Exercise $4$ Evaluating performance of a mock biometric$-$based authentication system

Imagine you are using a biometrics$-$based user authentication system and would like to evaluate its performance for a specific user $($let$$s say, Alice$).$ Below are some data you obtained from some earlier experiments:

$$ For $100$ biometric templates Alice produces for login attempts, the distance between those templates and the enrolled template of Alice is as follows:

$[0\mathrm{.}27,0\mathrm{.}079,0\mathrm{.}27,0\mathrm{.}36,0\mathrm{.}25,0\mathrm{.}3,0\mathrm{.}27,0\mathrm{.}17,0\mathrm{.}23,0\mathrm{.}12,0\mathrm{.}29,0\mathrm{.}085,0\mathrm{.}093,0\mathrm{.}12,0,0\mathrm{.}34,0\mathrm{.}23,0\mathrm{.}12,0\mathrm{.}34,0\mathrm{.}029,0\mathrm{.}19,0\mathrm{.}18,0\mathrm{.}23,0\mathrm{.}23,0\mathrm{.}11,0\mathrm{.}2,0\mathrm{.}18,0\mathrm{.}26,0\mathrm{.}31,0\mathrm{.}31,0\mathrm{.}11,0\mathrm{.}21,0\mathrm{.}079,0\mathrm{.}089,0\mathrm{.}2,0\mathrm{.}35,0\mathrm{.}12,0\mathrm{.}24,0\mathrm{.}18,0\mathrm{.}31,0\mathrm{.}091,0\mathrm{.}2,0\mathrm{.}26,0\mathrm{.}31,0\mathrm{.}35,0\mathrm{.}21,0\mathrm{.}051,0\mathrm{.}13,0\mathrm{.}094,0\mathrm{.}44,0\mathrm{.}14,0\mathrm{.}27,0\mathrm{.}18,0\mathrm{.}29,0\mathrm{.}12,0\mathrm{.}06,0\mathrm{.}058,0\mathrm{.}25,0\mathrm{.}18,0\mathrm{.}18,0\mathrm{.}34,0\mathrm{.}23,0\mathrm{.}22,0\mathrm{.}36,0\mathrm{.}12,0\mathrm{.}27,0\mathrm{.}28,0\mathrm{.}18,0\mathrm{.}22,0\mathrm{.}083,0\mathrm{.}085,0\mathrm{.}21,0\mathrm{.}27,0\mathrm{.}46,0\mathrm{.}13,0\mathrm{.}22,0\mathrm{.}19,0\mathrm{.}0067,0\mathrm{.}16,0\mathrm{.}021,0\mathrm{.}28,0\mathrm{.}11,0\mathrm{.}21,0\mathrm{.}15,0\mathrm{.}23,0\mathrm{.}14,0\mathrm{.}25,0\mathrm{.}27,0\mathrm{.}37,0\mathrm{.}18,0,0\mathrm{.}12,0\mathrm{.}34,0\mathrm{.}093,0\mathrm{.}3,0\mathrm{.}21,0\mathrm{.}34,0\mathrm{.}0039,0\mathrm{.}18,0\mathrm{.}079]$

$$ For $100$ biometric templates Eve $($a mock attacker who pretended to impersonate Alice$)$ produced, the distance between those templates and the enrolled template of Alice is as follows:

$[1\mathrm{.}2,0\mathrm{.}77,0\mathrm{.}88,0\mathrm{.}39,0\mathrm{.}51,0\mathrm{.}55,0\mathrm{.}82,0\mathrm{.}54,0\mathrm{.}74,0\mathrm{.}19,0\mathrm{.}53,0\mathrm{.}44,0\mathrm{.}28,0\mathrm{.}7,0\mathrm{.}66,0\mathrm{.}61,0\mathrm{.}33,0\mathrm{.}83,0\mathrm{.}67,0\mathrm{.}54,0\mathrm{.}6,0\mathrm{.}55,0\mathrm{.}25,0\mathrm{.}54,0\mathrm{.}43,0\mathrm{.}4,0\mathrm{.}37,0\mathrm{.}49,0\mathrm{.}2,0\mathrm{.}79,0\mathrm{.}7,0\mathrm{.}6,0\mathrm{.}59,0\mathrm{.}44,0\mathrm{.}8,0\mathrm{.}57,0\mathrm{.}46,0\mathrm{.}87,0\mathrm{.}56,0\mathrm{.}48,0\mathrm{.}54,0\mathrm{.}43,0\mathrm{.}38,1\mathrm{.}1,0\mathrm{.}93,0\mathrm{.}66,0\mathrm{.}35,0\mathrm{.}43,0\mathrm{.}56,0\mathrm{.}76,0\mathrm{.}33,0\mathrm{.}13,0\mathrm{.}31,0\mathrm{.}67,0\mathrm{.}68,0\mathrm{.}69,0\mathrm{.}57,0\mathrm{.}64,0\mathrm{.}5,0\mathrm{.}77,0\mathrm{.}33,0\mathrm{.}69,0\mathrm{.}43,0\mathrm{.}53,0\mathrm{.}71,0\mathrm{.}81,0\mathrm{.}38,0\mathrm{.}85,0\mathrm{.}73,0\mathrm{.}59,0\mathrm{.}56,0\mathrm{.}56,0\mathrm{.}54,0\mathrm{.}6,0\mathrm{.}61,0\mathrm{.}77,0\mathrm{.}91,0\mathrm{.}69,0\mathrm{.}56,0\mathrm{.}73,0\mathrm{.}64,0\mathrm{.}39,0\mathrm{.}79,0\mathrm{.}66,0\mathrm{.}63,0\mathrm{.}7,0\mathrm{.}65,0\mathrm{.}41,0\mathrm{.}57,0\mathrm{.}57,0\mathrm{.}49,0\mathrm{.}94,0\mathrm{.}42,0\mathrm{.}5,0\mathrm{.}46,0\mathrm{.}37,0\mathrm{.}56,0\mathrm{.}55,0\mathrm{.}91,0\mathrm{.}55]$

Based on these data, now you need to calculate how to set a threshold to accept Alice with a small false reject rate $($FRR$)$ and reject Eve with a small $($if not zero$)$ false accept rate $($FAR$).$

Pick up a number of typical threshold values between the means of Alice$$s and Eve$$s distance values, write a programme to calculate the FAR and FRR$.$ Observe how they change with respect to the value of the threshold. A recommended set of thresholds are $[0\mathrm{.}25,0\mathrm{.}3,0\mathrm{.}35,0\mathrm{.}4,0\mathrm{.}45,0\mathrm{.}5,0\mathrm{.}55].$

$2$

Hint $1$: A smaller distance means two templates being more similar. Assume a user will be accepted if the distance is smaller $(<)$ than the threshold, but will be rejected if greater than or equal to $(>=)$ the threshold. Moving the $$equal to$(=)$ to the former will change the results to some extent but not significantly.

Hint $2$: If you are really struck with writing a program to do this exercise, use Excel to do the calculation instead.