Abstract

A previously unstudied optimization problem induced by noise-proof recognition of a quasi-periodic sequence, namely, by the recognition of a sequence \(Y\) of length \(N\) generated by a sequence \(U\) belonging to a given finite set \(W\) (alphabet) of sequences is considered. Each sequence \(U\) from \(W\) generates an exponentially sized set \(\mathcal{X}(U)\) consisting of all sequences of length \(N\) containing (as subsequences) a varying number of admissible quasi-periodic (fluctuational) repeats of \(U\). Each quasi-periodic repeat is generated by admissible transformations of U, namely, by shifts and extensions. The recognition problem is to choose a sequence \(U\) from \(W\) and to approximate \(Y\) by an element \(X\) of the sequence set \(\mathcal{X}(U)\). The approximation criterion is the minimum of the sum of the squared distances between the elements of the sequences. We show that the considered problem is equivalent to the problem of summing the elements of two numerical sequences so as to minimize the sum of an unknown number \(M\) of terms, each being the difference between the nonweighted autoconvolution of \(U\) extended to a variable length (by multiple repeats of its elements) and a weighted convolution of this extended sequence with a subsequence of Y. It is proved that the considered optimization problem and the recognition problem are both solvable in polynomial time. An algorithm is constructed and its applicability for solving model application problems of noise-proof processing of ECG- and PPG-like quasi-periodic signals (electrocardiogram- and photoplethysmogram-like signals) is illustrated using numerical examples.

中文翻译：

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包含未知数目的非线性扩展参考子序列的准周期序列的识别

摘要

通过防噪声识别准周期性序列，即诱导，通过识别一个序列的一个先前未研究的最优化问题\（Y \）长度的\（N \）由序列产生\（U \）属于考虑给定序列的有限集\(W\)（字母表）。每个序列\（U \）从\（W \）生成一个指数大小的组\（\ mathcal {X}（U）\）由长度的所有序列的\（N \）含有（作为子序列）的可变数量\(U\) 的准周期性（波动）重复。每个准周期重复是由U的容许变换产生的，即通过移位和扩展。识别问题是从\(W\)中选择一个序列\(U\)并通过序列集\(\mathcal{X}(U)\ )的一个元素\(X\)来近似\(Y\) )。近似标准是序列元素之间距离平方和的最小值。我们表明，所考虑的问题等价于对两个数值序列的元素求和以最小化未知数\(M\)项的总和，每个项都是\(U\ )扩展到可变长度（通过其元素的多次重复）以及此扩展序列与Y 子序列的加权卷积。证明了所考虑的优化问题和识别问题都是多项式时间可解的。构建了一种算法，并通过数值例子说明了其在解决ECG类和PPG类准周期信号（心电图类和光电容积图类信号）抗噪处理模型应用问题中的适用性。