*Exponentially Weighted Moving Averages of Counting Processes When the Time between Events… DOI: http://dx.doi.org/10.5772/intechopen.90873*

parameters of 0.85 and 0.02, respectively) is set to be equal to 400. The *ATS*<sup>0</sup> for the plans associated with other processes are equal to 300, 200, and 100, respectively. 100,000 events are employed to estimate the count mean for each process. In addition, 500 events are also considered for the burn-in period of the simulations. The performance of the devised plans is presented in T**ables 1–4**. The lowest ATS values are colored in the tables below in black bold text to make it easier to see which plans are more efficient in certain situations.

**Table 1** shows the performance results for the plans employed to monitor counting process where the in-control data are Weibull with scale and shape parameters of 0.035 and 1.25, respectively. *ATS*<sup>0</sup> (in days) for these plans are set approximately equal to 100. As shown in **Table 1**, for the outbreaks of larger magnitude, plans with larger smoothing parameter are superior. On the contrary,


#### **Table 1.**

*hc* is determined so that a desired in-control average run length (*ARL*0) is achieved. Appendix A provides models for establishing the thresholds for various values for 0*:*2≤ *scale*≤0*:*46, 0*:*6≤*shape*≤1*:*4, *α* ¼ 0*:*1 and *ARL*<sup>0</sup> ¼ 100, 200, 300 or 400*:* Given a *hc* and the process is out of control, we want the ARL to be as low as possible. Monitoring daily counts, the *ARL* is defined as the number of days before a signaled outbreak. Later we will use the ATS to assess the relative performance of control chart plans, because the ARL may vary according to the aggregation period we select for the counts. Note that a false outbreak is flagged by this EWMA statistic being significantly larger than expected given it is in-control.

For most of the real-world cases, TBE distribution may vary due to different circumstances while not experiencing an outbreak. For instance, if we aggregate the TBEs in daily intervals, occurring intervals for the events may vary based on the day of the week, and even working and non-working days may affect the distribution of TBEs. As a result, we often face nonhomogeneous count processes. Hence, we define an adaptive exponentially weighted move average (AEWMA) statistic for

*aei* ¼ max 0, ð Þ *αci=hc*ð*ARL*0, *scale*, *shape*, *α*Þ þ ð Þ 1 � *α aei*�<sup>1</sup> (2)

**3. Monitoring in-control nonhomogeneous counts**

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

where *aei* is the AEWMA statistic at time *t*, and *ae*<sup>0</sup> ¼

as defined earlier. To control the false discovery rate, we set

*E c*ð Þ*<sup>i</sup> =hc*ð Þ *ARL*0, *scale*, *shape*, *α* for days *i* ¼ 1, 2, … . The other notations are

*hc*ð Þ *ARL*<sup>0</sup> ¼ *c*, *scale*, *shape*, *α* so that a desired*ARL*<sup>0</sup> is achieved. An outbreak is

The *α*ð Þ 0<*α* <1 in Eq. (1) determines the level of memory of past observations in this average *aei*. Smaller *α* values retain more memory of past counts; therefore, small *α* values are efficient at retaining enough memory of past counts to have the power to flag smaller outbreaks. However, larger values of *α* are needed when there is a larger size outbreak, because a shorter range of memory is adequate to build enough power to detect the outbreak. In addition, the shorter memory for the EWMA average with larger values of *α* is less inclined to be influenced by too many

To assess the validity and applicability of our proposed adaptive method, we

10<*E c*ð Þ*<sup>i</sup>* <50 which limits the volume of simulations that are required to make a realistic judgment. This is also the range where the EWMA counts become competitive for small changes in the scale parameter (see [4] and Sparks et al., 2020). To do so, we consider four counting processes with daily aggregations. The shape parameters for the TBE distributions when the processes are in-control are equal to 0.85, 0.95, 1.15, and 1.25. The in-control scale parameter for each process is equal to 0.02, 0.025, 0.03, and 0.035, respectively. We intentionally imposed outbreaks in the simulated data to assess if the proposed method is capable of detecting them. In the simulation study, we assume that the outbreaks result in a decrease in the scale parameter. Various surveillance plans are devised to monitor each process. The *ATS*<sup>0</sup> for the plans associated with the first process (with shape and scale

employ simulations studies. We restrict our attention to plans that have

nonhomogeneous daily counts as

flagged approximately whenever *aei* >1.

past in-control counts.

**4. Simulation results**

**188**

*Performance of plans when the in-control TBEs are Weibull distributed with scale = 0.035 and shape = 1.25.*


#### **Table 2.**

*Performance of plans when the in-control TBEs are Weibull distributed with scale = 0.03 and shape = 1.15.*


approximately 300. Conclusions similar to those regarding **Tables 1** and **2** can be drawn from **Table 3**. It is clearly observed that the larger the magnitude of the outbreak, the larger the smoothing parameter should be. On the other hand, for the detection of small outbreaks, plans with larger values of *ATS*<sup>0</sup> need even smaller values for smoothing parameters than do the plans with smaller *ATS*0. As shown in **Table 1**, smoothing parameters equal to or larger than 0.08 work better for outbreak detection. As the *ATS*<sup>0</sup> increases, analogous to results presented in **Tables 2** and **3**, even smaller values for α are needed to devise a plan of larger detection power. Last but not the least, similar results can be driven from **Table 4**, which presents the performance of the monitoring plans applied to a counting process with underlying Weibull distribution for TBEs. The shape and the scale parameters of the aforementioned distribution are equal to 0.85 and 0.028, respectively.

*Exponentially Weighted Moving Averages of Counting Processes When the Time between Events…*

In this section, we apply our proposed method to a real-world example. The counting process to monitor is the number of presentations at Gold Coast University Hospital emergency department for a broad definition of influenza. The events

**5. Real-world example**

*DOI: http://dx.doi.org/10.5772/intechopen.90873*

**Figure 1.**

**191**

*Residual analysis for the fitted model.*

#### **Table 3.**

*Performance of plans when the in-control TBEs are Weibull distributed with scale = 0.025 and shape = 0.95.*


#### **Table 4.**

*Performance of plans when the in-control TBE are Weibull distributed with scale = 0.02 and shape = 0.85.*

for the early detection of the small outbreaks, smaller values for the smoothing parameter are preferred.

**Table 2** shows the performance results of the plans when the shape and the scale parameters of the Weibull distribution are equal to 1.15 and 0.03, respectively. The *ATS*<sup>0</sup> for the plans employed to monitor this counting process is set to be approximately 200. In the results indicated in **Table 1**, plans with smaller smoothing parameter work better in early detection of small outbreaks than those with larger smoothing parameter. For the larger outbreaks, the detection power of the plans increases as the smoothing parameters increases.

**Table 3** shows the performance results of the plans when the shape and the scale parameters of the Weibull distribution are equal to 0.95 and 0.025, respectively. The *ATS*<sup>0</sup> for the plans employed to monitor this counting process is set to be

*Exponentially Weighted Moving Averages of Counting Processes When the Time between Events… DOI: http://dx.doi.org/10.5772/intechopen.90873*

approximately 300. Conclusions similar to those regarding **Tables 1** and **2** can be drawn from **Table 3**. It is clearly observed that the larger the magnitude of the outbreak, the larger the smoothing parameter should be. On the other hand, for the detection of small outbreaks, plans with larger values of *ATS*<sup>0</sup> need even smaller values for smoothing parameters than do the plans with smaller *ATS*0. As shown in **Table 1**, smoothing parameters equal to or larger than 0.08 work better for outbreak detection. As the *ATS*<sup>0</sup> increases, analogous to results presented in **Tables 2** and **3**, even smaller values for α are needed to devise a plan of larger detection power. Last but not the least, similar results can be driven from **Table 4**, which presents the performance of the monitoring plans applied to a counting process with underlying Weibull distribution for TBEs. The shape and the scale parameters of the aforementioned distribution are equal to 0.85 and 0.028, respectively.
