Markov transition matrices , missing transitions for certain years
Make two assumptions:
(1) The initial state probability distribution (``ispd'') is *NOT* a
function of the
transition probability matrix (``tpm'').
(2) The boxes are stochastically independent of each other.
Both of these assumptions may be dubious. The second assumption is
the crucial one, and I would guess it to be *highly* dubious. However
without it, you simply can't get anywhere.
Subject to these assumptions the maximum likelihood estimates of the
entries of the tpm may be found as follows:
Count the number of times that any box is in state "i" at time "t" and
in state "j" at time "t+1". Count over all boxes and all times t = 1,
2, ..., m-1,
where you have observation over m years. (You have to stop at m-1
in order to be able to have observations at time t+1.)
Let this count be c_ij. Let c_i. be the sum over j of c_ij
Let the tpm be P = [p_ij].
Then the maximum likelihood estimate of p_ij is equal to c_ij/c_i.
[The only time that things can go wrong here is if state "i" never appears
in any box, at any time t < m. In such a case the p_ij (j = 1, 2, 3,
..., K, where
K is the number of states or species) are simply not estimable from the
available data. We never observe state i making a transition to *any*
state,
so we cannot estimate the probabilities of such transitions.]
Writing R code to effect this estimation procedure is easy and is left as an
exercise for the reader. :-)
cheers,
Rolf Turner
On 19/04/11 12:47, Abby_UNR wrote:
Hi all, I am working for nest box occupancy data for birds and would like to construct a Markov transition matrix, to derive transition probabilities for ALL years of the study (not separate sets of transition probabilities for each time step). The actual dataset I'm working with is 125 boxes over 14 years that can be occupied by 7 different species, though I have provided a slimmed down portion for this question... - A box can be in 1 of 4 "states" (i.e. bird species): 1,2,3,4 Included here are 4 "box histories" over 4 years (y97, y98, y99, y00) These are the box histories
b1<- c(1,1,4,2)
b2<- c(1,4,4,3)
b3<- c(4,4,1,2)
b4<- c(3,1,1,1)
boxes<- data.frame(rbind(b1,b2,b3,b4))
colnames(boxes)<- c("y97","y98","y99","y00")
boxes
y97 y98 y99 y00 b1 1 1 4 2 b2 1 4 4 3 b3 4 4 1 2 b4 3 1 1 1 My problem is that there are 16 possible transitions, but not all possible transitions occur at each time step. Therefore, don't think I could do something easy like create a table for each time step and add them together, for example:
t1.boxes<- table(boxes$y98, boxes$y97) t1.boxes
1 3 4
1 1 1 0
4 1 0 1
t2.boxes<- table(boxes$y99, boxes$y98) t2.boxes
1 4
1 1 1
4 1 1
t1.boxes and t2.boxes could not be added together to calculate the frequency
of each transition occurring because they are of different dimensions. I'm
not quite sure how to deal with this, I have attempted to write a function
(shown below), though I'm not sure if it is needed, I am a bit new the
programming world. If I could get some help either with the function or a
way around it that would be most appreciated! Thank you!
--------------
Function requires the commands already listed above:
FMAT<- matrx(0, nrow=4, ncol=4, byrow=TRUE)
#This is the matrix that will store the frequency of each possible
transition occurring over the 4 years
nboxes<- 4
nyears<- 4
for(row in 1:nboxes)
{
for(col in 1:(nyears-1))
{
FMAT[boxes[row,col+1], boxes[row,col]]<- boxes[boxes[row, col+1],
boxes[row,col]]
#This is the line of code I have been struggling with an am unsure about. I
have tried
#various versions of this and keep getting an assortment of error messages.
}
}
FMAT
}