Approximating the distributions of runs and patterns

Normal approximation

Given a pattern Λ, it is well known that the mean μ _W and the variance ${\sigma }_W^2$ are difficult to obtain via combinatoric arguments, especially when Λ is a compound pattern or the trials are Markov dependent. For example, as pointed out in Karlin (2005) and Kleffe and Borodovski (1992), approximate values of μ _W and ${\sigma }_W^2$ must sometimes be used. Since W(Λ) is finite Markov chain imbeddeble, (7) and (8), provide the exact values.

The limit in (6) is appropriate when the sequence of inter arrival times {W _i (Λ)} are i.i.d., which is the case for simple and compound patterns when the {X _i } are i.i.d. and counting is non-overlapping. When occurrences of Λ correspond to a delayed renewal process, which can occur for Markov dependent trials and/or overlapping counting, we could use the mean and variance of W₂(Λ) for the normalizing constants, which are easily obtained by modifying ξ₀ in (7) and (8). Even more general cases can be handled by making use of a functional central limit theorem for Markov chains (see, for example, (Meyn and Tweedie 1993, §17.4) and (Asmussen 2003, Theorem 7.2, pg. 30) for the details).

Poisson and compound poisson approximations

where $ℒ(·)$ denotes the distribution (law) of a random variable and d_TV(·,·) denotes the total variation distance.

The primary tool used to obtain μ _n and the bound ε _n is the Stein-Chen method (Chen 1975), and this method has been refined by various authors Arratia et al. (1990), Barbour and Eagleson (1983), Barbour and Eagleson (1984), Barbour and Eagleson (1987), Barbour and Hall (1984), Godbole (1990a), Godbole (1990b), Godbole (1991), Godbole and Schaffner (1993), and Holst et al. (1988). This method has also been extended to compound Poisson approximations for the distributions of runs and patterns and Barbour and Chryssaphinou (2001) provides an excellent theoretical review of these approximations.

Finding $\mathbb{E}{X}_n\left(\Lambda \right)$ and the bound ε _n is usually done on a case by case basis. For the mathematical details, the books (Barbour et al. 1992a) and (Balakrishnan and Koutras 2002) are recommended.

The distribution of N_n,k, the number of non-overlapping occurrences of k consecutive successes in n i.i.d. Bernoulli trials, is one of the most important in this area and one of the most studied in the literature. Reversing the roles of S (success) and F (failure), the reliability of consecutive-k-out-of-n system, denoted C(k,n : F), is given by $\mathbb{P}\{N_{n,k}=0\}$. Even in this simple case (i.e. Λ=S S⋯S), there are several ways to apply the Poisson approximation techniques. For example, (Godbole 1991, Theorem 2) shows that approximating N_n,k with a $P\left(\mathbb{E}{N}_{n,k}\right)$ distribution works well if certain conditions hold. Godbole and Schaffner (Godbole and Schaffner 1993, pg. 340) suggests an improved Poisson approximation for word patterns.

The primary difficulty in applying the Poisson approximation is the determination of the optimal parameter μ _n , which is higly dependent on the structure of the pattern Λ. In particular, if Λ is long and has several uneven overlapping sub-patterns, then finding μ _n by their method can be very tedious. In the sequel, we show that even the (asymptotic) best choice for μ _n for Poisson approximations does not perform well in the relative sense.

FMCI approximations

Approximations based on the FMCI approach depend on the spectral decomposition of the essential transition probability matrix N.

Let N be a w×w essential transition probability matrix associated with a finite Markov chain {Y _n :n≥0} corresponding to the distribution of the waiting time W(Λ). Let 1>λ₁≥|λ₂|≥⋯≥|λ _w | denote the ordered eigenvalues of N, repeated according to their algebraic multiplicities, with associated (right) eigenvectors ${\mathit{\eta }}_1^{\prime },{\mathit{\eta }}_2^{\prime },\cdots ,{\mathit{\eta }}_w^{\prime }$. When the geometric multiplicity of λ _i is less than its algebraic multiplicity, we will use vectors of 0’s for the unspecified eigenvectors. The fact that λ₁ can be taken as a positive real number and that η₁ can be taken to be non-negative are consequences of the Perron-Frobenious Theorem for non-negative matrices ( Seneta cf.1981).

Theorem 1

Let {X _i } be a sequence of i.i.d. trials taking values in , let Λ be a simple pattern of length ℓ with d×d essential transition probability matrix N and let X _n (Λ) be the number of non-overlapping occurrences of Λ in {X _i }. If N satisfies the FMCI approximation conditions then, for any fixed x≥0,

$\mathbb{P}\left\{X_n\right(\Lambda )=x\}\sim a^{x+1}\left(\genfrac{}{}{0.0pt}{}{n-x(\ell -1)}{x}\right){(1-{\lambda }_1)}^x{\lambda }_1^{n-x},$

(13)

where$a=\sum _{j=1}^g{a}_j\left({\mathit{\xi }}_0{\mathit{\eta }}_j^{\prime }\right)$. If g=1, as is usually the case, then a=a₁(ξ₀η 1′).

Given a pattern Λ, the approximation in (13) requires finding the Markov chain imbedding associated with the waiting time W(Λ), the essential transition probability matrix N as well as its eigenvalues and associated eigenvectors. Usually, these steps are rather simple and can be easily automated together with (13). Even for very large n and large ℓ, say n=1,000,000 and ℓ=50, the CPU time is negligible. Fu and Johnson (2009) also provide details on extending these results to compound patterns, overlapping counting and Markov dependent trials.

For the purpose of comparing these approximations, we prefer to write (13) as

$\begin{array}{ll}\mathbb{P}\left\{X_n\right(\Lambda )=x\}& \sim a^{x+1}{\left(\frac{1-{\lambda }_1}{{\lambda }_1}\right)}^x\left(\genfrac{}{}{0.0pt}{}{n-x(\ell -1)}{x}\right)exp\{nln{\lambda }_1\}\end{array}$

(14)

Note that the approximation havs three parts: a constant part; a polynomial in n of degree x; and a third (dominant) part which converges to 0 exponentially fast as n→∞.

More precisely, the FMCI approximation in (13) may be written as

$\begin{array}{ll}\mathbb{P}\left\{X_n\right(\Lambda )=x\}& =a^{x+1}{\left(\frac{1-{\lambda }_1}{{\lambda }_1}\right)}^x\left(\genfrac{}{}{0.0pt}{}{n-x(\ell -1)}{x}\right)\\ \times exp\{nln{\lambda }_1\}\left[1+o\left({\left|\frac{{\lambda }_{g+1}}{{\lambda }_1}\right|}^{n/(x+1)-\ell }\right)\right].\end{array}$

(15)

Since |λ_g+1|<λ₁, the term |λ_g+1/λ₁|^{n/(x+1)−ℓ} tends to 0 exponentially as n→∞ and hence is negligible if n/(x+1)−ℓ is moderate or large (say ≥50).

Large deviation approximation

Fu et al. (2012) provide the following large deviation approximation for right-tail probabilities for the number of non-overlapping occurrences for simple patterns Λ. The reasons for providing only the right-tail large deviation approximation are (i) all of the above mentioned approximations fail to approximate the extreme right-tail probabilities and (ii) the FMCI approximation provides an accurate approximation for left-tail probabilities.

Theorem 3

Let {X _i } be a sequence of i.i.d. multi-state trials taking values in and let Λ be a simple pattern defined on . Then, for every fixed x, we have,

$\left(i\right)\underset{n\to \infty }{lim}R(x:E,F)=0;$

(20)

$\begin{array}{l}\left(\mathit{\text{ii}}\right)\underset{n\to \infty }{lim}R(x:E,P({\mu }_n\left)\right)=\left\{\begin{array}{cc}\infty ,& \text{if}{limsup}_n{\mu }_n/n<-ln{\lambda }_1;\\ c\left(x\right),& \text{if}{lim}_n{\mu }_n/n=-ln{\lambda }_1;\\ -\infty ,& \text{if}{liminf}_n{\mu }_n/n>-ln{\lambda }_1;\end{array}\right.\end{array}$

(21)

$\begin{array}{l}\left(\mathit{\text{iii}}\right)\underset{n\to \infty }{lim}R(x:E,N)=\left\{\begin{array}{cc}\infty ,& \text{if}{\mu }_W/2\underset{W}{\overset{2}{\sigma }}\le -ln{\lambda }_1;\\ -\infty ,& \text{if}{\mu }_W/2\underset{W}{\overset{2}{\sigma }}>-ln{\lambda }_1;\end{array}\right.\end{array}$

(22)

Proof

and hence (i) follows immediately from the definition of R(x:E,A) and Theorem 1.

and this completes the proof of (ii). Note also that, if ${limsup}_n{\mu }_n/n>-ln{\lambda }_1$ and ${liminf}_n{\mu }_n/n<-ln{\lambda }_1$, then ${lim}_{n}R(x:E,P({\mu }_n\left)\right)$ will not exist.

making it clear that (24) converges to ∞ if ${\mu }_W/2{\sigma }_W^2\ge -ln{\lambda }_1$ and 0 otherwise. Therefore, R(x:E,N)→∞ if ${\mu }_W/2{\sigma }_W^2\ge -ln{\lambda }_1$ and R(x:E,N)→−∞ if ${\mu }_W/2{\sigma }_W^2<-ln{\lambda }_1$ and the proof of (iii) is complete.

Theorem 3 (ii) implies that asymptotically (for fixed x and n→∞), the Poisson approximation performs poorly (in the relative sense) regardless of the value μ _n used. When Λ is simple and does not have overlapping sub-patterns, taking ${\mu }_n=\mathbb{E}{X}_n\left(\Lambda \right)$ is normally recommended for the Poisson approximation (cf. Arratia et al. 1990). In this case, non-overlapping and overlapping counting is equivalent. The following corollary shows that, for fixed x, the Poisson approximation will (asymptotically) always overestimate the exact probability in the following sense.

Corollary 1

for all fixed x.

Proof

which completes the proof.

Corollary 1 implies that, if ${\mu }_n\sim \mathbb{E}{X}_n\left(\Lambda \right)$, then the Poisson approximation will always overestimate the exact probability as n→∞. Together with Theorem 3 (ii), this implies that using μ _n ∼−n lnλ₁ results in the best Poisson approximation as n→∞.

Thus, R(x:E,N)→±∞ are both possible depending on x, the pattern, and the probability structure of the {X _i }.

Reliability of C(k,n:F) systems

The FMCI approximation performs very well for the parameters tested here. As expected, the Poisson and compound Poisson approximations perform well when n q ^k is relatively small. When the reliability of the system is relatively low, the Poisson and compound Poisson approximations begin to degrade.

Approximating the distribution of N _n,k

Recall that N_n,k is the number of non-overlapping occurrences of k consecutive successes in {X _i } (i.e. N_n,k=X _n (Λ) with Λ=S S⋯S of length k). By reversing the roles of success and failure, the reliability of C(k,n : F) systems can be related to the distribution of N_n,k. In this section we present some examples of approximating $\mathbb{P}\{N_{n,k}=x\}$ with the approximations FMCI, Normal, Poisson and LD.

Figure 1 shows the relative error R(x:E,A) in these approximations for (a) N_2000,4; (b) N_5000,4; and (c) N_250000,6 when the probability of success is p=0.3. On all of the figures, the top axis is on a standard z-scale making use of the asymptotic mean and variance of X _n (Λ) — namely,

$z=\frac{x-n/{\mu }_W}{\sqrt{n{\sigma }_W^2{\mu }_W^{-3}}}.$

We notice that the Finite Markov chain imbedding approximation (FMCI) performs very well in the left tail of the distribution in all cases. Its performance degrades as x gets large but its performance is more consistent than both the Poisson and Normal approximations in this case. The large deviation approximation performs well in the right tail in all cases. In (c), the FMCI approximation performs very well throughout most of the support. The Poisson approximations also perform well over most of the x considered. The normal approximation performs well in the neighbourhood of $\mathbb{E}{X}_n\left(\Lambda \right)$ but not in the tails.

As the probability of success p increases, the FMCI approximation still performs very well in the left tail, but it’s performance tends to degrade more quickly as x increases. The Poisson approximations also quickly degrades as p increases since $\mathbb{E}{N}_{n,k}$ increases. For larger p, the Normal approximation tends to work better near the mean. In the far left tail, the FMCI approximation is preferred and in the far right tail, the LD approximation is preferred.

Biological sequences

Sequences of DNA nucleotides are of great interest (as are sequences of amino acids and other biological sequences). Figure 2 shows the relative errors for approximating $\mathbb{P}\left\{X_n\right(\Lambda )=x\}$ with Λ=A C G (n=1,000 and 10,000) and Λ=C A T T A G (n=500,000). We see that the FMCI approximation again performs very well in the left tail, although, in (b), the performance degrades somewhat as x gets large. The large deviation approximation performs very well in the right tail, especially when x is greater than 3 standard deviations above the mean. While it is difficult to give a rule of thumb, the FMCI approximation seems to perform very well when $x\le \mathcal{O}\left(n^{1/2}\right)$. The normal approximation works best within a few standard deviations of the mean and performs best in this region when $\mathbb{E}{X}_n\left(\Lambda \right)$ is relatively large.