A heuristic threshold policy for fault detection and diagnosis in multivariate statistical quality control environments

Abstract

In this paper, a heuristic threshold policy is developed to detect and classify the states of a multivariate quality control system. In this approach, a probability measure called belief is first assigned to the quality characteristics and then the posterior belief of out-of-control characteristics is updated by taking new observations and using a Bayesian rule. If the posterior belief is more than a decision threshold, called minimum acceptable belief determined using a heuristic threshold policy, then the corresponding quality characteristic is classified out-of-control. Besides using a different approach, the main difference between the current research and previous works is that the current work develops a novel heuristic threshold policy, in which in order to save sampling cost and time or when these factors are constrained, the number of the data gathering stages is assumed limited. A numerical example along with some simulation experiments is given at the end to demonstrate the application of the proposed methodology and to evaluate its performances in different scenarios of mean shifts.

Appendices

Appendix 1: Conditional mean and variance of the variables

Conditional mean of variables gr and sm can be evaluated using the following equation.

$$ \left( {{\mu_{\mathrm{sm}}},{\mu_{\mathrm{gr}}}\left| {{{{\left( {\mu_j } \right)}}_{{j\ne \mathrm{gr}\text{,}\mathrm{sm}}}}} \right.} \right)=\left( {{\mu_{\mathrm{sm}}},{\mu_{\mathrm{gr}}}} \right)+{\bf b}_{{\bf 2}}^{\prime}\left( {{{{\left( {{{{\bf X}}_{{\bf kj}}}} \right)}}_{{j\ne \mathrm{gr},\mathrm{sm}}}}-{{{\left( {{{\varvec{\mu}}_{{\bf j}}}} \right)}}_{{j\ne \mathrm{gr}\text{,}\mathrm{sm}}}}} \right) $$

(A1.1)

Where,

$$ \begin{array}{*{20}{c}} {\mathbf{b}_{2}^{\prime }={{\Sigma }_{{\mathbf{xX}}}}\Sigma _{{\mathbf{XX}}}^{{-1}}\,\text{and}} \hfill \\ {\Sigma =\left[ {\begin{array}{*{20}{c}} {{{\Sigma }_{{\mathbf{XX}}}}} \hfill & {{{\Sigma }_{{\mathbf{xX}}}}} \hfill \\ {{{\Sigma }_{{\mathbf{xX}}}}} \hfill & {{{\Sigma }_{{\mathbf{xx}}}}} \hfill \\ \end{array}} \right]} \hfill \\ \end{array} $$

Σ :

The covariance matrix of the process

Σ _xx :

Submatrix of the covariance matrix Σ for variables j = gr, sm

Σ _xX :

Submatrix of the covariance matrix Σ between variables j = gr, sm and j ≠ gr, sm

Σ _XX :

Submatrix of the covariance matrix Σ for variables j ≠ gr, sm

Further, the conditional covariance matrix of variables j = gr, sm on variables j ≠ gr, sm is obtained as ${{\varvec{\varSigma}}_{{\bf xx}}}-\varvec{\varSigma}_{{\bf xX}}^{{\bf T}}\varvec{\varSigma}_{{\bf XX}}^{{-{\bf 1}}}{{\varvec{\varSigma}}_{{\bf xX}}}$.

Appendix 2: Evaluating the optimal value of d _gr,sm(k)

Assume ${{\left( {\mu_j } \right)}_{{j\in \left\{ {1,2, \ldots, m} \right\}}}}=0$ and ${{\left( {\sigma_j } \right)}_{{j\in \left\{ {1,2, \ldots, m} \right\}}}}=1$. Then,

$$ \begin{array}{*{20}{c}} {\Pr \left\{ {\left\{ {{{e}^{{0.5{{{({{T}_{{k,\text{gr}\left| {sm} \right.}}})}}^{2}}}}}\geqslant h\left( {{{d}_{{\text{gr,sm}}}}(k)} \right){{e}^{{0.5{{{({{T}_{{k,\text{sm}}}})}}^{2}}}}}} \right\}} \right\}=} \hfill \\ {\Pr \left\{ {0.5{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}\geqslant \ln \left( {h\left( {{{d}_{{\text{gr,sm}}}}(k)} \right)} \right)+0.5{{{({{T}_{{k,\text{sm}}}})}}^{2}}} \right\}} \hfill \\ {=\Pr \left\{ {{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}-{{{({{T}_{{k,\text{sm}}}})}}^{2}}\geqslant 2\ln \left( {h\left( {{{d}_{{\text{gr,sm}}}}(k)} \right)} \right)} \right\}} \hfill \\ \end{array} $$

Now since (T _k , _sm ,T _{k,gr| sm}) follow a standard normal distribution ${{\left( {\mu_j } \right)}_{{j\in \left\{ {\mathrm{gr}\text{,}\mathrm{sm}} \right\}}}}=0$ and ${{\left( {\sigma_j } \right)}_{{j\in \left\{ {\mathrm{gr}\text{,}\mathrm{sm}} \right\}}}}=1$, hence (T _{k,gr| sm})² and (T _k ,_sm)² follow a χ ² distribution with one degree of freedom. Then, if we assume that (T _k,sm)² is approximately equal to its mean, we have

$$ {{({T_{{_{{k,\mathrm{sm}}}}}})}^2}\propto E\left( {{T_{{_{{k,\mathrm{sm}}}}}}^2} \right)=E{{\left( {{T_{{_{{k\text{,}\mathrm{sm}}}}}}} \right)}^2}+\mathrm{Var}\left( {{T_{{_{{k,\mathrm{sm}}}}}}} \right)=1 $$

(A2.1)

Thus,

$$ \begin{array}{*{20}{c}} {\Pr \left\{ {{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}-({{T}_{{k,\text{sm}}}})2\geqslant 2\ln \left( {h\left( {{{d}_{{\text{gr},\text{sm}}}}(k)} \right)} \right)} \right\}} \hfill \\ {\propto \Pr \left\{ {{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}-E(T_{{k,\text{sm}}}^{2})\geqslant 2\ln \left( {h\left( {{{d}_{{\text{gr},\text{sm}}}}(k)} \right)} \right)} \right\}} \hfill \\ {=\Pr \left\{ {{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}-1\geqslant 2\ln \left( {h\left( {{{d}_{{\text{gr},\text{sm}}}}(k)} \right)} \right)} \right\}} \hfill \\ {=\Pr \left\{ {{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}\geqslant 2\ln \left( {h\left( {{{d}_{{\text{gr},\text{sm}}}}(k)} \right)} \right)+1} \right\}} \hfill \\ \end{array} $$

(A2.2)

Now, since ${{\left( {{T_{{k,\left. {\mathrm{gr}} \right|\mathrm{sm}}}}} \right)}^2}\propto {\chi^2}(1)$, we have

$$ \Pr \left\{ {{{{({T_{{k\text{,}\mathrm{gr}\left| {\mathrm{sm}} \right.}}})}}^2}\geqslant 2\ln \left( {h\left( {{d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)} \right)} \right)+1} \right\}=\int_{{2\ln \left( {h\left( {{d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)} \right)} \right)+1}}^{\infty } {\frac{{{e^{{-\frac{t}{2}}}}{t^{{-\frac{1}{2}}}}}}{{\varGamma \left( {\frac{1}{2}} \right){2^{{\frac{1}{2}}}}}}} \mathrm{dt} $$

(A2.3)

Hence,

$$ \Pr \left\{ {{{{({T_{{k,\mathrm{gr}\left| {\mathrm{sm}} \right.}}})}}^2}-{{{({T_{{_{{k,\mathrm{sm}}}}}})}}^2}\geqslant 2\ln \left( {h\left( {{d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)} \right)} \right)} \right\}\simeq \int_{{2\ln \left( {h\left( {{d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)} \right)} \right)+1}}^{\infty } {\frac{{{e^{{-\frac{t}{2}}}}{t^{{-\frac{1}{2}}}}}}{{\varGamma \left( {\frac{1}{2}} \right){2^{{\frac{1}{2}}}}}}} \mathrm{dt} $$

(A2.4)

Similarly

$$ \begin{array}{*{20}{c}} {\Pr \left\{ {{{B}_{{\text{gr,sm}}}}\left( {\text{gr};{{O}_{k}}} \right)\geqslant {{d}_{{\text{gr,sm}}}}(k)} \right\}} \hfill \\ {=\Pr \left\{ {{{e}^{{0.5{{{({{T}_{{k,\text{gr}\left| {\text{sm}} \right.}}})}}^{2}}}}}\geqslant r\left( {{{d}_{{\text{gr,sm}}}}(k)} \right){{e}^{{0.5{{{({{T}_{{k,\text{sm}}}})}}^{2}}}}}} \right\}} \hfill \\ {\simeq \int_{{2\ln \left( {r\left( {{{d}_{{\text{gr,sm}}}}(k)} \right)} \right)+1}}^{\infty } {\frac{{{{e}^{{-\frac{t}{2}}}}{{t}^{{-\frac{1}{2}}}}}}{{\Gamma \left( {\frac{1}{2}} \right){{2}^{{\frac{1}{2}}}}}}} \text{dt}} \hfill \\ \end{array} $$

(A2.5)

Replacing the above equations in Eq. () results in

$$ \begin{array}{*{20}{c}} {V_{{i,j}}^{*}(N)\propto \left( {{{B}_{{\text{gr,sm}}}}\left( {\text{gr};{{O}_{k}}} \right)-\alpha V_{{i,j}}^{*}\left( {N-1} \right)} \right)\int_{{\left( {2\ln \left( {r\left( {{{d}_{{\text{gr,sm}}}}(k)} \right)} \right)+1} \right)}}^{\infty } {\frac{{{{e}^{{-\frac{t}{2}}}}{{t}^{{-\frac{1}{2}}}}}}{{\Gamma \left( {\frac{1}{2}} \right){{2}^{{\frac{1}{2}}}}}}} \text{dt}+} \hfill \\ {\left( {{{B}_{{\text{gr,sm}}}}\left( {\text{sm};{{O}_{k}}} \right)-\alpha V_{{i,j}}^{*}\left( {N-1} \right)} \right)\int_{{\left( {2\ln \left( {h\left( {{{d}_{{\text{gr},\text{sm}}}}(k)} \right)} \right)+1} \right)}}^{\infty } {\frac{{{{e}^{{-\frac{t}{2}}}}{{t}^{{-\frac{1}{2}}}}}}{{\Gamma \left( {\frac{1}{2}} \right){{2}^{{\frac{1}{2}}}}}}} \text{dt}+\alpha V_{{i,j}}^{*}\left( {N-1} \right)} \hfill \\ \end{array} $$

(A2.6)

Now by solving the equation $\frac{{\delta {V_{i,j }}(N)}}{{\delta {d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)}}=0$, the following equation is obtained.

$$ \begin{array}{*{20}c} \left( {{B_{\mathrm{gr}\text{,}\mathrm{sm}}}\left( {\mathrm{gr};{O_k}} \right)-\alpha V_{gr,sm}^{*}\left( {N-1} \right)} \right)\frac{1}{{\sqrt{{\left( {\ln \left( {r\left( {{d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)} \right)} \right)+1} \right)}}}}= \hfill \\ -\left( {{B_{\mathrm{gr}\text{,}\mathrm{sm}}}\left( {\mathrm{sm};{O_k}} \right)-\alpha V_{\mathrm{gr}\text{,}\mathrm{sm}}^{*}\left( {N-1} \right)} \right)\frac{1}{{\sqrt{{\left( {\ln \left( {h\left( {{d_{\mathrm{gr}\text{,}\mathrm{sm}}}(k)} \right)} \right)+1} \right)}}}} \hfill \\\end{array}$$

(A2.7)

Finally, the optimal value of d _gr,sm(k) is determined by solving this equation numerically or by a search algorithm.