Abstract of master's thesis theme
A research purpose is to find of that minimum size of weight error, which does not have influence on exactness of equalization of networks of poligonometry, and also on the development of some universal graph which would reflect influencing of errors of weights of measurings taking into account concrete network configuration.
Many famous scientists were engaged in research of this problem. The algorithms developed by them lean against the followings generalized criteria of exactness of all of geodesic network on the whole:
- generalized dispersion (det K, where K is correlation matrix of the equated results of measurings or parameters);
- middle weight from the elements of correlation matrix of K;
- spectral norm of matrix of K.
On the basis of the studied materials I want to make an effort to offer other method of research of the problem.
I convey researches on the program Mathcad on the example of simple network of poligonometry of the broken form. The co-ordinates of starting points A and B, direction angles aAC and aBD and middle root-sum-square uncertainties of lengths of lines (5 mm) and corners (2”), are known in this network..
This network is equalized by a parametric method and result matrixes A (coefficients of equalizations of amendments) (Fig. 1), P (weights) and vector L (free members of equalization of amendments) (Fig. 2), utilized in further calculations.
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| Fig. 1. Matrix of coefficients of equalizations of amendments |
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| Fig. 2. Weight matrix and free members vector of equalization of amendments |
To probe influencing weight errors of the measurings I offer to use the matrix of erroneous weights Psl (Fig. 4), which appears by addition of weight matrix P and matrix of errors to the weights DÐ (Fig. 3), generated casual appearance.
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(1) |
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| Fig. 3. Matrix of errors to the weights |
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| Fig. 4. Matrix of erroneous weights |
Using the formula of searching a vector-decision from the parametric equalization method, it is possible to find a vector-decision for network by two methods: with true matrix and erroneous weight matrix. For achievement of research purpose it is necessary to find distinction of these two vectors, to know, as far as the truth values will be displaced by entered errors (Fig. 5).
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(2) |
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| Fig. 5. Distortions of results vector because of weight errors |
To estimate influencing weight errors, it is necessary to know dispersions of these errors, and to find the covariance matrix of errors of results because of weight errors.
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(3) |
We shell find the covariance matrix of truth values co-ordinates Mx.
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(4) |
The matrices Mx and K X are presented on Fig. 6 and Fig. 7.
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| Fig. 6. Covariance matrix of truth values co-ordinates |
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| Fig. 7. Covariance matrix of errors of results because of weight errors |
Thus, we have to find what part does deviation make because of weight errors in the fatal error. We shell find the relation of diagonal elements of matrices ÊDÕ and Mx.
The eventual result of research is the graph, which shows a dependence of maximal weight error because of number of times, in which casual weight vector can differ from true weight vector.
To build the graph, I generated the aggregate of casual weight vectors, which differs from true weight vector on different values (from 0,05 to 100). After necessary calculations, I have got some matrix. The first column of this matrix is a number of times, in which casual vector higher than true vector, and second column is a value of the respective part of error because of weight in a fatal error.
The got graph is presented on a Fig. 8.
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| Fig. 8. - Graph of dependence maximal error because of weight from the number of times, in which casual vector higher than true vector |
On this figure the graph is an aggregate of points. We shell conduct the line of trend with these points.
I have learnt, that weight rejection, which less than 10%, does not have an influence on the results equalization. If in this network the maximal error of element of weight vector distorts the conformable element of true vector less than 4,4 times, errors of weight vector does not have an influence on the results equalization and estimation of exactness.
Further I see the purpose in the search ways of finding some limiting value of part error, which would render minimum influence on the results of equalization and estimation of exactness networks poligonometry. And also I plan to probe influence of weight errors on equalization of networks poligonometry by different configuration and to draw possible general conclusions.
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Authoress
Zakharova Helen |
© DonNTU 2008 Zakharova
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