On the generalized Karhunen-Loeve expansion (Corresp.)
On the generalized Karhunen-Loeve expansion (Corresp.)
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关于广义的 Karhunen-Loeve 展开 (Corresp.)
DOI：
10.1109/tit.1967.1054021
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
K. Fu
中科院分区：
文献类型：
--
作者：
Y. Chien;K. Fu
在最近的一篇论文中，Watanabe（1965）建立了两个关于随机过程的随机函数的Karhunen-Loeve展开式的最优性质的有用定理。由此产生的坐标系（Karhunen-Loeve系统）被认为是最佳的，在这个意义上，i）最小化通过有限数量的项近似无限级数的展开而产生的均方误差，ii）最小化在整个系综的坐标系数的概率分布上定义的熵函数。然后将这些优化特征有效地应用于语音识别问题的输入数据预处理。然而，在这些性质的发展考虑的情况下，随机函数来自相同的随机过程，因此，在展开的坐标系数被视为非随机的。然后出现的问题是，如果随机函数是一个以上的随机过程的实现，在许多预处理问题的情况下，在模式识别和信号检测的最佳性能是否将举行。这封信的目的是为了表明。所引用的最优性质可以通过定义广义Karhunen-Loeve展开来保留，该广义Karhunen-Loeve展开考虑了两个或更多个随机过程生成随机函数的可能性。导出了保证这种展开式存在的必要条件。这些结果的应用表明在排序和选择的顺序识别（决策）问题的特征测量。
In a recent paper Watanabe (1965) has established two useful theorems concerning the optimum properties of the Karhunen-Loeve expansion for the random functions of a stochastic process. The resulting coordinate system (Karhunen-Loeve system) was found to be optimum in the sense of i) minimizing the mean-square error committed by approximating the expansion of an infinite series by a finite number of terms, ii) minimizing the entropy function defined over the probability distribution of the coordinate coefficients for the entire ensemble. These optimum features were then effectively applied to the preprocessing of input data for the speech recognition problem. However, in the development of these properties considerations were only given to the situation where the random functions come from the same stochastic process, and consequently the coordinate coefficients in the expansion were treated to be nonrandom. The question then arises as to whether the optimum properties will hold if the random functions are realizations of more than one stochastic process as in the case of many preprocessing problems in pattern recognization and signal detection. The purpose of this correspondence is to show. that the cited optimum properties can be retained by defining a generalized Karhunen-Loeve expansion which considers the possibility of two or more stochastic processes generating the random functions. Necessary conditions are derived to assure the existence of such an expansion. Applications of these results are indicated in the ranking and selection of feature measurements for the sequential recognition (decision) problems.