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متد کمترین مربعات با استفاده از جبر ماتریس ها حالت (معادلات مشاهدات ) | ||
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| مقاله 8، دوره 30، شماره 0 - شماره پیاپی 1000194، دی 1353 اصل مقاله (460 K) | ||
| نویسنده | ||
| مهندس علی اصغر شریفی* | ||
| عنوان مقاله [English] | ||
| Least Squares Adjustment with the use of Matrix Algebra Method A: Observation Equations Summary | ||
| چکیده [English] | ||
| Statistics has proved that when you increase number of observations the mean of accidental errors goes to zero, in other words with statistical terms we can say the expected value of accidental errors is zero n ?i E{?}=lim =0 . n Increasing the number of observed quantities will cause another problem, we will have more equations than it is necessary to solve for unknown parameters. Here is the point that mathematical solution differs from computation in experimental sciences. The later method is the least squares solution. In this me thod all the observed quantities are used in such a way that if we find differences between observed quantities and the adjusted values of these quantities (residuals), the sum of squares of residuals be minimum. We can summarize this method by it's two major characteristics: 1-The solution is unique and independent of the path, and the order of equations. 2 - The sum of squares of residuals is minimum. The classic method of least squares (as we call it) can be found in such books like geodesy, mathematical statistics, and numerical analysis. With the application of matrix algebra we present the method of least squares adjustment in different way, however the principle is the same, but it does have some advantages. Let us summarize weaknesses of the classical method as bellow: 1 - With classical method computation is slow and uneconomical. 2 - The simultaneous adjustment of large networks is very difficult and actualy imposible, therefore the network have to be devided in blocks, and adjust each block respect to the previous one. This will cause a kind of controversy with the first principle, because the final results depend on the path and the starting point of adjustment. 3 - It is difficult to analyse the accuracy of computation, and the accuracy of the adjusted values. 4 - Enforcing the effect of correlated observations (the effect of covariances) is difficult and actually imposible. In the method of least squares adjustment with the use of matrix algebra most of the computations are systematic kind wich can be done by electronic computers very fast. There are only two parts which shoud have to be found or estimated by us: 1-To estimate a weight matrix (P) or variance-covariance matrix of observations ( Lb= . P ) by the informations we have about instrument, operator, and outer circumstances. 2-To find and write down a suitable mathematical structure (mathematical model). MATHEMATICAL MODEL Mathematical model is some functions that exist between observations and parameters. The mathematical model for observation equations is a special kind, and that is when each observed quantity can be written as a function of some parameters. La=f(Xa) Where X, is a vector of adjusted parameters with u components (number of parameters), and La is avector of adjusted observations with n (number of observations) components. The other notations that we are going to use are: Xo , the initial approximate values of parameters chosen arbitrarily before adjustment. The differences of Xa, and Xo , or the variation of parameters are X=X -X The measured quantities are denoted by vector L with n components using the following subscripts: Lb , the measured values, La' the adjusted values, Lo , the values computed from values X, through the mathematical model Lo=f(xo). The differences of these sets are denoted as : L=LO-Lb V=La-Lb residuals. We can rewrite the mathematical model (I) interms of L, and X, Lb+V=F(Xo-X) After expanding it we get Lb+V=F(XO)+AX where A is a matrix with n rows and u columns, the elements of A are partial derivatives of measured quantities with respect to the parameters. A X =XO Since Lo=F(Xo), and L=Lo-Lb equation (2) can be wirtten as V=AX+L The principle of lea | ||
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