On Limited Length Binary Strings with an Application in Statistical Control

In a 0 1 sequence of Markov dependent trials we consider a statistic which counts strings of a limited length run of 0s between subsequent 1s. Its probability mass function is used to determine the chance that a stochastic process remains or not in statistical control. Illustrative numerics are presented.


INTRODUCTION AND PRELIMINARIES
Let {X i } i≥1 be a sequence of binary random variables (RVs) with values ordered on a line.For n ≥ 2 and two non-negative integer numbers k and l, 0 ≤ k ≤ l ≤ n -2, we consider the statistic (random variable) M n;k,l which enumerates strings (binary patterns) of a limited (constrained) length equal to d+2, k ≤ d ≤ l; i.e. strings which consist of a zero run (consecutive 0s) of length at least equal to k and at most equal to l between two subsequent ones.
The counting of such constrained (k, l) strings is considered in the overlapping sense; that is, a 1 which is not at either end of the sequence may contribute towards counting two possible strings, the one which ends with the occurrence of it and the next one which starts with it.

with (2)
* Address correspondence to this author at the Department of Mathematics, University of Patras, 26500 Patras, Greece; Tel: 0030-2610-996738; Email: makri@math.upatras.gr Readily, for n < k + 2, M n;k,l = 0. We notice that throughout the article we apply the conventions for a > b; that is, an empty sum (product) is to be interpreted as a zero (unity).The support (range set) of M n;k,l is: where by we denote the greatest integer less than or equal to a real number x.
A statistic related to M n;k,l is the waiting time W ;k,l until the m-th, m ≥ 1, occurrence of a constrained (k, l) string.It is defined and connected to M n;k,l as follows: (4) Hence, Eq. ( 4) offers an alternative way of obtaining results for the waiting time RV W m;k,l through formulae established for the string enumerative RV M n;k,l and vise versa.
The study, via M ;k,l , of constrained (k, l) strings where a 1 is followed by at least k and at most l 0s before the next 1 in a 0 -1 sequence covers as particular cases strings with zero run length at most equal (M n;0,l , d ≤ l), exactly equal (M n;k,k , d = k) and at least equal (M n;k,n-2 , d ≥ k) to a non-negative integer number.Some relevant contributions on the subject are the works of Sarkar et al. [2], Sen and Goyal [3], Holst [4,5], Huffer et al. [6], Eryilmaz and Zuo [7], Eryilmaz and Yalcin [8], Dafnis et al. [9], and Makri and Psillakis [10].Moreover, M n;0,0 is the Ling's [11] RV which counts overlapping runs of 1s of length 2 (with overlapping part of length at most 1), see e.g.Balakrishnan and Koutras [12].
In Section 2 of the present paper we first introduce the required notation and second we recall a recent result of Makri and Psillakis [1] which refers to the probability mass function (PMF) of M n;k,l defined on a 0 -1 Markov chain.The presented expressions for the PMF are then used in an application case study which is discussed in Section 3 and it refers to statistical process control.Throughout the article, δ i,j denotes the Kronecker delta function of the integer arguments i and j.

PMF OF M n;k,l FOR MARKOV DEPENDENT TRIALS
In this Section we provide the PMF The 0 -1 sequence n ≥ 2 on which M n;k,l is defined, is generated by a time-homogeneous first order Markov chain (MRKV) with one step transition probability matrix P = (p i,j ) and initial probability vector p , n ≥ 2 of independent and identically distributed (IID) RVs with a common probability of 1s, p = P(X i = 1) = 1 -P(X i = 0) = 1 -q, i = 1,2,...,n is a particular MRKV sequence with p 00 = p 10 = q, p 01 = p 11 = p and = (q, p).
We next present two combinatorial results (Lemma 1 and Corollary 1) which then are used for the calculation of f n;k,l (x) given by Proposition 1.
Lemma 1 (Makri et al. [13]) be the number of allocations of α indistinguishable balls into m distinguishable cells, i specified of which have capacity k 1 -1 and each of r-i specified cells has capacity We note that the number C i,r-i (α, m, k 1 -1, k 2 -1) equivalently gives the number of integer solutions of the equation ) (p corollary, the following result.

A NOTE FOR APPLICATION: STATISTICAL PROCESS CONTROL
Suppose we observe values of a stochastic process {Y i } i≥1 which operates under chance causes in or out an acceptable zone of interest (ZI); i.e. the process is in statistical control.We are interested in remaining the process in ZI, 0 , p x ∈ R(M n;k,ℓ ), otherwise a risky or an awkward situation of the process might appear.For instance, a ZI might be a control zone in a statistical control chart, a trading zone in a stock or exchange market, a comfort zone in a patient mechanical support system, a zone of acceptable items with specific characteristics in an assembly line, etc.
We denote by 0 and 1 the occurrence of the value of the process in and out of ZI, respectively.We assume that each value of the process is out of ZI with probability p 01 or p 11 if the previous value was in or out of ZI, respectively.Therefore, the indicator RVs X i = 0, if ; 1, otherwise constitute a 0 -1 Markov chain {X i } i≥1 with transition probability matrix and initial probability vector p (1) = (1 -p 1 , p 1 ).In practise, ZI and p 01 , p 11 might be evaluated from a reference sample in Phase I (retrospective) analysis of the process whereas the monitoring of the process, we are interested in, is considered as Phase II (prospective) analysis of the process (see, e.g.Chakraborti et al. [15]).
Following Dafnis and Philippou [16] we interpret the occurrence of two subsequent 1s separated by at least k and at most l 0s as a sign that the process leaves its ZI and it enters in a risky situation for which additional care has to be paid.In such a case the number of values in ZI is not enough to compensate for the others out of it.What values of k and l should depend on the under study process.
Accordingly, the event of having m (≥1) or more constrained (k, l) strings in a stream of n future values of the process could serve as an index that the process will not any more be bounded in its ZI.The risk of experiencing of such an event would be measured by P(M n;k,l ≥m) = P(W m;k,l ≤n).Therefore, the distributions and the characteristics of the related RVs M n;k,l and W m;k,l would be of importance in understanding the future process behavior.Usually in practice, we consider a detector that counts such events and consequently sends an alarm signal to a process monitoring system.The detector's tolerance (the false alarm probability) is taken to be at most equal to γ (0 < γ < 1) so that an expected number, γ100%, of false alarms happen.For a given γ a critical value of m (if there is such a value for the selected γ and the process parameters n, k, l and p 1 , p 01 , p 11 ) m γ is: m γ = min{m ≥ 1: P(M n;k,l ≥m) = γ* ≤ γ}.The probability γ* is the largest real number which does not exceed γ.It may not be equal (in fact it is rare to be equal) to the assigned probability γ, as it refers to the discrete RV M n;k,l .
As a numerical example we consider Markov dependent trials with p 1 = 0.5, p 01 = 0.45 and p 11 = 0.90.That is, we assume that the process has 50-50 chance to be initially in or out its ZI and the occurrence of a value out of ZI implies that the process continues to remain out of ZI with large probability, in fact twice the probability that the process leaves its ZI.The latter one is assumed to be a little smaller than 0.5.
For a reasonable value of the detector's tolerance γ = 0.05 and for large enough values of n, we present in Table 1 critical values m γ , and exact probabilities γ* = P(M n;k,l ≥m γ ) for several constrained (k, l) strings of type at most (AM), at least (AL), at least -at most (AL-AM) and equal (EQ).For comparison we have included in the Table the respective m γ and γ* for IID trials with p = 0.5.The entries of Table 1 suggest that in order to have at most 5 false alarms in a total of 100 alarms in a stream of n future values of a process, defined on MRKV with the prementioned values p 01 , p 11 and p 1 , we have to take larger m γ when we use a (0, 2) string than m γ of a (2, 2) string.This is so, because a (2, 2) string is one of the strings counted as a (0, 2) string and therefore as a more strict pattern is a less frequently appearing string, as a warning pattern, than a (0, 2) string.Analogous arguments/interpretations are also suggested for the other presented types of strings as well as when someone compares MRKV and IID trials.

CONCLUSION
In this study the RV M n;k;l counting a flexible class of binary strings of a limited length is considered.A potential application of its PMF in describing the probabilistic behavior of a binary stochastic process is discussed.The stochastic process is assumed to operate under chance in or out an acceptable zone of interest.Such a zone can be a control chart zone, a trading zone of a stock market, etc.An index connected to the number of occurrences of limited length binary strings is properly defined via the PMF of M n;k;l .Accordingly, it is then used to determine whether or not the process is or not under statistical control.Numerical results clarify the implementation of several types of binary strings used in the control process.