Residual Control Chart for Binary Response with Multicollinearity Covariates by Neural Network Model
Abstract
:1. Introduction
2. Statistical Methods
2.1. Generalized Linear Model and Neural Network Model for Binary Response Data
2.2. Dimension Reduction by Principal Component Analysis
2.3. Dimension Reduction by Functional Principal Component Analysis
2.4. New Binary response statistical process control Procedure
- Apply the (functional) principal component analysis in input variables and obtain the principal components from (8).
- Fit the binary response regression model by using the binary response variable y and the (functional) principal components through probit link function, logit link function, and neural network regression models, respectively.
- Obtain the deviance residuals from each model.
3. Illustrated Examples
3.1. Simulation Study
3.2. Real Data Analysis
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Probit | Logit | Neural Network | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Case | ||||||||||
In-control | ARL | 2.590 | 520.152 | NA | 2.586 | 536.938 | NA | 2.453 | 322.806 | NA |
Center | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.000 | 0.000 | 0.000 | |
LCL | −1.063 | −2.202 | −3.340 | −1.063 | −2.201 | −3.340 | −0.455 | −0.911 | −1.366 | |
UCL | 1.215 | 2.354 | 3.492 | 1.215 | 2.354 | 3.492 | 0.455 | 0.911 | 1.366 | |
CI Length | 2.278 | 4.555 | 6.833 | 2.278 | 4.555 | 6.833 | 0.911 | 1.821 | 2.732 | |
Coverage | 0.610 | 1.000 | 1.000 | 0.610 | 1.000 | 1.000 | 0.591 | 0.998 | 1.000 | |
One Inflated | ARL | 3.532 | 291.492 | NA | 3.528 | 302.546 | NA | 3.094 | 62.599 | 429.500 |
Center | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.000 | 0.000 | 0.000 | |
LCL | −0.934 | −2.002 | −3.069 | −0.934 | −2.002 | −3.069 | −0.421 | −0.842 | −1.263 | |
UCL | 1.200 | 2.268 | 3.335 | 1.200 | 2.268 | 3.335 | 0.421 | 0.842 | 1.263 | |
CI Length | 2.135 | 4.269 | 6.404 | 2.135 | 4.269 | 6.404 | 0.842 | 1.684 | 2.527 | |
Coverage | 0.717 | 0.997 | 1.000 | 0.717 | 0.998 | 1.000 | 0.677 | 0.981 | 1.000 | |
Zero Inflated | ARL | 2.177 | NA | NA | 2.178 | NA | NA | 2.197 | 429.347 | NA |
Center | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.000 | 0.000 | 0.000 | |
LCL | −1.154 | −2.320 | −3.486 | −1.154 | −2.320 | −3.486 | −0.469 | −0.938 | −1.406 | |
UCL | 1.179 | 2.345 | 3.512 | 1.179 | 2.345 | 3.512 | 0.469 | 0.938 | 1.406 | |
CI Length | 2.333 | 4.665 | 6.998 | 2.333 | 4.665 | 6.998 | 0.938 | 1.875 | 2.813 | |
Coverage | 0.536 | 1.000 | 1.000 | 0.536 | 1.000 | 1.000 | 0.555 | 0.999 | 1.000 |
Probit | Logit | Neural Network | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Case | ||||||||||
In-control | ARL | 2.9 | NA | NA | 2.9 | NA | NA | 3.0 | NA | NA |
Center | 0.076 | 0.076 | 0.076 | 0.076 | 0.076 | 0.076 | 0.000 | 0.000 | 0.000 | |
LCL | −1.074 | −2.225 | −3.375 | −1.074 | −2.225 | −3.375 | −0.482 | −0.964 | −1.446 | |
UCL | 1.226 | 2.376 | 3.527 | 1.226 | 2.376 | 3.527 | 0.482 | 0.964 | 1.446 | |
CI Length | 2.301 | 4.601 | 6.902 | 2.301 | 4.601 | 6.902 | 0.964 | 1.927 | 2.891 | |
Coverage | 0.617 | 1.000 | 1.000 | 0.617 | 1.000 | 1.000 | 0.604 | 1.000 | 1.000 | |
One Inflated | ARL | 4.467 | NA | NA | 4.467 | NA | NA | 4.533 | 423.833 | NA |
Center | 0.139 | 0.139 | 0.139 | 0.139 | 0.139 | 0.139 | 0.001 | 0.001 | 0.001 | |
LCL | −0.935 | −2.009 | −3.084 | −0.935 | −2.009 | −3.084 | −0.442 | −0.884 | −1.326 | |
UCL | 1.214 | 2.288 | 3.362 | 1.214 | 2.288 | 3.362 | 0.443 | 0.885 | 1.328 | |
CI Length | 2.149 | 4.298 | 6.446 | 2.149 | 4.298 | 6.446 | 0.885 | 1.769 | 2.654 | |
Coverage | 0.725 | 1.000 | 1.000 | 0.725 | 1.000 | 1.000 | 0.723 | 0.999 | 1.000 | |
Zero Inflated | ARL | 2.100 | NA | NA | 2.100 | NA | NA | 2.400 | NA | NA |
Center | 0.016 | 0.016 | 0.016 | 0.016 | 0.016 | 0.016 | 0.000 | 0.000 | 0.000 | |
LCL | −1.160 | −2.335 | −3.510 | −1.160 | −2.335 | −3.510 | −0.495 | −0.990 | −1.484 | |
UCL | 1.191 | 2.366 | 3.542 | 1.191 | 2.366 | 3.542 | 0.495 | 0.989 | 1.484 | |
CI Length | 2.351 | 4.701 | 7.052 | 2.351 | 4.701 | 7.052 | 0.990 | 1.979 | 2.969 | |
Coverage | 0.527 | 1.000 | 1.000 | 0.527 | 1.000 | 1.000 | 0.520 | 1.000 | 1.000 |
Cl.thickness | Cell.size | Cell.shape | Marg.adhesion | Epith.c.size | Bare.nuclei | Bl.cromatin | Normal.nucleoli | Mitoses | |
---|---|---|---|---|---|---|---|---|---|
Cl.thickness | 1.000 | 0.642 | 0.653 | 0.488 | 0.524 | 0.593 | 0.554 | 0.534 | 0.355 |
Cell.size | 0.642 | 1.000 | 0.907 | 0.707 | 0.754 | 0.692 | 0.756 | 0.719 | 0.465 |
Cell.shape | 0.653 | 0.907 | 1.000 | 0.686 | 0.722 | 0.714 | 0.735 | 0.718 | 0.447 |
Marg.adhesion | 0.488 | 0.707 | 0.686 | 1.000 | 0.595 | 0.671 | 0.669 | 0.603 | 0.425 |
Epith.c.size | 0.524 | 0.754 | 0.722 | 0.595 | 1.000 | 0.586 | 0.618 | 0.629 | 0.481 |
Bare.nuclei | 0.593 | 0.692 | 0.714 | 0.671 | 0.586 | 1.000 | 0.681 | 0.584 | 0.349 |
Bl.cromatin | 0.554 | 0.756 | 0.735 | 0.669 | 0.618 | 0.681 | 1.000 | 0.666 | 0.354 |
Normal.nucleoli | 0.534 | 0.719 | 0.718 | 0.603 | 0.629 | 0.584 | 0.666 | 1.000 | 0.437 |
Mitoses | 0.355 | 0.465 | 0.447 | 0.425 | 0.481 | 0.349 | 0.354 | 0.437 | 1.000 |
Comp.1 | Comp.2 | Comp.3 | Comp.4 | Comp.5 | Comp.6 | Comp.7 | Comp.8 | Comp.9 | |
---|---|---|---|---|---|---|---|---|---|
Standard deviation | 2.430 | 0.875 | 0.734 | 0.680 | 0.617 | 0.550 | 0.543 | 0.511 | 0.297 |
Proportion of Variance | 0.656 | 0.085 | 0.060 | 0.051 | 0.042 | 0.034 | 0.033 | 0.029 | 0.010 |
Cumulative Proportion | 0.656 | 0.741 | 0.801 | 0.853 | 0.895 | 0.928 | 0.961 | 0.990 | 1.000 |
Probit | Logit | Neural Network | |||||||
---|---|---|---|---|---|---|---|---|---|
Center | 0.022 | 0.022 | 0.022 | 0.038 | 0.038 | 0.038 | −0.002 | −0.002 | −0.002 |
LCL | −0.381 | −0.785 | −1.188 | −0.368 | −0.773 | −1.178 | −0.123 | −0.244 | −0.365 |
UCL | 0.426 | 0.829 | 1.233 | 0.443 | 0.849 | 1.254 | 0.119 | 0.240 | 0.361 |
CL Length | 0.807 | 1.614 | 2.421 | 0.811 | 1.622 | 2.433 | 0.242 | 0.485 | 0.727 |
Probit | Logit | Neural Network | |||||||
---|---|---|---|---|---|---|---|---|---|
Center | 0.095 | 0.095 | 0.095 | 0.095 | 0.095 | 0.095 | −0.003 | −0.003 | −0.003 |
LCL | −1.034 | −2.163 | −3.293 | −1.034 | −2.163 | −3.293 | −0.470 | −0.937 | −1.403 |
UCL | 1.225 | 2.354 | 3.484 | 1.225 | 2.354 | 3.483 | 0.464 | 0.930 | 1.397 |
CL Length | 2.259 | 4.518 | 6.776 | 2.259 | 4.518 | 6.776 | 0.933 | 1.867 | 2.800 |
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Kim, J.-M.; Wang, N.; Liu, Y.; Park, K. Residual Control Chart for Binary Response with Multicollinearity Covariates by Neural Network Model. Symmetry 2020, 12, 381. https://doi.org/10.3390/sym12030381
Kim J-M, Wang N, Liu Y, Park K. Residual Control Chart for Binary Response with Multicollinearity Covariates by Neural Network Model. Symmetry. 2020; 12(3):381. https://doi.org/10.3390/sym12030381
Chicago/Turabian StyleKim, Jong-Min, Ning Wang, Yumin Liu, and Kayoung Park. 2020. "Residual Control Chart for Binary Response with Multicollinearity Covariates by Neural Network Model" Symmetry 12, no. 3: 381. https://doi.org/10.3390/sym12030381
APA StyleKim, J.-M., Wang, N., Liu, Y., & Park, K. (2020). Residual Control Chart for Binary Response with Multicollinearity Covariates by Neural Network Model. Symmetry, 12(3), 381. https://doi.org/10.3390/sym12030381