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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Simple linear regression ： ウィキペディア英語版 Simple linear regression In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable. In other words, simple linear regression fits a straight line through the set of points in such a way that makes the sum of squared ''residuals'' of the model (that is, vertical distances between the points of the data set and the fitted line) as small as possible. The adjective ''simple'' refers to the fact that the outcome variable is related to a single predictor. The slope of the fitted line is equal to the correlation between and corrected by the ratio of standard deviations of these variables. The intercept of the fitted line is such that it passes through the center of mass of the data points. Other regression methods besides the simple ordinary least squares (OLS) also exist (see linear regression). In particular, when one wants to do regression by eye, one usually tends to draw a slightly steeper line, closer to the one produced by the total least squares method. This occurs because it is more natural for one's mind to consider the orthogonal distances from the observations to the regression line, rather than the vertical ones as OLS method does. ==Fitting the regression line== Suppose there are data points The function that describes x and y is : $y\_i\; =\; \backslash alpha\; +\; \backslash beta\; x\_i\; +\; \backslash varepsilon\_i.$ The goal is to find the equation of the straight line : $y\; =\; \backslash alpha\; +\; \backslash beta\; x,$ which would provide a "best" fit for the data points. Here the "best" will be understood as in the least-squares approach: a line that minimizes the sum of squared residuals of the linear regression model. In other words, (the -intercept) and (the slope) solve the following minimization problem: : $\backslash text\backslash min\_\; Q(\backslash alpha,\backslash beta),\; \backslash qquad\; \backslash text\; Q(\backslash alpha,\backslash beta)\; =\; \backslash sum\_^n\backslash varepsilon\_i^\; =\; \backslash sum\_^n\; (y\_i\; -\; \backslash alpha\; -\; \backslash beta\; x\_i)^2\backslash$ By using either calculus, the geometry of inner product spaces, or simply expanding to get a quadratic expression in and , it can be shown that the values of and that minimize the objective function 〔Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in ''Mathematics of Statistics'', Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285〕 are : $\backslash begin$ \hat\beta &= \frac (x_-\bar)(y_-\bar) } (x_-\bar)^2 } \\() &= \frac} - \frac1n \sum_^^ - \frac1n (\sum_^ \\() &= \frac\bar }^2 } \\ &= \frac \\ &= r_ \frac, \\() \hat\alpha & = \bar - \hat\beta\,\bar, \end where is the sample correlation coefficient between and , the sample standard deviation of , and the sample standard deviation of . A horizontal bar over a quantity indicates the average value of that quantity. For example: :$\backslash overline\; =\; \backslash tfrac\; \backslash sum\_^n\; x\_iy\_i.$ Substituting the above expressions for $\backslash hat\backslash alpha$ and $\backslash hat\backslash beta$ into : $f\; =\; \backslash hat\backslash alpha\; +\; \backslash hat\backslash beta\; x,$ yields : $\backslash frac\; =\; r\_\; \backslash frac$ This shows that is the slope of the regression line of the standardized data points (and that this line passes through the origin). It is sometimes useful to calculate from the data independently using this equation: :$r\_\; =\; \backslash frac\backslash bar\; \}^2)\; (\backslash overline\; -\; \backslash bar^2\; )\}\; \}$ The coefficient of determination (R squared) is equal to $r\_^2$ when the model is linear with a single independent variable. See sample correlation coefficient for additional details. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Simple linear regression」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.