Algorithm 1: Visual abnormal event detection via online least squares one-class support vector machine (LS-OC-SVM) and sparse online LS-OC-SVM.

Require

n training frames ${\left\{I_i\right\}}_{i=1}^n$and the corresponding optical flow ${\left\{O{P}_i\right\}}_{i=1}^n$.

Compute the covariance matrix of each frame. $$\begin{array}{ccc}\left\{O{P}_1,O{P}_2,\mathrm{\dots },O{P}_n\right\}& \to & \left\{{\mathit{C}}_1,{\mathit{C}}_2,\mathrm{\dots },{\mathit{C}}_n\right\}\end{array}$$ (46) (a)Online strategy: Applying LS-OC-SVM on the small subset of training samples to calculate the coefficient matrix. $$\left\{{\mathit{C}}_1,{\mathit{C}}_2,\mathrm{\dots },{\mathit{C}}_m\right\},1\le m\ll n\stackrel{online}{\to }\text{coefficient matrix}\left[\mathit{K}\right]{\left[\mathit{\alpha }-\rho \right]}^{\top }$$ (47) (b)Sparse online strategy: Applying LS-OC-SVM to train the initial dictionary, C, offline. $${\mathit{C}}_{\mathcal{D}}=\left\{{\mathit{C}}_1,{\mathit{C}}_2,\mathrm{\dots },{\mathit{C}}_m\right\},1\le m\ll n\stackrel{\mathit{\text{offline}}}{\to }\text{coefficient matrix}\left[\mathit{K}\right]{\left[\mathit{\beta }-\rho \right]}^{\top }$$ (48) (a)Online strategy: Applying online LS-OC-SVM on the remaining samples to calculate the coefficient matrix. $$\left\{{\mathit{C}}_{m+1},{\mathit{C}}_{m+2},\mathrm{\dots },{\mathit{C}}_n\right\},\left[\mathit{K}\right]\stackrel{\mathit{\text{online}}}{\to }\text{coefficient matrix}\left[\mathit{K}\right]{\left[\mathit{\alpha }-\rho \right]}^{\top }$$ (49) (b)Sparse online strategy: Applying sparse online LS-OC-SVM on the remaining samples to calculate the coefficient matrix and to update the dictionary. $$\begin{array}{l}\left\{{\mathit{C}}_{\mathcal{D}},{\mathit{C}}_K\right\},m<k\le n\stackrel{\mathit{\text{sparse online}}}{\to }\text{coefficient matrix}{\left[\mathit{\beta }-\rho \right]}^{\top }\hfill \\ \{\begin{array}{lll}{\mathit{C}}_{\mathcal{D}}:={\mathit{C}}_{\mathcal{D}}\cup {\mathit{C}}_k,\hfill & \text{if}\hfill & {\mathit{ϵ}}_t\ge {\mu }_0\hfill \\ {\mathit{C}}_{\mathcal{D}}:={\mathit{C}}_{\mathcal{D}},\hfill & \text{if}\hfill & {\mathit{ϵ}}_t<{\mu }_0\hfill \end{array}\hfill \end{array}$$ (50) Each frame Cn+l is classified via LS-OC-SVM.