Statistical Process Control on the Circle: A Review and Some New Results

Abstract

Statistical process control (SPC) is an important and mature area of statistics which finds application in many diverse disciplines. Until quite recently, virtually all developments and applications were restricted to data in Euclidean spaces. In this paper, we review some SPC methodology that has been developed to deal with data lying on the unit circle. With regard to parametric methods, a review of sequential probability ratio tests and Page-type CUSUMs is provided. A Shiryaev–Roberts-type sequential procedure is also proposed. Nonparametric rank-based CUSUMs for detecting location and/or concentration changes are also reviewed.

Appendix: Continuous Time Analog of the S-R Procedure

Denote by $\mu _{0}$ the targeted shift and by $\mu$ the actual shift occurring at the changepoint $\tau$. The process underlying likelihood ratio is

$$\begin{aligned} C_{n}=\left\{ \begin{array}{lc} \sum \nolimits _{j=1}^{n}\left[ \cos (Z_{j}-\mu _{0})-\cos (Z_{j})\right] &{} n<\tau \\ &{} \\ C_{\tau -1}+\sum \nolimits _{j=\tau }^{n}\left[ \cos (Z_{j}-\mu _{0}-\mu )-\cos (Z_{j}-\mu )\right] &{} n\ge \tau \end{array} \right. \end{aligned}$$

(5)

We will need the following facts in the development which follows, namely

$$\begin{aligned} E[\cos (Z)]=A(\kappa ),\ E[\sin (Z)]=0,\ var[\sin (Z)]=\ \frac{A(\kappa )}{\kappa }. \end{aligned}$$

Let m denote the number of observations accruing per unit time. A non-degenerate continuous time version of the underlying process can be constructed by letting $m\rightarrow \infty$ and simultaneously letting $\mu _{0}$ and $\mu$ $\rightarrow 0$ at an appropriate rate. Toward this, set

$$\begin{aligned} n=[mt],\ \tau =[m\gamma ]\ \text {for }t,\gamma >0 \end{aligned}$$

and

$$\begin{aligned} \mu _{0}=\frac{\theta _{0}}{\sqrt{m}},\ \mu =\frac{\theta }{\sqrt{m}}. \end{aligned}$$

For $n<\tau$, we then find upon expanding the deterministic trigonometric coefficients that

$$\begin{aligned} C_{[mt]}&=\left( \cos (\frac{\theta _{0}}{\sqrt{m}})-1\right) \sum \nolimits _{j=1}^{[mt]}\cos (Z_{j})+\sin (\frac{\theta _{0}}{\sqrt{m}})\sum \nolimits _{j=1}^{[mt]}\sin (Z_{j}) \\&\\&=\left( -\frac{\theta _{0}^{2}}{2}+O(\frac{1}{m})\right) \frac{1}{m}\sum \nolimits _{j=1}^{[mt]}\cos (Z_{j})+\theta _{0}\left( 1+O(\frac{1}{\sqrt{m}})\right) \frac{1}{\sqrt{m}}\sum \nolimits _{j=1}^{[mt]}\sin (Z_{j}). \end{aligned}$$

Letting $m\rightarrow \infty$, it follows from the strong law of large numbers that for every finite $T<\gamma$,

$$\begin{aligned} \frac{1}{m}\sum \nolimits _{j=1}^{[mt]}\cos (Z_{j})\rightarrow _{a.s}A(\kappa )t \end{aligned}$$

uniformly on [0, T] and from Donsker’s theorem (Billingsley [1], Theorem 8.2) that

$$\begin{aligned} \frac{1}{\sqrt{m}}\sum \nolimits _{j=1}^{[mt]}\sin (Z_{j})\rightarrow _{\mathcal {L}}B_{t},\ 0\le t<T \end{aligned}$$

with a standard Brownian motion B. Thus, for $t<\gamma$ the continuous time version of $C_{n}$ is

$$\begin{aligned} \tilde{C}_{t}=-\frac{\theta _{0}^{2}}{2}A(\kappa )t+\theta _{0}\sqrt{\frac{A(\kappa )}{\kappa }}B_{t} \end{aligned}$$

(6)

Next, for $T\ge t\ge \gamma ,$

$$\begin{aligned} C_{[mt]}-C_{[m\gamma ]}=\sum \nolimits _{j=\left[ m\tau \right] }^{\left[ nt \right] }\left[ \cos (Z_{j}-\mu _{0}-\mu )-\cos (Z_{j}-\mu )\right] \end{aligned}$$

and a computation similar to the preceding one leads to the continuous time version

$$\begin{aligned} \tilde{C}_{t}=\tilde{C}_{\gamma }+\frac{1}{2}\theta _{0}(2\theta -\theta _{0})A(\kappa )(t-\gamma )+\theta _{0}\sqrt{\frac{A(\kappa )}{\kappa }}\left( B_{t}-B_{\gamma }\right) \end{aligned}$$

(7)

Putting (6) and (7) together, we find after some algebraic manipulation that the continuous time version of $C_{n}$ from (5) is

$$\begin{aligned} \tilde{C}_{t}=-\frac{\theta _{0}^{2}}{2}A(\kappa )t+\theta A(\kappa )(t-min\{t,\gamma \})_{.}+\theta _{0}\sqrt{\frac{A(\kappa )}{\kappa }}B_{t} \end{aligned}$$

for all $t>0$. This is in the same form as the general change point process $\xi _{t}$ formulated by Srivastava and Wu ([34], next to last paragraph page 649). Hence, their results and those of Pollak and Siegmund [24] are also applicable in our situation.