The primary objects of this essay are twofold (a) to expose certain tacit insidious potential sources of confusion which seem to permeate many writings on the Method of Least Squares, and (b) to deduce compact rigorous formulas for all of the important cases that can arise in the adjusting of a straight line to a set of observed points in two dimensions.
The first problem solved in detail is: To determine the parameters (X,Y) of the equation Xx+Yy+1=0 so as to make the sum of the weighted squares of the “residuals” a minimum with respect to a set of observed data x=xi, y=yi; i=1, 2, 3,…n. Two weights, wx and wy, are used. Several new formulas are obtained in this connection.
Then the geometrical interpretation of the preceding pure analysis is presented. It is shown that (under the stated conditions) the representative straight line always passes through the centroid of the observed points. The transformation from an unequally weighted (wx>wy or wx<wy) plane to an equally weighted (wx=wy) plane is explained.
The classical rules for the formation of normal equations from observational expressions involving the parameters linearly are interpreted and applied to illustrative cases. The method of approximation to be followed when the parameters are not involved linearly in the original functions is discussed and employed in concrete examples.
The last general problem solved in full is: To find formulas for the parameters and other determining quantities of the equation of a straight line which passes through the fixed point (xo,yo) and among the points of an observed set (xi,yi) in such a manner as to make the sum of the weighted squares of the residuals a minimum. This line does not pass through the centroid of (xi,yi), in general.
Finally, the meaning of the term “curve fitting” is subjected to mild analysis.
© 1923 Optical Society of AmericaFull Article | PDF Article
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