Phase II Monitoring

3.3.1 Monitoring Statistics

In this study, we focus on Phase II monitoring. The purpose of Phase II analysis is to detect shifts in the process parameters as quickly as possible. In most Phase II studies, it is usually assumed that the in-control process distribution has been characterized, either from prior experiences or estimated from the Phase I analysis. In our study, we do not require any assumptions about the process distribution because of the nonparametric nature of data depth. Hence we only assume that a set of m in-control profiles is available. We first apply a smoothing technique to each of the m profiles to filter out noise, and then apply principal component analysis to the smoothed profiles. Denote the p×1 data vector of the i-th smoothed profile by x_i, i = 1, . . . , m, and the sample covariance matrix of {x_i, i = 1, . . . , m}

by S. Calculate the eigenvalues and eigenvectors of S. The eigenvector vr corresponding to the r-th largest eigenvalue λ_r is the r-th principal component. S_ir ≡ v_r⁰x_i is the PC-scores of the r-th principal component of the i-th profiles, where r = 1, . . . , p and i = 1, . . . , m. We could consider selecting the first k principal components for which the total variation explained by the chosen principal components,P_k

r=1λr/P_p

r=1λr, reaches a desired

level. Alternatively, it is also a useful proposal that we consider the ability of each principal component in capturing a particular feature of the profiles. If a particular mode of process change could be caught more easily by certain principal components, then we select these particular principal components. That is, for detecting different modes of process change, we might choose different principal components. Denote the k × 1 score vector (S_i1, . . . , S_ik)⁰ by s_i, i = 1, . . . , m. Calculating simplicial data depth for many profiles is fairly time-consuming, especially when k is large. For simplicity, with these principal components, we only consider k = 2 in the paper. We then simplify the m in-control profiles to m scores vectors s_i, i = 1, . . . , m. The resulting principal components will be used to compute the PC-scores of incoming profiles in Phase II on-line process monitoring as follows. For each incoming profile in Phase II monitoring, we first smooth and then project it onto the k principal components chosen earlier to obtain the k × 1 PC-scores vector s^∗_j, j = 1, 2, . . ..

Denote the set of scores vectors {s_i, i = 1, . . . , m} by {X₁, . . . , X_m} and {s^∗_j, j = 1, 2, . . .}

by {Y₁, Y₂, . . .}. Then we compute the simplicial depth values of {X₁, . . . , X_m} by equation (4) and {Y₁, Y₂, . . .} by equation (5), which are viewed as a measure of centrality relative to the points {X₁, . . . , X_m}. We denote the simplicial depth values by {SD(X_i), i = 1, . . . , m}

and {SD(Y_j), j = 1, 2, . . .}. Set the desired in-control false-alarm rate at a. Consider three monitoring statistics corresponding to the three control charts, r-chart, Q-chart, and DDMA-chart.

• r-chart: According to the definition of r-value by equation (7), we have the r-values of {Y₁, Y₂, . . .} by comparing the magnitude of {SD(Y_j), j = 1, 2, . . .} with respect to {SD(X_i), i = 1, . . . , m}. For the incoming samples, denote the monitoring statistics of r-values by {r(Y₁), r(Y₂), . . .}.

• Q-chart: Additionally, we can monitor the Q-values by applying the idea of ¯X-chart to the r-values {r(Y₁), r(Y₂), . . .}. Assume the subgroup size is q. Then the first monitoring statistic of Q-values is [r(Y₁)+· · ·+r(Y_q)]/q, the second monitoring statistic of Q-values is [r(Y_q+1) + · · · + r(Y_2q)]/q, and so on. For the incoming samples, denote the monitoring statistics of Q-values by {Q₁, Q₂, . . .}.

• DDMA-chart: In fact, the DDMA-chart is also a type of r-chart. The difference is

that we compute the moving averages of {X₁, . . . , X_m} and {Y₁, Y₂, . . .} to get new sets { ˜X₁, . . . , ˜X_m−q+1} and { ˜Y₁, ˜Y₂, . . .} with the length of moving window q. Then the following monitoring procedure is exactly the same as that of the r-chart: compute the simplicial depth values {SD( ˜X_i), i = 1, . . . , m − q + 1} of { ˜X₁, . . . , ˜X_m−q+1} by equation (11) and {SD( ˜Y_j), j = 1, 2, . . .} of { ˜Y₁, ˜Y₂, . . .} by equation (12), respectively.

Compare the magnitude of {SD( ˜Y_j), j = 1, 2, . . .} with respect to {SD( ˜X_i), i = 1, . . . , m − q + 1} to get the r-values {r( ˜Y₁), r( ˜Y₂), . . .} by equation (13), but referring to them as the DDMA-values here. For the incoming samples, the monitoring statistics are {r( ˜Y₁), r( ˜Y₂), . . .}.

3.3.2 Control Limits

Assume the false-alarm rate is set at a, the control limits of the three control charts are given below:

• r-chart: The r-chart monitors the r-values {r(Y₁), r(Y₂), . . .} with the LCL = a.

• Q-chart: The Q-chart monitors the Q-values {Q₁, Q₂, . . .} with LCL set under two different conditions: (i) when q is large, the LCL is set as 0.5 − z_a[(1/m + 1/q)/12]^1/2; and (ii) when q is relatively small and a ≤ 1/q!, the LCL is set as (q!a)^1/q/q.

• DDMA-chart: The DDMA-chart monitors the DDMA-values {r( ˜Y₁), r( ˜Y₂), . . .} with LCL = a.

We will evaluate the performances of the proposed Phase II monitoring schemes described above in terms of ARL. Assuming the probability of detecting the shift by a control chart is p, the value 1/p is the ARL of the chart. In this study, the probability p can not be obtained analytically, so we estimate the probability p by simulation.

4 Simulation and Comparative Studies

4.1 Generating Data

The comparative study is conducted with the aspartame example given in Kang and Albin [7]

as an example. Since there are no available data, we use two methods based on the form Y = I + Me^{N (x−1)2} + ² to generate the in-control aspartame profiles in the following.

The idea is to perturb the parameters I, M, N randomly to create allowable profile-to-profile variations for the in-control profile-to-profiles. We first define the setting of the parameters with µ_I = 1, σ_I = 0.2, µ_M = 15, σ_M = 1, µ_N = −1.5, σ_N = 0.3, σ_² = 0.3, p = 19, and x = 0.64, 0.80, . . . , 3.52. Both of x and y values are scaled variables, not the actual temperature levels and the amount of aspartame dissolved in the dissolving process.

1. The first method is to model the in-control aspartame profiles as following MVN (µ₀, Σ), where µ₀ = (µ₀₁, . . . , µ_0p)⁰ with

µ_0i = µ_I+ µ_Me^µN(xi−1)2, i = 1, . . . , p (14) and Σ is the covariance matrix as follows. For i, j = 1, . . . , p

Cov (Y_i, Y_j) = σ²_I+ (µ²_M + σ_M² ) + [e^{µN[(xi−1)2+(xj−1)2]+σ2N [(xi−1)2+(x2 j −1)2]2}

−µ²_Me^{µN(xi−1)2+σ2N (xi−1)}

4

2 +µN(xj−1)²+^{σ2N (xj −1)}

4

2 + σ_²²δ_ij, (15) where δ_ij = 1 if i = j and δ_ij = 0 if i 6= j. This method was adopted by Shiau, Huang, Lin, and Tsai [19] because their profile monitoring schemes are developed under the Gaussian assumption.

For out-of-control profiles, we can generate the “location-shifted” aspartame profiles as following MVN (µ, Σ), where µ = (µ₁, . . . , µ_p)⁰ with

µi = (µI+ ασI) + (µM + βσM)e^{(µN+γσN)(xi−1)2}, i = 1, . . . , p. (16) Then the shift on the mean of Y is δ ≡ µ − µ₀.

2. The second method directly generates the parameters (I, M, N ), where I ∼ N(µ_I, σ²_I), M ∼ N(µ_M, σ_M² ), N ∼ N(µ_N, σ_N²), ² ∼ N(0, σ_²²), and all the random parameters are

independent of each other. This model is also referred to as the random-coefficients model. Then use the following random-effect model to generate the in-control aspar-tame profiles with already generated (I, M, N, ²):

Yi = I + Me^{N (xi−1)2} + ²i, i = 1, . . . , p. (17)

We can generate the “location-shifted” aspartame profiles with the parameters (I, M, N, ²), where I ∼ N(µ_I + ασ_I, σ_I²), M ∼ N(µ_M + βσ_M, σ²_M), and N ∼ N(µ_N + γσ_N, σ_N²). In the same way, we can also generate the “scale-shifted” aspartame profiles with the parameters (I, M, N ), where I ∼ N(µ_I, (ασ_I)²), M ∼ N(µ_M, (βσ_M)²), and N ∼ N(µN, (γσN)²). The advantage of this method is that it is intuitive and interpretable.

However, the profiles generated in this way are no longer Gaussian.