In studying the performances of the monitoring schemes, researchers usually assume µ and Σ are known in Phase II monitoring. In this study, we evaluate the performances of the proposed monitoring schemes in terms of the average run length (ARL). Several versions of control limits are considered as below.
1. T2 Chart:
For T2 statistic, we consider another version and refer to the original T1 statistic in (12) as
Since T11 has poor power when n0 is large, we consider a T2 statistic by adding only the tanderized scores of effective components as
T12 =
K
X
k=1
λ−1k A2k, (20)
where K is the number of the “effective” principal components. It is obvious that T11 and T12 have chi-squre distributions with degrees of freedom n0 and K, respectively.
Then the control limits for T11 and T12 are χ2n0,1−α and χ2K,1−α, respectively.
2. Combined Chart:
The control limit of T2 is Zα0/2, where α0 = 1 − (1 − α)1/n0.
3. Total Squared Score (TSS) Chart:
Since Figure 3 shows that the distribution of T3 is not close to cχ2df, we study two versions of control limits: one is the empirical (1 − α) quantitle and the other is cχ2df,1−α. For the empirical quantitle, we take the eigenvalues {λk}nk=10 of the smooth VDP data as the true eigenvalues/eigenvectors and generate 1,000,000 set of {Ak}nk=10 with Ak ∼ N (0, λk), to obtain 1,000,000 T3, and hence the empirical (1−α) quantitle of T3. We refer to the T2 control chart with this control limit as the T31 chart in simulation studies. For our simulation study, U CL31 is 8262.107. We refer to T2 chart with the control limit cχ2df,1−α as the “T32”, where c and df are estimated by eigenvalues and eigenvectors of the smoothed VDP.
On the other hand, if the parameters are unknown, we need to use some in-control historical data to estimate these parameters. Jensen et al. (2006b) mentioned that the effect of parameter estimation on control chart properties should not be ignored. Thus we consider the case that eigenvalues/eigenvectors are estimated from in-control profiles to investigate the effect of the estimation error. For this cases, the control limits are obtained as below. The T2 and the Combined Chart have exact control limits as given above. Fore versions of control limits for T3 are considered. The control limit for T31 and T32 are as the same as above. Assume that we have m in-control profiles available. Apply PCA to them to obtain eigenvalues and eigenvectors, generate 1,000,000 set of scores as before with these estimated eigenvalues to obtain 1,000,000 T30s. Let the control limit be the
Since these estimated parameters depends on this particular set of m profiles, the performance of the control chart may not be the average case. Thus we will repeat the study, say b times to coverage sampling bias. The last version of the TSS Chart is referred to as “T34” chart, in which the control limit is cχ2df,1−α with c and f as in (17) but using the setimated eigenvalues for {λk}nk=10 .
4 Simulation Studies
4.1 Data Generation
In our simulation studies, we simulate data based on the vertical board density profiles.
The vertical board density profiles from Walker and Wright (2002) consist of 24 profiles of vertical density, each profile has 314 measurements. These data set is available at http://bus.utk.edu/stat/walker/VDP/Allstack.TXT. First, we smooth each profile and then apply SVD to the sample covariance matrix of VDP data to get the “original”
eigenvalues and eigenvectors, {(λk, βk)}, and treat them as the population version. Note that monitoring the functional data can be reduced to monitoring the discrete profile data as shown in Section 2.4. Thus, we only need to generate discrete profile data. Table 1 gives 23 eigenvalues of the smoothed VDP data. To generate a new profile as a data vector, we use:
Y = µ +
23
X
k=1
Akβk, (21)
where Ak ∼ N (0, λk), µ is the mean vector of the smoothed VDP data and is assumed known. Akis the k-th PC score of a profile. When the process is in control, the score Ak is distributed as N (0, λk). For an out-of-control process, we let the score Ak be distributed as N (δ√
λk, λk). In our study, δ ranges from 1 to 12.
In our simulation study, two methods are considered in Phase II monitoring. One uses the “true” eigenvalues and eigenvectors of the smoothed VDP, the other uses the estimated
eigenvalues and eigenvectors from m profiles. And each of the m profiles in our study is of the form (21).
For simulation study with “true” eigenvalues and eigenvectors of the smoothed VDP data, we only need to generate {Ak}nk=10 1,000,000 times for each shift in PC1, PC2, and PC23. For simulation study with estimated eigenvalues and eigenvectors, we first generate m profiles by equation (21). Apply PCA to these m profiles to obtain estimated eigenvalues ˆλ1, . . . , ˆλ23 and eigenvectors ˆβ1, . . . , ˆβ23. Generate 1,000,000 profiles of (21).
Since the mean vector µ of the process is known in Phase II monitoring, without loss of generality, we assume µ=0. Project profiles onto these estimated eigenvectors to get their PC scores. We use these principal component scores to evaluate the performances of the three charts constructed with the statistics T1, T2, and T3 in Phase II monitoring in terms of the detecting powers. Repeat the above procedure 1,000 times to average the detecting power so that the simulation results can represent an average case, not biased by the particular m profiles used for constructing the control limits. In our study, the false alarm rate α is set at 0.0027 and m=200, 300, 400, 500, 600.