$A$	Projection matrix for manifold feature extraction	$T_{ti}$	Matching matrix for historical transition modes
$D$	A $n\times n$ diagonal matrix	$\widetilde{T_{ti}}$	The first ω column vectors of $T_{ti}$
$D_d$	Distance vector between $T_{online }$ and $\widetilde{T_{ti}}$	$T_{\alpha }{}^2$	The confidence limit for $T^2$
$D_{ii}$	Importance of each sample	$t_s$	The t-statistic of the ADF test
$F$	Low dimensional manifold = $\left[f_1,f_2,\dots ,f_n\right]$	$W$	Weight matrix
$H$	Window length for the first stage	$W_{ij}$	Relationship between two samples
$H^{*}$	Smoothness matrix of the first stage	$X$	Sample set = ${\left[x_m\left(1\right),x_m\left(2\right),\dots ,x_m\left(\mathrm{n}\right)\right]}^T$
$I$	Unit matrix	$X^{*}$	Sample set after dynamic expansion
$k$	Minimum number of generalized eigenvalues	$\overline{X}$	Trend variable
$L$	Window length for the second stage	$\tilde{X}$	Time series = ${[x_1,x_2,\dots ,x_n]}^T$
$L_P$	A Laplacian matrix defined = $D-W$	${\overline{X}}_{new}$	Sample dataset to be reidentified and tested
$l$	Number of time lags	$x^T\left(t\right)$	The observation vector at moment t
$m$	Number of variables	$\sum$	The covariance matrix of F
$m_i$	The matching value between $T_{online}$ and $\widetilde{T_{ti}}$	$\beta$	Coefficient of time trend
$n$	Number of samples	${\gamma }_i$	A trending term
$P$	The projection matrix when using PCA	$\delta$	Coefficient of presenting process roots
$ℝ$	Set of real numbers	$\widehat{\delta }$	The estimated value of $\delta$
$SP{E}_{\alpha }$	The confidence limit for the $SPE$	${\widehat{\sigma }}_{\delta }$	Standard errors
$T$	The feature data extracted by DLPPCA	${\epsilon }_t$	White noise sequence
$T_{online}$	The online matching matrix	$\eta$	An intercept constant called drift