LIII. On lines and planes of closest fit to systems of points in space

K Pearson - The London, Edinburgh, and Dublin philosophical …, 1901 - Taylor & Francis
K Pearson
The London, Edinburgh, and Dublin philosophical magazine and journal …, 1901Taylor & Francis
(1)-] IN many physical, statistical, and biological investil_ gations it is desirable to represent
a system of points in phme, three, or higher dhnensioned space by the" best-fits straight line
or plane. Analytically this consists in taking y= ao+ alx, or Z= ao+ alx+ b, y, or z= ao+ a~ xl+
a2x 2+ aaxa+ 9 9 9+ at, xn, where y, x, z, xl, x,~,.., x~ are variables, and determining the"
best" values for the constants a0, a,, bl, a0, al, a2, as, 9 9 9 as in relation to the observed
corresponding values of the variables.] n nearly all the cases dealt with in the text-books of …
(1)-] IN many physical, statistical, and biological investil_ gations it is desirable to represent a system of points in phme, three, or higher dhnensioned space by the" best-fits straight line or plane. Analytically this consists in taking y= ao+ alx, or Z= ao+ alx+ b, y, or z= ao+ a~ xl+ a2x 2+ aaxa+ 9 9 9+ at, xn, where y, x, z, xl, x,~,.., x~ are variables, and determining the" best" values for the constants a0, a,, bl, a0, al, a2, as, 9 9 9 as in relation to the observed corresponding values of the variables.] n nearly all the cases dealt with in the text-books of least squares, the variables on the right of our equations are treated as the independent, those on the left as the dependent variables. The result of this treatment is that we get one straight line or plane if we treat some one variable as independent, and a quite different one if we treat another variable as the independent variable. There is no paradox about this; it is, in ihct, an easily understood and most important feature of the theory of a system of correlated variables. The most probable value of y for a given value of x, say, is not given by the s~ me relation as the most probable value of x tbr a given value of y. Or, to take a concrete example, the most probable stature of a man with a given length of leg I being s, the most probable length of leg for a man of stature s will not be I. The" best-fitting" lines and planes for the cases of z up to n variables for a correlated system are given in my memoir on regression t. They depend upon a determination of the means~ standard-deviations, and correlation-coefficients of the system. In such cases the values of the independent variables are supposed to be accurately known, and the probable value of the dependent variable is a~ certained.(2) In many cases of physics and biology, however, the" independent" variable is subject to just as much deviation or error as the~'dependent" variable. We do not, for example, know x accurately and then proceed to find y, but both x and y are found by experiment or observation. We observe x and y and seek tbr a unique functional relation between them. Men of given stature may have a variety
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