1 Introduction
In some studies, harvest (recovery strata) start after the run has started and terminate prior to the run ending. For example, consider the following recovery matrix where releases and recoveries have been stratified on a weekly basis:
## Tagging SW22 SW23 SW24 SW25 SW26 SW27 SW28 SW29 SW30 SW31 SW32 SW33 SW34 SW35 SW36 SW37 Applied
## 1 SW22 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10
## 2 SW23 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 100
## 3 SW24 0 0 0 51 2 0 0 0 0 0 0 0 0 0 0 0 525
## 4 SW25 0 0 0 10 45 0 0 0 0 0 0 0 0 0 0 0 403
## 5 SW26 0 0 0 0 169 64 9 0 0 0 0 0 0 0 0 0 849
## 6 SW27 0 0 0 0 0 139 41 5 0 0 0 0 0 0 0 0 742
## 7 SW28 0 0 0 0 0 0 155 31 3 1 0 0 0 0 0 0 675
## 8 SW29 0 0 0 0 0 0 0 266 32 5 0 0 0 0 0 0 916
## 9 SW30 0 0 0 0 0 0 0 0 33 49 3 0 0 0 0 0 371
## 10 SW31 0 0 0 0 0 0 0 0 0 33 36 0 0 0 0 0 296
## 11 SW32 0 0 0 0 0 0 0 0 0 0 39 8 0 0 0 0 234
## 12 SW33 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 39
## 13 SW34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 97
## 14 SW35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 61
## 15 SW36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26
## 16 SW37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3
## 17 CatchComm 0 0 0 1869 5394 5131 5668 6733 1780 1828 2493 157 0 0 0 0 NAThe bottom line is the total recoveries (tagged and untagged) from a commercial harvest. In this case, the commercial harvest did not start until statistical week SW25 and ended in SW33 but the run started earlier and ended later than the commercial harvest.
1.1 Fit with the current data.
We now fit the BTSPAS model using the current data
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## *** Start of call to JAGS
## Working directory: /Users/cschwarz/Dropbox/SPAS-Bayesian/BTSPAS/vignettes
## Initial seed for JAGS set to: 308093
## Random number seed for chain 895595
## Random number seed for chain 624580
## Random number seed for chain 439377
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 32
## Unobserved stochastic nodes: 96
## Total graph size: 897
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## Initializing model
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## *** Finished JAGS ***## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUEOn the surface, the fit looks fine:
but the spline remains very large in the first 3 weeks leading to unrealistic estimates of the run in the first 3 weeks and an unrealistic estimate of the total run:
## mean sd 2.5% 97.5%
## Ntot 207788 84023 133226 387826
## U[1] 29243 55082 102 127046
## U[2] 23161 22111 977 80308
## U[3] 21366 13700 4497 53247
## U[4] 19260 2938 13932 25432
## U[5] 18487 1541 15590 21598
## U[6] 20061 1535 17200 23236
## U[7] 19892 1399 17336 22831
## U[8] 16975 1279 14514 19592
## U[9] 10003 1745 6604 13512
## U[10] 7906 1064 5776 10040
## U[11] 6942 1400 4538 10004
## U[12] 3385 1111 1716 5980
## U[13] 2492 1724 474 6692
## U[14] 1587 1614 72 5401
## U[15] 1012 1400 5 4423
## U[16] 669 1460 0 3672
## Utot 202441 84023 127879 382479The problem is that without a commercial catch in the first 3 and last 3 weeks, there is no information about the probability of capture for those weeks and BTSPAS simply interpolates the spline from the middle of the data to the first 3 and last 3 weeks. The interpolation for the last 3 weeks isn’t too bad – the spline is already on a downwards trend and so this is continued. However, the interpolation back for the first 3 weeks is not very realistic
1.2 Forcing the run curve to zero.
It is possible to “force” BTSPAS to interpolate the first 3 and last 3 weeks down to zero by adding ``fake’’ data. In particular, we pretend that in the first 3 and last 3 weeks, that a commercial catch of 1 fish occurred and it was tagged. You also need to ensure that enough fish were tagged and released to accommodate the fake data.
The revised recovery matrix is:
## Tagging SW22 SW23 SW24 SW25 SW26 SW27 SW28 SW29 SW30 SW31 SW32 SW33 SW34 SW35 SW36 SW37 Applied
## 1 SW22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10
## 2 SW23 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 100
## 3 SW24 0 0 1 51 2 0 0 0 0 0 0 0 0 0 0 0 525
## 4 SW25 0 0 0 10 45 0 0 0 0 0 0 0 0 0 0 0 403
## 5 SW26 0 0 0 0 169 64 9 0 0 0 0 0 0 0 0 0 849
## 6 SW27 0 0 0 0 0 139 41 5 0 0 0 0 0 0 0 0 742
## 7 SW28 0 0 0 0 0 0 155 31 3 1 0 0 0 0 0 0 675
## 8 SW29 0 0 0 0 0 0 0 266 32 5 0 0 0 0 0 0 916
## 9 SW30 0 0 0 0 0 0 0 0 33 49 3 0 0 0 0 0 371
## 10 SW31 0 0 0 0 0 0 0 0 0 33 36 0 0 0 0 0 296
## 11 SW32 0 0 0 0 0 0 0 0 0 0 39 8 0 0 0 0 234
## 12 SW33 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 39
## 13 SW34 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 97
## 14 SW35 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 61
## 15 SW36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 26
## 16 SW37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3
## 17 CatchComm 1 1 1 1869 5394 5131 5668 6733 1780 1828 2493 157 1 1 1 1 NANotice how “fake” recoveries were added to the diagonal entries for the first and final weeks of the data including “fake” harvest.
Because the fake data values are very small, it has little impact on the total run size, but a recovery of 1 tagged fish in a commercial harvest of 1 fish is not consistent with a very large run size and so this forces the run curve down at these points as seen in the revised fit:
##
##
## *** Start of call to JAGS
## Working directory: /Users/cschwarz/Dropbox/SPAS-Bayesian/BTSPAS/vignettes
## Initial seed for JAGS set to: 388835
## Random number seed for chain 120932
## Random number seed for chain 79161
## Random number seed for chain 202631
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 32
## Unobserved stochastic nodes: 96
## Total graph size: 897
##
## Initializing model
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## *** Finished JAGS ***## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUE## [1] TRUENotice that in the revised fit, the run curve is forced to 0 at the start and end of the study:
The estimates of total run size and the weekly estimates of the runsize are also more sensible:
## mean sd 2.5% 97.5%
## Ntot 126876 3935 119430 134994
## U[1] 0 2 0 2
## U[2] 11 36 0 89
## U[3] 497 623 11 2218
## U[4] 16687 2867 11435 22653
## U[5] 18806 1725 15577 22271
## U[6] 20237 1644 17183 23703
## U[7] 20166 1490 17522 23334
## U[8] 17462 1362 14795 20150
## U[9] 9348 1941 5751 13505
## U[10] 7536 1239 5123 9970
## U[11] 8845 1705 5656 12324
## U[12] 1831 652 854 3317
## U[13] 97 130 3 435
## U[14] 4 13 0 36
## U[15] 0 2 0 2
## U[16] 0 1 0 0
## Utot 121529 3935 114083 129647