**3.2. Background**

In a multivariate quality control environment, suppose there are *m* correlated quality charac‐ teristics whose means are being monitored simultaneously. Further, assume there is only one observation on the quality characteristics at each iteration of the data gathering process, where the goal is to detect the variable with the maximum mean shift. Let *xki* be the observation of the *i* th quality characteristic, *i* =1, 2, ..., *m*, at iteration *k*, *k* =1, 2, ..., and define the observation vector *xk* = *xk* 1, *xk* 2...., *xkm <sup>T</sup>* and observation matrix *Ok* =(*x***1**, *x***2**, ..., *xk* ). After taking a new observation, *xk* , define *Bi* (*xk* , *Ok* **-1**), the probability of variable *i* to be in an out-of-control state, as

$$B\_i(\mathbf{x}\_{k'}, \mathbf{O}\_{k\cdot 1}) = \Pr\{OOC\_i \Big| \mathbf{x}\_{k'}, \mathbf{O}\_{k\cdot 1}\big| \},\tag{32}$$

where *OOC* stands for out-of-control. This probability has been called the belief of variable *i* to be in out-of-control condition given the observation matrix up to iteration *k* −1 and the observation vector obtained at iteration *k*.

Assuming the observations are taken independently at each iteration, to improve the belief of the process being in an out-of-control state at the *kth* iteration, based on the observation matrix *Ok* **-1** and the new observation vector *xk* , we have

$$\Pr\{\mathbf{x}\_{k}\big|\mathrm{COC}\_{i},\mathbf{O}\_{k\cdot 1}\} = \Pr\{\mathbf{x}\_{k}\big|\mathrm{OOC}\_{i}\}\tag{33}$$

Then, using the Bayesian rule the posterior belief is:

Therefore the approximate optimal value of *di*, *<sup>j</sup>*

136 Dynamic Programming and Bayesian Inference, Concepts and Applications

**statistical quality control environments**

**3.1. Introduction**

**3.2. Background**

observation, *xk* , define *Bi*

observation vector obtained at iteration *k*.

*Ok* **-1** and the new observation vector *xk* , we have

phase I.

the *i*

as

( ) { ( ) ( )} \* 12

**3. An application for fault detection and diagnosis in multivariate**

In this section, a heuristic threshold policy is applied in phase II of a control charting procedure to not only detect the states of a multivariate quality control system, but also to diagnose the quality characteristic(s) responsible for an out-of-control signal. It is assumed that the incontrol mean vector and in-control covariance matrix of the process have been obtained in

In a multivariate quality control environment, suppose there are *m* correlated quality charac‐ teristics whose means are being monitored simultaneously. Further, assume there is only one observation on the quality characteristics at each iteration of the data gathering process, where the goal is to detect the variable with the maximum mean shift. Let *xki* be the observation of

th quality characteristic, *i* =1, 2, ..., *m*, at iteration *k*, *k* =1, 2, ..., and define the observation vector *xk* = *xk* 1, *xk* <sup>2</sup>...., *xkm <sup>T</sup>* and observation matrix *Ok* =(*x***1**, *x***2**, ..., *xk* ). After taking a new

( , ) Pr{ , },

where *OOC* stands for out-of-control. This probability has been called the belief of variable *i* to be in out-of-control condition given the observation matrix up to iteration *k* −1 and the

Assuming the observations are taken independently at each iteration, to improve the belief of the process being in an out-of-control state at the *kth* iteration, based on the observation matrix

(*xk* , *Ok* **-1**), the probability of variable *i* to be in an out-of-control state,

*i i B k k-1* = *OOC k k-1 x O x O* (32)

Pr{ , } Pr{ } *xO x <sup>k</sup> OOCi i k-1 k* = *OOC* (33)

\* (*s*) can be determined from following equation,

, ,, , *i j ij ij d s Max d s d s* = (31)

$$\begin{aligned} \operatorname{B}\_{i}(\mathbf{x}\_{k}, \mathbf{O}\_{k\cdot 1}) &= \Pr[\text{OOC}\_{i} \big| \mathbf{x}\_{k}, \mathbf{O}\_{k\cdot 1}] = \frac{\Pr[\text{OOC}\_{i}, \mathbf{x}\_{k}, \mathbf{O}\_{k\cdot 1}]}{\Pr[\mathbf{x}\_{k}, \mathbf{O}\_{k\cdot 1}]} = \\ \frac{\Pr[\text{OOC}\_{i}, \mathbf{x}\_{k}, \mathbf{O}\_{k\cdot 1}]}{\sum\_{j=1}^{m} \Pr[\text{OOC}\_{j}, \mathbf{x}\_{k}, \mathbf{O}\_{k\cdot 1}]} &= \frac{\Pr[\text{OOC}\_{i} \big| \mathbf{O}\_{k\cdot 1}] \Pr[\mathbf{x}\_{k}] \text{OOC}\_{i}, \mathbf{O}\_{k\cdot 1}]}{\sum\_{j=1}^{m} \Pr[\text{OOC}\_{j} \big| \mathbf{O}\_{k\cdot 1}] \Pr[\mathbf{x}\_{k}] \text{OOC}\_{j}, \mathbf{O}\_{k\cdot 1}]} \end{aligned} \tag{34}$$

Since the goal is to detect the variable with the maximum mean shift, only one quality characteristic can be considered out-of-control at each iteration. In this way, there are *m*−1 remaining candidates for which *m*−1 quality characteristics are in-control. Hence, one can say that the candidates are mutually exclusive and collectively exhaustive. Therefore, using the Bayes' theorem, one can write equation (34) as

$$\begin{aligned} \mathbf{B}\_{i}(\mathbf{x}\_{k}, \mathbf{O}\_{k-1}) &= \frac{\Pr\{\mathbf{O}\mathbf{O}\mathbf{C}\_{i} \Big| \mathbf{O}\_{k-1}\} \Pr\{\mathbf{x}\_{k}\Big| \mathbf{O}\mathbf{O}\mathbf{C}\_{i}\}}{\sum\_{j=1}^{m} \Pr\{\mathbf{O}\mathbf{O}\mathbf{C}\_{j} \Big| \mathbf{O}\_{k-1}\} \Pr\{\mathbf{x}\_{k}\Big| \mathbf{O}\mathbf{O}\mathbf{C}\_{j}\}} = \frac{\mathbf{B}\_{i}(\mathbf{x}\_{k-1}, \mathbf{O}\_{k-2}) \Pr\{\mathbf{x}\_{k}\Big| \mathbf{O}\mathbf{O}\mathbf{C}\_{i}\}}{\sum\_{j=1}^{m} \mathbf{B}\_{j}(\mathbf{x}\_{k-1}, \mathbf{O}\_{k-2}) \Pr\{\mathbf{x}\_{k}\Big| \mathbf{O}\mathbf{O}\mathbf{O}\mathbf{C}\_{j}\}} \tag{35}$$

When the system is in-control, we assume the *m* characteristics follow a multinormal distri‐ bution with mean vector *μ* = *μ*1, *μ*2, ..., *μm <sup>T</sup>* and covariance matrix

$$
\boldsymbol{\Sigma} = \begin{bmatrix}
\sigma\_1^2 & \sigma\_{12} & \dots & \sigma\_{1m} \\
\sigma\_{21} & \sigma\_2^2 & \dots & \sigma\_{2m} \\
\cdot & \cdot & \cdot & \cdot \\
\sigma\_{m1} & \sigma\_{m2} & \cdot & \sigma\_m^2 \\
\end{bmatrix} \tag{36}
$$

In out-of-control situations, only the mean vector changes and the probability distribution along with the covariance matrix remain unchanged. In latter case, equation (35) is used to calculate the probability of shifts in the process mean *μ* at different iterations. Moreover, in order to update the beliefs at iteration *k* one needs to evaluate Pr{*xk* |*OOCi* }.

The term Pr{*xk* |*OOCi* } is the probability of observing *xk* if only the *i* th quality characteristic is out-of-control. The exact value of this probability can be determined using the multivariate normal density, *A* exp( −<sup>1</sup> 2 (*xk* <sup>−</sup>*μ***1***i*)*<sup>T</sup> <sup>Σ</sup>* **-1** (*xk* −*μ***1***i*)), where *μ***1***i* denotes the mean vector in which only the *i* th characteristic has shifted to an out-of-control condition and *A* is a known constant. Since the exact value of the out-of-control mean vector *μ***1***i* is not known a priori, two approx‐ imations are used in this research to determine Pr{*xk* |*OOCi* }. Note that we do not want to determine the exact probability. Instead, the aim is to have an approximate probability (a belief) on each characteristic being out-of-control. In the first approximation method, define *ICi* to be the event that all characteristics are in-control, and let Pr{*xk* | *ICi* } be the conditional probability of observing *xk* given all characteristics are in-control. Further, let *xk* **'** = *μ*01, ..., *xki* , *μ*0*i*+1, ..., *μ*0*<sup>m</sup> <sup>T</sup>* in the aforementioned multivariate normal density, so that Pr{*xk* | *ICi* } can be approximately evaluated using Pr{*xk* | *ICi* } =Pr{*x*' *<sup>k</sup>* | *ICi* }, where Pr{*x*' *<sup>k</sup>* | *ICi* } <sup>=</sup> *<sup>A</sup>* exp( <sup>−</sup><sup>1</sup> 2 (*xk* **'** <sup>−</sup>*μ***0**)*<sup>T</sup> <sup>Σ</sup>* **-1** (*xk* **'** −*μ***0**)). Note that this evaluation is proportional to exp( −<sup>1</sup> 2 ( *xki* <sup>−</sup> *<sup>μ</sup>*0*<sup>i</sup> σi* ) 2 ), and since it is assumed that characteristic *i* is under control, no matter the condition of the other characteristics, this approximation is justifiable.

**3.3. The proposed procedure**

of being out-of-control.

**Step I**

**Step II**

**Step III** Set *k* =0 **Step IV**

**Step V**

**Step VI**

**Step VII**

work of the proposed decision-making process follows.

Using the maximum entropy principle, initialize *Bi*

maximum number of decision making stages*N* .

Obtain the order statistics on the posterior beliefs *Bi*

Furthermore, let *Bgr*(*Ok* ) = *B*(*m*)(*Ok* ) and *Bsm*(*Ok* ) = *B*(*m*−1)(*Ok* ).

correct choice between the variables *i* and *j*, where *di*, *<sup>j</sup>*

control, where *N* decision-making steps are available. Define *V* (*N* , *di*, *<sup>j</sup>*

Obtain an observation of the process.

Estimate the posterior beliefs, *Bi*

*B*(1)(*Ok* ) < *B*(2)(*Ok* ) <...< *B*(*m*)(*Ok* ).

variables *i* and *j*. Then, we have:

where " ≜ " means "defined as."

Assuming a limited number of the data gathering stages, *N* , to detect and diagnose charac‐ teristic(s), a heuristic threshold policy-based model is developed in this Section. The frame‐

Using Dynamic Programming Based on Bayesian Inference in Selection Problems

Define *i* =1, 2, ..., *m* as the set of indices for the characteristics, all of which having the potential

variable to be out-of-control. In other words, at the start of the decision-making process all variables have an equal chance of being out-of-control. Set the discount rate*α*, the maximum probability of correct selection when *N* decision making stages remains*V* (*N* ), and the

(*Ok* ) (for*i* =1, 2, ..., *m*), using equation (35).

Assume the variables with the indices *i* = *gr* and *j* =*sm* are the candidates of being out-of-

*CS* the event of correct selection and event *Ei*, *<sup>j</sup>* the existence of two out-of-control candidate

(*Ok* ) such that

( , ) , , , ( ) Pr{ } Pr { } *V N d k CS E CS i j ij ij* = @ (38)

(*O*0) =1 / *m* as the prior belief of the *i*

http://dx.doi.org/10.5772/57423

(*k*)) the probability of

(*k*) is the acceptable belief. Also, define

*th*

139

In the second approximation method, we assume Pr{*xk* |*OOCi* }<sup>∝</sup> <sup>1</sup> Pr{*xk* | *I Ci* } . Although it is obvious that Pr{*xk* |*OOCi* } is not equal to <sup>1</sup> Pr{*xk* | *I Ci* } , since we only need a belief function to evaluate Pr{*xk* |*OOCi* } and also we do not know the exact value of out-of-control mean vector, this approximation is just used to determinePr{*xk* |*OOCi* }. Moreover, it can be easily seen that the closer the value of the *i* th characteristic is to its in-control mean the smaller is Pr{*xk* |*OOCi* } as expected. We thus let

$$\Pr\{\mathbf{x}\_{k}\big|\text{OOC}\_{i}\} \approx \frac{1}{\Pr\{\mathbf{x}\_{k}\big|\text{IC}\_{i}\big|}}=\text{Rexp}\left(\bigvee\_{i}\big{\frac{\chi\_{ii}-\mu\_{0i}}{\sigma\_{i}}\big{}}^{2}\right);\quad i=1,2,...,m,\tag{37}$$

where *R* is a sufficiently big constant number to ensure the above definition is less than one. The approximation to Pr{*xk* |*OOCi* } in equation (37) has the following two properties:


Niaki and Fallahnezhad [8] defined another equation for the above conditional probability and showed that if a shift occurs in the mean of variable *i*, then*Limk*→*<sup>∞</sup> Bi* (*x<sup>k</sup>* , *O<sup>k</sup>* <sup>−</sup>1)= *Bi* =1. They

proposed a novel method of detection and classification and used simulation to compare its performances with that of existing methods in terms of the average run length for different mean shifts. The results of the simulation study were in favor of their proposed method in almost all shift scenarios. Besides using a different equation, the main difference between the current research and Niaki and Fallahnezhad [8] is that the current work develops a novel heuristic threshold policy, in which to save sampling cost and time or when these factors are constrained, the number of the data gathering stages is limited.
