Lippert Ross A, Huang Haiyan, Waterman Michael S
Informatics Research, Celera Genomics, Rockville, MD 20878, USA.
Proc Natl Acad Sci U S A. 2002 Oct 29;99(22):13980-9. doi: 10.1073/pnas.202468099. Epub 2002 Oct 8.
When comparing two sequences, a natural approach is to count the number of k-letter words the two sequences have in common. No positional information is used in the count, but it has the virtue that the comparison time is linear with sequence length. For this reason this statistic D(2) and certain transformations of D(2) are used for EST sequence database searches. In this paper we begin the rigorous study of the statistical distribution of D(2). Using an independence model of DNA sequences, we derive limiting distributions by means of the Stein and Chen-Stein methods and identify three asymptotic regimes, including compound Poisson and normal. The compound Poisson distribution arises when the word size k is large and word matches are rare. The normal distribution arises when the word size is small and matches are common. Explicit expressions for what is meant by large and small word sizes are given in the paper. However, when word size is small and the letters are uniformly distributed, the anticipated limiting normal distribution does not always occur. In this situation the uniform distribution provides the exception to other letter distributions. Therefore a naive, one distribution fits all, approach to D(2) statistics could easily create serious errors in estimating significance.
在比较两个序列时，一种自然的方法是计算这两个序列共有的k字母单词的数量。计数时不使用位置信息，但它的优点是比较时间与序列长度呈线性关系。因此，这个统计量D(2)以及D(2)的某些变换被用于EST序列数据库搜索。在本文中，我们开始对D(2)的统计分布进行严格研究。利用DNA序列的独立性模型，我们通过斯坦因方法和陈 - 斯坦因方法推导出极限分布，并确定了三种渐近情形，包括复合泊松分布和正态分布。当单词大小k较大且单词匹配很少时会出现复合泊松分布。当单词大小较小时且匹配很常见时会出现正态分布。本文给出了大单词大小和小单词大小具体含义的明确表达式。然而，当单词大小较小时且字母均匀分布时，预期的极限正态分布并不总是出现。在这种情况下，均匀分布是其他字母分布的例外。因此，对D(2)统计采用一种天真的、一种分布适用于所有情况的方法在估计显著性时很容易产生严重错误。
