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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 Isotonic regression ： ウィキペディア英語版 Isotonic regression In numerical analysis, isotonic regression (IR) involves finding a weighted least-squares fit $x\backslash in\; \backslash Bbb^n$ to a vector $a\backslash in\; \backslash Bbb^n$ with weights vector $w\backslash in\; \backslash Bbb^n$ subject to a set of non-contradictory constraints of the kind $x\_i\; \backslash ge\; x\_j$. Such constraints define partial order or total order and can be represented as a directed graph $G=(N,E)$, where N is the set of variables involved, and E is the set of pairs (i, j) for each constraint $x\_i\; \backslash ge\; x\_j$. Thus, the IR problem corresponds to the following quadratic program (QP): :$\backslash min\; \backslash sum\_^n\; w\_i\; (x\_i\; -\; a\_i)^2$ $\backslash textx\_i\backslash ge\; x\_j~\; \backslash text\; (i,j)\backslash in\; E.$ In the case when $G=(N,E)$ is a total order, a simple iterative algorithm for solving this QP is called the pool adjacent violators algorithm (PAVA). Best and Chakravarti (1990) have studied the problem as an active set identification problem, and have proposed a primal algorithm in O(n), the same complexity as the PAVA, which can be seen as a dual algorithm. IR has applications in statistical inference, for example, to fit of an isotonic curve to mean experimental results when an order is expected. A benefit of isotonic regression is that it does not assume any form for the target function, such as linearity assumed by linear regression. Another application is nonmetric multidimensional scaling, where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order. Isotonic regression is also sometimes referred to as ''monotonic regression''. Correctly speaking, ''isotonic'' is used when the direction of the trend is strictly increasing, while ''monotonic'' could imply a trend that is either strictly increasing or strictly decreasing. Isotonic regression under the $L\_p$ for $p>0$ is defined as follows: :$\backslash min\; \backslash sum\_^n\; w\_i\; |x\_i\; -\; a\_i|^p$ $\backslash mathrmx\_i\backslash ge\; x\_j~\; \backslash text\; (i,j)\backslash in\; E.$ == Simply ordered case == To illustrate the above, let $x\_1\; \backslash leq\; x\_2\; \backslash leq\; \backslash ldots\; \backslash leq\; x\_n$, and $f(x\_1)\; \backslash leq\; f(x\_2)\; \backslash leq\; \backslash ldots\; \backslash leq\; f(x\_n)$, and $w\_i\; \backslash geq\; 0$. The isotonic estimator, $g^$ *, minimizes the weighted least squares-like condition: :$\backslash min\_g\; \backslash sum\_^n\; w\_i\; (g(x\_i)\; -\; f(x\_i))^2$ Where $g$ is the unknown function we are estimating, and $f$ is a known function. Software has been developed in the R statistical package for computing isotone (monotonic) regression. 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「Isotonic 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.