Least squares, method theory

It is very important to obtain accurately the first 2 constants (Ci and C2 or Ki and K ) from the experimental data in order to compare them with those calculated from the molecular-statistical theory. Here, Cl and C2 constants were determined from the experimental isotherm by the least squares method with the aid of a computer. However, for practical purposes, it is necessary to describe the isotherms to a higher adsorption level. From Figures 3 and 6, it can be seen that in order to reproduce the isotherm up to — 75% saturation of the adsorbent and to calculate a at different values of p and T, it is sufficient to determine only 2 more constants, C3 and C4. 

The treatment of statistics is focused on explicit applications of both linear and nonlinear least-squares methods, rather than on the alphabet soup (F, Q, R, T, etc.) of available tests. However, within that rather narrow framework, many practical aspects of error analysis and curve fitting are considered. They are chosen to illustrate the now almost two centuries old dictum of de Laplace that the theory of probability is merely common sense confirmed by calculation. 

The coefficients (/30.. . . / i ) were unknown, and they could be calculated by the least squares method as suggested by the regression theory in order to obtain the best estimate. This calculation was performed on a UNIVAC 1106 computer by the stepwise method perfected by M. A. Efroym-... 

It is a calibration method in conventional photogrammetry to solve Eq. (7) with the theory of the least squares method. On the other hand, in the case where distances are used as geodetic data, the distance Dij between the points Pi and Pj is written... 

Solving Eqs. (7) and (10) simultaneously by the least squares method is the theory of the method for simultaneous adjustment. Assuming the correction value for the ground coordinates is the difference between the most probable value and the approximate value, the least squares equation can, in this case, be written as follows ... 

Huyberechts, S., A. Halleux, and P, Kruys,Bu//. Soc. Chim. Beiges, 64, 203 (1955). Linnik, Yu. V., Me tod Naimenshikh Kvadratov i Osnovy Matematichesko-Statisticheskoi Teorii Obrabotki Nablyudenii (Method of Least Squares and Principles of Mathematico-Statistical Theory of Data Processing), Gos. Izdatelstvo Fiz. Mat. Literatury, Moscow, 1958. 

Bennett KP, Embrechts MJ. An optimization perspective on kernel partial least squares regression. In Suykens JAK, Horvath G, Basu S, Micchelli J, Vandewalle J, editors. Advances in learning theory methods, models and applications. Amsterdam lOS Press, 2003. p. 227-50. 

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