In n independent Bernoulli observations with success probability p and failure probability q = 1—p, values are denoted by 1 (success with probability p) or 0. A crossing consists of two consecutive different values, and a run of length l consists of l successive observations, delimited by a crossing or the first or last observation. The possible values of the number C of crossings are c = 0, …, n-1, and the possible values for the length L of the longest run are l = 1, …, n. The joint probabilities of L and C for given n are denoted by P_n(L = l,C = c).

The iterative procedure involves conditioning on the first observation denoted by S, with values 1 for success (probability p) and 0 for failure (probability q). The iterative procedure computes the conditional probabilities Pn(L=l,C=c|S=1),Pn(L=l,C=c|S=0)

This conditioning on the first observation is an essential part of the procedure. One way to see that this is reasonable is to consider the case when p is close to 1. Then most observations are successes, most runs are success runs and the conditional joint distribution of runs and crossings is quite different dependent on the first observation. It is sufficient to be able to compute these conditional distributions, because the unconditional joint distribution is Pn(L=l,C=c)=Pn(L=l,C=c|S=1)·p+Pn(L=l,C=c|S=0)·q

For the iterative procedure to work it is also necessary to take another variable into account, the first crossing. More precisely, we denote the end position of the first crossing by F, with values F = 2, …, n. An additional value F = 1 denotes, by convention, the case of no crossing. The joint probabilities for C and L conditional on S are partitioned by further conditioning on F as detailed below.

It is important to underline that the conditioning variables S and F are not parameters of the distribution but useful constructions that help break down the iterative procedure in manageable parts.

This article first describes the setting for the iterative computation procedure and its initial stage. Next, we introduce conditioning on the starting position and partitioning on the position of the first crossing. Finally, the joint distribution of the number of crossings and the longest run conditional on these variables is given. The computation procedure is different in two cases that are subsequently described.

After the exposition of the iterative procedure follows a brief comment on the simpler symmetric case, and the precision of the procedure is discussed. The resulting joint distribution is described in a simple case, and procedures for checking are briefly commented. Next follows sections on generalizations and applications and on limitations of the procedures in crossrun together with a brief description of ongoing work to address these limitations. Finally, a Conclusions section summarizes the article. Two appendices are included, one giving details on the "times" representation in which the joint distributions are actually stored, as described in the section on precision, and one on the code for the main function crossrunbin and for the analyses in this article.

First, we present the starting point of the iterative procedure, the conditional probabilities in the rather redundant case with only one observation. If n = 1, then 0 is the only possible value of C and 1 the only possible value of L, therefore P₁(C = 0,L = 1|S = 1) = P₁(C = 0,L = 1|S = 0) = 1.

In this case the joint distribution of (C,L) is degenerate and takes the value (0,1) with probability one. Moving to more than one observation, the next step is presenting the conditional distribution of the end position F of the first crossing, conditional on the starting position S.

If the first value is 1 (success), no crossing means that all the remaining n-1 values are also 1, therefore P_n(F = 1|S = 1) = pⁿ⁻¹. Similarly, P_n(F = 1|S = 0) = qⁿ⁻¹. Next, if f = 2, …, n and the first value is a success, F = f means that the sequence starts with a success, then f-2 more successes and then one failure. Therefore, Pn(F=f|S=1)=pf−2·q,Pn(F=f|S=0)=qf−2·p,f=2,…,n where the last formula is based on a similar argument conditional on S = 0. In the following, arguments will in many cases be given for S = 1 only, and similar results for S = 0 will be stated with no explicit arguments. By symmetry, these results will simply involve replacing p by q. In the next step the formulas in this section will be used for partitioning the joint conditional probabilities for C and L given S, by the position F of the first crossing.

Partitioning on F we have Pn(L=l,C=c|S=1)=∑f=1nPn(L=l,C=c|S=1,F=f)·Pn(F=f|S=1) where, as shown in the previous section, Pn(F=f|S=1)=pf−2qiff≥2andPn(F=1|S=1)=pn−1

The formulas for P_n(L =l,C = c|S = 0) are the same, just interchanging p and q. This implies that the joint probabilities of C and L conditional on S may be computed if it is possible to compute all the joint probabilities of C and L conditional on S and F. This is the next step.

First, if there is no crossing (F = 1) the entire sequence constitutes one single run, therefore Pn(C=0,L=1|S=1,F=1)=Pn(C=0,L=1|S=0,F=1)=1 and all other conditional probabilities are 0. Thus, the matrices of joint probabilities of C and L conditional on F = 1 together with each value of S, are matrices with all components equal to 0, except for a 1 in the upper right corner.

If crossings do occur (f = 2, …, n), the conditional probabilities Pn(C=c,L=l|S=1,F=f),Pn(C=c,L=l|S=0,F=f) are more complicated. The key to computing these probabilities is to recognize that, except for the initial run of f-1 observations, the remaining observations constitute n-(f-1) = n+1-f identical and independent Bernoulli observations with success probability p, they represent the same setting as for all n observations, just a shorter sequence. Further, these n+1-f observations are also conditional on a fixed value of their first observation, only that this fixed value is the opposite as in the entire sequence. This is because the last n+1-f observations start with the observation after the first crossing.

We now have to distinguish between two cases. In case 1, the f-1 observations in the initial run, before the first crossing, are at least as many as the last n+1-f ones. In case 2, the initial run is shorter:

Case 1: f-1 ≥ n+1-f

Case 2: f-1 < n+1-f

The two cases are illustrated in Fig 2 below. In both cases the observations (red) from the end of the first crossing constitute a runchart of its own, only shorter and starting on the opposite side. In Case 1 (top), the initial run of f-1 = 14 observations is longest, l = f-1 = 14. The number of crossings in the last n+1-f = 10 observations is between 0 and 9, and the total number of crossings is one more, 1 ≤ c ≤ 10. In Case2 (bottom) the initial run of f-1 = 8 observations may or may not be longest, l ≥ 8. It is a longest run precisely if no run within the last n+1-f = 16 (red) observations is longer than 8. If all observations in the last 16 observations are below the median, l = 16, thus for the whole sequence 8 ≤ l ≤ 16. The number of crossings in the last part (red, 16 observations) is between 0 and 16–1 = 15, and the number of crossings in the whole sequence is one more, 1° ≤ c ≤ 16. What is shown here in an example is the core of the general argument that follows.

Top: f = 15, the initial run is necessarily longest. Bottom: f = 9, the initial run may or may not be longest.

This is the simplest case. Here, the first f-1 observations constitute a run of length f-1, and no run in the last n+1-f observations may be longer than that. Therefore, the longest run is f-1, and the non-zero probabilities P_n(C = c,L = l|S = 1,F = f) are confined to the vertical strip l = f-1. And, in fact, to only a part of this strip. First, there is at least one crossing, from time f-1 to f. Also, any further crossings are within the last observations, and may be any number between 0 and (n+1-f)– 1. The total number of crossings may therefore be any number between 1 and n+1-f, which means that the non-zero probabilities P_n(C = c,L = l|S = 1,F = f) are confined to the strip l = f-1, 1 ≤ c ≤ n+1-f.

The non-zero probabilities P_n(C = c,L = l|S = 1,F = f), l = f−1,1≤c≤n+1−f are somehow determined by what happens within the last n+1-f observations. More specifically, the last n+1-f observations constitute a sequence of the same type as the original n observations, only shorter, and with the starting observation fixed on the opposite side of the central line. To put it into a formula, Pn(C=c,L=f−1|S=1,F=f)=Pn+1−f(C=c−1|S=0) where C = c-1 is because the crossing from f-1 to f is just before the last n+1-f observations. Similarly, Pn(C=c,L=f−1|S=0,F=f)=Pn+1−f(C=c−1|S=1)

The probabilities on the right-hand side of these formulas are for a lower number of observations and are therefore already computed in the iterative procedure.

Note that the n+1-f observations after the initial run start on opposite side of the centre line. Therefore, it is necessary to compute conditional probabilities conditional on starting values both above and below the centre line in the iterative procedure, they cannot be computed separately. The computations are a bit more complicated in the second case, when the initial run is the shorter part, but the main idea is the same.

As in case 1, the total number of crossings is between 1 and n+1-f. As to the longest run L, it cannot be shorter than f-1 or longer than n+1-f, and it is necessary to distinguish between values l = f-1 and l ≥.f. A longest run f-1 in the entire sequence means that all runs in the last n+1-f observations have length l ≤ f-1. Therefore Pn(C=c,L=f−1|S=1,F=f)=Pn+1−f(C=c−1,L≤f−1|S=0) (and similarly conditional on S = 0). For longer runs in which f ≤ l ≤ n+1-f, the longest run has to be within the last n+1-f observations and we have Pn(C=c,L=l|S=1,F=f)=Pn+1−f(C=c−1,L=l|S=0) (and similarly conditional on S = 0). All these conditional probabilities, based on a shorter sequence, have already been computed in an iterative computation procedure.

For p = 0.5 there is a symmetry between crossings up or down, and between success and failure runs. Therefore, conditioning on the first observation is not necessary, although it is still necessary to partition on the first crossing F. Also, by an induction argument following the iterative procedure, all these probabilities are integer multiples of 0.5ⁿ⁻¹ and, in fact, represent a partition of the binomial coefficients in the distribution of C, by the values l = 1, …, n of L.

To enhance precision, computations have been performed in the R package Rmpfr [6], an R interface to the GNU MPFR library [7]. Preliminary investigations pointed to precision problems above values about 50 for sequence length n without this increased precision, but no such problems up to n = 100 when using Rmpfr. To further enhance precision, probabilities are represented in the times representation, they have been multiplied by m^n-1 where m is a multiplier with default value 2. Thereby very small numbers are avoided, at least to some extent, and the numbers computed are integers in the symmetric case. The joint probabilities, using the times representation, in the symmetric case for n = 16 are shown in Table 1 below in this representation:

The corresponding joint probabilities are obtained by dividing these integers by 2ⁿ⁻¹ = 2¹⁵ = 32768, for instance P₁₆(C = 5,L = 6) = 741/32768 = 0.023. The highest joint probability is P₁₆(C = 7,L = 4) = 2716/32768 = 0.083. It is also seen that a high proportion of the joint probabilities consists of zeroes, and except for some very small numbers the joint probabilities are concentrated within a narrow sloping band, as also commented in the crossrun vignette, these are fairly general phenomena. For comparison the joint distribution for n = 16 is also shown in Table 2 below for p = 0.6, a case where observations tend to stay above the midline. These probabilities are still shown in the "times" representation, they are multiplied by 2ⁿ⁻¹ = 32768, and are shown with one decimal digit: