Chapter 1 Introduction
3.3.3 Schnabel and Schumacher-Eschmeyer Estimators
The Schnabel estimate exploits capture information better than its two-sample prede- cessors. It stores the captures, the number of individuals already encountered, and the individuals re-encountered at each occasion. The Schnabel and Schumacher-Eschmeyer estimators are quite similar in their results, as shown in Figure 3.9 (Schnabel on the left and Schumacher-Eschmeyer on the right) when the population is sampled but once.
Both estimators exploit the same underlying assumptions using different regression tech- niques. The estimates are shown with the solid line and the cumulative count with the dashed line. The colours represent the various cadence strategies applied to the model A simulation.
We observe that the simulated sources (N = 100) are captured within 2000 days, and all cadence strategies converge to the true population size from about 500 days (±20%) and
300~ - - - ~ every 5 observations grouped
200
100
o ~---
300~- - - ~ every 10 observations grouped
200
100
o ~---
300~ - - - -~
200
100
O 4000 4500
every 20 observations grouped
5000 5500 6000 Time (days)
6500 7000 7500 8000
□ Captures Running mean of Chapman index
+
Chapman index(a) (b)
(c) (d)
Figure 3.9: Schnabel and Schumacher-Eschmeyer estimates (solid lines), and the cumulative count (dashed lines) when sampled from a simulated population model for various cadences above a 0.2 detection threshold. Both estimators for the various sampling strategies tend to converge to the true population size of N = 100 faster than the cumulative count with the exception of the 7 to 14 day cadence. The estimators’ convergence is non-monotonic in cases where there is aliasing present between the model median orbital period and the sampling cadence. Convergence to the true population size is reached within 5% after
∼2000 days. The error on all estimates were consistently 1.
300~ - - - ~
200
100
20 50 100 200
I
-
7-14d-
15-30d-
30-G0d-
60-90d-
60-120d-
90-120d-
120-240d__ ,._
_s_ .... ,-'
_, ,.
Model B 500 1000 2000 Time after First Observation (days)
300~ - - - ~
-
7-14d-
15-30d-
30-G0d-
60-90d-
60-120d-
90-120d-
120-240d200
100
Model F 20 50 100 200 500 1000 2000
Time after First Observation (days)
300~ - - - -~ - - - ~
200
100
20 50 100 200
- 7-14d
- 15-30d - 30-G0<l
- 60-90d
- 60-120d - 90-120d - 120-240d
__,._
_, ,.
--••"' ,-'
__ r-~
Model B 500 1000 2000 Time after First Observation (days)
300~ - - - -~ - - - ~
- 7-14d
- 15-30d - 30-G0<l
- 60-90d
- 60-120d - 90-120d - 120-240d
200
100
Model F 20 50 100 200 500 1000 2000
Time after First Observation (days)
3.3. Implementation of different estimators on Models A to F 67
(a) (b)
(c) (d)
Figure 3.10: Schnabel (solid lines), and the cumulative count (dashed lines), as plotted in Figure 3.9, for Model B along with their 95% confidence intervals.
The two highest and two lowest cadences are shown.
400
300
~ 200
100
300
~ 200
100
- 7-14d
- 15-30d
- 30-GOd
- 60-90d
- 60-120d - 90-120d - 120-240d
Model B 20 50 100 200 500 1000 2000
Time after First Observation (days)
20
__ f_,-J- r---J
---
--·
- 7-14d
- 15-30d - 30-GO<l
- 60-90d
- 60-120d - 90-120d - 120-240d
Model B 50 100 200 500 1000 2000 Time after First Observation (days)
400
- 7-14d
- 15-30d
- 30-GO<l
- 60-90d
- 60-120d - 90-120d
300 - 120-240d
~ 200
100
Model B 20 50 100 200 500 1000 2000
Time after First Observation (days)
400
- 7-14d
- 15-30d
- 30-GO<l
- 60-90d
- 60-120d - 90-120d
300 - 120-240d
~ 200
100
Model B 20 50 100 200 500 1000 2000
Time after First Observation (days)
beyond. The model A 15 to 30-day cadence strategy converges much faster compared to other cadences. Schnabel and Schumacher-Eschmeyer reach within 5% of the true population size around 200 days because of significant aliasing of the sampling strategy with the common orbital periods. Model A has a median orbital period around 132 days, with the bulk in the range of 100 to 150 days, hence sampled in the 0.1Porbit to 0.3Porbit range. The periastron outbursts (σ = 0.1Porbit) are of similar duration to the 15 to 30-day cadence for Model A, which allows for efficient capturing of new sources and periodic recapture of known sources. The high cadences tend to underestimate the population size, whereas the low cadences tend to overestimate early on (though often stillwithin 95% confidence; cf. Model B cadences plotted in in Figure 3.10); since many new individuals are encountered without recapturing a large portion of previous observations. However, they too converge to within 5-10% of the true size between 200 and 500 days.
Each simulated HMXB model was resampled by drawing a new set of cadences from the seven distributions a total of 1000 times. The median of the Schnabel estimates and their 75% confidence intervals, obtained from the resampling of the population models, were plotted as a function of observation number k for models A through F in Figures 3.11 and 3.12 (for brightness thresholds of 0.2 and 0.5). Similar to Figures 3.4 and 3.5, the highest cadence of 7 to 14 days converges slower to the true population size, and even underestimates the cumulative count at later times, because of the much lower associated capture probability. Observations at high cadence also tend to violate independence between population measurements, which the homogeneous Schnabel and Schumacher-Eschmeyer estimators are ill-equipped to correct for. Multiple of the chosen cadences have a spread of 30 days, and these seem to consistently display similar capture probabilities compared to the lower p of the 7-14 and 15-30 day cadence.
The high-cadenced sampling relative to the orbital period is ultimately unreliable in estimating the population size when inspected across the various models. It follows that the Schnabel and Schumacher estimators perform best from data with a large spread cadence distribution, and that it optimises estimator efficiency and accuracy.
Figure 3.11: Schnabel estimates as a function of observation numberk for each simulatedHMXB model (threshold=0.2). The cadences are shown in their respective colours for the cumulative
count N ck.
200 200
■ 7-14cl ■ 7-14d
175 ♦ l'.i-:J0d 175 ♦ 15-30d
•
30-G0d•
30-G0clT 60-90d T 60-90d
150
...
61l-120d 150...
61l-121ld•
lJll-120d•
911-12lld125
•
120-2°I 0d125
•
120-210d~ 100 ~100
75 75
50 50
25 25
Model A Model D
00 5 10 15 20 25 30 00 5 10 15 20 25 30
Observation number k Observation number k
200 200
■ 7-14d ■ 7-14cl
175 ♦ 15-30cl 175 ♦ 15-30d
•
:30-G0cl•
30-G0dT 60-CJ0cl T 60-lJlld
150
...
G0-120cl 150...
G0-120cl•
90-120d•
90-120d125
•
120-240d 125•
120-240cl~ 100 ~100
75 75
50 50
25 25
Model C Model D
00 5 10 15 20 25 30 00 5 10 15 20 25 30
Observation number k Observation number k
200 200
■ 7-Md ■ 7-Ucl
175 ♦ l'.i-:30d 175 ♦ 15-30cl
•
30-G0<l•
30-G0dT fill-CJfkl T fill-9lld
150
...
G0-120cl 150...
G0-120cl•
90-120d•
90-120d125
•
120-240d 125•
120-240d~ 100 ~100
75 75
50 50
25 25
Model E Model F
00 5 1() 15 20 25 30 00 5 1() 15 20 25 30
Observation number k Observation number k
Figure 3.12: Schnabel estimates as a function of observation numberk for each simulatedHMXB model (threshold=0.5). The cadences are shown in their respective colours for the cumulative
count N ck. 175
150 125
~ 100 75 50
25
5
• 30-G0d
T 60-90d
.A. 61l-120d
• lJll-120d
• 120-2°I0d
Model A
10 15 20 25
Observation number k
30
200- - - - 175
150 125
~ 100 75 50
25 ■
5
■ 7-14d
♦ 15-30cl
• :30-G0cl
T 60-CJ0cl
.6. G0-120cl
• 90-120d
• 120-240d
Model C
10 15 20 25
Observation number k
30
200~ - - - ~ 175
150 125
~ 100 75 50
25
5
■ 7-Md
♦ l'.i-:30d
• 30-G0<l
T fill-CJ(kl
.6. G0-120cl
• 90-120d
• 120-240d
Model E
10 15 20 25
Observation number k
30
175 150 125
~100 75 50 25
5
• 30-G0cl
T 60-90d
.A. 61l-121ld
• 911-12lld
• 120-210d
Model D
10 15 20 25
Observation number k
30
200- - ~ - - - - 175
150 125
~100 75 50 25
5
■ 7-14cl
♦ 15-30d
• 30-G0d
T 60-lJlld
.6. G0-120cl
• 90-120d
• 120-240cl
Model D
10 15 20 25
Observation number k
30
200~ - - - ~ 175
150 125
~100 75 50 25
■ 5
■ 7-Ucl
♦ 15-30cl
• 30-G0d
T fill-9lld
.6. G0-120cl
• 90-120d
• 120-240d
Model F
10 15 20 25
Observation number k
30
3.3. Implementation of different estimators on Models A to F 71