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50 Science Manual – Using Video Images for Fisheries Monitoring

Appendix 2: Model for calculating

uncertainty

Estimation of numbers of fish from a resistivity counter, image counting

system (IC) and ground-truthing (GT)

Counts from the Manley Hall counter (Dee) over four days are as follows:

Counter IC GT Day 1 Day 2 Day 3 Day 4 Total

Yes No No 4 6 1 1 12

Yes No Yes 3 7 2 1 13

Yes Yes Yes 11 10 12 5 38

No No Yes 0 0 0 0 0

No Yes Yes 1 0 1 1 3

It was assumed that the GT was 100 per cent efficient, that the counter and IC

operated with a constant underlying efficiency over the four days, and that the

counter generated false positives at a constant underlying rate. A simple

probabilistic model for the data is:

Observed count = True count + False positives

True count ~ Binomial(Counter-efficiency, GT)

False positives ~ Poisson(

False-positive rate)

IC ~ Binomial(

IC-efficiency, GT)

Fitting this model to the above data provides estimates of the three unknown

parameters.

Parameter Estimate Standard error Lower 95%CL Upper 95% CL

Counter-efficiency (proportion) 0.93 0.03 0.85 0.98

IC-efficiency (proportion) 0.75 0.06 0.63 0.85

False-positive rate (counts/day) 3.00 0.87 1.55 4.92

The counter is 93 per cent efficient (85-98 per cent), the IC is 75 per cent efficient

(63-85 per cent), and the counter generates false positives at an average rate of

three per day (1.55 to 4.92). Had the initial calibration been undertaken using IC but

not GT, it would not have been possible to estimate the three parameters separately.

Having calibrated the model on counter + IC + GT data, it can then be applied to

days when just the counter is running:

Day Count Truth Estimate Standard error Lower 95%CL Upper 95% CL

5 18 15 16.15 2.45 11.52 21.20

6 23 17 21.20 2.58 16.41 26.62

7 15 15 13.12 2.38 8.55 18.02

8 7 7 5.06 2.17 0.98 9.56

Or days when both the counter and IC are running:

Day Count IC Truth Estimate Standard error Lower 95%CL Upper 95% CL

9 18 12 15 16.23 1.78 13.11 20.05

10 23 10 17 19.08 2.09 14.92 23.22

11 15 13 15 15.63 1.50 13.29 19.03

12 7 6 7 7.52 1.09 6.08 10.15

1 2 ... 53 54 55 56 57 58 59 60 61 62 63 ... 74 75

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