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Research Article

Tr(R²) control charts based on kernel density estimation for monitoring multivariate variability process

Muhammad Mashuri1 Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, IndonesiaCorrespondence[email protected]

https://orcid.org/0000-0001-9348-4507 View further author information

Haryono Haryono1 Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, IndonesiaView further author information

Diaz Fitra Aksioma1 Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, IndonesiaView further author information

Wibawati Wibawati1 Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia

https://orcid.org/0000-0002-9477-9718 View further author information

Muhammad Ahsan1 Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia

https://orcid.org/0000-0003-3444-2766 View further author information

Hidayatul Khusna1 Department of Statistics, Institut Teknologi Sepuluh Nopember, Surabaya, 60111, Indonesia

https://orcid.org/0000-0001-9889-2884 View further author information

Yuegang Yuegang Tan2 School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan, ChinaView further author information

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Figures & data

Figure 1. Tr(R2) Statistics for p = 3 with: a) $r_{i k} = 0.1$ , b) $r_{i k} = 0.3$ , c) $r_{i k} = 0.5$ , and d) $r_{i k} = 0.9$

Table 1. List of kernel function (Härdle & Linton, Citation1994)

Display Table

Table 2. Control limit of the proposed chart with $μ = 0$ and $Σ = I$ for p = 3 to 10 for in-control ARL 370

Display Table

Table 3. Control limit of the proposed chart with $μ = 0$ and $Σ = I$ for p = 11 to 30 for in-control ARL 370

Display Table

Table 4. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 10 for various number of quality characteristics p

Display Table

Table 5. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 20 for various number of quality characteristics p

Display Table

Table 6. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 30 for various number of quality characteristics p

Display Table

Table 7. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 50 for various number of quality characteristics p

Display Table

Table 8. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 100 for various number of quality characteristics p

Display Table

Figure 2. ARLs of Proposed chart with p = 3 for various n.

Table 9. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.005$ and n = 10 for various number of quality characteristics p

Display Table

Table 10. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.01$ and n = 10 for various number of quality characteristics p

Display Table

Table 11. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.05$ and n = 10 for various number of quality characteristics p

Display Table

Figure 3. Monitoring ZA fertilizer production dataset using Tr(R²) control chart

Figure 3. Monitoring ZA fertilizer production dataset using Tr(R2) control chart

Figure 4. Monitoring ZA fertilizer production dataset using conventional GV control chart

Table

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Tr(R²) control charts based on kernel density estimation for monitoring multivariate variability process

Table 1. List of kernel function (Härdle & Linton, Citation1994)

Table 2. Control limit of the proposed chart with $μ = 0$ and $Σ = I$ for p = 3 to 10 for in-control ARL 370

Table 3. Control limit of the proposed chart with $μ = 0$ and $Σ = I$ for p = 11 to 30 for in-control ARL 370

Table 4. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 10 for various number of quality characteristics p

Table 5. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 20 for various number of quality characteristics p

Table 6. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 30 for various number of quality characteristics p

Table 7. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 50 for various number of quality characteristics p

Table 8. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 100 for various number of quality characteristics p

Table 9. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.005$ and n = 10 for various number of quality characteristics p

Table 10. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.01$ and n = 10 for various number of quality characteristics p

Table 11. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.05$ and n = 10 for various number of quality characteristics p

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Tr(R2) control charts based on kernel density estimation for monitoring multivariate variability process

Figures & data

Table 1. List of kernel function (Härdle & Linton, Citation1994)

Table 2. Control limit of the proposed chart with μ=0 and Σ=I for p = 3 to 10 for in-control ARL 370

Table 3. Control limit of the proposed chart with μ=0 and Σ=I for p = 11 to 30 for in-control ARL 370

Table 4. ARLs of the proposed chart with μ=0, Σ=I and n = 10 for various number of quality characteristics p

Table 5. ARLs of the proposed chart with μ=0, Σ=I and n = 20 for various number of quality characteristics p

Table 6. ARLs of the proposed chart with μ=0, Σ=I and n = 30 for various number of quality characteristics p

Table 7. ARLs of the proposed chart with μ=0, Σ=I and n = 50 for various number of quality characteristics p

Table 8. ARLs of the proposed chart with μ=0, Σ=I and n = 100 for various number of quality characteristics p

Table 9. ARLs of the proposed chart with μ=0, Σ=I, α=0.005 and n = 10 for various number of quality characteristics p

Table 10. ARLs of the proposed chart with μ=0, Σ=I, α=0.01 and n = 10 for various number of quality characteristics p

Table 11. ARLs of the proposed chart with μ=0, Σ=I, α=0.05 and n = 10 for various number of quality characteristics p

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Tr(R²) control charts based on kernel density estimation for monitoring multivariate variability process

Table 2. Control limit of the proposed chart with $μ = 0$ and $Σ = I$ for p = 3 to 10 for in-control ARL 370

Table 3. Control limit of the proposed chart with $μ = 0$ and $Σ = I$ for p = 11 to 30 for in-control ARL 370

Table 4. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 10 for various number of quality characteristics p

Table 5. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 20 for various number of quality characteristics p

Table 6. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 30 for various number of quality characteristics p

Table 7. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 50 for various number of quality characteristics p

Table 8. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ and n = 100 for various number of quality characteristics p

Table 9. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.005$ and n = 10 for various number of quality characteristics p

Table 10. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.01$ and n = 10 for various number of quality characteristics p

Table 11. ARLs of the proposed chart with $μ = 0$ , $Σ = I$ , $α = 0.05$ and n = 10 for various number of quality characteristics p