Dynamic Foraging Patterns in Antarctic Humpbacks
Ben Weinstein
2017-07-31 11:43:37
## Source: local data frame [12 x 2]
##
## Animal max(timestamp, na.rm = T)
## (int) (time)
## 1 112699 2012-06-17 03:57:31
## 2 121207 2013-05-09 18:49:00
## 3 121208 2013-02-18 07:52:00
## 4 121210 2013-05-05 07:44:00
## 5 123224 2013-05-24 12:13:00
## 6 123232 2013-09-28 07:28:00
## 7 123236 2013-03-18 11:26:00
## 8 131127 2016-07-15 07:58:36
## 9 131130 2016-04-30 00:30:06
## 10 131132 2016-05-10 19:44:39
## 11 131133 2016-07-05 20:26:44
## 12 131136 2016-06-30 18:49:35
0.1 By Month
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
1.3.1 After filitering
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1],Month[i,g,t]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t],Month[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1],Month[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
#for each month
for (m in 1:Months){
#Intercepts
alpha[1,m] ~ dbeta(1,1)
alpha[2,m] ~ dbeta(1,1)
gamma[1,m] ~ dnorm(gamma_mu,gamma_tau) ## gamma for state 1
dev[m] ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2,m] <- gamma[1,m] * dev[m]
}
##Behavioral States
#Hierarchical structure across months
#Switching among states in inv.logit space
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)
#Variance in state change per month
alpha_tau[1] ~ dunif(0,5)
alpha_tau[2] ~ dunif(0,5)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
#spatial autocorrelation priors, we know the true state is high autocorrelation
gamma_mu~dnorm(0.8,100)
#reasonable variance keeps the chains from wandering.
gamma_tau ~ dunif(100,500)
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
## user system elapsed
## 334.943 13.404 115238.6771.4 Check Gelman Rubin
1.5 Chains
## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1491782 79.7 3205452 171.2 3205452 171.2
## Vcells 1714636332 13081.7 4032004780 30761.8 3457182926 26376.3## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1491975 79.7 3205452 171.2 3205452 171.2
## Vcells 1835596070 14004.5 4032004780 30761.8 3457182926 26376.3
1.6 Change in autocorrelation over time
2 Change in transition probabilities over time
2.1 Parameter Summary
## parameter par mean lower upper
## 1 alpha alpha[1,1] 0.80272609 0.709781302 0.88275731
## 2 alpha alpha[2,1] 0.18417316 0.109886082 0.28596504
## 3 alpha alpha[1,2] 0.85779609 0.769663126 0.92651316
## 4 alpha alpha[2,2] 0.14277837 0.074662328 0.23242453
## 5 alpha alpha[1,3] 0.84809249 0.756291365 0.92646800
## 6 alpha alpha[2,3] 0.08231248 0.041061798 0.13314298
## 7 alpha alpha[1,4] 0.71974044 0.552429628 0.91616349
## 8 alpha alpha[2,4] 0.18694099 0.062148052 0.33259023
## 9 alpha alpha[1,5] 0.72416208 0.553740143 0.86467130
## 10 alpha alpha[2,5] 0.07342636 0.033334537 0.13045811
## 11 alpha alpha[1,6] 0.64232640 0.444304376 0.82654295
## 12 alpha alpha[2,6] 0.17339837 0.060332138 0.34555909
## 13 alpha_mu alpha_mu[1] 0.02486589 -2.725346674 2.69872887
## 14 alpha_mu alpha_mu[2] -0.01029971 -2.662084854 2.60340546
## 15 alpha_tau alpha_tau[1] 2.51146610 0.245123592 4.77690829
## 16 alpha_tau alpha_tau[2] 2.52500326 0.258015652 4.76923839
## 17 gamma gamma[1,1] 0.92994797 0.874182895 0.99815805
## 18 gamma gamma[2,1] 0.15544185 0.019085260 0.31478130
## 19 gamma gamma[1,2] 0.84353009 0.787270981 0.90370236
## 20 gamma gamma[2,2] 0.07979238 0.005575408 0.20071307
## 21 gamma gamma[1,3] 0.82789181 0.749898718 0.90894382
## 22 gamma gamma[2,3] 0.06339788 0.004223703 0.16736089
## 23 gamma gamma[1,4] 0.92005535 0.786832814 1.04616744
## 24 gamma gamma[2,4] 0.07579794 0.004179654 0.23179100
## 25 gamma gamma[1,5] 0.84989878 0.741810880 0.96045772
## 26 gamma gamma[2,5] 0.36474752 0.187056218 0.53299185
## 27 gamma gamma[1,6] 0.90467546 0.793598913 1.03364088
## 28 gamma gamma[2,6] 0.36325160 0.091319031 0.65163358
## 29 gamma_mu gamma_mu 0.87402906 0.806961545 0.94251461
## 30 gamma_tau gamma_tau 287.28682070 116.488343696 476.90033397
## 31 theta theta[1] 0.02360181 0.005872277 0.04146352
## 32 theta theta[2] 3.04564301 2.780647705 3.28326050
3 Behavioral Prediction
3.1 Spatial Prediction
3.2 By individual
3.3 Autocorrelation in behavior
3.3.1 As single timeline
4 Simulated tracks
4.1 Behavioral description
4.1.1 Proportion of states by month
4.1.2 Distance between bouts
5 Behavior duration
## Source: local data frame [177 x 7]
## Groups: Animal, Track, Bout, phistate [177]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Traveling 1 0.6059028 January
## 2 1 2 1 Traveling 1 0.4953819 January
## 3 1 3 1 Traveling 1 0.6483912 January
## 4 1 4 1 Traveling 1 1.5586921 January
## 5 1 5 1 Traveling 1 0.6500116 January
## 6 1 7 1 Traveling 2 0.9866435 February
## 7 1 8 1 Traveling 2 0.5460301 February
## 8 1 10 1 Traveling 2 0.9215972 February
## 9 1 11 1 Traveling 2 2.6218171 February
## 10 1 12 1 Traveling 2 0.6120602 February
## .. ... ... ... ... ... ... ...## Source: local data frame [193 x 7]
## Groups: Animal, Track, Bout, phistate [193]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 6 1 Area-restricted Search 1 2.6031250 January
## 2 1 9 1 Area-restricted Search 2 3.3496065 February
## 3 2 1 2 Area-restricted Search 2 0.1513079 February
## 4 2 1 4 Area-restricted Search 2 0.1564352 February
## 5 2 1 6 Area-restricted Search 2 0.1554977 February
## 6 2 1 8 Area-restricted Search 2 0.1646644 February
## 7 2 2 2 Area-restricted Search 3 0.1447917 March
## 8 2 2 4 Area-restricted Search 3 11.9918056 March
## 9 2 2 6 Area-restricted Search 3 1.9623843 March
## 10 2 2 8 Area-restricted Search 3 1.9876620 March
## .. ... ... ... ... ... ... ...## Source: local data frame [370 x 7]
## Groups: Animal, Track, Bout, phistate [370]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Traveling 1 0.6059028 January
## 2 1 2 1 Traveling 1 0.4953819 January
## 3 1 3 1 Traveling 1 0.6483912 January
## 4 1 4 1 Traveling 1 1.5586921 January
## 5 1 5 1 Traveling 1 0.6500116 January
## 6 1 6 1 Area-restricted Search 1 2.6031250 January
## 7 1 7 1 Traveling 2 0.9866435 February
## 8 1 8 1 Traveling 2 0.5460301 February
## 9 1 9 1 Area-restricted Search 2 3.3496065 February
## 10 1 10 1 Traveling 2 0.9215972 February
## .. ... ... ... ... ... ... ...
5.1 Proportion of time allocation
## Month Traveling Area-restricted Search PropF TotalTime
## 1 January 116.31559 139.96898 0.5461467 256.28457
## 2 February 222.67517 359.08926 0.6172417 581.76443
## 3 March 83.89773 216.83172 0.7210192 300.72946
## 4 April 50.95119 166.57176 0.7657664 217.52295
## 5 May 22.33618 83.91296 0.7897754 106.24914
## 6 June 13.01757 49.52854 0.7918724 62.546115.2 Number of bouts
6 Time spent in grid cell
6.1 All years
6.2 Add in sea ice
