Approximation of sojourn-times via maximal couplings: motif frequency distributions

Abstract

Sojourn-times provide a versatile framework to assess the statistical significance of motifs in genome-wide searches even under non-Markovian background models. However, the large state spaces encountered in genomic sequence analyses make the exact calculation of sojourn-time distributions computationally intractable in long sequences. Here, we use coupling and analytic combinatoric techniques to approximate these distributions in the general setting of Polish state spaces, which encompass discrete state spaces. Our approximations are accompanied with explicit, easy to compute, error bounds for total variation distance. Broadly speaking, if ${\mathsf{T}}_n$ is the random number of times a Markov chain visits a certain subset ${\mathsf{T}}$ of states in its first $n$ transitions, then we can usually approximate the distribution of ${\mathsf{T}}_n$ for $n$ of order $(1-\alpha )^{-m}$, where $m$ is the largest integer for which the exact distribution of ${\mathsf{T}}_m$ is accessible and $0\le \alpha \le 1$ is an ergodicity coefficient associated with the probability transition kernel of the chain. This gives access to approximations of sojourn-times in the intermediate regime where $n$ is perhaps too large for exact calculations, but too small to rely on Normal approximations or stationarity assumptions underlying Poisson and compound Poisson approximations. As proof of concept, we approximate the distribution of the number of matches with a motif in promoter regions of C. elegans. Mathematical properties of the proposed ergodicity coefficients and connections with additive functionals of homogeneous Markov chains as well as ergodicity of non-homogeneous Markov chains are also explored.

Appendices

Appendix A: Parameter estimation associated with the DA-motif

Here we describe how we estimated the parameters of the fifth-order models in Sect. 3.2. Since the training sequences (i.e. promoter regions) are short in comparison to the length of the model (e.g. the dat-1 promoter region contains only 417 base-pairs but a fifth-order Markov model on the DNA-alphabet has 1,024 possible prefixes of length $5$), we applied the following incremental approach over each promoter region. Letting the index $\ell$ denote memory order under consideration, first estimate the four parameters associated with $\ell =0$, i.e. the memoryless model. Subsequently, for each $1\le \ell \le 5$, estimate the model parameters of the order $\ell$ model as follows: For each prefix of length $\ell$, say $w_1\cdots w_\ell$ with $w_i\in \{\text{ A, } \text{ C, } \text{ G, } \text{ T }\}$, compute the relative frequencies of the words $w_1\cdots w_\ell \text{ A }, w_1\cdots w_\ell \text{ C }, w_1\cdots w_\ell \text{ G }$ and $w_1\cdots w_\ell \text{ T }$ in the training sequence. If the prefix $w_1\cdots w_\ell$ does not appear or if any of the relative frequencies is equal to 1 (i.e. we estimate transitions from $w_1\cdots w_\ell$ to be deterministic) then assign to $w_1\cdots w_\ell$ the transition probabilities of $w_2\cdots w_\ell$ from the model of order $(\ell -1)$. Otherwise, take those relative frequencies to be the transition probabilities.

Appendix B: A heuristic approach to choosing $m$ and $k$

It is not obvious for general Markov chains how one should choose the parameters $k$ and $m$ in Theorem 3 in order to minimize computation. The problem is difficult because the amount of work required to compute the approximation depends on the parameter $\alpha _k(p)$, but $p_{u}^k$ is expensive to compute for large $k$. We suggest an approach that chooses $k$ and $m$ to satisfy a given error bound.

First, suppose $k$ is fixed and we would like to choose $m$ to guarantee the total variation distance is below a given bound, $0<\epsilon <1$. One may compute a value of $m$ to satisfy $\mathbb{P }(L_{(n\,\mathrm{div }\,k)}> m)\le \epsilon$ exactly by recursion, or one may use an approximation.

Arratia et al. (1990) give an extreme value approximation to the longest run of heads in a Bernoulli sequence with length $n$. For our application, the total variation distance between our approximation and the actual distribution is

$$\begin{aligned} d_{TV}\le \mathbb{P }({\mathsf{L}}_n\ge m+1) = 1-\mathbb{P }({\mathsf{L}}_n<m+1) \approx 1-e^{-(1-\alpha )^t} \end{aligned}$$

where $t=\log _{1/(1-\alpha )}((n-1)\alpha +1)-m-1$. Consequently, given $\alpha$, we should choose

$$\begin{aligned} m = \lceil \log _{1/(1-\alpha )}((n-1)\alpha +1) - \log \log (1/(1-\epsilon ))/\log (1-\alpha )-1\rceil . \end{aligned}$$

(66)

Now we turn to choosing $k$ and $m$, jointly. The algorithm will attempt to minimize computation based on a few observations. First, the quantity $k\cdot m$ is the largest number of consecutive transitions of the Markov chain considered by the approximating random variable. Our heuristic approach is to search for the first (in $k$) minimum of this quantity. The minimum must exist since for $k>n/2$ we always have $m=1$.

Another observation is that if $\alpha _k(p)$ increases with $k$, the remaining computational effort of applying the approximation with $p_{u}^k$ is lower than with $p_{u}^{k-1}$. Finally, $k\cdot m<n$ (ideally $k\cdot m\ll n$) is required for our approximation to be an improvement over the transfer matrix method, in the case that the algorithm we present below terminates with $k\cdot m\approx n$; the transfer matrix method can be applied with $p_{u}^k$ and the computation is not wasted. The program below increments $k$ until the first minimum is found. The final loop to recalculate $\mathbb{P }(L_{(n \,\mathrm{div }\,k)} > m)$ exactly via recursion is only necessary if one uses an approximation to find $m$.