knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) library(SPAS)
There are several types of problem that can occur that make the use of SPAS problematic.
If the row in the movement matrix are proportional, then there an infinite number of solutions. Note it is the singularity of the movement matrix that is problematic; the number of tags releases and not recovered (last column of the input data) can be proportional without affecting the fit.
The model may converge to different solutions depending on your set up. There is no automatic "flagging" of this situation and you need to be vigilant. A diagnostic tool is the condition number of $XX'$ where $X$ is the movement matrix. If the rows are exactly proportional (or more generally if the rows are colinear) the condition number will be $+\infty$.
The solution is to pool the row(s) that are proportional.
Here are some examples:
test.data.csv <- textConnection(" 160 , 120 , 72 , 82 , 3592 80 , 60 , 36 , 41 , 532 7960 , 9720 , 6264 , 7934 , 0 ") test.data <- as.matrix(read.csv(test.data.csv, header=FALSE, strip.white=TRUE)) test.data mod..1 <- SPAS.fit.model(test.data, model.id="No restrictions", row.pool.in=1:2, col.pool.in=1:4) SPAS.print.model(mod..1)
In this case, it appears that the model has converged, but this is ONE of many possible solutions and cannot be relied upon.
The condition number is very large (!) as expected and shown in the above output:
# Compute the condition number of XX' XX <- test.data[1:2, 1:4] %*% t(test.data[1:2, 1:4]) XX cat("\n\nCondition number is\n") kappa(XX)
We will either physically or logically pool rows:
mod..2 <- SPAS.fit.model(test.data, model.id="No restrictions", row.pool.in=c(1,1), col.pool.in=1:4, row.physical.pool=FALSE) SPAS.print.model(mod..2)
This has solved the colinearity problem.
Of course with real data, it is highly unlikely that the rows in the recovery matrix will be exactly proportional.
The model may converge to different solutions depending on your set up. There is no automatic "flagging" of this situation and you need to be vigilant. The condition number of $XX'$ may be useful.
The solution is to pool the row(s) that are proportional.
Here are some examples where the entries in row 2 are modified slightly to make the rows only approximately proportional
test.data.csv <- textConnection(" 160 , 120 , 72 , 82 , 3592 75 , 62 , 38 , 35 , 532 7960 , 9720 , 6264 , 7934 , 0 ") test.data <- as.matrix(read.csv(test.data.csv, header=FALSE, strip.white=TRUE)) test.data mod..2 <- SPAS.fit.model(test.data, model.id="No restrictions", row.pool.in=1:2, col.pool.in=1:4) SPAS.print.model(mod..2) # Compute the condition number of XX' XX <- test.data[1:2, 1:4] %*% t(test.data[1:2, 1:4]) XX cat("\n\nCondition number is\n") kappa(XX)
Now there is only one solution, but the estimate is very sensitive to small changes in the data. Notice that the condition number is still very large.
In theory this should have no influence on the fit (see the section on pooling columns).
test.data.csv <- textConnection(" 160 , 120 , 72 , 82 , 0, 3592 100 , 45 , 39 , 90 , 0, 532 7960 , 9720 , 6264 , 7934 , 0, 0 ") test.data <- as.matrix(read.csv(test.data.csv, header=FALSE, strip.white=TRUE)) test.data mod..3 <- SPAS.fit.model(test.data, model.id="No restrictions", row.pool.in=1:2, col.pool.in=1:5) SPAS.print.model(mod..3) XX <- test.data[1:2, 1:5] %*% t(test.data[1:2, 1:5]) XX cat("\n\nCondition number is\n") kappa(XX)
Notice that the column of 0's does not affect the fit and has no impact on the condition number of $XX'$.
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