Control charts for high-dimensional time series with estimated in-control parameters

Rostyslav Bodnara Department of Statistics, European University Viadrina, Frankfurt an der Oder, GermanyView further author information

Taras Bodnarb Department of Mathematics, Stockholm University, Stockholm, SwedenCorrespondence[email protected]

https://orcid.org/0000-0001-7855-8221 View further author information

Wolfgang Schmida Department of Statistics, European University Viadrina, Frankfurt an der Oder, GermanyView further author information

Figures & data

Figure 1. Probabilities $P (T_{1, t}^{*} \leq x), P (T_{2, t}^{*} \leq x), P (T_{3, t}^{*} \leq x), P (T_{4, t}^{*} \leq x), P (T_{6, t}^{*} \leq x), P (T_{7, t}^{*} \leq x), P (T_{8, t}^{*} \leq x),$ and $P (T_{9, t}^{*} \leq x)$ as functions of x for t = 5, p = 20, and $N_{0} \in {100, 250, 500} .$ The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 1. Probabilities P(T1,t*≤x),P(T2,t*≤x),P(T3,t*≤x),P(T4,t*≤x),P(T6,t*≤x),P(T7,t*≤x),P(T8,t*≤x), and P(T9,t*≤x) as functions of x for t = 5, p = 20, and N0∈{100,250,500}. The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 2. Probabilities $P (T_{1, t}^{*} \leq x), P (T_{2, t}^{*} \leq x), P (T_{3, t}^{*} \leq x), P (T_{4, t}^{*} \leq x), P (T_{6, t}^{*} \leq x), P (T_{7, t}^{*} \leq x), P (T_{8, t}^{*} \leq x),$ and $P (T_{9, t}^{*} \leq x)$ as functions of x for t = 5, p = 50, and $N_{0} \in {100, 250, 500} .$ The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 2. Probabilities P(T1,t*≤x),P(T2,t*≤x),P(T3,t*≤x),P(T4,t*≤x),P(T6,t*≤x),P(T7,t*≤x),P(T8,t*≤x), and P(T9,t*≤x) as functions of x for t = 5, p = 50, and N0∈{100,250,500}. The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 3. Probabilities $P (T_{1, t}^{*} \leq x), P (T_{2, t}^{*} \leq x), P (T_{3, t}^{*} \leq x), P (T_{4, t}^{*} \leq x), P (T_{6, t}^{*} \leq x), P (T_{7, t}^{*} \leq x), P (T_{8, t}^{*} \leq x),$ and $P (T_{9, t}^{*} \leq x)$ as functions of x for t = 5, p = 500, and $N_{0} \in {2000, 5000, 10000} .$ The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 3. Probabilities P(T1,t*≤x),P(T2,t*≤x),P(T3,t*≤x),P(T4,t*≤x),P(T6,t*≤x),P(T7,t*≤x),P(T8,t*≤x), and P(T9,t*≤x) as functions of x for t = 5, p = 500, and N0∈{2000,5000,10000}. The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 4. Probabilities $P (T_{1, t}^{*} \leq x), P (T_{2, t}^{*} \leq x), P (T_{3, t}^{*} \leq x), P (T_{4, t}^{*} \leq x), P (T_{6, t}^{*} \leq x), P (T_{7, t}^{*} \leq x), P (T_{8, t}^{*} \leq x),$ and $P (T_{9, t}^{*} \leq x)$ as functions of x for t = 5, p = 1,000, and $N_{0} \in {2000, 5000, 10000} .$ The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 4. Probabilities P(T1,t*≤x),P(T2,t*≤x),P(T3,t*≤x),P(T4,t*≤x),P(T6,t*≤x),P(T7,t*≤x),P(T8,t*≤x), and P(T9,t*≤x) as functions of x for t = 5, p = 1,000, and N0∈{2000,5000,10000}. The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 5. Probabilities $P (T_{Mah, t}^{*} \leq x)$ (left column) and $P (T_{MahInf, t}^{*} \leq x)$ (right column) as functions of x for t = 5. We set $p \in {20, 50}$ with $N_{0} \in {100, 250, 500}$ and $p \in {500, 1000}$ with $N_{0} \in {2000, 5000, 10000} .$ The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Figure 5. Probabilities P(TMah,t*≤x) (left column) and P(TMahInf,t*≤x) (right column) as functions of x for t = 5. We set p∈{20,50} with N0∈{100,250,500} and p∈{500,1000} with N0∈{2000,5000,10000}. The red plot corresponds to the density of the distribution in the case without misspecification; that is, the standard normal distribution.

Table 1. ARLs of the $T_{1, t}^{}, T_{3, t}^{}, T_{4, t}^{}, T_{6, t}^{}, T_{7, t}^{}, T_{Mah, t}^{},$ and $T_{MahInf, t}^{*}$ MEWMA control charts for $r \in {0.1, 0.2, \dots, 1.0}$ when the in-control process is the 20-dimensional VAR(1) process.

Display Table

Table 2. ARLs of the $T_{1, t}^{}, T_{3, t}^{}, T_{4, t}^{}, T_{6, t}^{}, T_{7, t}^{}, T_{Mah, t}^{},$ and $T_{MahInf, t}^{*}$ MEWMA control charts for $r \in {0.1, 0.2, \dots, 1.0}$ when the in-control process is the 100-dimensional VAR(1) process.

Display Table

Control charts for high-dimensional time series with estimated in-control parameters

Table 1. ARLs of the $T_{1, t}^{}, T_{3, t}^{}, T_{4, t}^{}, T_{6, t}^{}, T_{7, t}^{}, T_{Mah, t}^{},$ and $T_{MahInf, t}^{*}$ MEWMA control charts for $r \in {0.1, 0.2, \dots, 1.0}$ when the in-control process is the 20-dimensional VAR(1) process.

Table 2. ARLs of the $T_{1, t}^{}, T_{3, t}^{}, T_{4, t}^{}, T_{6, t}^{}, T_{7, t}^{}, T_{Mah, t}^{},$ and $T_{MahInf, t}^{*}$ MEWMA control charts for $r \in {0.1, 0.2, \dots, 1.0}$ when the in-control process is the 100-dimensional VAR(1) process.

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Control charts for high-dimensional time series with estimated in-control parameters

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Table 1. ARLs of the T1,t*,T3,t*,T4,t*,T6,t*,T7,t*,TMah,t*, and TMahInf,t* MEWMA control charts for r∈{0.1,0.2,…,1.0} when the in-control process is the 20-dimensional VAR(1) process.

Table 2. ARLs of the T1,t*,T3,t*,T4,t*,T6,t*,T7,t*,TMah,t*, and TMahInf,t* MEWMA control charts for r∈{0.1,0.2,…,1.0} when the in-control process is the 100-dimensional VAR(1) process.

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Table 1. ARLs of the $T_{1, t}^{}, T_{3, t}^{}, T_{4, t}^{}, T_{6, t}^{}, T_{7, t}^{}, T_{Mah, t}^{},$ and $T_{MahInf, t}^{*}$ MEWMA control charts for $r \in {0.1, 0.2, \dots, 1.0}$ when the in-control process is the 20-dimensional VAR(1) process.

Table 2. ARLs of the $T_{1, t}^{}, T_{3, t}^{}, T_{4, t}^{}, T_{6, t}^{}, T_{7, t}^{}, T_{Mah, t}^{},$ and $T_{MahInf, t}^{*}$ MEWMA control charts for $r \in {0.1, 0.2, \dots, 1.0}$ when the in-control process is the 100-dimensional VAR(1) process.