ARL Estimation of the Control Chart of Log Likelihood Ratios’ Sum for Markov Sequence
To evaluate the surveillance performance of a control chart with the charting statistic of the sum of log likelihood ratios in the statistical process control (SPC), in this paper, we give the proof procedure based on Markov chains for the asymptotic estimation of the average run length (ARL) for this kind of chart. The out-of-control is approximately equal to 1 for any fixed in-control with a negative control limit. By the equivalence between limit distribution of a sum and that of a suprema sum of Markov chain, we derive the estimation of with a large enough positive control limit. Numerical experiments are conducted to confirm our results.
The main aim of SPC is to detect an abrupt change in the observation of time series as soon as possible after the change has happened. The study, found in , is the first to show the design of the control chart for quickly detecting possible changes in the underlying process. Subsequently, a great number of control charts have been proposed and come into a wider use in many fields, such as environmental control [2, 3], biostatistics [4, 5], clinical medicine [6, 7], economics and finance , industrial quality and process control [9, 10], process monitoring [11–16], public health [7, 14, 16], and social network .
Obviously, the research on constructing various control charts for surveillance has never stopped since Shewhart presented this method. The existing theoretical studies in statistical monitoring, on the whole, can be classified into three areas. The first category is the optimal economic or economic-statistical design, where, in most cases, the net sum of all costs is minimized [18–21]. The ideas of the second category are the choice of the statistical parameters of a control chart to minimize the out-of-control for a given in-control or a probability of false alarm, and it is named as the optimal statistical design [16, 22–25]. Recently, the theoretical approximation to ARL was proposed in . Detecting changes in distribution of the optimal control charts is the last category, which can be regarded as a kind of optimal stopping time [27–29]. This metric for defining optimality is based on that it has smallest out-of-control among all control charts with either a given probability of a false alarm no greater than a preset level, or a given false alarm rate no less than a given value. So, this more mathematical area is thus related to statistical design. In the following argument, we will focus on the third type of these bibliographies mentioned above.
Consider the Gaussian observation sequence , whose distribution may change at time . Let and be the prechange probability density function and the postchange probability density function of , respectively. Denote the postchange joint probability distribution, expectation, and variance by , respectively. Especially, when the change time , we suppose that a change never occurs. Frisén  showed that there exists a positive value such that the control chart with the charting statistic of the sum of log likelihood ratios (SLR),is optimal in the sense,when the change is the case of a shift in the mean of , where denotes the control chart test, is the change-point time, and is the constant control limit such that . Although the optimality of has been proved in , there were few surveys and bibliographies which considered the performance of the control chart for monitoring the change point in the process.
Therefore, in this paper, we regard the ARL as the criterion of optimality of a control chart. This is because the ARL is the connection among itself, control limit, and statistical properties of the observation sequence . In Section 2, we prove that the out-of-control is approximately equal to for a given as the control limit is negative. In Section 3, we apply the equivalence between limit distribution of a sum and that of a suprema sum of Markov chain to obtain the estimation of when the control limit is some large enough positive constant. Finally, we conduct some numerical experiments to verify our theoretical analysis in Section 4.
2. Estimation of with a Negative Control Limit
In this section, we consider a constant control limit to construct a control chart and estimate the corresponding in-control and out-of-control . Let the observations be a time-homogeneous Markov chain with the discrete state space . Here, we consider only detecting the change at the initial time τ = 1, and the Markov chain is positive recurrence with the prechange transition probability and the postchange transition probability . For convenience, we use instead of for . Similar to the notations defined in Section 1, we use , and to denote, respectively, the prechange joint probability distribution, expectation, and variance of .
For ease of the subsequent analysis processing, we choose a large enough positive number to cut out the first terms of the sequence . In other words, the sequence tends to when goes to infinity. The optimality of the control chart for the Markov chain appeared in . Now, we define the charting statistics as
Let for ; then, for . For any state , let be the time of th visit of state and be the number of times for visiting from time 0 to time . The cumulative sums of and on the th block of time are, respectively, defined byfor and . Then, both and are sequences of independent and identically distributed (i.i.d.) random variables.
Throughout this paper, we assume that(i) if and only if , as well as has no atom with respect to for . Here, we define 0/0 = 1.(ii).
The sum of is written by . Set , , and and . Moreover, for , let
Then, for large , we have
Now, we define a control chart in the following with the charting statistics of the sum of log likelihood ratios for detecting the change in distribution of the Markov sequence :for some constant .
Now, we are ready to estimate . Based on Section 3 presented in , is the average run length until an alarm is signaled, so and are equivalent. Consequently, we use the estimator of to substitute in the subsequent analysis. The main result is proposed in the following theorem.
Theorem 1. Let be a time-homogeneous Markov chain satisfying conditions (i) and (ii). Then, for a given and , there exists a negative number such thatas .
Proof. Choose the control limit and note thatwhere . Let . Using the law of total expectation and Markov property, we obtainSince is a concave function, by Jensen’s inequality, we haveSubstituting (13) into (12) and combining (11), we can get that for any . Similarly, we can prove for any . Hence, and ; then, . Let for some , where denotes the integer part of . Then,The total probability formula tells us thatEquations (4) and (7) yield thatNote that and almost everywhere converges to 0 as therefore,Notice that, for a large , the relationship between probability distribution function and density function of standard normal distribution isCombining (16)–(18) and the central limit theorem of Markov chain (see ), we can get thatPut (19) into (15) and combine to yield thatFor , by the Markov inequality and , we haveInequality (21) implies that the value of tends to 0 as . Put (20) and (21) into (14); then, there exists some constant such thatfor a large .
Using Theorem 5.1.7 in , there exists a nonpositive number such thatholds for a large enough . Thus, we can choose a negative number satisfying and .
Next, we prove (10). For any , let . Similar to (20), we can get thatNote that the second and third terms on the right-hand side of the last inequality in (24) tends to 0 as goes to infinity.
This completes the proof.
3. Estimation of with a Positive Control Limit
In the subsequent discussion, we use the equivalence between limit distribution of a sum and that of a suprema sum of Markov chain to estimate the out-of-control when the control limit is a sufficiently large positive constant. The main result is presented in the following theorem.
Theorem 2. Assume that is a homogeneous Markov chain. For any state ifthenfor large and , where and , are defined in (5) and (6), respectively. The sign denotes the infinitesimal of higher order and .
Proof. It follows from (8) that, for ,The estimation of is reduced to the estimation of . According to Theorem 2 in , we have the following conclusion:Then, we use (28) to calculate the value of . Let , where and denotes the real number field. When becomes sufficiently large, by (5), (27), and (28), we haveLet , and we havefor the large enough and . When goes to infinity, we deal with the second term on the right-hand side of (30) as follows:Note that, for a large , we haveWe obtain the right-hand side of (30) as follows:as tends to infinity. Combining (30), (33), and , we obtainTo prove the inequality on the left-hand side of (26), let , and goes to infinity; then, we use (27) to yield thatwhere is the distribution function and . Similar to the derivation of (33), let tend to infinity; then, we obtain the estimation of the second term on the right-hand side of (35) asFor the first term on the right-hand side of (35), because the function is monotonically decreasing in the interval with respect to and the distribution function is monotonically nondecreasing in , we havefor and . By (35)–(37), we obtainfor a large enough . Combine (34) and (38) to yield (26). This completes the proof.
4. Numerical Experiment
In this section, we perform two numerical experiments to verify our theoretical results. In our first numerical experiment, let be a sequence of i.i.d. Gaussian random variables with the prechange and the postchange probability densities and . It is assumed that the standard case of a shift in the mean of from to is considered.
These results suggest that the value of is equal to 1 with a negative control limit and a given , which is predicted in Theorem 1.
In the second experiment, we choose and . Let be a homogeneous Poisson process with the prechange and the postchange parameters and ; then, the corresponding prechange and postchange transition probabilities are defined aswhere and are arbitrary states from the state space and for .
The results are obtained and presented in Table 2, which show the ratio of and equals approximately 2.38 as . It implies that the value of and the control limit are the infinite of the same order, which is predicted by Theorem 2.
The data used to support the findings of the study are generated by Matlab.
Conflicts of Interest
The authors declare no conflict of interest.
Yan Zhu was supported by NSFC, Grant no. 11801353.
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