Interpolating run early and late

Carl James Schwarz

2020-09-04

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      NA

The bottom line is the total recoveries (tagged and untagged) from a commercial harvest. In this case, the commerical harvest did not start until statistical week SW25 and ended in SW33 but the run started earlier and ended later than the commerical harvest.

1.1 Fit with the current data.

We now fit the BTSPAS model using the current data

## 
## 
## *** Start of call to JAGS 
## Working directory:  /Users/cschwarz/Dropbox/SPAS-Bayesian/BTSPAS/vignettes 
## Initial seed for JAGS set to: 661607 
## Random number seed for chain  582324 
## Random number seed for chain  424815 
## Random number seed for chain  911807 
## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 32
##    Unobserved stochastic nodes: 96
##    Total graph size: 895
## 
## Initializing model
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## [1] TRUE
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## [1] TRUE
## [1] TRUE
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## [1] TRUE
## [1] TRUE
## [1] TRUE
## [1] TRUE

On 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  205328 85279 132582 384186
## U[1]   27923 53018     44 126771
## U[2]   22501 26908    744  77048
## U[3]   21007 13950   4339  53651
## U[4]   19230  2954  13815  25510
## U[5]   18544  1560  15646  21814
## U[6]   20058  1564  17187  23283
## U[7]   19909  1386  17394  22791
## U[8]   17003  1296  14518  19599
## U[9]    9978  1749   6550  13557
## U[10]   7919  1067   5838  10042
## U[11]   6910  1401   4391   9946
## U[12]   3385  1109   1665   5948
## U[13]   2446  1595    512   6470
## U[14]   1541  1459    102   5213
## U[15]    952  1235      8   4163
## U[16]    675  3417      0   3451
## Utot  199981 85279 127235 378839

The 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 accomodate 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      NA

Notice 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 commerical 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: 624849 
## Random number seed for chain  321164 
## Random number seed for chain  19375 
## Random number seed for chain  719718 
## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 32
##    Unobserved stochastic nodes: 96
##    Total graph size: 895
## 
## 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] TRUE

Notice 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  126974 3961 119555 135124
## U[1]       0    5      0      2
## U[2]      12   35      0     96
## U[3]     528  638      8   2270
## U[4]   16728 2893  11598  22997
## U[5]   18775 1728  15614  22287
## U[6]   20207 1605  17198  23402
## U[7]   20209 1475  17521  23267
## U[8]   17443 1366  14735  20143
## U[9]    9361 1978   5713  13610
## U[10]   7467 1259   5026   9969
## U[11]   8934 1695   5850  12403
## U[12]   1855  693    829   3498
## U[13]    101  137      2    462
## U[14]      5   15      0     41
## U[15]      0    1      0      2
## U[16]      0    1      0      0
## Utot  121627 3961 114208 129777