Probability modelling for DNA sequence evolution is well established and
provides a rich framework for understanding genetic variation between samples
of individuals from one or more populations. We show that both classical and
more recent models for coalescence (with or without recombination) can be
described in terms of the so-called phase-type theory, where complicated and
tedious calculations are circumvented by the use of matrices. The application
of phase-type theory consists of describing the stochastic model as a Markov
model by appropriately setting up a state space and calculating the
corresponding intensity and reward matrices. Formulae of interest are then
expressed in terms of these aforementioned matrices. We illustrate this by a
few examples calculating the mean, variance and even higher order moments of
the site frequency spectrum in the multiple merger coalescent models, and by
analysing the mean and variance for the number of segregating sites for
multiple samples in the two-locus ancestral recombination graph. We believe
that phase-type theory has great potential as a tool for analysing probability
models in population genetics. The compact matrix notation is useful for
clarification of current models, in particular their formal manipulation
(calculation), but also for further development or extensions.

采用相态理论，描述了共聚模型和考虑到重组后的共聚模型，并且计算了仅仅考虑到固定重组点和超过一个重组点的共聚模型中的因子频率谱、非固定重组点的两点祖先基因图计算量的平均数和方差。

人口遗传学中的相位类型分布

Phase-type distributions in population genetics

One of the ubiquitous representation of long DNA sequence is dividing it into
shorter k-mer components. Unfortunately, the straightforward vector encoding of
k-mer as a one-hot vector is vulnerable to the curse of dimensionality. Worse
yet, the distance between any pair of one-hot vectors is equidistant. This is
particularly problematic when applying the latest machine learning algorithms
to solve problems in biological sequence analysis. In this paper, we propose a
novel method to train distributed representations of variable-length k-mers.
Our method is based on the popular word embedding model word2vec, which is
trained on a shallow two-layer neural network. Our experiments provide evidence
that the summing of dna2vec vectors is akin to nucleotides concatenation. We
also demonstrate that there is correlation between Needleman-Wunsch similarity
score and cosine similarity of dna2vec vectors.

本文提出了一种基于 word2vec 模型的新方法，用于训练变长 k-mers 的分布式表示。实验表明 dna2vec 向量求和类似于核苷酸串联，并且 Needleman-Wunsch 相似度得分与 dna2vec 向量的余弦相似度之间存在相关性。

dna2vec: 可变长度 k-mer 的一致向量表示

dna2vec: Consistent vector representations of variable-length k-mers

In this paper, we introduce the on-line Viterbi algorithm for decoding hidden
Markov models (HMMs) in much smaller than linear space. Our analysis on
two-state HMMs suggests that the expected maximum memory used to decode
sequence of length $n$ with $m$-state HMM can be as low as $\Theta(m\log n)$,
without a significant slow-down compared to the classical Viterbi algorithm.
Classical Viterbi algorithm requires $O(mn)$ space, which is impractical for
analysis of long DNA sequences (such as complete human genome chromosomes) and
for continuous data streams. We also experimentally demonstrate the performance
of the on-line Viterbi algorithm on a simple HMM for gene finding on both
simulated and real DNA sequences.

本文提出一种在线维特比算法，用于在远小于线性空间的情况下解码隐马尔可夫模型，该算法计算长度为 $n$，具有 $m$ 个状态的 HMM 所需的最大内存使用量期望可以低至 $\Theta (m\log n)$，用于基因发现的简单 HMM 模型的实验结果表明该算法的性能表现出色。