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首页> 外文期刊>Quality Control and Applied Statistics >Influence of the sampling interval, decision limit and autocorrelation on the average run length in CUSUM charts
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Influence of the sampling interval, decision limit and autocorrelation on the average run length in CUSUM charts

机译：采样间隔，决策极限和自相关对CUSUM图表中平均游程长度的影响

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Suppose that the system is in a satisfactory state wheny_t = #mu#_0 + a_t (1)where {a_t} is a series of normally, independently and identically distributed random variables with mean zero and variance #sigma#_a~2 = #sigma#_y~2 often referred to as white noise. The authors also suppose that a malfunction is characterized by an increased d = D#sigma#_a in the mean from #mu#_0 to #mu#_0 + d = #mu#_1, where the quantity D is called the standardized increase in the mean.For formal detection, it is convenient to use the centered (one-sided) CUSUMQ* =sum (y_i -#mu#-bar),where#mu#-bar = (#mu#_0 + #mu#_+1)/2 = #mu#_0 +d/2When the mean is equal to #mu#_0, the quantity d/2 will be subtracted from each deviation, producing a downward trend in the centered CUSUM. However, if the mean level is greater than #mu#_0 + d/2, then the centered CUSUM will tend to increase. In practice, it is often necessary to monitor the mean #mu#_0 for negative as well as positive values of d, so that a two-sided CUSUM may be used and arranged as a combination of two-sided CUSUMs.For given standardized increase in the mean D and the sampling interval S, the standardized decision interval H = h/#sigma#_a will determine the characteristics of the CUSUM chart. A quantity that describes the performance of such schemes is the run length (RL) time before an alarm is triggered. This is a random variable that varies from one run to another, and its mean is called the average run length (ARL). When the system is in a satisfactory state, the probability distribution for the RL is often close to a geometric distribution (Refs. 1, 2) 50 that it can be completely determined from its ARL. On this basis, it has become customary to define the CUSUM scheme in terms of the ARL for false alarms (L_a) when the process is in the acceptable condition #mu# = #mu#_0, and in terms of the ARL for valid alarms (L_t) when the process is in the rejectable condition #mu# = #mu#_1. A specific example is illustrated. Section 2 considers the effects that reducing the sampling interval and simultaneously increasing the decision interval have on the ARL and the quantiles of the distribution. Section 3 analyzes how the resulting ARL and quantiles change when there is a moderate autocorrelation between consecutive observations. Concluding remarks are presented in Section 4. (23 refs.)
机译：假设系统在y_t =＃mu＃_0 + a_t（1）时系统处于令人满意的状态，其中{a_t}是一系列具有均值为零和方差的正态，独立且均等分布的随机变量＃sigma＃_a〜2 = #sigma ＃_y〜2通常称为白噪声。作者还假设故障的特征是d = D＃sigma＃_a从＃mu＃_0到＃mu＃_0 + d =＃mu＃_1的平均值增加，其中量D称为对于形式检测，使用居中（单侧）CUSUMQ * = sum（y_i-＃mu＃-bar）很方便，其中＃mu＃-bar =（＃mu＃_0 +＃mu＃_ +1）/ 2 =＃mu＃_0 + d / 2当平均值等于＃mu＃_0时，将从每个偏差中减去数量d / 2，从而使居中居中值呈下降趋势。但是，如果平均水平大于＃mu＃_0 + d / 2，则居中的CUSUM将趋于增加。在实践中，通常需要监视d的负值和正值的平均值＃mu＃_0，以便可以使用双面CUSUM并将其作为双面CUSUM的组合进行排列。在平均值D和采样间隔S中，标准化决策间隔H = h /＃sigma＃_a将确定CUSUM图表的特征。描述此类方案的性能的数量是触发警报之前的运行时间（RL）时间。这是一个随机变量，从一次运行到另一次运行都不同，其平均值称为平均运行长度（ARL）。当系统处于令人满意的状态时，RL的概率分布通常接近可以从其ARL完全确定的几何分布（参考文献1、2）50。在此基础上，习惯的做法是，当过程处于可接受的条件＃mu＃=＃mu＃_0时，根据误警报的ARL（L_a）定义CUSUM方案，并根据有效警报的ARL定义CUSUM方案（L_t），当进程处于可拒绝状态＃mu＃=＃mu＃_1时。示出了具体示例。第2节考虑了减少采样间隔并同时增加决策间隔对ARL和分布分位数的影响。第三部分分析了当连续观察之间存在适度的自相关时，结果的ARL和分位数如何变化。结束语在第4节中提供（23个参考）。

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