Online Detectors Check missing logic

[KaraMelih]

Thanks to @Sheshuk we have noticed a bug in the coincidence system.

Normally, in here

SNEWS_Coincidence_System/snews_cs/cs_stats.py

Lines 4 to 31 in 8803365

	def ncr(n, r):
	f = math.factorial
	return int(f(n) / f(r) / f(n - r))
	def cache_false_alarm_rate(cache_sub_list, hb_cache):
	""" Assume false alarm rate of 1 per week
	returns the combined false alarm rate in years
	meaning; if there are 8 active detectors, each with false alarm rate of 1/week
	We would get a false alarm with 2-fold coincidence every X years
	The formula;
	n = number of detectors
	r = number of coincidences
	C(n,r) = \frac{n!}{r!(n-r)!}
	R_{combined} = C(n,r)+1 \times F_{im, d1} * F_{im, d2} ... * F_{im, dn} \times δt^{r-1}
	"""
	seconds_year = 31_556_926
	seconds_week = 604800
	single_imitation_freq = 1 / seconds_week # 1/week in seconds
	online_detectors = len(hb_cache.Detector.unique()) # n
	coincident_detectors = len(cache_sub_list['detector_name']) # r
	time_window = 10 # seconds
	combinations = ncr(online_detectors, coincident_detectors)
	combined_imitation_freq = (combinations + 1) * np.power(single_imitation_freq, coincident_detectors) * np.power(
	time_window, coincident_detectors - 1)
	comb_Fim_year = combined_imitation_freq / seconds_year
	return 1/comb_Fim_year

We compute the number of detectors available at the time of the alert to compute false-alarm probability of such a trigger. There are 2 problems at the moment, one was already discussed;

1. We check the number of available detectors at the time when we receive the alerts not at the time where the neutrino events were detected. (See picture)
   

2. We depend on the heartbeat messages to come frequently, however, if there are no heartbeats registered at the time of Observation message, the heartbeat cache will be empty and the FAR calculation thinks that there is a coincidence between 2 detectors out of 0 available detectors (since no HB is registered).

For the second issue, we talked about registering a heartbeat for each observation message but not yet implemented. Currently, when the heartbeat cache is empty, FAR tries to compute a factorial with 0 detectors and crashes before sending the message. We can do a quick-patch to ignore FAR when cache is empty. However, we need to fix the real issue.