Doğrusal uzaylarda en iyi yaklaşım ve jeodezik uygulamaları

Abstract

En küçük karelerle kestirim yöntemi Jeodezide olduğu gibi diğer uygulamalı bilimlerde de, 19 uncu yüzyılın başından bu yana bilinmekte ve yaygın kullanılmaktadır. Ancak en bilinen ve yaygın yöntem olması nedeniyle, genellikle diğer kestirim yöntemleri gözardı edilmekte ve kestirimin yalnızca en küçük karelerle yapıldığı yanlış kanısı uyan maktadır. Bunun yanısıra, en küçük karelerin klasik yöntem ile bulunan ve anlamları açık olmayan çözüm formülleri en küçük kareleri soyut ol maktan ileri götürememektedir. Bu çalışmada, kestirim problemi doğrusal uzaylarda en iyi yakla şım olarak ele alınmakta ve ayrıca en küçük karelerin, norm tanımına bağlı oluşturulan, sonsuz sayıdaki kestirim yönteminden yalnızca biri olduğu gösterilmektedir. En küçük kareler geometrik ve diagramatik yaklaşımla ele alın makta ve Hilbert uzaylarında somut çözümleri verilmektedir. Geometrik yaklaşımda; Hilbert uzayında tanımlı izdüşüm teoremi ile, en küçük ka relerin minimum norm koşulunu sağlayan bir çözüm verdiği kanıtlanmak tadır. Ayrıca en küçük kareler, geniş bir görüş açısından ve bütün olarak diagramatik yaklaşımla incelenmekte ve çözümlerin kolayca bulun masında kullanılan diagramlar hazırlanmaktadır. Son olarak, günümüzde etkin biçimde kullanılan genel invers'in Hilbert uzaylarında geometrik yorumu verilmektedir.

This study is entitled "Best Approximation in Linear Spaces and its Geodetic Applications" since estimation theory is detterministicaly investigated with the help of functional analysis avoiding the probabilistic approach. It can be undertood from the title that the main tools of this study are linear spaces, and their specialised types such as metric spaces, normed linear spaces, Banach spaces, inner product spaces, Euclide Spaces and Hubert spaces. As it is well - known that a linear space consists of a set of elements (vectors, points-} x,y,z.... And two operations, addition of elements and multiplication of an element by a scalar are defined in a linear space. A metric space can be formed by defining a one-valued, non-negative, real function p(x,y), namely metric func tion for a pair of elements x,y£X.. Furthermorş, a normed linear space is obtained with the definition of a fonctional for ¥x6X called norm, |jx

, instead of metric function in X. In fact, a metric space is also a normed linear space since. a metric function can be assumed as a norm or vice versa. However,.we are mainly concerned with a complete normed linear space X, called Banach space, wherein each Cauchy series {x } has a limit in the sense of lim {x } ->. x. n n.*¦ °° x6X In respect of best approximation, the concept of completeness is the most important property of a Banach space. Let X be a normed linear space, then two examples of Banach spaces % and L with suitable norms are given below, ; P p llxfl { Z |x|P> i=l i/p x6X Kp<« l|x|!.={/ |x|p dx}1^ x?X y a The functional, called inner product ?dot product, scalar product), on a linear space X, (x,y) x,y£X defines an inner product space. Finite and infinite dimensionals of complete inner product spaces are called Euclide and Hilbert spaces respectively. However, there is no agreement between mathematicians on the dimension of a Hilbert space, and a finite dimensional complete inner product space is accepted as Hilbert space as well, vixi One of the main objectives of this study is the best approximation in Banach spaces and in Hubert spaces with particular interest. After a brief explanation of the required spaces, now the best approximation in a normed linear space can be expressed as follows; Let Y be a subspace of a normed linear space X. Then given x£X\Y exists y?Y, called best approximant of x, satisfies

x-y

= min

x-y

y?Y the minimum norm (distance) condition. The best approximation problem can be investigated with particular regard to. the existence. of best approximations. the uniqueness of best approximations. the characterization of best approximations. the construction of methods for determining best approximations. However here,. the problem is considered in detail only with regards to the existence and the uniqueness of best approximations. Furthermore, the computation methods of best approximations in Banach spaces R!P and RT investigated and applicable formulae are derived. In addition, a test levelling network is adjusted separately in Banach spaces RS1 and in Hubert spaces R£, only to make the best a approximation transparent. A.new outlier detection method with the best approximation in R1!? is improved as well. Comparisons ix between the new method and data snooping given by Baarda (1968) point out that the new method can be succesfully applied in practice. Best approximation in Hubert spaces is the main topic that. investigated in two different approaches; geometrical and diagramatical. "The ualassical solutions of various least squares estimation methods, namely adjustment of indirect observations, adjustment of condition observations, combined adjustment and unified (generalized) adjustment, are also derived separately by using geometrical and diagramatical approaches. The geometrical approach depends upon the projection theorem that may be expressed as follows; Classical Projection Theorem; Let. M be a closed subspace of a Hubert space H. Corresponding to x 6 M, there exists a unique element m 6 M such that x - mo

= min

x - m| m ? M Furthermore, a necessary and sufficient condition that m ? M be a unique minimizing element is that (x-m ) be orthogonal to M. ° Another expression of the projection theorem using an auxilary concept the linear variety is given below. Restatement of Projection Theorem \ Let M be a closed subspace of a Hubert space H.Let x be a fixed element in H and let V be the linear variety x + M. Then there is a unique element x £ V of minimum norm. Furthermore, x is orthogonal to M. °. I In addition, the property that a Hubert space H can be represented as-, the. direct sum of a subspace M and of its orthocomplement M (H » M © M ) is also essential for the geometrical approach. Similarly, ¥xcH can be represented as the sum of two orthogonal elements m?M and m*ÇM as follows; m + m where (m,m ) = 0 The solutions, identical to classical solutions, for various kinds of the best approximation in Hubert spaces are derived by using the projection theorem and the benefits of the above kind representations. It is also proved that the classical solutions satisfy the minimum norm (distance) condition. Then it can be said that the geometrical approach makes the abstract classical least squares solutions concrete. Contrary to the geometrical approach, the diagramatic approach provides a bird's-eye view on least squares estimation. Furthermore, the diagramatical. approach is flexible since'the solutions depending upon the data available can be easily derived...The diagramatic approach is outlined below. Let X and Y be two Hubert spaces. 'The diagrams used in this approach are constituted with linear operators providing transformations mutually between each other of X, Y, XX, Yx. Xx and Y* denote -dual spaces of X and Y respectively. As it is well-known that a dual space Xx consists of all linear functionals f which are linear operators defined between a Hilbert space X and the space of real numbers R. The Frechet-Riesz Theorem expressed below may be -. accepted as the key of the diagramatic approach. xi The Frechet-Riesz Theorem: If f?Hx be a linear functional on a Hubert space H, there exists a unique element y?H such that for all x?H f (x) = (y, x) Furthermore, we have l|f|İHX=lly|lH and every y determines a unique linear functional in this way. i The element Y H corresponding to f?H is called the representer. The relation between a linear functional f and its representer y may be restated as follows; y = J"1 f f = JH y x -1 where JT is a duality operator from H to H. JH denotes the inverse operator of JH. In the diagramatic approach, Jjj is accepted as identical to the weight matrix of unknowns or observations P which is assumed as a linear operator. The definition of a suitable duality operator is essential in order to create a Hubert space which match to a special least squares estimation. Therefore, the duality operator may be considered as the key operator. In the diagramatic approach, first, various least squares estimation methods, considered geometricaly as well, are investigated. Then, identical solutions to the classical solutions are derived by using the linear operators. Next, related diagrams are formed and the same formulae are derived diagramaticaly. Finally, various kinds of generalized inverses (g-inverses), namely reflexive, minimum norm, least squares and minimum norm least squares are investigated geometrically in Hubert spaces.