Performance of the Zone Control Chart for Detecting Prespecified Quadratic Changes in Linear Profiles

Abstract

In profile monitoring, one possible and challenging purpose is detecting changes away from the ‘normal’ profile toward one of several prespecified ‘bad’ profiles. In this paper, to detect the prespecified quadratic changes in linear profiles, the zone control chart is suggested to be constructed based on the Student’s t-statistic. The performance of the zone control charts with three different combinations of scores are investigated and compared with alternative control charts. Simulation results show that the zone charts are effective and stable, and perform much better on detecting small to moderate shifts of quadratic changes in linear profile.

This research was supported in part by grants 71401123 from the National Natural Science Foundation of China (NSFC).

Appendix

In this section, the Markov chain method is given for the zone control chart for finding the tuning parameter L to achieve a desired ARL₀. For illustration, the zone chart with scores (0, 2, 3, 6) is considered here as is in [20]. In profile monitoring, the charting procedure starts with an initial score of $U_{0} = + 0$ and $L_{0} = - 0$, i.e., (U ₀, L ₀) = (+0, −0). Then at time j, all possible IC ordered pair cumulative scores are (U _j , L _j ) = (+0, −0), (+0, −5), (+0, −4), (+0, −3), (+0, −2), (+2, −0), (+3, −0), (+4, −0) and (+5, −0). Let these ordered pairs correspond with the transient states 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively, and let state 10 be the absorbing state. Then the transition probability matrix R from the current state to the next state can be easily obtained for the zone control chart with scores (0, 2, 3, 6) and is presented in Table 6, where the probabilities in the transition probability matrix R can be calculated as follows based on the control limits and the cdf of the Student’s t distribution with n − 3 degrees of freedom, $F_{t} (\,)$. Since the regions above the central line CL are symmetric to those regions below CL, we have

$$ p1 = p4 = F_{t} ({\text{UCL}}_{1} ) - F_{t} ({\text{CL}}), $$

$$ p2 = p5 = F_{t} ({\text{UCL}}_{2} ) - F_{t} ({\text{UCL}}_{1} ), $$

$$ p3 = p6 = F_{t} ({\text{UCL}}_{3} ) - F_{t} ({\text{UCL}}_{2} ). $$

Table 6 Transition probability matrix R

It should be noted that the transition probability matrix R for the zone chart depends on the choice of the scores S _k , k = 1, …, 4. However, it still can be obtained following this procedure.

When the control limits are determined or the tuning parameter L is set, the IC ARL can be obtained using the following equation as shown in [20]:

$$ {\text{ARL}}_{0} = {{\mathbf{s}}^{\prime}}(I - R)^{ - 1} {\mathbf{1}}, $$

(11)

where s′ = (1, 0, …, 0) is the initial probability vector having a unity in the first element and zeros elsewhere, I is the identity matrix and 1 is a vector having all elements unity.