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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 predictor function ： ウィキペディア英語版 Linear predictor function In statistics and in machine learning, a linear predictor function is a linear function (linear combination) of a set of coefficients and explanatory variables (independent variables), whose value is used to predict the outcome of a dependent variable. Functions of this sort are standard in linear regression, where the coefficients are termed regression coefficients. However, they also occur in various types of linear classifiers (e.g. logistic regression, perceptrons, support vector machines, and linear discriminant analysis), as well as in various other models, such as principal component analysis and factor analysis. In many of these models, the coefficients are referred to as "weights". ==Basic form== The basic form of a linear predictor function $f(i)$ for data point ''i'' (consisting of ''p'' explanatory variables), for ''i'' = 1, ..., ''n'', is :$f(i)\; =\; \backslash beta\_0\; +\; \backslash beta\_1\; x\_\; +\; \backslash cdots\; +\; \backslash beta\_p\; x\_,$ where $\backslash beta\_0,\; \backslash ldots,\; \backslash beta\_p$ are the coefficients (regression coefficients, weights, etc.) indicating the relative effect of a particular explanatory variable on the outcome. It is common to write the predictor function in a more compact form as follows: *The coefficients ''β''0, ''β''1, ..., ''β''''p'' are grouped into a single vector ''β'' of size ''p'' + 1. *For each data point ''i'', an additional explanatory pseudo-variable ''x''''i''0 is added, with a fixed value of 1, corresponding to the intercept coefficient ''β''0. *The resulting explanatory variables ''x''''i0'', ''x''''i''1, ..., ''x''''ip'' are then grouped into a single vector ''xi'' of size ''p'' + 1. This makes it possible to write the linear predictor function as follows: :$f(i)=\; \backslash boldsymbol\backslash beta\; \backslash cdot\; \backslash mathbf\_i$ using the notation for a dot product between two vectors. An equivalent form using matrix notation is as follows: :$f(i)=\; \backslash boldsymbol\backslash beta^\; \backslash mathbf\_i\; =\; \backslash mathbf^\_i\; \backslash boldsymbol\backslash beta$ where $\backslash boldsymbol\backslash beta$ and $\backslash mathbf\_i$ are assumed to be a ''p''-by-1 column vectors (as is standard when representing vectors as matrices), $\backslash boldsymbol\backslash beta^$ indicates the matrix transpose of $\backslash boldsymbol\backslash beta$ (which turns it into a 1-by-''p'' row vector), and $\backslash boldsymbol\backslash beta^\; \backslash mathbf\_i$ indicates matrix multiplication between the 1-by-''p'' row vector and the ''p''-by-1 column vector, producing a 1-by-1 matrix that is taken to be a scalar. An example of the usage of such a linear predictor function is in linear regression, where each data point is associated with a continuous outcome ''y''''i'', and the relationship written :$y\_i\; =\; f(i)\; +\; \backslash varepsilon\_i\; =\; \backslash boldsymbol\backslash beta^\backslash mathbf\_i\backslash \; +\; \backslash varepsilon\_i,$ where $\backslash varepsilon\_i$ is a ''disturbance term'' or ''error variable'' — an unobserved random variable that adds noise to the linear relationship between the dependent variable and predictor function. 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 predictor function」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.