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.

scholarly journals Variability of linearly evaluated traits of type of simmental bull dams

2007 ◽  
Vol 23 (5-6-1) ◽  
pp. 201-208 ◽  
Author(s):  
V. Pantelic ◽  
Z. Skalicki ◽  
M.M. Petrovic ◽  
S. Aleksic ◽  
B. Miscevic ◽  
...  
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Official Journal ◽  
Body Frame ◽  
Body Type ◽  
Method Of Least Squares ◽  
Evaluation Result ◽  
Secondary Traits ◽  
Method Of Evaluation ◽  
Selection Of

One of the important factors of successful milk production is group of secondary traits: health, longevity, type and milking traits. These traits have become increasingly significant. In order for productive life of cows used in production of milk to be as long and as successful as possible, special attention must be directed to traits of body type and constitution. In selection of cows into the category of bull dams all heads are measured - exterior traits, and evaluated linearly for traits of type or body frame, body muscular characteristics/muscularity, form (appearance) and udder according to Regulation on method of evaluation of traits of breeding livestock (Official journal of RS, No. 21,1996). The effects of rearing region and year of measuring, i.e. evaluating were analyzed in detail using the Least Squares Method (Harvey 1987). Average values of linear evaluations of investigated heads obtained by method of Least Squares were: body frame 7,85; muscularity 7,74; form 7,66; udder 7,62. The effect of region on linear evaluation was highly significant (P<0,01). Evaluation year also influenced highly significantly (P<0,01) obtained evaluation result for form, significantly (P<0,05) obtained evaluation result for body frame and muscularity, but no effect was established on evaluation result for udder.

scholarly journals Method of determining the distance to the object by analyzing its image blur

Vestnik MGSU ◽  
2015 ◽  
pp. 140-151 ◽  
Author(s):  
Aleksey Alekseevich Loktev ◽  
Daniil Alekseevich Loktev
Keyword(s):  
Least Squares ◽  
Statistical Approach ◽  
Evaluation Method ◽  
Least Squares Method ◽  
Focal Length ◽  
Error Variance ◽  
Method Of Least Squares ◽  
Automated Control ◽  
Image Blur

In modern integrated monitoring systems and systems of automated control of technological processes there are several essential algorithms and procedures for obtaining primary information about an object and its behavior. The primary information is characteristics of static and moving objects: distance, speed, position in space etc. In order to obtain such information in the present work we proposed to use photos and video detectors that could provide the system with high-quality images of the object with high resolution. In the modern systems of video monitoring and automated control there are several ways of obtaining primary data on the behaviour and state of the studied objects: a multisensor approach (stereovision), building an image perspective, the use of fixed cameras and additional lighting of the object, and a special calibration of photo or video detector.In the present paper the authors develop a method of determining the distances to objects by analyzing a series of images using depth evaluation using defocusing. This method is based on the physical effect of the dependence of the determined distance to the object on the image from the focal length or aperture of the lens. When focusing the photodetector on the object at a certain distance, the other objects both closer and farther than a focal point, form a spot of blur depending on the distance to them in terms of images. Image blur of an object can be of different nature, it may be caused by the motion of the object or the detector, by the nature of the image boundaries of the object, by the object’s aggregate state, as well as by different settings of the photo-detector (focal length, shutter speed and aperture).When calculating the diameter of the blur spot it is assumed that blur at the point occurs equally in all directions. For more precise estimates of the geometrical parameters determination of the behavior and state of the object under study a statistical approach is used to determine the individual parameters and estimate their accuracy. A statistical approach is used to evaluate the deviation of the dependence of distance from the blur from different types of standard functions (logarithmic, exponential, linear). In the statistical approach the evaluation method of least squares and the method of least modules are included, as well as the Bayesian estimation, for which it is necessary to minimize the risks under different loss functions (quadratic, rectangular, linear) with known probability density (we consider normal, lognormal, Laplace, uniform distribution). As a result of the research it was established that the error variance of a function, the parameters of which are estimated using the least squares method, will be less than the error variance of the method of least modules, that is, the evaluation method of least squares is more stable. Also the errors’ estimation when using the method of least squares is unbiased, whereas the mathematical expectation when using the method of least modules is not zero, which indicates the displacement of error estimations. Therefore it is advisable to use the least squares method in the determination of the parameters of the function.In order to smooth out the possible outliers we use the Kalman filter to process the results of the initial observations and evaluation analysis, the method of least squares and the method of least three standard modules for the functions after applying the filter with different coefficients.

Automatic Point Clouds Registration Based on the Method of Least Squares

2009 ◽  
Vol 419-420 ◽  
pp. 305-308
Author(s):  
Fan Wen Meng ◽  
Lu Shen Wu ◽  
Qing Jin Peng
Keyword(s):  
Coordinate System ◽  
Least Squares ◽  
Least Squares Method ◽  
Local Coordinate System ◽  
Surface Profile ◽  
Point Clouds ◽  
Fringe Projection ◽  
Method Of Least Squares ◽  
Surface Patches ◽  
Euclidean Distances

An object has to be measured to recover its 3D shape in reverse engineering applications. The object surface is sampled point by point using a fringe projection. The method of least squares is used to match overlapping surfaces to estimate transformation parameters between a local coordinate system and the template coordinate system. The Gauss–Markoff model can minimize the sum of squares of Euclidean distances between surfaces for matching arbitrarily oriented 3D surface patches. This research uses the least squares method for the registration of point clouds. A relief example shows the feasibility of the proposed method. It takes about 4 seconds for the registration of 1531209 points with the error less than 0.03mm, and the iteration number is only 20. The surface profile is complete and smooth after the registration, which can meet the requirement of surface reconstruction.

scholarly journals TREATMENT OF GEODETIC SURVEY DATA AS FUZZY VECTORS

2015 ◽  
Vol XL-3/W3 ◽  
pp. 29-33
Author(s):  
G. Navratil ◽  
E. Heer ◽  
J. Hahn
Keyword(s):  
Normal Distribution ◽  
Least Squares ◽  
Survey Data ◽  
Measurement Errors ◽  
Least Squares Method ◽  
Measurement Data ◽  
Geodetic Survey ◽  
Present Measurement ◽  
A Value ◽  
Light Signals

Geodetic survey data are typically analysed using the assumption that measurement errors can be modelled as noise. The least squares method models noise with the normal distribution and is based on the assumption that it selects measurements with the highest probability value (Ghilani, 2010, p. 179f). There are environment situations where no clear maximum for a measurement can be detected. This can happen, for example, if surveys take place in foggy conditions causing diffusion of light signals. This presents a problem for automated systems because the standard assumption of the least squares method does not hold. A measurement system trying to return a crisp value will produce an arbitrary value that lies within the area of maximum value. However repeating the measurement is unlikely to create a value following a normal distribution, which happens if measurement errors can be modelled as noise. In this article we describe a laboratory experiment that reproduces conditions similar to a foggy situation and present measurement data gathered from this setup. Furthermore we propose methods based on fuzzy set theory to evaluate the data from our measurement.

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.

Total Least-Squares Methods for Active View Registration of Three-Dimensional Line Laser Scanning Data

2004 ◽  
Vol 127 (1) ◽  
pp. 50-56 ◽  
Author(s):  
F. Xi ◽  
D. Nancoo ◽  
G. Knopf
Keyword(s):  
Least Squares ◽  
Laser Scanning ◽  
Least Squares Method ◽  
Three Dimensional ◽  
Measurement Data ◽  
Ordinary Least Squares ◽  
Total Least Squares ◽  
Position Measurement ◽  
Point Position ◽  
Ordinary Least Squares Method

In this paper a method is proposed to register three-dimensional line laser scanning data acquired in two different viewpoints. The proposed method is based on three-point position measurement by scanning three reference balls to determine the transformation between two views. Since there are errors in laser scanning data and sphere fitting, the two sets of three-point position measurement data at two different views are both subject to errors. For this reason, total least-squares methods are applied to determine the transformation, because they take into consideration the errors both at inputs and outputs. Simulations and experiment are carried to compare three methods, namely, ordinary least-squares method, unconstrained total least-squares method, and constrained total least-squares method. It is found that the last method gives the most accurate results.

Why we should give up the sin2ψ method

2009 ◽  
Vol 24 (S1) ◽  
pp. S16-S21 ◽  
Author(s):  
Balder Ortner
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Linear Equations ◽  
Single System ◽  
Stress Determination ◽  
Method Of Least Squares ◽  
X Ray ◽  
Stress Tensors ◽  
Course Of Action ◽  
Simultaneous Linear Equations

The sin2ψ method can be formulated as a single system of simultaneous linear equations. Using this it is easy to show that the sin2ψ method is not a least-squares method. It further helps to compare the accuracies of the stress tensors obtained by the sin2ψ method and the method of least squares. Quantitative comparisons have been made for different fictitious measurements. It is shown that the unnecessary loss in accuracy by using the sin2ψ method is quite significant and by no means negligible. The same course of action has been applied to compare the so-called Dölle-Hauk method with a least-squares method; the result is similar. Some other methods for X-ray stress determination, most often similar to the sin2ψ method, and their shortcomings are also discussed briefly, together with the corresponding, more effective and more versatile least-squares method.

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