Understanding the sigmoid scale ($A_TP$-$A_FP$) in the original paper

[AndrewLawrence80]

According to the paper Section II.C, the weights $A_{TP}$ and $A_{FP}$ in the scoring function are bounded to $0\leq A_{TP} \leq 1$ and $-1\leq A_{FP}\leq 0$, but to satisfy the following scoring function
$$\sigma^A(y)=(A_{TP}-A_{FP})\dfrac{1}{1+e^{5y}}-1$$
to $\sigma^A(0)=0$ only allows for $A_{TP}=1$ and $A_{FP}=-1$
should $\sigma^A$ goes with another hyper-parameter $\alpha$ to meet the satisfaction as follows
$$\sigma^A(y)=\alpha(A_{TP}-A_{FP})\dfrac{1}{1+e^{5y}}-1$$
and to control the reward gained by how early or late the anomaly detected, replace fixed scale 5 with another hyper-parameter $\beta$ as follows
$$\sigma^A(y)=\alpha(A_{TP}-A_{FP})\dfrac{1}{1+e^{\beta y}}-1$$