Alignment-Free Sequence Comparison (II): Theoretical Power of Comparison Statistics

2.1. The model and the count statistics

The alternative model I renders the two sequences dependent through a common motif which is randomly distributed across the two sequences. As in Reinert et al. (2009), we model the background sequence as independent identically distributed (IID) random variables taking different letters from finite alphabet \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A}$$ \end{document} with probability \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$p_a (a\in {\cal A})$$ \end{document} . For notational convenience, we also denote \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$p_a^{(0)} = p_a$$ \end{document} . For nucleotide sequences, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A} = \{A , C , G , T \}$$ \end{document} and for amino acid sequences, the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A}$$ \end{document} is the set of 20 amino acids. In general, we assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A}$$ \end{document} contains L letters and write \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A} = \{0 , 1 , 2 , \cdots , L - 1 \}$$ \end{document} . For the motif instances, we use the model in Zhai et al. (2010), which is more general than the model used in Reinert et al. (2009), where fixed motifs were used. In this article and in Zhai et al. (2010), a position weight matrix (PWM) is used to describe the distribution of the nucleotides at the different positions of a motif (Stormo, 2000). For a given motif of length M, and at the m-th position of the motif, the probability that the base takes value a from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A}$$ \end{document} is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$p_a^{(m )} , m = 1 , 2 , \cdots , M$$ \end{document} . The motif instances are randomly distributed across the sequence with density 1 − λ (0 < λ < 1). That is, at each position in the sequence which is not already covered by an instance of a motif, with probability λ, a base with the background distribution is generated, and with probability 1 −λ, an instance of the motif of length M is generated based on the PWM for the motif. Once an instance of a motif is generated, we move to the end of the instance of the motif to repeat this process.

For the model in more detail, see Zhai et al. (2010). The sequences with random motif instances were modeled by an HMM (Rabiner, 1989). The underlying Markov chain (MC) of each sequence is denoted as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Q_1Q_2 \cdots Q_i \cdots Q_{n + k - 1}$$ \end{document} (i is the position index of the sequence with length n + k − 1) which take values in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{0 , 1 , 2 , \cdots , M \}$$ \end{document} . The 0 indicates that the state of the sequence is the background sequence while m (1 ≤ m ≤ M) indicates the state at the m-th position of the motif. Under each state, the emission probability of each letter from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal A}$$ \end{document} is denoted as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$p^{(m)}_a \ (a \in{\cal A} \ {\rm and} \ m = 0 , 1 , 2 , \cdots , M)$$ \end{document} . The transition matrix for the underlying MC \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Q_1Q_2 \cdots Q_i \cdots Q_{n + k - 1}$$ \end{document} is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$T = (t_{mm^{\prime}}) _{(M + 1) \times (M + 1)}$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$t_{00} = t_{M0} = \lambda , \ t_{01} = t_{M1} = 1 - \lambda , \ t_{m , m + 1} = 1 , \ m = 1 , 2 , \cdots , M - 1$$ \end{document} , and all the other t's are 0. The MC has as stationary distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\pi = \frac {1} {\lambda + M (1 - \lambda)} (\lambda , 1 - \lambda , 1 - \lambda , \cdots , 1 - \lambda)$$ \end{document} (Zhai et al., 2010). Therefore, in stationarity, the expected fraction of the sequence that is covered by the motif instances is M(1 − λ)/(λ + M(1 − λ)). Unless λ is close to 1, the expected fraction of the sequence covered by inserted motif instances can be unrealistically large (Table S1; for Supplementary Material, see www.liebertonline.com/cmb). Hence we only study values of λ which are no smaller than 0.9.

Now we consider two sequences of length n + k − 1 generated by the above HMM, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\bf A} = A_1A_2 \cdots A_{n + k - 1}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\bf B} = B_1B_2 \cdots B_{n + k - 1}$$ \end{document} . We let the sequence length be n + k − 1 for notational simplicity in the remainder of the paper. Given a k-tuple \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\bf w} = (w_1 , w_2 , \ldots , w_k) \in {\cal A}^k$$ \end{document} , let X _w and Y _w be the numbers of occurrences of w within A and B, respectively; within each sequence, the occurrences could overlap. Assume that the Markov process starts in the stationary distribution. Based on Proposition 2.2 in Zhai et al. (2010), the means of X _w (n) and Y _w (n) can be calculated as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {\mathbb E}_{\lambda}X_{\bf w} = {\mathbb E}_{\lambda}Y_{\bf w} = n P_{\lambda} ({\bf w}), \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_{\lambda}{({\bf w})} = \sum\nolimits_{m = 0}^{M} \alpha_{k}^{({\bf w})} (m)$$ \end{document} is the probabiltiy of the word w under the alternative model I. The \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\alpha_i^{({\bf w})} (m) = P (A_j = w_j , j = 1 , 2 , \cdots , i; Q_i = m) , i = 1 , 2 , \cdots , k$$ \end{document} , are calculated recursively using the standard forward procedure for calculating the probability of an observation sequence based on HMM (Zhai et al., 2010; Rabiner, 1989) for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$i = 1 , 2 \cdots$$ \end{document} : \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha_{i + 1}^{({\bf w})} (0) & = (\alpha_i^{({\bf w})} (0) + \alpha_i^{({\bf w})} (M)) \lambda p^{(0)}_{w_{i + 1}} ,\\ \alpha_{i + 1}^{({\bf w})} (1) & = (\alpha_i^{({\bf w})} (0) + \alpha_i^{({\bf w})} (M)) (1 - \lambda) p^{(1)}_{w_{i + 1}} , \\ \alpha_{i + 1}^{({\bf w})} (m) & = \alpha_i^{({\bf w})} (m - 1) p^{(m)}_{w_{i + 1}} , \qquad (m = 2 , 3 , \ldots , M), \end{align*} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha_1^ {({\bf w})} (0) & = \frac {\lambda p^ {(0)} _ {w_1}} {\lambda + M (1 - \lambda)} , \\ \alpha_1^ {({\bf w})} (m) & = \frac {(1 - \lambda) p^ {(m)} _ {w_1}} {\lambda + M (1 - \lambda)} , \qquad (1 \leq m \leq M). \end{align*} \end{document}

In particular, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_1 ({\bf w}) = p_{\bf w} = p_{w_1} p_{w_2} \cdots p_{w_k}$$ \end{document} .

2.2. The expectations of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^*_2$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^S_2$$ \end{document} under alternative model I

It is easy to see that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$E_ \lambda ({\widetilde X}_{\bf w}) = n (P_ \lambda ({\bf w}) - p_{\bf w})$$ \end{document} , where P_λ(w) is the probability of word w under the alternative model I. However, for the mean of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac {{\tilde X} _ {\bf w} {\tilde Y} _ {\bf w}} {\sqrt {{\tilde X} _ {\bf w} ^2 + {\tilde Y} _ {\bf w} ^2}}$$ \end{document} , it is in general only known that it is non-negative, and when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\tilde X}_{\bf w}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\tilde Y}_{\bf w}$$ \end{document} are IID, the mean is zero if and only if the distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\tilde X}_{\bf w}$$ \end{document} is symmetric (Novak, 2007). Note that the two sequences A and B are independent under the alternative model I. Then, we have the following theorem.

2.3. The approximate distributions of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^S_2$$ \end{document} under alternative model I

The variances of D₂ and its variants are complicated. Under the null model of IID sequences, upper and lower bounds for the variance of D₂ were first explored in Lippert et al. (2002). In Kantorovitz et al. (2007b), an explicit formula for the variance of D₂ is given in the IID case. To study the power of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^S_2$$ \end{document} in detecting the relationship between two sequences, we explore the approximate distributions of these statistics as the sequence length goes to infinity. Note that the distributions of D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} under the null model when \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(p^{(0)}_A , p^{(0)}_C , p^{(0)}_G , p^{(0)}_T) = (1 / 4 , 1 / 4 , 1 / 4 , 1 / 4)$$ \end{document} have been carefully studied in Reinert et al. (2009). Therefore, in the rest of the article, we assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(p^{(0)}_A , p^{(0)}_C , p^{(0)}_G , p^{(0 )}_T)\neq (1 / 4 , 1 / 4 , 1 / 4 , 1 / 4)$$ \end{document} .

For 0 < λ < 1, the values of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \sigma_ {\lambda} ^2 ({\bf w}) = \lim_ {n \rightarrow \infty} \frac {{\rm Var} (X_ {\bf w})} {n} \ {\rm and} \ \sigma_ {\lambda } ({\bf w} , {\bf w} ^ {\prime})= \lim_ {n \rightarrow \infty} \frac {{Cov} (X_ {\bf w} , X_ {{\bf w} ^ {\prime}})} {n} \tag {1} \end{align*} \end{document}

can be calculated using the method in Zhai et al. (2010), Proposition 2.3; for λ = 1, the corresponding values can be found, for example, in Reinert et al. (2009), Corollary 6.1. We denote the asymptotic variance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\nolimits_{{\bf w} \in {\cal A}^k} P_{\lambda} ({\bf w})X_{\rm w} / {\sqrt n}$$ \end{document} in one sequence by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} (\Sigma_ \lambda)^2 = \sum_{{\bf w} \in {\cal A}^k} P^2_{\lambda} ({\bf w})\sigma_{\lambda}^2 ({\bf w})+ \sum_{{\bf w} \neq {\bf w}^{\prime}}P_{\lambda} ({\bf w})P_{\lambda} ({\bf w}^{\prime})\sigma_{\lambda} ({\bf w} , {\bf w}^{\prime}). \tag {2} \end{align*} \end{document}

The following theorem gives the approximate distributions of D₂ under the null and the alternative model I.

2.4. The power of detecting the relationship between two sequences under alternative model I using D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^{\rm S}_2$$ \end{document}

Knowing the asymptotic distributions of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^S_2$$ \end{document} under the null and the alternative models, we are able to approximate the power of detecting the relationships between two sequences using any of the three statistics. For notational simplicity, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} A (\lambda)= \sum_ {{\bf w} \in {\cal A} ^k} P_ {\lambda} ^2 ({\bf w}), \quad A^* (\lambda)= \sum_ {{\bf w} \in {\cal A} ^k} \frac {(P_ {\lambda} ({\bf w})- p_ {\bf w})^2} {p_ {\bf w}} , \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} A^S (\lambda)= \frac {1} {{\sqrt 2}} \sum_ {{\bf w} \in {\cal A} ^k} \mid P_ {\lambda} ({\bf w})- p_ {\bf w} \mid \end{align*} \end{document}

denote the (asymptotic) means of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^S_2$$ \end{document} under alternative model I. Let Φ(·) be the cumulative distribution for the standard normal distribution. From Theorems 2.2, 2.3, and 2.4, we can show the following theorem to hold.

3.1. A second HMM model for the sequence pair A and B

Alternative model II is again a HMM model for the sequence pair \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\bf A} = A_1A_2 \cdots A_{n + k - 1}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\bf B} = B_1B_2 \cdots B_{n + k - 1}$$ \end{document} . First, two IID sequences A and B′ are generated. From these two sequences we construct B as follows. We assume that at each position which is not already covered by a chosen segment, with probability λ, the original bases of the two sequences at the position are kept. With probability 1 − λ, a segment of length M from the first sequence is chosen, and the same segment in the second sequence is replaced by it. Then we move to the end of the segment to start this process again. Consider an underlying Markov chain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Q_1 Q_2 \cdots Q_i \cdots$$ \end{document} defined as follows. Each Q _i takes values in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{0 , 1 , 2 , \cdots , M \}$$ \end{document} , where Q _i  = 0 indicates that, at position i, A _i and B _i are the originally generated bases, whereas \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Q_{\rm i} =m\ (1 , 2 , \cdots , M)$$ \end{document} indicates that position i is at the m-th position of a segment which was copied from the first sequence to the second sequence. The transition matrix of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Q_1 Q_2 \cdots Q_i \cdots$$ \end{document} is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$T = (t_{mm^{\prime}})_{(M + 1)\times (M + 1 )}$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$t_{00} = t_{M0} = \lambda , \ t_{01} = t_{M1} = 1 - \lambda , \ t_{m , m + 1} = 1 , \ m = 1 , 2 , \cdots , M - 1$$ \end{document} , and all the other t's are 0. It is easy to see that the stationary distribution of this Markov chain is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\pi = \frac {1} {\lambda + M (1 - \lambda )} (\lambda , 1 - \lambda , 1 - \lambda , \cdots , 1 - \lambda)$$ \end{document} (see Proposition 2.1 in Zhai et al. [2010]).

Let C _i  = (A _i , B _i ) ^t . With p _a denoting the probability of letter a in the IID model, the emission probabilities are given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} P ((A_i , B_i)= (a , b)\mid Q_i = 0)= p_a p_b , \quad P ((A_i , B_i)= (a , b)\mid Q_i = m)= p_a I (a = b). \end{align*} \end{document}

Then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_1 C_2 \cdots C_i \cdots$$ \end{document} form a HMM.

3.2. The asymptotic distributions and power of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}_2^S$$ \end{document} for detecting relationships between sequences under alternative model II

Under alternative model II, the marginal distributions of the individual sequences are IID sequences and hence the means of X_w and Y_w are unchanged compared to the IID model. However, the two sequences depend on each other because they share some common segments. The following theorem shows an efficient way to calculate the covariance of the number of occurrences of word w in sequence A and the number of occurrences of word w′ in sequence B. These covariances are used to derive the limiting distributions of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} when the sequence length tends to infinity.

4.1. A program for calculating the power of detecting the relationships between two sequences under alternative model I

To facilitate the use of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} for sequence or genome comparison and for evaluation of statistical power for detecting the relationship between the sequences, a web-based online program (http://meta.cmb.usc.edu/d2) and a stand-alone R program were developed to calculate the power of sequence comparison using these statistics. We describe the program for the above model. However, the program can be easily extended to the general scenario of different background letter frequencies, sequence lengths, and motif densities as in the supplementary materials. The inputs of the program are:

The background nucleotide or amino acid frequencies \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$p^{(0 )}_l , l = 0 , 1 , \cdots L - 1$$ \end{document} of the two sequences A and B under study;

For each set of parameters, the program first calculates the mean \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_{\lambda} ({\bf w})= {\mathbb E}_{\lambda} (X_{\bf w} )$$ \end{document} for any word w and the covariance σ _λ (w, w′) = Cov(X _w , X_w′) for two words w and w′, related to sequence A. The corresponding quantities related to sequence B are also calculated. Secondly, the program calculates the approximate variance, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$2 (\Sigma_{\lambda})^2 , \ 2 (\Sigma^{*}_{\lambda})^2 , \ {\rm and} \ 2 (\Sigma^{S}_{\lambda})^2$$ \end{document} of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} using formulas derived in Theorems 2.2, 2.3, and 2.4, respectively. Thirdly, for the given type I error α, the threshold values z_α, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$z^*_{\alpha}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$z^S_{\alpha}$$ \end{document} for the corresponding statistics D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} in Theorem 2.5 are calculated. Since the cumulative distribution functions of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z^*_1$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z^S_1$$ \end{document} are not readily available, a simulation based method is used to obtain the threshold values. A large number of independent sequence pairs are simulated according to the specified letter frequencies and the sequence lengths, and the empirical distributions of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^S_2$$ \end{document} are estimated. The threshold values are estimated by the upper α% quantile of the simulated values of each statistic. Finally, the values of C(λ), C^*(λ), and C^S(λ), and thus the power using the corresponding statistics in Theorem 2.5 is calculated.

We use the program to study the power of detecting the relationship between related sequences under alternative model I using the different statistics. In Subsection 4.2, we present the results for the parameter sets used in Reinert et al. (2009) and compare the results derived using our program with the simulated quantities in previous studies. In Subsection 4.3, we present the power of the various statistics for comparing the relationships between sequences when any of the motifs with motif length at most 10 in JASPAR (Sandelin et al., 2004) are present in both sequences.

4.2. Comparison of theoretical mean, standard deviation, and power of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}_2^{\rm S}$$ \end{document} with their corresponding simulated values from Reinert et al. (2009)

In this subsection, we present some numerical results on the mean, standard deviation, and power of detecting the relationships between two sequences for the three statistics D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} under the alternative model I using the same set of parameters as in Reinert et al. (2009). The objective is to see how close the corresponding quantities calculated using our formulas approximate the true values. We let the background letter frequencies for the two sequences be p_A = p_T = \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac {1} {6}$$ \end{document} , p_C = p_G =  \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac {1} {3}$$ \end{document} . The inserted motif is “AGCCA” and the motif density 1 − λ = 0.01. The size of the k-tuple is k = 5. We used 10,000 simulations to find the threshold values z_0.05, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$z^*_{0.05}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$z^S_{0.05}$$ \end{document} . The type I error α was set at 0.05 and 0.01.

For scaled D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} defined respectively by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} ND_2 = {\sqrt n} \left(\frac {D_2} {n^2} - \sum_ {{\bf w} \in {\cal A} ^k} p_ {\bf w} ^2 \right) , \quad ND_2^* = D_2^* / {\sqrt n} , \quad ND_2^S = D_2^S / {\sqrt n} , \end{align*} \end{document}

from Theorems 2.2, 2.3, and 2.4, it can be seen that the (approximate) means are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\sqrt n} (A (\lambda)- A (1)), \ {\sqrt n} A^* (\lambda) \ {\rm and} \ {\sqrt n} A^S (\lambda)$$ \end{document} , respectively. Similarly, the approximate variance of ND₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$ND_2^*$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$ND_2^S$$ \end{document} are \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$2 (\Sigma_{\lambda})^2, \ 2 (\Sigma^*_{\lambda})^2, \ {\rm and} \ 2 (\Sigma^S_ \lambda)^2$$ \end{document} , respectively.

Table 1 shows the simulated mean and standard deviation of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$ND_2 , \ ND_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$ND_2^S$$ \end{document} , respectively, and their corresponding limits. Surprisingly, the approximate mean and standard deviation of ND₂ are within 15% of their limit even when the sequence length is just 1Kbp. For \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , the simulated mean is roughly the same as the theoretical limit and the simulated standard deviation is within 21% of its theoretical limit when the sequence length is at least 1Kbp. However, the simulated mean of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} is much smaller than its limit. The corrected mean for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} is very different from the simulated mean, too, probably because the difference between P_λ(w) − p _w for most 5-tuples are very small; both approximations do not work well in this parameter regime. Therefore, while we expect that the power formulas we derived should approximate the true power of D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} well even for sequences of over 1Kbp long the power formula for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} can significantly over-estimate the true power.

Table 2 shows the theoretical approximate power of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} calculated using our formulas and the simulated power with the same setting as in Table 1. The results show that the approximations are very close for D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} . However, the theoretical approximate power based on the first approximation significantly over-estimates, while the approximate power based on the second approximation significantly under-estimates the simulated power for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} , in the parameter regime we consider.

As the approximate power for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} is not accurate in the parameter regimes we have considered, in the following, we only show the results related to D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} using the theoretical approximate power. Figure 1 shows the values of C(λ) and C^*(λ) (upper panel) and the power of D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} for detecting the relationships between pairs of sequences (lower panel) as a function of sequence length and the word length k when λ = 0.99. It should be noted that the power is a decreasing function of the values of C's and the smaller the values of C, the higher the power of the corresponding statistic is. From the left panel related to D₂, it can be seen that, when k = 2 or 3, the value of C actually increases and that the power 1 − Φ(C(λ)) decreases with the sequence length. For given sequence length and word size k, the power of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} is generally higher than the power of D₂. All these conclusions are consistent with the simulation studies in Reinert et al. (2009). Comparing the two figures in the lower panel of Figure 1 here with Figures 1 and 2 in Reinert et al. (2009), respectively, we can see that the the theoretical power is slightly higher than the simulated power, but the difference is generally small, less than 10% in all the situations considered.

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FIG. 1.

The values of C(λ) and C^*(λ) (upper panels) and the power of D2 and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} (lower panels) for detecting the relationships between sequence pairs related through alternative model I for different values of word size k = 2, 3, 4, 5 and sequence length n. The parameters were set at p_A = p_T = 1/6, p_C = p_G = 1/3, λ = 0.99, and type I error α = 0.05.

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FIG. 2.

The values of B(λ) and B^*(λ) for λ = 0.93, 0.99 and k = 2, 3, 4, 5. Dashed lines refer to B and solid lines to B^*; triangles refer to λ = 0.93 and circles to λ = 0.99. B(0.99), dash line with circle points; B(0.99), dash line with triangle points; B^*(0.99), solid line with circle points; B^*(0.99), solid line with triangle points.

Simulation studies can only explore the influence of a relatively small range of parameter sets on the power of the different tests. With the theoretical results presented in this paper, we are able to explore a much larger parameter space. Theorem 2.5 shows that the power of D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} is mainly determined by B(λ) and B^*(λ), respectively. The higher the values of B's, the more powerful the test is. Therefore, we also plot the values of B(λ) and B^*(λ) for k = 2, 3, 4, 5 and λ = 0.93 or 0.99 (Fig. 2). Again it is shown that B^*(λ) is generally larger than B(λ) indicating that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} is generally more powerful than D₂. We note that both B and B^* decrease when λ increases. The smaller λ is, the larger is the probability of inserting a motif, and the eaiser it is to detect a difference from the null model.

4.3. The power of D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}_2^*$$ \end{document} for comparing two sequences when motifs in JASPAR are present

Since the approximate distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} in Theorem 2.4 requires very long sequences and the resulting formula for calculating the power of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} significantly over-estimates the true power, we will not consider \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} in the following. We next investigate whether the relative performance of D₂ and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} for comparing the relationships between two sequences holds for a large class of motifs. To achieve this objective, we downloaded the transcription factor (TF) binding sites in the database JASPAR CORE (Sandelin et al., 2004) as motifs and studied the power of detecting the relationship between two sequences if such motifs are present in the sequences. The same letter frequencies for the background as in Reinert et al. (2009) are used. The theoretical formulas obtained in this paper make such large scale comparisons possible. Due to the long computational time required when the motif length is large, we only consider motifs with length at most 10.

A total of 323 transcription factor binding profiles with length at most 10 from JASPAR CORE (Sandelin et al., 2004) (October 12, 2009 version) are currently available. These motifs represent the most abundant publicly available knowledge regarding nucleotide sequence motifs. The corresponding PWMs are used to insert motifs as in alternative model I. Based on these assumptions, we can calculate the values of B(λ), B^*(λ), C(λ), C^*(λ), and the corresponding power for different values of word length k and motif density 1 − λ. The resulting figures and the corresponding letter frequencies in each position for all the motifs are presented in the supplementary material. From this large-scale study, we can conclude that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} is more powerful than D₂ in more than 90% of the motifs. An example motif profile “MA0003” for which D₂ is more powerful than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} is given in Figure 3. Note that in this motif, the overall frequencies of (A, C, G, T) in the motif are (0.11, 0.40, 0.40 0.09).

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FIG. 3.

The sequence LOGO of motif “MA0003”.

We then calculate the mean overall letter frequencies of (A, C, G, T) in those motifs for which D₂ is more powerful than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} for at least three of the k = 2, 3, 4, 5 (λ = 0.93) and the corresponding frequencies are (0.08, 0.22, 0.57, 0.13). On the other hand, the mean overall letter frequencies of (A, C, G, T) in the other motifs are (0.30, 0.22, 0.25, 0.23). Under the background sequence model with (A, C, G, T) frequencies equal to (1/6, 1/3, 1/3, 1/6), in general, the GC content of the motifs for which D₂ is more powerful than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} is higher than that of the other motifs under the background model considered in this article. If the background sequence model is changed, the PWM of motifs for which D₂ outperforms \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} should also change. As a general rule, D₂ outperforms \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} if the letter frequencies in a motif are close to the background letter frequencies.

Since we found that, in most situations, the power of D₂ can be even smaller than the type I error, whereas the power of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} always approaches 1 for sequence length tending to infinity, we do not suggest using D₂ in general situations even if it can potentially perform well in some special cases.

4.4. The power of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\rm D}_2^{\rm S}$$ \end{document} for detecting the relationships between two sequences under alternative model II

Previous simulation studies have shown that, under alternative model II, the power of D₂ is less than 0.4 and decreases with sequence length n when the word size k is 2 to 6. Actually, we can show that when n is large, the power of D₂ is always less than 0.5 for any parameter set. Note that Theorem 3.3 shows that the power of D₂ is approximately \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P ({\widetilde Z}_ \lambda \geq {\widetilde z}_{\alpha} )$$ \end{document} . Since \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\widetilde z}_{\alpha}$$ \end{document} is positive and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\widetilde Z}_{\lambda}$$ \end{document} is approximately normally distributed, the power is less than 0.5 when the sequence length is large for any set of parameters. However, similar arguments will not work for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} since the distributions of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\widetilde Z}^*_{\lambda}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\widetilde Z}^S_{\lambda}$$ \end{document} are not normal when λ < 1. This shows that D₂ does not have enough power to detect the relationship between sequences under alternative model II. So we will not study D₂ further under alternative model II. Previous simulation studies also showed that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} is less powerful than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} . So we now concentrate on further understanding \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} .

Theorems 3.2 and 3.3 show that the power of detecting the relationships between two sequences related through the alternative model II using any of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} reaches its plateau quickly as the sequence length increases, and the limit is generally much smaller than 1. Theorems 3.2 and 3.3 justify the simulation results that the simulated power by any of the statistics tends to a limit which is typically less than 1 when sequence length goes to infinity (Reinert et al., 2009), which was quite intriguing at the time of the simulation studies. Let T be any one of statistics D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^*_2$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} . It is theoretically shown here that the primary reason for the power of T to be stable with respect to sequence length n is that there exist constants a_n and b_n such that U_λ,n = a_n(T − b_n) approximates non-degenerate random variables U_λ under both the null model (λ = 1) and the alternative model λ < 1. Although U_λ is stochastically decreasing with respect to λ, the power of the test approaches a constant P(U_λ ≥ u_α), where P(U₁ ≥ u_α) = α. In order for the power of T to increase with respect to sequence length n and to finally reach 1, we need that (1) U_1,n approximates a non-degenerate random variable U₁ under the null model (λ = 1), and (2) U_λ,n tends to infinity as n tends to infinity.

Another interesting observation from previous simulation studies is that the power of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} seems to increase with the length, k, of word pattern used (see Figure 8 in Reinert et al. (2009)). In order to explain this phenomenon, we study the mean \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} as a function of word length k. We are aware that in general the power of a test depends on the distributions of the test statistics under the null and the alternative hypothesis, not just the mean and/or the variance. However, as an explanation to the intriguing observation, we try to see if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\mathbb E} (D^*_2)$$ \end{document} increases with k when other parameters are fixed. Theorem 3.1 (d) shows that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} (ED_2^*) = \sum_ {{\bf w} \in {\cal A} ^k} \left[ \frac {\gamma_0 ({\bf w})+ 2 \sum_ {j = 1} ^ {k - 1} \gamma_j ({\bf w} , {\bf w} )} {p_ {\bf w}} \right] - (2k - 1)= S (\lambda , k). \end{align*} \end{document}

Figure 4 shows the relationship between S(λ, k) and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lambda \in (0.9 , 1)$$ \end{document} for k = 2, 4, 6, 8, 10. It can be seen that S(λ, k) increases with k for any \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lambda \in (0.9 , 1.0)$$ \end{document} , as does the discrepancy between S(λ, k) and S(1, k) for λ < 1. As our statistic is based on comparing the means of the counts under the two models, this partially explains that the power of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} increases with word length k.

OPEN IN VIEWER

FIG. 4.

The values of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S (\lambda , k)= \lim\nolimits_{n \rightarrow \infty} {\mathbb E} (D_2^*)$$ \end{document} as a function of motif density λ and word length k, λ = 0.9 to 1.0 by step 0.01, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$k = 2 , \cdots , 10$$ \end{document}

A.1. Proofs of Theorems 2.2–2.5 under alternative model I

A.2. Proofs of Theorems 3.1, 3.2, and 3.3

Theorem 5.1

Under alternative model I for the two sequences as described above, the expectations of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D^S_2$$ \end{document} can be calculated as follows. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {\mathbb E} (D_2)& = n_X n_Y A^g (\lambda_X , \lambda_Y), \\ {\mathbb E} (D_2^ {*})& = \sqrt {n_X n_Y} A^ {g*} (\lambda_X , \lambda_Y), \\ and \ \lim_ {n \rightarrow \infty} \frac {{\mathbb E} (D_2^ {S} )} {\sqrt {n_X n_Y}} & = A^ {gS} (\lambda_X , \lambda_Y). \end{align*} \end{document}

The limiting distributions of D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} under the general model are given as follows.

Theorem 5.2

Assume that in the background model not all letters are equally likely.

a. Suppose λ_X = λ_Y = 1 (the null model that the sequences are independent). Then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} \left({n_X n_Y} \right) ^ \frac {1} {4} \bigg(\frac {D_2} {n_X n_Y} - \sum_ {{\bf w} \in {\cal A} ^k} p^X_ {\bf w} p^Y_ {\bf w} \bigg) = Z^g_1 , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z^g_1$$ \end{document} has normal distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal N} (0 , {\sqrt {C_{YX}}} (\Sigma^X_1)^2 + {\sqrt {C_{XY}}} (\Sigma^Y_1)^2)$$ \end{document} . Here the asymptotics is valid when the sequence length tends to infinity with alphabet size, motif length, and word length kept fixed.

b. Suppose 0 < λ_X, λ_Y < 1 (the alternative model I). Then \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} \left({n_X n_Y} \right) ^ \frac {1} {4} \bigg(\frac {D_2} {n_X n_Y} - A^g (\lambda_X , \lambda_Y)\bigg)= Z^g_ {\lambda_X , \lambda_Y} , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z^g_{\lambda_X , \lambda_Y}$$ \end{document} has normal distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal N} (0 , {\sqrt {C_{YX}}} (\Sigma^X_{\lambda_X})^2 + {\sqrt {C_{XY}}} (\Sigma^Y_{\lambda_Y})^2 )$$ \end{document} . Here the asymptotics is valid when the sequence length tends to infinity with alphabet size, motif length, and word length kept fixed.

For \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} , we have:

Theorem 5.3

a. Suppose λ_X = λ_Y = 1 (the null model that the sequences are independent). Then, in distribution, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} D^*_2 = Z^ {g*} _1 = \sum_ {{\bf w} \in {\cal A} ^k} \frac {Z^ {(g1 )} _ {\bf w} Z^ {(g2 )} _ {\bf w}} {\sqrt {p^X_ {\bf w}} \sqrt {p^Y_ {\bf w}}} , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{Z^{(g1 )}_{\bf w} , {\bf w} \in {\cal A}^k \}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{Z^{(g2 )}_{\bf w} , {\bf w} \in {\cal A}^k \}$$ \end{document} are independent and have mean 0 normal distributions (with non-trivial covariance matrix).

b. Suppose 0 < λ < 1 (the alternative model I), and that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac {(P_{\lambda} ({\bf w})- p_{\bf w})} {p_{\bf w}}$$ \end{document} is not constant in w. Then, in distribution, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} \left({n_X n_Y} \right) ^ \frac {1} {4} \left(\frac {D^*_2} {{\sqrt n_X n_Y}} - A^ {g*} (\lambda_X , \lambda_Y)\right) = Z_ {\lambda_X , \lambda_Y} ^ {g*} , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z_{\lambda_X , \lambda_Y}^{g*}$$ \end{document} has normal distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal N} \bigg(0 , \sqrt{C_{YX}} (\Sigma^{X*}_{\lambda_X})^2 + \sqrt{C_{XY}} (\Sigma^{Y*}_{\lambda_Y})^2 \bigg)$$ \end{document} .

In order to state the limit distribution for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} , we let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} u_ {\bf w} = \frac {C_ {YX} (P^Y_{\lambda_Y} ({\bf w})- p^Y_ {\bf w})^3} {\left\{C_ {XY} (P^X_{\lambda_X} ({\bf w})- p^X_ {\bf w})^2 + C_ {YX} (P^Y_ {\lambda_Y} ({\bf w})- p^Y_ {\bf w})^2 \right\} ^ \frac {3} {2}} , \\ v_ {\bf w} = \frac {C_ {XY} (P^X_ {\lambda_X} ({\bf w})- p^X_ {\bf w})^3} {\left\{C_ {XY} (P^X_ {\lambda_X} ({\bf w})- p^X_ {\bf w})^2 + C_{YX} (P^Y_ {\lambda_Y} ({\bf w})- p^Y_ {\bf w})^2 \right\} ^ \frac {3} {2}} , \end{align*} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \left(\Sigma^{XS}_{\lambda_X} \right) ^2 & = \sum_{{\bf w} \in {\cal A}^k} (u_{\bf w})^2 (\sigma^X_{\lambda_X} ) ^2 ({\bf w})+ \sum_{{\bf w} \neq {\bf w}^{\prime}} u_{\bf w} u_{{\bf w}^{\prime}} \sigma^X_{\lambda_X} ({\bf w} , {\bf w}^{\prime}), \\ \left(\Sigma^{YS}_{\lambda_Y} \right) ^2 & = \sum_{{\bf w} \in {\cal A}^k} (v_{\bf w})^2 (\sigma^Y_{\lambda_Y} ) ^2 ({\bf w})+ \sum_{{\bf w} \neq {\bf w}^{\prime}} v_{\bf w} v_{{\bf w}^{\prime}} \sigma^Y_{\lambda_Y} ({\bf w} , {\bf w}^{\prime}). \end{align*} \end{document}

The following theorem gives the approximate distribution of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} under the null and the alternative models for the general situation.

Theorem 5.4

a. Suppose λ_X = λ_Y = 1 (the null model that the sequences are independent). Then, in distribution, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} \frac {D^S_2} {\left({n_X n_Y} \right) ^ \frac {1} {4}} = Z^ {gS} _1 = \sum_ {{\bf w} \in {\cal A} ^k} \frac {Z^ {(g1 )} _ {\bf w} Z^ {(g2 )} _ {\bf w}} {\sqrt {\sqrt {C_ {XY}} (Z^ {(g1 )} _ {\bf w})^2 + \sqrt {C_ {YX}} (Z^{(g2 )} _ {\bf w})^2}} , \tag {15} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{Z^{(g1 )}_{\bf w} , {\bf w} \in {\cal A}^k \}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{Z^{(g2 )}_{\bf w} , {\bf w} \in {\cal A}^k \}$$ \end{document} are independent and have mean 0 normal distribution.

b. Suppose 0 < λ_X, λ_Y < 1 (the alternative model I), and assume that both \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P^X_{\lambda_X} ({\bf w})- p^X_{\bf w}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P^Y_{\lambda_Y} ({\bf w})- p^Y_{\bf w}$$ \end{document} are not constant in w. Then, in distribution, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lim_ {n \rightarrow \infty} \left({n_X n_Y} \right) ^ \frac {1} {4} \left(\frac {D^S_2} {\sqrt {n_X n_Y}} - A^ {gS} (\lambda_X , \lambda_Y)\right) = Z^ {gS} _ {\lambda_X , \lambda_Y} , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z^{gS}_{\lambda_X , \lambda_Y}$$ \end{document} has normal distribution \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $${\cal N} (0 , \sqrt{C_{YX}} (\Sigma^{XS}_{\lambda_X})^2 + \sqrt{C_{XY}} (\Sigma^{YS}_{\lambda_Y})^2 )$$ \end{document} .

The proof of Theorem 5.4 is sketched as follows. Similarly as for (14), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \frac {D_2^S} {\left({n_X n_Y} \right) ^ \frac {1} {4}} = \sum_{{\bf w} \in {\cal A} ^k} \frac {\left({\widetilde X} _ {\bf w} / \sqrt {n_X} \right) \left({\widetilde Y} _ {\bf w} / \sqrt {n_Y} \right)} {\sqrt {\sqrt {C_ {XY}} {\widetilde X} _ {\bf w} ^2 / n_X + \sqrt {C_ {YX}} {\widetilde Y} _ {\bf w} ^2 / n_Y}}. \end{align*} \end{document}

For part (a), under the null hypothesis, we have that, in distribution, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {\widetilde X}_{\bf w} / \sqrt{n_X} \rightarrow Z^{(g1 )}_{\bf w} , \quad {\widetilde Y}_{\bf w} / \sqrt{n_Y} \rightarrow Z^{(g2)}_{\bf w}. \end{align*} \end{document}

For part (b), we can write \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \frac {{\widetilde X} _{\bf w}} {n_X} = \frac {X_ {\bf w}} {n_X} - P^X_ {\lambda_X} ({\bf w})+ P^X_ {\lambda_X} ({\bf w})- p^X_{\bf w} , \end{align*} \end{document}

Then we use Taylor expansion for the function g _w (x, y) given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} g_ {\bf w} (x , y)= \frac {(x + P^X_ {\lambda_X} ({\bf w})- p^X_{\bf w})(y + P^Y_ {\lambda_Y} ({\bf w})- p^Y_ {\bf w} )} {\sqrt {C_ {XY} (x + P^X_ {\lambda_X} ({\bf w})- p^X_ {\bf w})^2 + C_{YX} (y + P^Y_ {\lambda_Y} ({\bf w})- p^Y_ {\bf w})^2}} , \end{align*} \end{document}

at (x, y) = (0, 0), as well as (14).

From Theorems 5.2, 5.3, and 5.4, we are able to calculate the power of detecting the relationships between sequences A and B under the general model.

Theorem 5.5

Assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(P^X_{\lambda_X} ({\bf w})- p^X_{\bf w})^2 / p^X_{\bf w}, \ (P^Y_{\lambda_Y} ({\bf w})- p^Y_{\bf w})^2 / p^Y_{\bf w} \ and \ P^X_{\lambda_X} ({\bf w})- p^X_{\bf w}$$ \end{document} are not constant in w. Then, for any given type I error α, the power of detecting the relationship between two sequences A and B against the null model that λ_X = λ_Y = 1 using D₂, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^*$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$D_2^S$$ \end{document} can be approximated by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$1 - \Phi (C^g (\lambda_X, \lambda_Y)), \ 1 - \Phi \left(C^g (\lambda_X , \lambda_Y)\right) , 1 - \Phi \left(C^{g*} (\lambda_X , \lambda_Y)\right), \ and \ 1 - \Phi \left(C^{gS} (\lambda_X , \lambda_Y)\right)$$ \end{document} , respectively, where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} C^g (\lambda_X , \lambda_Y)& = - \left({n_X n_Y} \right) ^ \frac {1} {4} B^g (\lambda_X , \lambda_Y)+ z^g_ {\alpha} / \sqrt {\sqrt {C_ {YX}} (\Sigma^X_{\lambda_X})^2 + \sqrt {C_ {XY}} (\Sigma^Y_ {\lambda_Y})^2} , \\ C^ {g*} (\lambda_X , \lambda_Y)& = - \left({n_X n_Y} \right) ^ \frac {1} {4} B^ {g*} (\lambda_X , \lambda_Y)+ z_{\alpha} ^ {g*} / \left(\left({n_X n_Y} \right) ^ \frac {1} {4} \sqrt {\sqrt {C_ {YX}} (\Sigma^ {X*} _ {\lambda_X})^2 + \sqrt {C_{XY}} (\Sigma^ {Y*} _ {\lambda_Y})^2} \right) , \\ C^ {gS} (\lambda_X , \lambda_Y)& = - \left({n_X n_Y} \right) ^ \frac {1} {4} B^ {gS} (\lambda_X , \lambda_Y)+ z_{\alpha} ^ {gS} / \sqrt {\sqrt {C_ {YX}} (\Sigma^ {XS}_{\lambda_X})^2 + \sqrt {C_ {XY}} (\Sigma^ {YS} _ {\lambda_Y})^2} \end{align*} \end{document}

and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} B^g (\lambda_X , \lambda_Y)& = \frac {A^g (\lambda_X , \lambda_Y)- A^g (1 , 1 )} {\sqrt {\sqrt {C_ {YX}} (\Sigma^X_ {\lambda_X})^2 + \sqrt {C_ {XY}} (\Sigma^Y_ {\lambda_Y})^2}} , \\ B^ {g*} (\lambda_X , \lambda_Y)& = \frac {A^ {g*} (\lambda_X , \lambda_Y)} {\sqrt {\sqrt {C_ {YX}} (\Sigma^ {X*} _ {\lambda_X})^2 + \sqrt {C_ {XY}} (\Sigma^ {Y*} _ {\lambda_Y})^2}} ,\\ B^ {gS} (\lambda_X , \lambda_Y)& = \frac {A^ {gS} (\lambda_X , \lambda_Y )} {\sqrt {\sqrt {C_ {YX}} (\Sigma^ {XS} _ {\lambda_X})^2 + \sqrt {C_{XY}} (\Sigma^ {YS} _ {\lambda_Y})^2}}. \end{align*} \end{document}

Here, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$z^g_{\alpha} , z_{\alpha}^{g*}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$z_{\alpha}^{gS}$$ \end{document} are the upper α quantile of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Z^g_1, \, Z^{g*}_1, \, Z^{gS}_1$$ \end{document} from Theorems 5.2, 5.3, and 5.4, respectively.

The alternative model II can equally be extended to the situation of different letter frequencies in the two sequences; we omit the details here.