scholarly journals The Use of Decomposition SVD to Approximate a Surface

2017 ◽  
Author(s):  
Edward Preweda
Keyword(s):  
Least Squares ◽  
Covariance Matrix ◽  
Classical Method ◽  
Interval Estimation ◽  
Point Of View ◽  
Method Of Least Squares ◽  
Practical Point ◽  
Estimated Parameters ◽  
Main Emphasis ◽  
Special Value

This paper describes the procedure of determining the parameters of the approximating function of the surface using a distribution of special value. From a practical point of view, an important issue is to determine the covariance matrix of the estimated parameters. Interval estimation was carried out and a methodology to obtain an optimal equation of approximating surface was presented. The main emphasis was given to the elimination of these parameters functions that generate unnecessary disturbance. The results obtained using the approximate decomposition SVD were compared with those obtained by a classical method of least squares. They have been used a variety of software, including software written by author, also packages Matlab and Statistica. The main purpose of discussion is to solve sample tasks for better understanding and expand the use of decomposition SVD in geodetic issues.

scholarly journals On the reliability of errors-in-variables models

2012 ◽  
Vol 16 (1) ◽  
pp. 69-81 ◽  
Author(s):  
Burkhard Schaffrin ◽  
Sibel Uzun
Keyword(s):  
Markov Model ◽  
Least Squares ◽  
Covariance Matrix ◽  
Total Least Squares ◽  
Maximum Effect ◽  
Errors In Variables ◽  
Reliability Measures ◽  
Estimated Parameters ◽  
Individual Observation ◽  
Original Approach

Reliability has been quantified in a simple Gauss–Markov model (GMM) by Baarda (1968) for the application to geodetic networks as the potential to detect outliers – with a specified significance and power – by testing the least-squares residuals for their zero expectation property after an adjustment assuming “no outliers”. It was shown that, under homoscedastic conditions, the so-called “redundancy numbers” could very well serve as indicators for the “local reliability” of an (individual) observation. In contrast, the maximum effect of any undetectible outlier on the estimated parameters would indicate “global reliability”. This concept had been extended successfully to the case of correlated observations by Schaffrin (1997) quite a while ago. However, no attempt has been made so far to extend Baarda’s results to the (homoscedastic) errors-in-variables (EIV) model for which Golub and van Loan (1980) had found their – now famous – algorithm to generate the total least-squares (TLS) solution, together with all the residuals. More recently, this algorithm has been generalized by Schaffrin and Wieser (2008) to the case where a truly – not just elementwise –weighted TLS solution can be computed when the covariance matrix has the structure of a Kronecker–Zehfuss product. Here, an attempt will be made to define reliability measures within such an EIV-model, in analogy to Baarda’s original approach.

scholarly journals XII.—On the Law of Frequency of Error

1865 ◽  
Vol 24 (1) ◽  
pp. 139-145 ◽  
Author(s):  
Tait
Keyword(s):  
Least Squares ◽  
Point Of View ◽  
Method Of Least Squares ◽  
Natural Manner ◽  
The Real ◽  
Real Nature ◽  
The Subject ◽  
The Law

It has always appeared to me that the difficulties which present themselves in investigations concerning the Frequency of Error, and the deduction of the most probable result from a large number of observations by the Method of Least Squares (which is an immediate consequence of the ordinary “Law of Error”), are difficulties of reasoning, or logic, rather than of analysis. Hence I conceive that the elaborate analytical investigations of Laplace, Poisson, and others, do not in anywise present the question in its intrinsic simplicity. They seem to me to be necessitated by the unnatural point of view from which their authors have contemplated the question. It is, undoubtedly, a difficult one; but this is a strong reason for abstaining from the use of unnecessarily elaborate analysis, which, however beautiful in itself, does harm when it masks the real nature of the difficulty it is employed to overcome. I believe that, so far at least as mathematics is concerned, the subject ought to be found extremely simple, if we only approach it in a natural manner.

scholarly journals Creation of a Program for Finding an Approximating Function of the Obtained Experimental Results by the Method of Least Squares

2021 ◽  
Vol 7 (12) ◽  
pp. 367-372
Author(s):  
N. Kadyrkulova ◽  
V. Zhulev
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Measurement Data ◽  
Point Of View ◽  
Main Task ◽  
Method Of Least Squares ◽  
Empirical Formulas ◽  
Physical Experiments ◽  
Mathematical Relationships ◽  
Selection Of

When solving engineering and economic problems, it is often necessary to obtain mathematical relationships between various parameters characteristic of a given problem. As a rule, all physical experiments are reduced to measuring the dependence of a certain quantity u on one or several other quantities z1, z2,…, zn. The main task of using the least squares method as an approximation method from the point of view of approximate recovery of a function from its known values at a number of points is the selection of empirical formulas that allow an analytical presentation of the obtained experimental measurement data. This article discusses the problems of obtaining data and approximating a function by the least squares method using OOP.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

Tertiary Nitrification Pilot Plants on Parisian Waste Water

1990 ◽  
Vol 22 (1-2) ◽  
pp. 347-352 ◽  
Author(s):  
C. Paffoni ◽  
B. Védry ◽  
M. Gousailles
Keyword(s):  
Waste Water ◽  
Metropolitan Area ◽  
Fixed Bed ◽  
Point Of View ◽  
Operating Conditions ◽  
Research Center ◽  
Daily Output ◽  
Practical Point ◽  
Fixed Bed Reactors

The Paris Metropolitan area, which contains over eight million inhabitants, has a daily output of about 3 M cu.meters of wastewater, the purification of which is achieved by SIAAP (Paris Metropolitan Area Sewage Service) in both Achères and Valenton plants. The carbon pollution is eliminated from over 2 M cu.m/day at Achères. In order to improve the quality of output water, its tertiary nitrification in fixed-bed reactors has been contemplated. The BIOFOR (Degremont) and BIOCARBONE (OTV) processes could be tested in semi-industrial pilot reactors at the CRITER research center of SIAAP. At a reference temperature of 13°C, the removed load is approximately 0.5 kg N NH4/m3.day. From a practical point of view, it may be asserted that in such operating conditions as should be at the Achères plant, one cubic meter of filter can handle the tertiary nitification of one cubic meter of purified water per hour at an effluent temperature of 13°C.

scholarly journals On the Design of Outdoor Leader-Follower UAV-Formation Controllers from a Practical Point of View

IEEE Access ◽  
2021 ◽  
pp. 1-1
Author(s):  
Lucas Vago Santana ◽  
Alexandre Santos Brandao ◽  
Mario Sarcinelli-Filho
Keyword(s):  
Point Of View ◽  
Practical Point

scholarly journals Functional ratings in sports

2020 ◽  
Vol 16 (3) ◽  
pp. 183-191
Author(s):  
Brad Lowery ◽  
Abigail Slater ◽  
Kaison Thies
Keyword(s):  
Least Squares ◽  
Method Of Least Squares ◽  
Sports Teams ◽  
New Model ◽  
College Basketball ◽  
Division 1

AbstractIn this paper, we present a new model for ranking sports teams. Our model uses all scoring data from all games to produce a functional rating by the method of least squares. The functional rating can be interpreted as a team average point differential adjusted for strength of schedule. Using two team’s functional ratings we can predict the expected point differential at any time in the game. We looked at three variations of our model accounting for home-court advantage in different ways. We use the 2018–2019 NCAA Division 1 men’s college basketball season to test the models and determined that home-court advantage is statistically important but does not differ between teams.

Quantitative Analysis of Microstructure and Modeling of Sintering

2009 ◽  
Vol 624 ◽  
pp. 1-18 ◽  
Author(s):  
Jean Marc Chaix
Keyword(s):  
Image Analysis ◽  
Input Data ◽  
3D Imaging ◽  
Point Of View ◽  
Size Distributions ◽  
Practical Point ◽  
The Past ◽  
Modeling Techniques ◽  
Mean Size ◽  
Sintering Mechanisms

Microstructure is the key scale to understand and describe sintering mechanisms and their consequences at the macroscopic level. As modeling techniques are continuously developing, the need for input data and comparison with more and more accurate descriptions of the evolution is expected to create a growing demand for quantitative microstructure data. Image analysis is the classic way to get these data. This paper reviews the practical use and progresses of this old technique in the sintering literature during the past and recent years. The place of basic tools and more recent ones, such as 3D imaging, are discussed from a practical point of view accounting from sintering models needs: mean size and size distributions in pores and grains, homogeneity, sintering trajectories…

scholarly journals Historical Note on the Method of Least Squares

Nature ◽  
1872 ◽  
Vol 6 (136) ◽  
pp. 101-102
Author(s):  
ASAPH HALL
Keyword(s):  
Least Squares ◽  
Historical Note ◽  
Method Of Least Squares
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