Predictive ratio CUSUM (PRC): A Bayesian approach in online change point detection of short runs

Figures & data

Table 1. The PRC scheme using an initial conjugate prior in a power prior, for some of the univariate distributions typically used in SPC/M, belonging to the k-PREF.

Likelihood
Prior	OOC State	Interpretation	$\log (L_{n+1})$
$\mathit{P}(\theta ·s_i)$	$k·\theta$	k > 1: $(k-1)100\%$ increase in θ	$({\widehat{c}}_n+x_{n+1})\hspace{0.17em}\log \frac{{\widehat{d}}_n+s_{n+1}}{{\widehat{d}}_n/k+s_{n+1}}-{\widehat{c}}_n·\mathit{logk}$
G(c, d)		k < 1: $(1-k)100\%$ decrease in θ
${\widehat{c}}_n=c+{\sum }_{j=1}^{N_D}{w}_j{d}_j,$ ${\widehat{d}}_n=d+{\sum }_{j=1}^{N_D}{w}_j{s}_j$
$\mathit{Bin}(N_i,\theta )$	$k·E_{\theta |\mathit{X}}\left(\frac{\theta }{1-\theta }\right)$	k > 1: $(k-1)100\%$ increase in expected odds of θ	$\log \frac{B(k·{\widehat{a}}_n+{\widehat{b}}_n,N_{n+1})·B({\widehat{a}}_n,x_{n+1})}{B({\widehat{a}}_n+{\widehat{b}}_n,N_{n+1})·B(k·{\widehat{a}}_n,x_{n+1})}$
Beta(a, b)		k < 1: $(1-k)100\%$ decrease in expected odds of θ
${\widehat{a}}_n=a+{\sum }_{j=1}^{N_D}{w}_j{d}_j,$ ${\widehat{b}}_n=b+{\sum }_{j=1}^{N_D}{w}_j{N}_j-{\sum }_{j=1}^{N_D}{w}_j{d}_j$
$\mathit{N}\mathit{Bin}(r,\theta )$	$k·E_{\theta |\mathit{X}}\left(\frac{\theta }{1-\theta }\right)$	k > 1: $(k-1)100\%$ increase in expected odds of θ	$\log \frac{B(k·{\widehat{a}}_n+{\widehat{b}}_n,r+x_{n+1})·B({\widehat{a}}_n,r)}{B({\widehat{a}}_n+{\widehat{b}}_n,r+x_{n+1})·B(k·{\widehat{a}}_n,r)}$
Beta(a, b)		k < 1: $(1-k)100\%$ decrease in expected odds of θ
${\widehat{a}}_n=a+r{\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{b}}_n=b+{\sum }_{j=1}^{N_D}{w}_j{d}_j$
$\mathit{N}(\theta ,{\sigma }^2)$	$\theta +k·\sigma$	k > 0: increase in θ size of $k·\sigma$	$\left(z_{n+1}-\frac{k}{2}·\frac{\sigma }{\sqrt{{\widehat{\sigma }}_n^2+{\sigma }^2}}\right)·\frac{k·\sigma }{\sqrt{{\widehat{\sigma }}_n^2+{\sigma }^2}}$
N$({\mu }_0,{\sigma }_0^2)$		k < 0: decrease in θ size of $k·\sigma$
${\widehat{\mu }}_n=\frac{{\sigma }^2{\mu }_0+{\sigma }_0{}^2{\sum }_{j=1}^{N_D}{w}_j{d}_j}{{\sigma }^2+{\sigma }_0{}^2{\sum }_{j=1}^{N_D}{w}_j},$ ${\widehat{\sigma }}_n^2=\frac{{\sigma }_0{}^2{\sigma }^2}{{\sigma }^2+{\sigma }_0{}^2{\sum }_{j=1}^{N_D}{w}_j},$ $z_{n+1}=\frac{x_{n+1}-{\widehat{\mu }}_n}{\sqrt{{\widehat{\sigma }}_n^2+{\sigma }^2}}$
$\mathit{N}(\mu ,{\theta }^2)$	$k·{\theta }_2^2$	k > 1: $(k-1)100\%$ increase in ${\theta }^2$	$({\widehat{a}}_n+1/2)·\log \frac{2{\widehat{a}}_n+z_{n+1}^2}{2{\widehat{a}}_n+z_{n+1}^2/k}-\log \sqrt{k}$
IG(a, b)		k < 1: $(1-k)100\%$ decrease in ${\theta }^2$
${\widehat{a}}_n=a+\frac{{\sum }_{j=1}^{N_D}{w}_j}{2},$ ${\widehat{b}}_n=b+\frac{{\sum }_{j=1}^{N_D}{w}_j{(d_j-\mu )}^2}{2},$ $z_{n+1}=\frac{x_{n+1}-\mu }{\sqrt{{\widehat{b}}_n/{\widehat{a}}_n}}$
$\mathit{N}({\theta }_1,{\theta }_2^2)$	${\theta }_1+k·{\widehat{\theta }}_2$	k > 0: increase in θ size of $k·{\widehat{\theta }}_2$	$({\widehat{a}}_n+1/2)·\log \frac{2{\widehat{a}}_n+z_{n+1}^2}{2{\widehat{a}}_n+{(z_{n+1}-k·{\widehat{\lambda }}_n/({\widehat{\lambda }}_n+1))}^2}$
NIG$({\mu }_0,\lambda ,a,b)$		k < 0: decrease in θ size of $k·{\widehat{\theta }}_2$
$\mathit{N}({\theta }_1,{\theta }_2^2)$	$k·{\theta }_2^2$	k > 1: $(k-1)100\%$ increase in ${\theta }_2^2$	$({\widehat{a}}_n+1/2)·\log \frac{2{\widehat{a}}_n+z_{n+1}^2}{2{\widehat{a}}_n+z_{n+1}^2/k}-\log \sqrt{k}$
NIG$({\mu }_0,\lambda ,a,b)$		k < 1: $(1-k)100\%$ decrease in ${\theta }_2^2$
${\widehat{\mu }}_n=\frac{\lambda {\mu }_0+{\sum }_{j=1}^{N_D}{w}_j{d}_j}{\lambda +{\sum }_{j=1}^{N_D}{w}_j},$ ${\widehat{\lambda }}_n=\lambda +{\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{a}}_n=a+\frac{{\sum }_{j=1}^{N_D}{w}_j}{2},$ ${\widehat{b}}_n=b+\frac{\lambda {\mu }_0^2+{\sum }_{j=1}^{N_D}{w}_j{d}_j^2}{2}-\frac{{(\lambda {\mu }_0+{\sum }_{j=1}^{N_D}{w}_j{d}_j)}^2}{2(\lambda +{\sum }_{j=1}^{N_D}{w}_j)},$ ${\widehat{\theta }}_2=\sqrt{\frac{{\widehat{b}}_n}{{\widehat{a}}_n}},$ $z_{n+1}=\frac{x_{n+1}-{\widehat{\mu }}_n}{\sqrt{\frac{({\widehat{\lambda }}_n+1)·{\widehat{b}}_n}{{\widehat{\lambda }}_n·{\widehat{a}}_n}}}$
$\mathit{G}(\alpha ,\theta )$	$k·\theta$	k > 1: $(k-1)100\%$ increase in θ	$({\widehat{c}}_n+\alpha )·\log \frac{{\widehat{d}}_n+x_{n+1}}{{\widehat{d}}_n/k+x_{n+1}}-{\widehat{c}}_n·\mathit{logk}$
G(c, d)		k < 1: $(1-k)100\%$ decrease in θ
${\widehat{c}}_n=c+\alpha {\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{d}}_n=d+{\sum }_{j=1}^{N_D}{w}_j{d}_j$
$\mathit{Weibull}(\theta ,\kappa )$	$k·{\theta }^{\kappa }$	k > 1: $(k-1)100\%$ increase in ${\theta }^{\kappa }$	$({\widehat{a}}_n+1)·\log \frac{{\widehat{b}}_n+x_{n+1}^{\kappa }}{{\widehat{b}}_n+x_{n+1}^{\kappa }/k}-\mathit{logk}$
IG(a, b)		k < 1: $(1-k)100\%$ decrease in ${\theta }^{\kappa }$
${\widehat{a}}_n=a+{\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{b}}_n=b+{\sum }_{j=1}^{N_D}{w}_j{d}_j^{\kappa }$
$\mathit{IG}(\alpha ,\theta )$	$k·\theta$	k > 1: $(k-1)100\%$ increase in θ	$({\widehat{c}}_n+\alpha )·\log \frac{{\widehat{d}}_n·x_{n+1}+1}{{\widehat{d}}_n·x_{n+1}/k+1}-{\widehat{c}}_n·\mathit{logk}$
G(c, d)		k < 1: $(1-k)100\%$ decrease in θ
${\widehat{c}}_n=c+\alpha {\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{d}}_n=d+{\sum }_{j=1}^{N_D}\frac{w_j}{d_j}$
$\mathit{Pareto}(m,\theta )$	$k·\theta$	k > 1: $(k-1)100\%$ increase in θ	$({\widehat{c}}_n+1)·\log \frac{{\widehat{d}}_n+\log (x_{n+1}/m)}{{\widehat{d}}_n/k+\log (x_{n+1}/m)}-{\widehat{c}}_n·\mathit{logk}$
G(c, d)		k < 1: $(1-k)100\%$ decrease in θ
${\widehat{c}}_n=c+{\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{d}}_n=d+{\sum }_{j=1}^{N_D}\frac{w_j}{\log (d_j/m)}$
$\mathit{logN}(\theta ,{\sigma }^2)$	$\theta +k·\sigma$	k > 0: increase in θ size of $k·\sigma$	$\left(z_{n+1}-\frac{k}{2}·\frac{\sigma }{\sqrt{{\widehat{\sigma }}_n^2+{\sigma }^2}}\right)·\frac{k·\sigma }{\sqrt{{\widehat{\sigma }}_n^2+{\sigma }^2}}$
N$({\mu }_0,{\sigma }_0^2)$		k < 0: decrease in θ size of $k·\sigma$
${\widehat{\mu }}_n=\frac{{\sigma }^2{\mu }_0+{\sigma }_0{}^2{\sum }_{j=1}^{N_D}{w}_j\hspace{0.17em}\log \hspace{0.17em}(d_j)}{{\sigma }^2+{\sigma }_0{}^2{\sum }_{j=1}^{N_D}{w}_j},$ ${\widehat{\sigma }}_n^2=\frac{{\sigma }_0{}^2{\sigma }^2}{{\sigma }^2+{\sigma }_0{}^2{\sum }_{j=1}^{N_D}{w}_j},$ $z_{n+1}=\frac{\log (x_{n+1})-{\widehat{\mu }}_n}{\sqrt{{\widehat{\sigma }}_n^2+{\sigma }^2}}$
$\mathit{logN}(\mu ,{\theta }^2)$	$k·{\theta }^2$	k > 1: $(k-1)100\%$ increase in ${\theta }^2$	$({\widehat{a}}_n+1/2)·\log \frac{2{\widehat{a}}_n+z_{n+1}^2}{2{\widehat{a}}_n+z_{n+1}^2/k}-\log \sqrt{k}$
IG(a, b)		k < 1: $(1-k)100\%$ decrease in ${\theta }^2$
${\widehat{a}}_n=a+\frac{{\sum }_{j=1}^{N_D}{w}_j}{2},$ ${\widehat{b}}_n=b+\frac{{\sum }_{j=1}^{N_D}{w}_j{(\log (d_j)-\mu )}^2}{2},$ $z_{n+1}=\frac{\log (x_{n+1})-\mu }{\sqrt{{\widehat{b}}_n/{\widehat{a}}_n}}$
$\mathit{logN}({\theta }_1,{\theta }_2^2)$	${\theta }_1+k·\widehat{{\theta }_2}$	k > 0: increase in θ size of $k·{\widehat{\theta }}_2$	$({\widehat{a}}_n+1/2)·\log \frac{2{\widehat{a}}_n+z_{n+1}^2}{2{\widehat{a}}_n+{(z_{n+1}-k·{\widehat{\lambda }}_n/({\widehat{\lambda }}_n+1))}^2}$
NIG$({\mu }_0,\lambda ,a,b)$		k < 0: decrease in θ size of $k·{\widehat{\theta }}_2$
$\mathit{logN}({\theta }_1,{\theta }_2^2)$	$k·{\theta }^2$	k > 1: $(k-1)100\%$ increase in ${\theta }_2^2$	$({\widehat{a}}_n+1/2)·\log \frac{2{\widehat{a}}_n+z_{n+1}^2}{2{\widehat{a}}_n+z_{n+1}^2/k}-\log \sqrt{k}$
NIG$({\mu }_0,\lambda ,a,b)$		k < 1: $(1-k)100\%$ decrease in ${\theta }_2^2$
${\widehat{\mu }}_n=\frac{\lambda {\mu }_0+{\sum }_{j=1}^{N_D}{w}_j\hspace{0.17em}\log \hspace{0.17em}(d_j)}{\lambda +{\sum }_{j=1}^{N_D}{w}_j},$ ${\widehat{\lambda }}_n=\lambda +{\sum }_{j=1}^{N_D}{w}_j,$ ${\widehat{a}}_n=a+\frac{{\sum }_{j=1}^{N_D}{w}_j}{2},$ ${\widehat{b}}_n=b+\frac{\lambda {\mu }_0^2+{\sum }_{j=1}^{N_D}{w}_j{(\log (d_j))}^2}{2}-\frac{{(\lambda {\mu }_0+{\sum }_{j=1}^{N_D}{w}_j\hspace{0.17em}\log \hspace{0.17em}(d_j))}^2}{2(\lambda +{\sum }_{j=1}^{N_D}{w}_j)},$ ${\widehat{\theta }}_2=\sqrt{\frac{{\widehat{b}}_n}{{\widehat{a}}_n}},$ $z_{n+1}=\frac{\log (x_{n+1})-{\widehat{\mu }}_n}{\sqrt{\frac{({\widehat{\lambda }}_n+1)·{\widehat{b}}_n}{{\widehat{\lambda }}_n·{\widehat{a}}_n}}}$

${\mathit{D}}_n=(\mathit{Y},{\mathit{X}}_n)=(y_1,\dots ,y_{n_0},x_1,\dots ,x_n)$ is the vector of historical and current data, $\mathit{w}=({\alpha }_0,\dots ,{\alpha }_0,1,\dots ,1)$ are the weights corresponding to each element d_j of ${\mathit{D}}_n$ and $N_D=n_0+n.$

Figure 1. PRC flowchart 1. A parallelogram corresponds to an input/output information, a decision is represented by a rhombus and a rectangle denotes an operation after a decision making. In addition, the rounded rectangles indicate the beginning and end of the process.

${}^{★}$For the likelihoods with two unknown parameters and total prior ignorance (i.e. initial reference prior and ${\alpha }_0=0$ in the power prior) we need n = 3 to initiate PRC, while for all other cases, PRC starts right after x₁ becomes available.

Figure 2. The FWER(k) at each time point $k=2,3,\dots ,50,$ the probability of successful detection, $\mathit{PSD}(\omega )$ and the truncated conditional expected delay, $\mathit{tCED}(\omega )$ for shifts at locations $\omega =\{11,26,41\},$ of SSC, CBF and PRC, under a reference (CBF_r, PRC_r) or a moderately informative (CBF_mi, PRC_mi) prior. The results refer to Normal data with step changes for the mean of size $\{1{\theta }_2,1.5{\theta }_2\},$ Normal data with inflated standard deviation of size $\{50\%,100\%\},$ Poisson data with rate increase of size $\{50\%,100\%\}$ and Binomial data with increase for the odds of size $\{50\%,100\%\}.$