Proposed monitoring scheme - 利用計分檢定統計量之指數加權移動平均控制圖來監控一般線性輪廓

¯¯

θ=θ^∗,θj=θ^∗_j

= n0

2 µσˆ²

σ^∗2 − 1

¶₂ + nj

2 µσˆ²_j

σ^∗2_j − 1

¶2

= (n₀ + n_j)³n0nj

´ µ σˆ² − ˆσ_j² n₀σˆ²+ n_jσˆ²_j

¶2

· 1_{ˆ_σ_j_<ˆ_σ}. (2.19) When the process is in control at time j, the distribution of W_0,j^∗ depends only on p, n0, and nj. Set

W¯_0,j^∗ ≡ W_0,j^∗ − E_θ,θ_j(W_0,j^∗ )|_θ=˜_θ,θ_j_=˜_θ_j q

Varθ,θj(W_0,j^∗ )|_θ=˜_θ,θ_j_=˜_θ_j , (2.20) where both E_θ,θ_j(W_0,j^∗ )|_θ=˜_θ,θ_j_=˜_θ_j and Var_θ,θ_j(W_0,j^∗ )|_θ=˜_θ,θ_j_=˜_θ_j are given in Appendix D.1.

2.6 Proposed monitoring scheme

The EWMA sequence is defined as





 U ≡ 0,

where λ is a smoothing parameter in (0, 1], and ˜W_j = ¯W_j in subsection 2.2, or ˜W_j = ¯W_j^∗ in subsection 2.3, ˜W_j = ¯W_0,j in subsection 2.4, ˜W_j = ¯W_0,j^∗ in subsection 2.5. Then, standardize the statistic and define the control chart statistic as

U_j^∗ ≡ U_j − E_θ(U_j)

pVar_θ(U_j) , (2.22)

where

E_θ(U_j) = 0,

and

Varθ(Uj) = λ [1 − (1 − λ)^2j] 2 − λ , for known θ case;

E_θ(U_j) = 0,

and

Var_θ(U_j) = λ [1 − (1 − λ)^2j]

2 − λ + 2λ² X

0≤k1<k2≤j−1

(1 − λ)^k¹^+k²Cov( ˜W_j−k₁, ˜W_j−k₂),

for unknown θ case. Because ˜W_j−k₁ and ˜W_j−k₂ are correlative, so the calculation of Cov( ˜W_j−k₁, ˜W_j−k₂) will be very complex when X_js are not design matrix. Therefore,

here recommend X0 and Xj as

X0 =





 X

...







(mn)×p

, Xj =





 X

...







(mjn)×p

, j ≥ 1,

where X is a n×p design matrix, then Cov( ˜Wj−k1, ˜Wj−k2) will only depend on n, p, m, mj−k1, mj−k1

Out-of-control signal occurs at time j if

U_j^∗ > C,

where C > 0 is chosen by a specified in-control ARL (≡ ARL₀). C only depend on ARL₀, λ, p, n₁, n₂, . . . for known θ case and depend on ARL₀, λ, p, n, m, m₁, m₂, . . . for unknown θ case.

3 A Simulation Study

In this Section, we explain how to simulate the proposed EWMA control chart for comparison with the Kim et al. (2003) and Zou et al. (2007). Consider the case of constrained EWMA control chart with known parameters first. If the process is in control, in equation (2.11), the Hj1 follows chi-squared distribution with p degrees of freedom, H_j2 follows chi-squared distribution with n_j− p degrees. Use this property to generate test statistic and find control limit C.

Step 1 : Choose the specific ARL₀ and smoothing parameter λ, here given ARL₀=200 and λ=0.2 and 50,000 simulations, The same assumptions with Kim et al. (2003) and Zou et al. (2006)

Step 2 : Generate 200 Hj1 and Hj2, and calculated 200 W_j^∗, Uj, and U_j^∗, for j = 1, 2, . . . , 200

Step 3 : c ≡ max{U₁^∗, U₂^∗, . . . , U₂₀₀^∗ } .

Step 4 : Repeat Step (2) ∼ Step (3) 50,000 times, obtain 50,000 c, and make them to sort (c₍₁₎ < c₍₂₎ < . . . < c_(50000)).

Step 5 : Choose the median of the 50,000 c as the control limit. Use this control limit, make 50,000 time simulation, then obtain 50,000 Run Length, and compute the ARL.

Step 6 : Use bisection method if the ARL > 200 then make control limit as the smaller

It can be easily generate the out-of-control W_j^∗ by equation (2.11), and use the con-trol limit which obtained previously to calculate the out-of concon-trol ARL (ARL₁) with 50,000 simulations.

Next, consider the case of constrained EWMA control chart with unknown param-eters. The procedure is same as case of constrained EWMA control chart with known parameters except the generation of U^∗. In subsection 2.5, W_0,j^∗ consists of ˆβ, ˆβ_j, ˆσ, and ˆσj, where

β ∼ Nˆ _p(β, σ²(X₀^TX₀)⁻¹), βˆj ∼ Np(βj, σ_j²(X_j^TXj)⁻¹),

σ ∼ σ²/n₀χ²_n₀_−p, ˆ

σ_j ∼ σ_j²/n_jχ²_n_j_−p,

and they are independent respectively. W_0,j^∗ and W_0,j^∗ 0 are depends on p, n₀, n₁, . . .(j 6=

j⁰), and therefore the X₀ and X_j must be design matrix. Then generate U^∗ through above property.

We consider the case from Kim et al. (2003) and Zou et al. (2007), the simplest case of model (2.2) with p = 2, in-control parameters β ≡ (β₀, β₁)^T = (3, 2)^T, σ = 1, fixed

Xj,

X_j ≡ X =





 1 2 1 4 1 6 1 8







, j ≥ 1,

and

X₀ =





 X

...







4m×p

4 Conclusions

4.1 Monitoring performance comparisons

Compared proposed EWMA chart with the KMW, ZTW and Huang (2012) EWMA chart by out of control ARL (ARL₁), the ARL₁ of three control chart with known pa-rameters are given in Table 5.1∼5.3. The process parameter β0 is changed to β0+ δ0σ, and β₁ is changed to β₁+ δ₁σ in Table 5.1 and Table 5.2. The ARL₁ of different ratio with σ and σ_j are presented in Table 5.3.

First focus on the KMW and ZTW, In Table 5.1 and Table 5.2. The proposed EWMA chart is favorable for detecting large shift in β₀ and β₁, but when shift is mod-erate or small, proposed EWMA chart has a significant adverse, it maybe cause by the property of score test and LR test. In the case of large shift or nj, the performance should be good, on the other hand, the case of small shift and nj, have the opposite result. The n_j is fixed to 4, and n_j − p = 2, therefore, this result is not surprising.

In the table 5.3. show that performance in all change case of the proposed EWMA chart are superior to others chart. In equation (2.7) and (2.8) show that changes of β₀ and β₁only affect the H_j1, but changes of σ² affect not only H_j1 but also H_j2, therefore, the proposed EWMA chart is more sensitive when σ² shifted.

Table 5.4 presented ARL1 with process parameters β and σ change simultane-ously. And the unknown in-control process parameters case also presented. When

rameters case.

4.2 Modify monitoring scheme (1)

In Huang (2012), he proposed an EWMA control chat based on LR test statistic, it is similar with the method we proposed. It should be similar results in theory, but according to Table 5.1∼5.3, they are a little difference. The HEWMA is better than proposed EWMA when the β shifted, and it is worse than proposed EWMA when the σ shifted.

Here proposed an improved method, we transform the score statistic to make it more sensitive for the detection of β.

Wj = ( ˆβ_j− β)^TX_j^TX_j( ˆβ_j− β) σ₀²

Φ⁻¹ Ã

F_χ²_nj

Ã( ˆβ_j − β)^TX_j^TX_j( ˆβ_j − β)

σ² +n_jσˆ²_j σ²

!!#₂ ,

for two-sided case, and

W_j^∗ = ( ˆβ_j − β)^TX_j^TX_j( ˆβ_j − β) σ²

+aF_χ⁻¹2 1

Ã F_χ²_nj

Ã( ˆβ_j− β)^TX_j^TX_j( ˆβ_j− β)

σ² +n_jσˆ_j² σ²

detection of β, and therefore the ARL1 performance with 0 < a < 1 are given in Table 5.6 and Table 5.7. When a the smaller, the performance will be almost the same as with the HEWMA or better. We found very bad performance for the detection of σ shifted in table 5.8. When the emphasis on the sift of β, we recommend the use this method, and when the emphasis on the shift of σ, we recommend using the methods mentioned earlier.

4.3 Modify monitoring scheme (2)

In the previous subsection, even the use of the modified statistic β shifted ARL₁ performance still worse than KMW and ZTW. And think it is ( ˆβ_j − β)^TX_j^TX_j( ˆβ_j − β)/σ² terms caused. It makes all the factors of β changes into a single value. Then propose a improve method. First use the MEWMA Chart. Let

W₀ ≡ 0_p+1×1

where X_j^T(y_j− X_jβ)/σ² and Φ⁻¹

³ F_χ²

nj (ky_j − X_jβk²/σ²)

are not independent but

Cov_θ_j_=θ

Then make the two-sided score test statistic as

U_j = and the one-sided score test statistic as

U_j^∗ = where a ∈ (0, ∞). Similar with before, the greater a is, and the more sensitive for the

In table 5.9 and table 5.10 shows the simulation results of the β shifted with two-sided case. The ARL₁ performance of the improve method is very close with ZTW and KMW, and when the constant a is getting smaller even better than their. In table 5.11 shows the simulation results of the σ shifted with two-sided and one-sided case. When constant a = 1, one-sided case more sensitive than two-sided, and the they are better than ZTW and KMW, but slightly worse than HEMWA. And the greater constant a is, the worse performance becomes. This improve method can be seen here in the case of whether β shifted or σ shifted has good, and to focus on detection of shifts in the β or σ by by adjusting constant a.

5 Future Work

In this paper, the MEWMA performance seems better than the performance of EWMA based on score test statistics. In future work, it can focus on MEWMA. Con-sider the nonlinear model. e.g., y_j = u_j(X_j; β_j) + σ_jε_j, where u_j(· ; ·) is a known function for j = 1, 2, · · · . May also consider the error term is not a normal distribution.

e.g., t-distribution. And considered the MEWMA method based on score test statistics will have a good performance.

References

[1] Castagliola, P., Celano, G., and Fichera, S. (2006) Monitoring process variability using EWMA. Springer Handbook of Engineering Statistics, 291-325. Ch.17.

[2] Huang, W-C. (2012) An exponentially weighted moving average control chart based on likelihood ratio test statistics for monitoring general linear profiles. Master the-sis, National Chiao Tung University, Hsinchu, Taiwan.

[3] Jensen, W. A., Champ, C. W., and Woodall, W. H. (2006) Effects of parameter estimation on control chart properties: a literature review. Journal of Quality Technology, 38, 156-167.

[4] Jones, L. A., Champ, C. W., and Rigdon, S. E. (2001) The performance of exponen-tially weighted moving average charts with estimated parameters. Technometrics, 43, 156-167.

[5] Kim, K., Mahmoud, M., and Woodall, W. H. (2003) On the monitoring of linear profiles. Journal of Quality Technology, 35, 317-328.

[6] Kots, Balakrishnan and Johnson. (2000) Continuous multivariate distributions vol-ume 1 second edition.

[7] Page, E. S. (1954) Continuous inspection schemes. Biometrika, 41, 100-115.

[9] Shewhart, W. A. (1931) Economic control of quality of manufactured product, American Society for Quality, Vol. 509.

[10] Zou, C., Lin, Y., and Wang, Z. (2009) Comparisons of control schemes for moni-toring the means of processes subject to drifts. Metrika, 70, 141-163.

[11] Zou, C., Tsung, F., and Wang, Z. (2007) Monitoring general linear profiles using multivariate exponentially weighted moving average schemes. Technometrics, 49, 395-408.

[12] Zou, C., Zhang, Y., and Wang, Z. (2006) A control chart based on a change-point model for monitoring linear profiles. Transactions, 38, 1093-1103.

[13] Zou, C., Zhang, J., and Wang, Z. (2010) A control chart based on likelihood ratio test for monitoring process mean and variability. Quality and Reliability Engineer-ing International, 26, 63-73.

Table 5.1: Out of control ARL comparisons between EWMA with known in-control parameters and constraint case, ZTW, KMW and HEWMA charts for shifts in β0, where θj = (β0+ δ0σ, β1, σ)^T and τ_j²= 4δ²₀ for j ≥ 1.

ARL1

δ0 τ_j² EWMA ZTW KMW HEWMA

0.1000 0.0400 178.7 131.5 133.7 171.1

0.2000 0.1600 133.1 59.9 59.1 113.1

0.3000 0.3600 86.7 29.6 28.3 67.2

0.4000 0.6400 53.4 17.2 16.2 38.1

0.5000 1.0000 32.9 11.5 10.7 22.1

0.6000 1.4400 20.5 8.5 7.9 13.8

0.8000 2.5600 9.1 5.8 5.1 6.1

1.0000 4.0000 4.8 4.1 3.8 3.4

1.5000 9.0000 1.7 2.6 2.4 1.5

2.0000 16.000 1.2 2.0 1.9 1.1

Table 5.2: Out of control ARL comparisons between EWMA with known in-control parameters and constraint case, ZTW, KMW and HEWMA charts for shifts in β1, where θj = (β0, β1+ δ1σ, σ)^T and τ_j²= 120δ₁² for j ≥ 1.

ARL1

δ₁ τ_j² EWMA ZTW KMW HEWMA

0.0250 0.0750 162.4 99.0 101.6 153.0

0.0375 0.1688 130.1 57.4 61.0 112.3

0.0500 0.3000 97.0 35.0 36.5 77.4

0.0625 0.4688 70.9 23.1 24.6 53.2

0.0750 0.6750 51.1 16.4 17.0 35.4

0.1000 1.2000 26.2 9.8 10.3 17.4

0.1250 1.8750 14.2 6.9 7.2 9.5

0.1500 2.7000 8.4 5.3 5.5 5.8

0.2000 4.8000 3.6 3.7 3.8 2.8

0.2500 7.5000 2.2 2.9 2.9 1.8

Table 5.3: Out of control ARL comparisons between EWMA with known in-control parameters and constraint case , ZTW, KMW and HEWMA charts for shifts in σ, where θj = (β0, β1, δσ)^T for j ≥ 1.

ARL1

δ EWMA ZTW KMW HEWMA

1.10 49.0 76.2 72.8 51.1

1.15 29.4 48.7 48.1 30.2

1.20 19.0 33.2 33.5 20.0

1.25 13.4 24.1 24.9 14.6

1.30 10.0 18.4 19.4 11.0

1.40 6.4 12.1 12.7 7.0

1.60 3.8 7.0 7.2 4.0

1.80 2.4 4.9 5.1 2.8

2.20 1.6 3.1 3.2 1.8

2.60 1.3 2.3 2.5 1.4

3.00 1.2 1.9 2.1 1.3

Table 5.4: ARLs with unknown in-control parameters and constraint case for shifts in θ with θ_j = (β^T_j, δσ)^T and τ_j²= τ²/δ²for j ≥ 1

m = 5 m = 25

m = 125 δ

m = ∞ 1.1⁰ 1.1¹ 1.1² 1.1³ 1.1⁴ 1.1⁵ 1.1⁶ 1.1⁷ 1.1⁸ 1.1⁹ 1.1¹⁰ 1.1¹¹ 1.1¹² 0.0² 200.0 71.6 31.9 14.9 9.5 5.3 4.1 3.0 2.1 1.9 1.7 1.5 1.4

200.0 54.9 20.2 9.9 6.0 3.9 2.9 2.3 1.9 1.5 1.4 1.3 1.2

200.0 50.1 18.7 9.4 5.4 3.5 2.7 2.0 1.8 1.4 1.3 1.3 1.2

200.0 49.0 18.0 8.9 4.8 3.3 2.5 1.9 1.7 1.3 1.3 1.2 1.2

0.2² 189.2 69.9 28.5 13.7 9.2 5.1 3.9 2.9 2.0 2.0 1.7 1.6 1.4

184.3 52.0 19.8 9.3 5.6 3.7 2.8 2.3 1.8 1.5 1.4 1.3 1.2

182.2 47.1 18.0 8.9 5.3 3.2 2.5 2.0 1.7 1.3 1.3 1.2 1.2

178.7 44.2 17.0 8.4 4.1 3.0 2.2 1.6 1.4 1.3 1.3 1.2 1.2

0.4² 150.1 62.1 27.3 12.9 8.8 4.9 3.7 2.7 2.0 1.9 1.6 1.5 1.4

140.1 54.3 18.0 9.1 5.4 3.5 2.7 2.2 1.7 1.4 1.3 1.3 1.2

136.8 49.2 17.0 8.5 4.8 3.1 2.5 2.0 1.7 1.3 1.3 1.2 1.2

133.1 40.1 14.1 7.9 3.9 2.6 2.0 1.5 1.4 1.3 1.3 1.2 1.2

0.6² 99.9 45.2 22.7 10.9 8.3 4.8 3.5 2.4 2.0 2.0 1.5 1.5 1.4

93.7 37.8 15.0 8.4 5.3 3.3 2.6 2.0 1.7 1.4 1.3 1.2 1.2

89.1 33.5 14.1 7.5 4.6 2.8 2.2 1.8 1.6 1.3 1.3 1.2 1.2

86.7 30.8 13.1 7.1 3.4 2.5 1.9 1.5 1.4 1.3 1.3 1.2 1.2

0.8² 70.2 43.1 20.9 10.1 7.1 4.1 3.3 2.2 1.9 1.8 1.5 1.4 1.4

58.2 26.9 12.9 7.3 4.5 2.8 2.5 2.0 1.6 1.4 1.3 1.2 1.2

55.1 24.2 11.5 6.7 3.8 2.5 2.1 1.7 1.6 1.3 1.3 1.2 1.2

53.4 22.1 10.8 6.3 3.2 2.3 1.8 1.5 1.4 1.3 1.3 1.2 1.2

τ² 1.0² 50.2 24.1 13.9 9.5 6.7 4.0 3.1 2.0 1.8 1.6 1.4 1.4 1.4

37.1 15.8 9.2 6.0 4.2 2.7 2.3 1.8 1.6 1.4 1.3 1.2 1.2

33.2 14.9 7.9 5.6 3.7 2.4 2.0 1.6 1.5 1.3 1.3 1.2 1.2

32.9 13.7 6.7 5.4 3.0 2.1 1.8 1.5 1.4 1.3 1.3 1.2 1.2

1.2² 30.5 14.2 10.5 7.8 5.8 3.8 3.0 2.0 1.8 1.6 1.4 1.4 1.3

25.6 10.1 7.2 5.1 3.5 2.6 2.1 1.8 1.6 1.4 1.3 1.2 1.2

21.9 9.2 6.4 4.8 3.2 2.3 1.9 1.6 1.5 1.3 1.3 1.2 1.2

20.5 8.5 5.8 4.4 2.8 2.0 1.7 1.5 1.4 1.3 1.3 1.2 1.1

1.4² 22.4 13.0 8.5 6.4 5.0 3.5 2.9 1.9 1.7 1.5 1.4 1.3 1.3

14.0 8.8 5.6 4.3 3.3 2.5 2.1 1.6 1.6 1.4 1.2 1.2 1.2

12.2 7.2 5.2 4.2 3.0 2.3 1.8 1.6 1.5 1.3 1.2 1.2 1.2

11.6 6.9 4.8 3.8 2.6 1.8 1.7 1.5 1.4 1.3 1.2 1.2 1.1

1.6² 12.9 11.1 7.7 5.5 4.6 3.4 2.8 1.7 1.7 1.5 1.4 1.3 1.3

9.8 5.5 4.7 3.8 3.2 2.4 1.9 1.6 1.5 1.3 1.2 1.2 1.2

9.6 5.3 4.4 3.5 2.9 2.1 1.7 1.8 1.4 1.3 1.2 1.2 1.2

9.1 4.9 4.0 3.3 2.5 1.7 1.6 1.5 1.4 1.3 1.2 1.2 1.1

1.8² 8.9 6.9 5.4 4.5 3.8 3.0 2.5 1.6 1.5 1.4 1.4 1.3 1.3

6.6 4.9 3.7 3.2 2.7 2.3 2.1 1.9 1.5 1.3 1.3 1.3 1.2

Table 5.5: ARLs with unknown in-control parameters case for shifts in θ with θj = (β^T_j, δσ)^T and τ_j²= τ²/δ² for j ≥ 1

m = 5 m = 25

m = 125 δ

m = ∞ 1.1⁰ 1.1¹ 1.1² 1.1³ 1.1⁴ 1.1⁵ 1.1⁶ 1.1⁷ 1.1⁸ 1.1⁹ 1.1¹⁰ 1.1¹¹ 1.1¹²

2.2² 5.4 4.3 3.6 3.3 3.1 2.6 2.4 2.1 1.9 1.8 1.6 1.5 1.3

4.1 3.6 2.7 2.3 2.2 2.0 1.8 1.7 1.6 1.5 1.4 1.2 1.2

3.8 3.3 2.5 2.3 2.2 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1

3.6 3.1 2.4 2.1 1.8 1.7 1.6 1.5 1.3 1.2 1.2 1.2 1.1

2.4² 4.5 3.8 3.1 2.8 2.6 2.4 2.2 1.8 1.8 1.7 1.5 1.4 1.3

3.5 3.0 2.2 2.1 2.0 1.7 1.7 1.6 1.5 1.4 1.3 1.2 1.2

3.2 2.9 2.2 2.0 1.9 1.7 1.7 1.5 1.5 1.4 1.3 1.2 1.1

3.0 2.2 2.0 1.9 1.6 1.6 1.6 1.5 1.3 1.2 1.2 1.2 1.1

2.6² 4.9 3.2 2.9 2.6 2.4 2.3 2.0 1.7 1.7 1.6 1.4 1.3 1.2

3.2 2.7 2.1 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.3 1.2 1.2

3.0 2.3 1.9 1.8 1.6 1.6 1.5 1.5 1.3 1.3 1.2 1.2 1.1

2.6 2.0 1.8 1.8 1.5 1.5 1.4 1.4 1.2 1.2 1.2 1.1 1.1

2.8² 3.5 3.1 2.4 2.2 2.0 2.0 1.9 1.6 1.6 1.5 1.4 1.3 1.2

2.8 2.1 1.9 1.7 1.6 1.6 1.5 1.4 1.4 1.3 1.2 1.2 1.2

2.4 1.9 1.7 1.6 1.6 1.5 1.5 1.4 1.3 1.3 1.2 1.1 1.1

2.2 1.6 1.5 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1

3.0² 3.2 2.9 2.2 2.0 1.9 1.9 1.8 1.6 1.5 1.4 1.3 1.3 1.3

2.5 1.9 1.7 1.6 1.5 1.4 1.4 1.4 1.3 1.3 1.2 1.2 1.1

2.1 1.8 1.6 1.4 1.4 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1

1.8 1.6 1.4 1.4 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.1 1.1

τ² 3.2² 2.9 2.7 2.1 1.9 1.8 1.8 1.7 1.5 1.5 1.4 1.3 1.3 1.3

2.1 1.9 1.6 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.1

1.9 1.7 1.5 1.4 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1.1

1.7 1.5 1.4 1.4 1.4 1.3 1.3 1.2 1.2 1.2 1.1 1.1 1.1

3.4² 2.7 2.1 1.9 1.8 1.7 1.7 1.6 1.5 1.5 1.4 1.4 1.3 1.2

1.8 1.6 1.4 1.3 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.1 1.1

1.7 1.5 1.3 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1

1.5 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1

3.6² 2.0 1.7 1.7 1.7 1.6 1.6 1.5 1.4 1.4 1.3 1.3 1.3 1.2

1.6 1.5 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1

1.5 1.4 1.3 1.3 1.3 1.3 1.2 1.2 1.2 1.2 1.1 1.1 1.1

1.4 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1

3.8² 1.7 1.6 1.5 1.5 1.5 1.5 1.4 1.4 1.3 1.3 1.2 1.3 1.2

1.5 1.3 1.3 1.2 1.3 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1

Table 5.6: Out of control ARL comparisons between modified (1) EWMA with known in-control parameters case and HEWMA charts for shifts in β0, where θj = (β0+ δ0σ, β1, σ)^T and τ_j²= 4δ₀² for j ≥ 1.

ARL₁

δ0 τ_j² a=1 a=0.75 a=0.5 a=0.25 HEWMA

0.1000 0.0400 174.9 173.2 171.8 170.2 171.1 0.2000 0.1600 118.9 117.3 115.8 112.1 113.1

0.3000 0.3600 71.4 69.7 68.5 67.1 67.2

0.4000 0.6400 40.4 39.5 38.9 38.1 38.1

0.5000 1.0000 23.7 23.0 22.5 21.9 22.1

0.6000 1.4400 14.4 14.1 13.7 13.4 13.8

0.8000 2.5600 6.5 6.4 6.2 6.1 6.1

1.0000 4.0000 3.6 3.6 3.5 3.4 3.4

1.5000 9.0000 1.6 1.5 1.5 1.5 1.5

2.0000 16.000 1.1 1.1 1.1 1.1 1.1

Table 5.7: Out of control ARL comparisons between modified (1) EWMA with known in-control parameters case and HEWMA charts for shifts in β₁, where θ_j = (β₀, β₁+ δ₁σ, σ)^T and τ_j² = 120δ₁² for j ≥ 1.

ARL1

δ1 τ_j² a=1 a=0.75 a=0.5 a=0.25 HEWMA

0.0250 0.0750 153.6 152.4 152.3 152.2 153.0 0.0375 0.1688 115.6 113.8 112.6 112.3 112.3

0.0500 0.3000 81.6 79.9 78.7 77.6 77.4

0.0625 0.4688 55.8 54.5 53.8 52.8 53.2

0.0750 0.6750 37.7 37.3 36.4 35.3 35.4

0.1000 1.2000 18.6 18.0 17.6 17.2 17.4

0.1250 1.8750 10.0 9.7 9.5 9.3 9.5

0.1500 2.7000 6.1 5.9 5.8 5.6 5.8

0.2000 4.8000 2.9 2.8 2.8 2.7 2.8

0.2500 7.5000 1.8 1.8 1.7 1.7 1.8

Table 5.8: Out of control ARL comparisons between modified (1) EWMA with known in-control parameters case and HEWMA charts for shifts in σ, where θj= (β0, β1, δσ)^T for j ≥ 1.

ARL1

δ a=1 a=0.75 a=0.5 a=0.25 HEWMA

1.10 138.6 148.1 162.6 179.1 51.1 1.15 112.4 126.0 144.0 167.6 30.2

1.20 91.0 105.5 126.9 157.4 20.0

1.25 73.0 87.6 110.9 145.9 14.6

1.30 59.2 72.7 96.3 135.9 11.0

1.40 38.6 49.7 71.2 114.7 7.0

1.60 18.2 24.0 37.8 77.7 4.0

1.80 10.4 13.3 20.7 49.8 2.8

2.20 4.9 5.9 8.5 20.2 1.8

2.60 3.2 3.6 4.8 9.8 1.4

3.00 2.4 2.7 3.3 5.9 1.3

Table 5.9: Out of control ARL comparisons between modified (2) MEWMA with known in-control parameters case, HEWMA, ZTW and KMW charts for shifts in β0, where θj= (β0+ δ0σ, β1, σ)^T.

ARL1

unconstraint unconstraint unconstraint HEWMA ZTW KMW

δ0 a=1 a=0.75 a=0.5

0.1000 131.8 127.3 126.2 171.1 131.5 131.5

0.2000 59.7 58.0 57.1 113.1 59.9 59.1

0.3000 28.5 27.7 27.3 67.2 29.6 28.3

0.4000 16.5 16.0 15.7 38.1 17.2 16.2

0.5000 10.2 9.8 9.6 22.1 11.5 10.7

0.6000 7.5 7.3 7.1 13.8 8.5 7.9

0.8000 4.0 3.8 3.7 6.1 5.8 5.1

1.0000 2.8 2.7 2.6 3.4 4.1 3.8

1.5000 1.5 1.5 1.4 1.4 2.6 2.4

Table 5.10: Out of control ARL comparisons between modified (2) MEWMA with known in-control parameters case, HEWMA, ZTW and KMW charts for shifts in β1, where θj= (β0, β1+ δ1σ, σ)^T.

ARL1

unconstraint unconstraint unconstraint HEWMA ZTW KMW

δ1 a=1 a=0.75 a=0.5

0.0250 99.9 98.9 98.2 153.0 99.0 101.6

0.0375 57.3 56.3 55.9 112.3 57.4 61.0

0.0500 34.8 33.9 32.1 77.4 35.0 36.5

0.0625 22.9 22.0 21.5 53.2 23.1 24.6

0.0750 16.0 15.4 15.0 35.4 16.4 17.0

0.1000 8.5 8.2 8.0 17.4 9.8 10.3

0.1250 5.8 5.6 5.2 9.5 6.9 7.2

0.1500 4.5 4.5 4.3 5.8 5.3 5.5

0.2000 2.7 2.6 2.5 2.8 3.7 3.8

0.2500 1.7 1.7 1.6 1.8 2.9 2.9

Table 5.11: Out of control ARL comparisons between modified (2) MEWMA with known in-control parameters case, HEWMA, ZTW and KMW charts for shifts in σ, where θj = (β0, β1, δσ)^T for j ≥ 1.

ARL1

unconstraint unconstraint unconstraint constraint HEWMA ZTW KMW

δ a=1 a=0.75 a=0.5 a=1

1.10 53.1 58.1 66.1 50.0 51.1 76.2 72.8

1.15 32.2 38.2 45.0 30.1 30.2 48.7 48.1

1.20 20.8 24.3 29.4 19.8 20.0 33.2 33.5

1.25 15.2 17.8 21.8 14.0 14.6 24.1 24.9

1.30 11.0 12.6 15.4 10.0 11.0 18.4 19.4

1.40 6.9 7.8 9.3 6.4 7.0 12.1 12.7

1.60 3.9 4.3 4.8 3.8 4.0 7.0 7.2

1.80 2.8 3.0 3.2 2.4 2.8 4.9 5.1

2.20 1.7 1.8 2.0 1.6 1.8 3.1 3.2

2.60 1.4 1.5 1.7 1.3 1.4 2.3 2.5

3.00 1.2 1.3 1.4 1.2 1.3 1.9 2.1

Appendix

A.2

When the process is in control,

H_j1 ∼ χ²_p

and

H_j2 ∼ χ²_n_j_−p.

E_θ(W_j) = E_θ(H_j1+ 1

2n_j(H_j1+ H_j2− n_j)²) = p + Var(H_j1+ H_j2)

2n_j = p + 1.

Var_θ(W_j)

= 1

4n²_jE_θ¡

(H_j1² + H_j2² + 2H_j1H_j2− 2n_jH_j2+ n²_j)²¢

− (p + 1)²

= 1

4n²_jEθ(H_j1⁴ + H_j2⁴ + 6H_j1² H_j2² + 4H_j1³ Hj2+ 4Hj1H_j2³ − 4njH_j1² Hj2− 8njHj1H_j2²

−4n_jH_j2³ + 2n²_jH_j1² + 6n²_jH_j2² + 4n²_jH_j1H_j2− 4n³_jH_j2+ n⁴_j) − (p + 1)²

= 1

4n²_j[E_θ(H_j1⁴) + E_θ(H_j2⁴) + 6E_θ(H_j1²)E_θ(H_j2²) + 4E_θ(H_j1³)E_θ(H_j2)

+4E_θ(H_j1)E_θ(H_j2³) − 4n_jE_θ(H_j1²)E_θ(H_j2) − 8n_jE_θ(H_j1)E_θ(H_j2²) − 4n_jE_θ(H_j2³) +2n²_jE_θ(H_j1² ) + 6n²_jE_θ(H_j2² ) + 4n²_jE_θ(H_j1)E_θ(H_j2) − 4n³_jE_θ(H_j2) + n⁴_j ] − (p + 1)²

= 2p +8p nj

+ 12 nj

+ 2

where

E_θ(H_j1) = p,

Eθ(H_j1² ) = p(p + 2),

E_θ(H_j1³ ) = p(p + 2)(p + 4),

E_θ(H_j1⁴ ) = p(p + 2)(p + 4)(p + 6), Eθ(Hj2) = nj − p,

E_θ(H_j2² ) = (n_j − p)(n_j− p + 2),

E_θ(H_j2³ ) = (n_j − p)(n_j− p + 2)(n_j− p + 4),

Eθ(H_j2⁴ ) = (nj − p)(nj− p + 2)(nj− p + 4)(nj − p + 6)

B.1

and

Here

where

Here

C.1

Rewrite W_0,j in section 2.4 as

W_0,j = (y₀− X₀β)˜ ^TX₀(X₀^TX₀)⁻¹X₀^T(y₀− X₀β)˜

Let

Then

ε⁽¹⁾_0j = ε₀− X₀(X₀^TX₀+ X_j^TX_j)⁻¹(X₀^Tε₀+ X_j^Tε_j)

= ε0− X0(X₀^TX0+ X_j^TXj)^−1/2Z_0j⁽¹⁾

then kε⁽¹⁾_0j k² = kε₀k²− 2Z₀^T(X₀^TX₀)^1/2(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾ +k(X₀^TX₀)^1/2(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾k²

= H0+ kZ0− (X₀^TX0)^1/2(X₀^TX0+ X_j^TXj)^−1/2Z_0j⁽¹⁾k²

= H₀+ k(X₀^TX₀)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2

Z_0j⁽²⁾k².

And

cov

(X₀^TX₀)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2 Z_0j⁽²⁾

= (X₀^TX₀)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤₋₁

(X₀^TX₀)^−1/2

=©

(X₀^TX₀)^1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤

(X₀^TX₀)^1/2ª₋₁

=£

I_p+ (X₀^TX₀)^1/2(X_j^TX_j)⁻¹(X₀^TX₀)^1/2¤₋₁

=¡

I_p+ P_0jΛ⁻¹_0j P_0j^T¢₋₁

=£

P_0j(I_p+ Λ⁻¹_0j)P_0j^T¤₋₁

= P_0j(I_p+ Λ⁻¹_0j )⁻¹P_0j^T where P_0jΛ⁻¹_0jP_0j^T is eigendecomposition of (X₀^TX₀)^1/2(X_j^TX_j)⁻¹(X₀^TX₀)^1/2 and

P_0j⁻¹ = P_0j^T , Λ_0j ≡ diag{λ_0j1, · · · , λ_0jp}, λ_0j1≥ · · · ≥ λ_0jp≥ 0

and λ_0j1, · · · , λ_0jp are the eigenvalues of (X₀^TX₀)^1/2(X_j^TX_j)⁻¹(X₀^TX₀)^1/2, P_0j ≡ (P_0j1, · · · , P_0jp), P_0jk is the eigenvector of the eigenvalue λ_0jk, k = 1, 2, . . . , p.

Let

Z_0j ≡ (I_p+ Λ⁻¹_0j )^1/2P_0j^T(X₀^TX₀)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2 Z_0j⁽²⁾

∼ Np(Op×1, Ip), kZ_0j⁽²⁾k² = kZ0jk² then kε⁽¹⁾_0jk² = H0+ k (Ip + Λ⁻¹_0j)^−1/2Z0j k²= H0+

Xp k=1

Z_0jk² 1 + 1/λ_0jk

= H0+ Xp k=1

λ0jkZ_0jk²

1 + λ_0jk. (C.1)

And

ε⁽²⁾_0j ≡ ε_j − X_j(X₀^TX₀+ X_j^TX_j)⁻¹(X₀^Tε₀ + X_j^Tε_j)

= ε_j − X_j(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾

then kε⁽²⁾_0jk² = kεjk²− 2Z_j^T(X_j^TXj)^1/2(X₀^TX0+ X_j^TXj)^−1/2Z_0j⁽¹⁾ +k(X_j^TX_j)^1/2(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾k²

= H_j + kZ_j− (X_j^TX_j)^1/2(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾k²

= H_j + k(X_j^TX_j)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2 Z_0j⁽²⁾k²

= H_j + kZ_0j⁽²⁾k²− k(X₀^TX₀)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2 Z_0j⁽²⁾k²

= H_j + kZ_0jk² − Xp

k=1

λ_0jkZ_0jk² 1 + λ_0jk

= H_j + Xp k=1

Z_0jk²

1 + λ_0jk. (C.2)

Hence

kε⁽¹⁾k² + kε⁽²⁾k² = H + H + Xp

Z² = H + H + kZ k² ∼ χ² . (C.3)

And

(ε⁽¹⁾_0j )^TX₀(X₀^TX₀)⁻¹X₀^T(ε⁽¹⁾_0j ) + (ε⁽²⁾_0j )^TX_j(X_j^TX_j)⁻¹X_j^T(ε⁽²⁾_0j )

= k(X₀^TX₀)^−1/2X₀^Tε⁽¹⁾_0jk²+ k(X_j^TX_j)^−1/2X_j^Tε⁽²⁾_0j k²

= kZ₀− (X₀^TX₀)^1/2(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾k² +kZ_j− (X_j^TX_j)^1/2(X₀^TX₀+ X_j^TX_j)^−1/2Z_0j⁽¹⁾k²

= k(X₀^TX₀)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2 Z_0j⁽²⁾k² +k(X_j^TX_j)^−1/2£

(X₀^TX₀)⁻¹+ (X_j^TX_j)⁻¹¤_−1/2 Z_0j⁽²⁾k²

= kZ_0j⁽²⁾k² = kZ_0jk² = Xp k=1

Z_0jk² . (C.4)

By (C.1), (C.2), (C.3), (C.4)

W_0,j = (n₀+ n_j)kZ_0jk² H₀ + H_j+ kZ_0jk² + (n0+ nj)³

2n₀n_j(H₀+ H_j+ kZ_0jk²)²

Hj+ Xp k=1

Z_0jk²

1 + λ_0jk −nj(H0+ Hj + kZ0jk²) n₀+ n_j

#₂ .

Let

D_0j0 ≡ H₀

H₀+ H_j + kZ_0jk², D_0jk ≡ Z_0jk²

H0+ Hj + kZ0jk², k = 1, 2, . . . , p,

D_0j,p+1 ≡ Hj

H₀+ H_j + kZ_0jk²,

(D_0j0, D_0j1, · · · , D_0jp, D_0j,p+1)^T ∼ Dirichlet

µn₀− p 2 ,1

2, · · · ,1

2,n_j − p 2

¶ .

And

E(D_0j0) = n₀ − p

n₀+ n_j − p, Var(D_0j0) = 2n_j(n₀− p)

(n₀+ n_j − p)²(n₀+ n_j− p + 2), E(D_0jk) = 1

n0+ nj − p, Var(D_0jk) = 2(n₀+ n_j− p − 1)

(n0+ nj− p)²(n0+ nj − p + 2), k = 1, 2, . . . , p.

E(D_0j,p+1) = n_j− p

n0+ nj − p, Var(D_0j,p+1) = 2n₀(n_j− p)

(n0+ nj − p)²(n0+ nj− p + 2).

Then

W_0,j = (n₀+ n_j) Xp k=1

D_0jk+(n₀+ n_j)³ 2n₀n_j

D_0j,p+1+ Xp

k=1

D_0jk

1 + λ_0jk − n_j n₀ + n_j

#₂ .

E(W_0,j) = (n₀+ n_j)p

n0+ nj − p+ (n₀+ n_j)³ 2n0nj

Where

Finally, E(W0,j) can be calculated by (C.5), (C.6).

For convenient, here calculate Var(W0,j) with the fixed X0 and Xj case in section 3.

X_j ≡ X_n×p, X₀ =

Then

and the first part of Var(W0,j) can be calculated using the previous method.

By the book ¿ Continuous multivariate distributions À p.488 , we have

E©

D^m_0jD_0j,p+1ⁿ ª

= (p/2)^[m][(n_j− p)/2]^[n]

(n_j/2)^[m+n] , where k^[m] ≡ k(k + 1) · · · (k + m − 1).

And here can use this method to calculate the second part of Var(W_0,j). Finally, we can calculate the Var(W_0,j).

D.1

E_θ(W_0,j^∗ ) and Var_θ(W_0,j^∗ )

E_θ(W_0,j^∗ ) = E_θ(W_0,j) − E_θ(W_0,j(θ^∗, θ^∗_j))

where Eθ(W0,j) is given in appendix C.1, and by equation (2.19) can know

E_θ(W_0,j(θ^∗, θ^∗_j))

j<_1/n0+1/nj^1/n0

! ,

Then

E_θ(W_0,j(θ^∗, θ_j^∗)) = (n₀+ n_j)³ 2n0nj

E_θ¡

(B_j− a_j)²· 1_B_j_<a_j¢

and

E_θ¡

B_j^k· 1_B_j_<a_j¢

= Z _aj

x^k+(n^j^−p)/2−1(1 − x)⁽ⁿ⁰^−p)/2−1

Γ ((nj− p)/2) Γ ((n0− p)/2) /Γ ((n0+ nj − 2p)/2))dx

= Γ ((n₀+ n_j − 2p)/2) Γ (k + (n_j − p)/2) Γ ((n_j − p)/2) Γ (k + (n₀+ n_j − 2p)/2)

· Z _aj

x^k+(n^j^−p)/2−1(1 − x)⁽ⁿ⁰^−p)/2−1

Γ (k + (n_j− p)/2) Γ ((n₀ − p)/2) /Γ (k + (n₀+ n_j − 2p)/2)dx

= Γ ((n₀+ n_j − 2p)/2) Γ (k + (n_j − p)/2)

Γ ((n_j − p)/2) Γ (k + (n₀+ n_j − 2p)/2)F (a_j), k = 1, 2, 3, · · · ,

where F (·) is the cdf of Beta(k + (nj − p)/2, (n0 − p)/2). Finally, Eθ(W_0,j^∗ ) can be calculated.

Var_θ(W_0,j^∗ ) = Var_θ(W_0,j) + Var_θ(W_0,j(θ^∗, θ^∗_j)) − Cov(W_0,j, W_0,j(θ^∗, θ_j^∗)).

Var_θ(W_0,j) is given in appendix C.1, and

Varθ(W0,j(θ^∗, θ_j^∗)) = Eθ(W_0,j² (θ^∗, θ^∗_j)) −£

Eθ(W0,j(θ^∗, θ_j^∗))¤₂

where Eθ(W0,j(θ^∗, θ^∗_j)) is calculated in the previous, and

Eθ(W_0,j² (θ^∗, θ_j^∗)) = (n₀+ n_j)⁶ 4n²₀n²_j Eθ

£(Bj− aj)⁴· 1Bj<aj

¤,

it can be calculated using the previous method. Cov(W_0,j, W_0,j(θ^∗, θ_j^∗)) is too compli-cated to calculate, so using the simple covariance to estimate it. Finally, Var (W^∗ )

在文檔中利用計分檢定統計量之指數加權移動平均控制圖來監控一般線性輪廓 (頁 27-65)