Optimal offset values (
This chapter deals with monitoring plans that exploit temporal predictable trends by adjusting the cumulative sum (CUSUM) plan to be efficient for their early detection. The adjustment involves changing the amount of memory the chart retains to detect persistent changes in location early. The focus is on steady-state situations when either the shift size is known in advance or when it is unknown. Several options are explored using simulation studies, and an example of application is considered.
- average run length
- early detection
- persistent trends
- statistical process control
The adaptive CUSUM of Sparks  exploits temporal predictable trends by adjusting its design to be efficient for the early detection of such trends. The adjustment involves changing the amount of memory the chart retains to detect persistent changes in location earlier.
In the zero-state case, Moustakides  proved that a step change of
Automation and sensor devices that measure very frequently means that data stream in these days in real-time, and therefore steady-state situations have now become more common than when the CUSUM was first advocated by Page . Most applications in environmental sciences are steady state since the process cannot be stopped. The majority of service processes, although can be stopped, are hardly ever stopped and restarted. Thus, they may be referred to as steady-state processes.
For this reason zero-state processes are less common, thus, revealing a scientific area that needs to be further researched.
2. Literature review
Sparks’s  adaptive CUSUM improved the CUSUM early detection performance by appropriately adjusting the reference value
This paper starts by introducing the conventional CUSUM and the adaptive CUSUM statistics. It derives the thresholds for the CUSUM plans in steady-state situations for high-sided signals only. Low-sided charts can be established by symmetry and two-sided charts can be applied by simultaneously applying two one-sided charts and halving the in-control ARL of the high-sided chart. The high-sided charts for steady-state situations are designed to deliver a specific in-control ARL of either 100, 200, 300, …, 1000 (see Appendix A). Monitoring plans are defined in Sparks . If the location is known in advance then this paper establishes the reference value closest to the best plan for the steady-state situation. A simulation study is carried out to find the CUSUM
Methods that compete with the adaptive CUSUM in terms of performance involve the simultaneous application of multiple CUSUMs with differing levels of memory [4, 12], combining Shewhart and CUSUM charts [8, 11], the adaptive EWMA  and multiple moving averages . Ryu et al.  assumes the shift is known and optimises the CUSUM plan without mentioning whether it is based on zero or steady state, and therefore it must be viewed as competing methodology. However, this paper’s contribution is on improving the out-of-control performance of the adaptive CUSUM plan in the steady-state situation and provides formulae to estimate the thresholds for the high-sided conventional CUSUM in steady-state situations.
3. CUSUM and adaptive CUSUM plans
The adaptive CUSUM allows the reference value to change over time
and flags an out-breaks whenever this exceeds a threshold of approximately 1. The challenge in practice is how to change
Sparks’s  plan was based on the hypothesis that the zero-state optimal setting was going to be optimal in the steady-state situation. This is however, not the case. The examples that illustrate this are reported in Figure 1(a)–(c).
Figure 1 plots the conventional CUSUM divided by its threshold (i.e., for
Figure 1(b) illustrates that fact that the CUSUM plan with
Figure 1(c) exemplifies that the CUSUM plans with
This begs the question of what reference values
4. Near optimal steady-state plans when the shift is known
A simulation study was carried out that started with running through 25 in-control observations before generating the out-of-control situations. This was designed to simulate a steady-state situation prior to the change point. The thresholds for this process are given in Appendix A for the standard normal distribution. There is no loss of generality by assuming mean of zero and variance of one, however the results only apply to normally distributed data. The smallest out-of-control ARLs for various scenarios are presented in Table 1 for in-control ARL = 200, and for in-control ARL = 800 in Table 2.
The reference value with the smallest out-of-control ARL is highlighted in bold text, e.g., for in-control ARL = 200 and a location shift of
The optimal reference value is reported in bold text in Table 2, for example, for in-control ARL = 800 and a location shift of
5. Improving adaptive CUSUM performance for the steady-state situation
The EWMA statistic in Sparks  and Jiang et al.  is used to forecast the change
where 0 <
In other words the local smoothed value
5.1. Attempts to improve on the adaptive plan of Sparks  in steady-state situations
Recall the adaptive CUSUM
Now the Signal-to-Noise Ratio, SNR, (
The threshold for this CUSUM is expected to be larger than 1. Therefore an increase in location is flagged when
Table 3 indicates that the user should select the EWMA weights to be 0.7 to improve on the traditional adaptive CUSUM plan when 0 <
6. Example of application
The example of application is the nitrogen dioxide (NO2) values at Liverpool (a suburb in the western part of Sydney, Australia). Nitrogen dioxide primary gets into the air from the burning of fuel. High exposure to this can cause respiratory problems such as asthma (see WHO ). Nitrogen dioxide reacts with other chemicals in the air to form both particulate matter and ozone (see ). Both of these are harmful to humans and possibly animals when inhaled.
The data was downloaded from New South Wales (Australia) Heritage Foundation website on air pollution. Data ranged from the beginning of 2010 to the end of March 2017 and were daily averages.
The data up to the end of August 2016 were used as training data to fit both the (in-control) mean and standard deviation of the normal distribution using gamlss library in R . The model had explanatory variables as time in days, day-of-the-week and harmonics. Harmonics are included because there were strong seasonal influences on nitrogen dioxide values at Liverpool. The qq-normal plot of standardised residuals of this model indicated that the normal assumption for the residuals was appropriate. This fitted model was then used to predict the mean and standard deviation for the period on 1 September 2016–31 March 2017 (taken as the expected value and standard deviation for in-control data).
The actual daily average nitrogen dioxide measures were standardised by subtracting their fitted mean and dividing by the fitted standard deviation. The adaptive CUSUM was then applied to these standardised scores to see if these values had increased significantly from expect during the period 1 September 2016–31 March 2017. The plan was designed to deliver an in-control ARL of 200. Whenever the chart flagged a significant increase the adaptive CUSUM was set equal to zero to see if the nitrogen dioxide levels remained significantly higher than expected.
Figure 2a adaptive CUSUM values as advocated in this paper for in-control ARL = 200 is plotted against the date for the period.
Figure 2b is the adaptive CUSUM of Sparks . Both signal an increase in nitrogen dioxides on 8 May 2016, but the adaptive CUSUM values signals again on the 18 May 2016 after starting the CUSUM again at zero. The traditional adaptive CUSUM of Sparks  failed to signal a second time (Figure 2b).
7. Conclusions and further work
Although the new adaptive CUSUM has promise, the
|In-control ARL||Fitted model for |