be as small as possible. Substitute this value of Σ2 in the type equations (viii.), and we find the required values of ,..., using (iv.). This is the complete analytical solution of the problem of drawing the best-fitting plane through a non-coplanar points. We see that it depends only on a knowledge of the means, standard-deviations, and correlations of the q variables. Whenever we may suppose that variation is due to errors of observation or measurement,-i. e., is not organic, but there exists a unique functional relation between the true values of the variables,--then, assuming it of the first degree, we may determine the best values of the constants in the manner given above. (3) A geometrical interpretation is of course to be found. from (viii.) bis. Consider the quadric o2x ̧x} 2 + σ 2x ̧‚xq2 + . . . + o2 xqx q2 + 20x ̧¤¤ ̧1x ̧æ ̧X ̧TM1⁄2 +...+2σxq-10xqXq_1XqXq—1Xq=e*, (xi.) where e is any line. Then this quadric will be "ellipsoidal" since the coefficients of x2... q2 are all positive quantities. Let R be its radius-vector measured in the direction 1, 2, . . . lq, or perpendicular to the plane from which we are measuring the residuals; then clearly: or U = ne1/R2, (xii.) Thus the inverse square of the radius of this "ellipsoid' measures the square of the mean square residual. We shall speak of the ellipsoid as the ellipsoid of residuals. Since Σ is to be a minimum, R must be a maximum; or we conclude: that the best-fitting plane is perpendicular to the greatest axis of the ellipsoid of residuals and the minimum mean square residual varies inversely as the length of this axis. A case of failure can only arise if the ellipsoid of residuals degenerates into an "oblate spheroid," i. e., when every plane through its shorter axis is one of "best fit," or into a sphere, when every plane through the centroid of the system of points is an equally good fit. This sphericity of distribution of points in space involves the vanishing of all the correlations between the variables and the equality of all their standarddeviations. It corresponds to isotropic inertia in the theory of moments in dynamics. (4) The theory of the best-fitting straight line need not Draw the plane perpendicular to this line through xị, x;', x... xq; i. e., 11x1 + lyx2 + 7zxz + . . . + lqtq = H, where H=4x+12x2' + lyxz' +...+1qrg'. Then if p be the perpendicular from any point in space on the line (xiii.): — { l1 (X 1 − x 1') + I2 (X2 — X2′) +13 (X3—X3') +...+lq(xq−xq') }2. Now x1, x2... xq′ and 41, 12 . . . lq, subject to the relation 122 + 122 + 132 +...+1=1, are the constants at our disposal. Sum p2 and differentiate to find when U=S( p2) is a minimum. We have for type equation ... S(xu—xu')—lu[S{}(x1−x1')+l2(X2−X2′ ) + . . . +lq (Xq−xg')}]=0, which show us that the straight line passes (as we have already noted) through the centroid of the system. We can accordingly take x,... q' to be that centroid, and we find : +212l2σx ̧σx, rx ̧x2+...+24q-140xq-1@rq".rq-17rq]· But the expression in square brackets is precisely the square of the mean square residual with regard to the plane, l1(X1−x1)+l2(X2−X2)+...+lq(Xq—ñq)=0, or Σ2. Thus we have: Σ12 =σ2x, + o2x2 + • . . + o3xq−Σ2. Now clearly o+o2, +...+o2xq is a constant. Hence ' will be a minimum when Σ2 is a maximum, or when the plane perpendicular to the best-fitting line is perpendicular to the least axis of the ellipsoid of residuals. Thus we find : That the line which fits best a system of n points in q-fold space passes through the centroid of the system and coincides in direction with the least axis of the ellipsoid of residuals. The mean square residual (which measures of course the closeness of the fit) is given by where R is the least radius of the ellipsoid of residuals. The direction-cosines of the line can be found from (ix.) by giving 2 the least value among the roots of (x.). Clearly the plane of best fit passes through the line of best fit, and is further perpendicular to the greatest radius, the maximum axis of the ellipsoid of residuals. (5) While the geometry of lines and planes of best fit is thus seen to be very simple from the standpoint of inertia ellipsoids, particularly from the consideration of the surface which, for the theory of errors, I have termed the ellipsoid of residuals, they most frequently occur, perhaps, in the case of correlated variations or errors, and it is thus of interest to consider them in relation to the ellipses and ellipsoids which arise as "contours" in correlation surfaces. Now take the case of two variables x and y only, the type-ellipse of the contours of the correlation surface is, when referred to its centroid as origin: Compare this with the ellipse of residuals o2xx12 + o2y y12+20x σy rxy x'y' = e1. Clearly if we take x'=y, y', and e=σ2roy the ellipse of residuals becomes the correlation type-ellipse. Further, x22+y'2= x2+y2, or the two ellipses have equal rays, but they are at right-angles to each other. Thus the best-fitting straight line for the system of points coincides in direction with the major axis of the correlation ellipse, and the mean square residual for this line == product of standard deviations semi-major axis of correlation ellipse' Phil. Mag. S. 6. Vol. 2. No. 11. Nov. 1901. 2 P The geometry of these results is indicated in the accompanying diagram: EE' is found by making S(y'-y)3 a minimum, The angle which AA' makes with Or is determined by GRESSION LINE: x on Y Physically the axes of the correlation type-ellipse directions of independent or uncorrelated variation. Hence the line of best fit is a direction of uncorrelated variation. are the (6) We turn to the correlation type-" ellipsoid" for 9 variables. It is *: where A11, 42, 412... Aq-19, Agg are the minors corresponding to the constituents marked by the same subscripts of the determinant: Now let us find the directions and magnitudes of the principal axes of this ellipsoid. We must make u2=x12 + x22 + ... +xq2 a maximum. Or if Q be an indeterminate multiplier, we στι have: +418 052 X3 ... + +418 xq=0, στη xq=0, +...+ Amore ... Multiply the last q-1 of these equations by r12, 13). respectively and add them to the first, then we know that: Phil. Trans. vol. clxxxvii. A, p. 302. |