On lines and planes of closest fit to systems of points in space.
On lines and planes of closest fit to systems of points in space.
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DOI：
10.1080/14786440109462720
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发表时间：
1901-07-01
影响因子：
1.6
通讯作者：
Pearson, Karl
中科院分区：
文献类型：
--
作者：
Pearson, Karl
(1)-] 在许多物理、统计和生物研究中，希望通过“最佳拟合直线或平面”来表示 phme、三维或更高维空间中的点系统。分析上，这包括取 y= ao+ alx，或 Z= ao+ alx+ b, y，或 z= ao+ a~ xl+ a2x 2+ aaxa+ 9 9 9+ at， xn，其中 y、x、z、xl、x、~、..、x~ 是变量，并确定常数 a0、a、bl、a0、al、a2、as、9 9 9 的“最佳”值，与观察到的变量相应值相关。] 在最小二乘教科书中涉及的几乎所有情况下，方程右侧的变量被视为自变量，左侧的变量被视为因变量。这种处理的结果是，如果我们将某个变量视为自变量，我们会得到一条直线或平面，而如果我们将另一个变量视为自变量，则得到一条完全不同的直线或平面。这并不矛盾，这是相关变量系统理论的一个易于理解且最重要的特征，例如，对于给定的 x 值，y 的最可能值并不是由与 x 的给定值的最可能值相同的关系给出的。或者，举一个具体的例子，给定腿长度 I 的人最可能的身高是 s，而身高 s 的人最可能的腿长度不是 I。在我的回归 t 回忆录中给出了相关系统的 z 至 n 个变量的情况的“最佳拟合”线和平面。它们取决于系统的平均值、标准差和相关系数的确定。自变量的概率应该是准确已知的，因变量的可能值是确定的。（2）然而，在物理学和生物学的许多情况下，“自”变量与“因”变量一样会受到同样多的偏差或误差的影响。例如，我们并不准确地知道x然后继续寻找y，但是x和y都是通过实验或观察找到的。我们观察 x 和 y 并寻求它们之间的独特函数关系。给定身材的男人可能有多种
(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