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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 Linear model ： ウィキペディア英語版 Linear model In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term is also used in time series analysis with a different meaning. In each case, the designation "linear" is used to identify a subclass of models for which substantial reduction in the complexity of the related statistical theory is possible. ==Linear regression models== For the regression case, the statistical model is as follows. Given a (random) sample $(Y\_i,\; X\_,\; \backslash ldots,\; X\_),\; \backslash ,\; i\; =\; 1,\; \backslash ldots,\; n$ the relation between the observations ''Yi'' and the independent variables ''Xij'' is formulated as :$Y\_i\; =\; \backslash beta\_0\; +\; \backslash beta\_1\; \backslash phi\_1(X\_)\; +\; \backslash cdots\; +\; \backslash beta\_p\; \backslash phi\_p(X\_)\; +\; \backslash varepsilon\_i\; \backslash qquad\; i\; =\; 1,\; \backslash ldots,\; n$ where $\backslash phi\_1,\; \backslash ldots,\; \backslash phi\_p$ may be nonlinear functions. In the above, the quantities ''εi'' are random variables representing errors in the relationship. The "linear" part of the designation relates to the appearance of the regression coefficients, ''βj'' in a linear way in the above relationship. Alternatively, one may say that the predicted values corresponding to the above model, namely :$\backslash hat\_i\; =\; \backslash beta\_0\; +\; \backslash beta\_1\; \backslash phi\_1(X\_)\; +\; \backslash cdots\; +\; \backslash beta\_p\; \backslash phi\_p(X\_)\; \backslash qquad\; (i\; =\; 1,\; \backslash ldots,\; n),$ are linear functions of the ''βj''. Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters ''βj'' are determined by minimising a sum of squares function :$S\; =\; \backslash sum\_^n\; \backslash left(Y\_i\; -\; \backslash beta\_0\; -\; \backslash beta\_1\; \backslash phi\_1(X\_)\; -\; \backslash cdots\; -\; \backslash beta\_p\; \backslash phi\_p(X\_)\backslash right)^2\; .$ From this, it can readily be seen that the "linear" aspect of the model means the following: : *the function to be minimised is a quadratic function of the ''βj'' for which minimisation is a relatively simple problem; : *the derivatives of the function are linear functions of the ''βj'' making it easy to find the minimising values; : *the minimising values ''βj'' are linear functions of the observations ''Yi''; : *the minimising values ''βj'' are linear functions of the random errors ''εi'' which makes it relatively easy to determine the statistical properties of the estimated values of ''βj''. 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「Linear model」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.