as “T” here. This form of threshold can also be applied to other types of distributions even there is no intersection between these two.[]{data-label="fig:example-threshold"}](./ExampleThreshold.pdf){width="\linewidth"}

Moreover, as is verified in experiments, the meta-distribution of anomalous data collections lies in the right-hand-side to the normal one on the real line. As shown in Fig \[fig:example-threshold\], the black curve ($PDF_{normal}(x)$ or in short $PDF_n(x)$) displays the probability density function (PDF) fitting those divergences calculated from normal data collections; the blue curve ($PDF_{anomalous}(x)$ or in short $PDF_a(x)$) displays the PDF derived from anomalous data collections. Threshold is chosen to minimize total errors(both false negative and false positive).

Suppose: $$\begin{aligned}
                PDF_n(x) &\approx \mathcal{N}(\mu_n, \sigma_n)\\
                PDF_a(x) &\approx \mathcal{N}(\mu_a, \sigma_a)
            \end{aligned}$$

Then the optimal threshold $T$ is calculated by E.q.(\[equ:equal-weight\]). The optimal threshold will minimize total errors and yield an optimal outcome. However, this is not accurate enough, since E.q.(\[equ:equal-weight\]) implicates an assumption that chances are the same for a new data collection to be either anomalous or not. If we can determine the probability for a new data collection to be anomalous in any segment of data sequence, the equation should be modified as E.q.(\[equ:linear-weight\]), where $\alpha$ is the anomaly probability.

$$\begin{aligned}
\label{equ:equal-weight}
                    T &= \mathop{\arg\min}_{T} \int_{0}^{T}PDF_{a}(x)dx +
                    \int_{T}^{\sup(D)}PDF_{n}(x)dx \nonumber\\
                    & \approx \mathop{\arg\min}_{T}
                    \int_{-\infty}^{T}
                    \frac{e^{-\frac{(x - \mu_a)^2}{2\sigma_a^2}}}{\sqrt{2\pi} \sigma_a}dx
                    + \int_{T}^{+\infty}
                    \frac{e^{-\frac{(x - \mu_n)^2}{2\sigma_n^2}}}{\sqrt{2\pi} \sigma_n}dx \nonumber\\
 