstrained variables on the arrival times, at least to the extent that those effects are common to the observations in question. A disadvantage of double differencing is that the system of linearized equations that must be iteratively solved generally is much larger than the undifferenced set of equations, in terms of both the number of rows and the number of nonzero elements. In this article, a procedure based on demeaning subsets of the system of equations for hypocenters and wavespeeds that preserves the advantages of double differencing is described; it is significantly more efficient for both wavespeed-only tomography and joint hypocenter location-wavespeed tomography. Tests suggest that such demeaning is more efficient than double differencing for hypocenter location as well, despite double-differencing kernels having fewer nonzeros. When these subsets of the demeaned system are appropriately scaled and simplified estimates of observational uncertainty are used, the least-squares estimate of the perturbations to hypocenters and wavespeeds from demeaning are identical to those obtained by double differencing. This equivalence breaks down in the case of general, observation-specific weighting, but tests suggest that the resulting differences in least-squares estimates are likely to be inconsequential. Hence, demeaning offers clear advantages in efficiency and tractability over double differencing, particularly for wavespeed tomography.

A new uncertain linear regression model based on slope mean

The least squares estimate can fully consider the given data and minimize the sum of squares of the residuals, and it can solve the linear regression equation of the imprecisely observed data effectively. Based on the least squares estimate and uncertainty theory, we first proposed the slope mean model, which is to calculate the slopes of expected value and each given data, and the average value of these slopes as the slope of the linear regression equation, substituted into the expected value coordinates, and we can get the linear regression equation. Then, we proposed the deviation slope mean model, which is a very good model and the focus of this paper. The idea of the deviation slope mean model is to calculate the slopes of each given data deviating from the regression equation, and take the average value of these slopes as the slope of the regression equation. Substituted into the expected value coordinate, we can get the linear regression equation. The deviation slope mean model can also be extended to multiple linear regression equation, we transform the established equations into matrix equation and use inverse matrix to solve unknown parameters. Finally, we put forward the hybrid model, which is a simplified model based on the combination of the least squares estimation and deviation slope mean model. To illustrate the efficiency of the proposed models, we provide numerical examples and solve the linear regression equations of the imprecisely observed data and the precisely observed data respectively. Through analysis and comparison, the deviation slope mean model has the best fitting effect. Part of the discussion, we are explained and summarized.

Total Least-Squares Determination of Body Segment Attitude

To examine segment and joint attitudes when using image based motion capture it is necessary to determine the rigid body transformation parameters from an inertial reference frame to a reference frame fixed in a body segment. Determine the rigid body transformation parameters must account for errors in the coordinates measured in both reference frames, a total least-squares problem. This study presents a new derivation that shows that a singular value decomposition based method provides a total least-squares estimate of rigid body transformation parameters. The total least-squares method was compared with an algebraic method for determining rigid body attitude (TRIAD method). Two cases were examined: Case 1 where the positions of a marker cluster contained noise after the transformation, and Case 2 where the positions of a marker cluster contained noise both before and after the transformation. The white noise added to position data had a standard deviation from zero to 0.002 m, with 101 noise levels examined. For each noise level 10000 criterion attitude matrices were generated. Errors in estimating rigid body attitude were quantified by computing the angle, error angle, required to align the estimated rigid body attitude with the actual rigid body attitude. For both methods and cases as the noise level increased the error angle increased, with errors larger for Case 2 compared with Case 1. The SVD based method was superior to the TRIAD algorithm for all noise levels and both cases, and provided a total least-squares estimate of body attitude.

Tracking—A

This chapter explains how to estimate an unobserved random variable or vector from available observations. This problem arises in many examples, as illustrated in Sect. 9.1. The basic problem is defined in Sect. 9.2. One commonly used approach is the linear least squares estimate explained in Sect. 9.3. A related notion is the linear regression covered in Sect. 9.4. Section 9.5 comments on the problem of overfitting. Sections 9.6 and 9.7 explain the minimum means squares estimate that may be a nonlinear function of the observations and the remarkable fact that it is linear for jointly Gaussian random variables. Section 9.8 is devoted to the Kalman filter, which is a recursive algorithm for calculating the linear least squares estimate of the state of a system given previous observations.

Implementation of GRACE and GRACE-FO observation covariance estimates at JPL

<p class="Standard">Developing meaningful uncertainty quantifications for GRACE or GRACE-FO derived products, e.g. water storage anomalies, requires a robust understanding of the information and noise content in the observables employed in their estimation.</p> <p class="Textbody">The stochastic models for GRACE and GRACE-FO K-Band, and GPS carrier phase and pseudorange observables employed in upcoming JPL solutions will be presented. Within these models, the time-domain correlations for each of the observations are estimated, and then applied in the least squares estimate of monthly gravity field solutions. Reproducing results from other groups, the resulting formal errors of monthly solutions are improved.</p> <p class="Standard">We compare this approach to the current state of the art at JPL, and show that noise content in the determined gravity field solutions is reduced. We further demonstrate the application of this method to data from the GRACE-FO Laser ranging interferometer.</p>

Implementation of GRACE and GRACE-FO observation covariance estimates at JPL

<p>Developing meaningful uncertainty quantifications for GRACE or GRACE-FO derived products, e.g. water storage anomalies, requires a robust understanding of the information and noise content in the observables employed in their estimation.</p><p>The stochastic models for GRACE and GRACE-FO K-Band, LRI, and GPS carrier phase and pseudorange observables employed in upcoming JPL solutions, along with notes on their implementation and development, will be presented. Within these models, the time-domain correlations for each of the observations are estimated, and then applied in the least squares estimate of monthly gravity field solutions. Reproducing results from other groups, the resulting formal errors of monthly solutions are improved.</p><p>It is envisioned that possible new Level 3 products can make these improved uncertainty quantifications accessible to the GRACE user community at large. Possible specifications for such products will be presented, and feedback from the community and discussion will be appreciated.</p>

Exponential Autoregressive Parameters Estimation

The purpose of the research was to compare the least squares estimatate and the least absolute deviation estimate depending on the probability distribution of the renewal process of the autoregressive equation. To achieve this goal, the sequence of observations of the exponential autoregressive process was repeatedly reproduced using computer simulation, and the least squares estimate and the least absolute deviation estimate were calculated for each sequence. The resulting estimation sequences were used to calculate the sample variances of the least squares estimate and the least absolute deviation estimate. The best estimate was the one with the lowest sample variance. The quantitative measure for the estimates comparison was the sample relative efficiency of estimates, defined as the inverse ratio of their sample variances. Normal distribution, contaminated normal distribution, i.e. Tukey distribution, with different values of the proportion and intensity of contamination, logistic distribution, Laplace distribution and Student distribution with different degrees of freedom, in particular, with one degree of freedom, that is, Cauchy distribution, were used as models of probability distribution of the renewal process. For each probability distribution, asymptotic values of the sample relative efficiency were obtained with an unlimited increase in the sample size of the observations of the autoregressive process. Findings of research show that the least absolute deviation estimate is better than the least squares estimate for Laplace distribution and the contaminated normal distribution with sufficiently large levels of the proportion and intensity of contamination. In other cases, the least squares estimate is preferable.