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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 arg max ： ウィキペディア英語版 arg max In mathematics, the argument of the maximum (abbreviated arg max or argmax) is the set of points of the given argument for which the given function attains its maximum value.〔For clarity, we refer to the input (''x'') as ''points'' and the output (''y'') as ''values;'' compare critical point and critical value.〕 In contrast to global maximums, which refer to a function's largest outputs, the arg max refers to the inputs which create those maximum outputs. == Definition == The arg max is defined by :$\backslash operatorname$ *_x f(x) := \. In other words, it is the set of points ''x'' for which ''f''(''x'') attains its largest value. This set may be empty, have one element, or have multiple elements. For example, if ''f''(''x'') is 1−|''x''|, then it attains its maximum value of 1 only at ''x'' = 0, so :$\backslash operatorname$ *_x (1-|x|) = \. The arg max operator is the natural complement of the max operator which, given the same function, returns the maximum value instead of the point or points that reach that value; in other words :$\backslash max\_x\; f(x)$ is the element in $\backslash .$ This set can contain no elements (in which case the maximum is undefined) or one element, but cannot contain multiple elements. Equivalently, if ''M'' is the maximum of ''f'', then the arg max is the level set of the maximum: :$\backslash operatorname$ *_x \, f(x) = \ =: f^(M) If the maximum is reached at a single point then this point is often referred to as ''the'' arg max, meaning we define the arg max as a point, not a set of points. So, for example, :$\backslash operatorname$ *_), since the maximum value of ''x''(10 − ''x'') is 25, which occurs for ''x'' = 5.〔Note that $x(10-x)\; =\; 25-(x-5)^2\backslash le\; 25$ with equality if and only if $x-5=0$.〕 However, in case the maximum is reached at many points, arg max is a ''set'' of points. Then, we have for example :$\backslash operatorname$ *_ \cos(x) = \ since the maximum value of cos(''x'') is 1, which occurs on this interval for ''x'' = 0, 2π or 4π. On the whole real line, the arg max is $\backslash .$ Note also that functions do not in general attain a maximum value, and hence will in general not have an arg max: $\backslash operatorname$ *_{x\in\mathbb{R}} x^3 is the empty set, as $x^3$ is unbounded on the real line. However, by the extreme value theorem (or the classical compactness argument), a continuous function on a compact interval has a maximum, and thus an arg max. 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「arg max」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.