Identifying a word (pattern) in a long sequence of letters is not an easy task. To achieve this objective, several models have been proposed under the assumption that the sequence of letters is described by a Markov chain. The Markovian hypothesis imposes restrictions on the distribution of the sojourn time in a state, which has geometric distribution in a discrete process. This is the main drawback when applying Markov chains to real problems. By contrast, semi-Markov processes are generalized. In semi-Markov processes, the sojourn time in a state can be governed by any distribution function. The goal of this article is to compute the first hitting time (position) of a word (pattern) in a semi-Markov sequence. To achieve this objective, we use the auxiliary prefix and backward chain. To give an example of the applications of the proposed model, the model is tested in a bacteriophage DNA sequence that is lacking the enzyme SmaI. We compute the probability that a word occurs for the first time after n nucleotides in a DNA sequence. The corresponding probability distribution, the mean waiting position, the variance, and rate of the occurrence of the word are obtained.

中文翻译：

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半马尔可夫假设下生物序列中单词的识别。

识别长字母序列中的单词（模式）并非易事。为了实现该目的，在假设字母序列由马尔可夫链描述的假设下，已经提出了几种模型。马尔可夫假设对状态下的停留时间分布施加了限制，该状态在离散过程中具有几何分布。这是将马尔可夫链应用于实际问题时的主要缺点。相反，半马尔可夫过程是广义的。在半马尔可夫过程中，状态下的停留时间可以由任何分布函数控制。本文的目的是计算半马尔可夫序列中单词（模式）的第一个击中时间（位置）。为了达到这个目的，我们使用辅助前缀和后向链。为了举例说明该模型的应用，SmaI。我们计算一个单词在DNA序列中的n个核苷酸之后首次出现的可能性。获得相应的概率分布，平均等待位置，方差和单词出现的速率。