amtThis vignette briefly introduces how one can fit a Step-Selection Function (SSF) with the amt package. We will be using the example data of one red deer from northern Germany and one covariate: a forest cover map.
First we load the required libraries and the relocation data (called deer)
library(lubridate)
library(raster)
library(amt)
data("deer")
deer## # A tibble: 826 x 4
## x_ y_ t_ burst_
## * <dbl> <dbl> <dttm> <dbl>
## 1 4314068. 3445807. 2008-03-30 00:01:47 1
## 2 4314053. 3445768. 2008-03-30 06:00:54 1
## 3 4314105. 3445859. 2008-03-30 12:01:47 1
## 4 4314044. 3445785. 2008-03-30 18:01:24 1
## 5 4313015. 3445858. 2008-03-31 00:01:23 1
## 6 4312860. 3445857. 2008-03-31 06:01:45 1
## 7 4312854. 3445856. 2008-03-31 12:01:11 1
## 8 4312858. 3445858. 2008-03-31 18:01:55 1
## 9 4312745. 3445862. 2008-04-01 00:01:24 1
## 10 4312651. 3446024. 2008-04-01 06:00:54 1
## # … with 816 more rowsIn order to continue, we need a regular sampling rate. To check the current sampling rate, we use summarize_sampling_rate:
summarize_sampling_rate(deer)## # A tibble: 1 x 9
## min q1 median mean q3 max sd n unit
## <table> <table> <table> <table> <table> <table> <dbl> <int> <chr>
## 1 5.964444 5.996667 6.005833 11.46179 6.016111 3923.997 137. 825 hourThe median sampling rate is 6h, which is what we aimed for.
Next, we have to get the environmental covariates. A forest layer is included in the package. Note, that this a regular RasterLayer.
data("sh_forest")
sh_forest## class : RasterLayer
## dimensions : 720, 751, 540720 (nrow, ncol, ncell)
## resolution : 25, 25 (x, y)
## extent : 4304725, 4323500, 3437725, 3455725 (xmin, xmax, ymin, ymax)
## crs : +proj=laea +lat_0=52 +lon_0=10 +x_0=4321000 +y_0=3210000 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
## source : memory
## names : sh.forest
## values : 1, 2 (min, max)Before fitting a step selection, the data well need to prepared. First, we change from a point representation to a step representation, using the function steps_by_burst, which in contrast to the steps function accounts for bursts.
ssf1 <- deer %>% steps_by_burst()Next, we generate random steps with the function random_steps. This function fits by default a Gamma distribution to the step lengths and a von Mises distribution to the turn angles, and then pairs each observed step with n random steps.
ssf1 <- ssf1 %>% random_steps(n = 15)As a last step, we have to extract the covariates at the end point of each step. We can do this with extract_covariates.
ssf1 <- ssf1 %>% extract_covariates(sh_forest) Since the forest layers is coded as 1 = forest and 2 != forest, we create a factor with appropriate levels. We also calculate the log of the step length and the cosine of the turn angle, which we may use later for a integrated step selection function.
ssf1 <- ssf1 %>%
mutate(forest = factor(sh.forest, levels = 1:2, labels = c("forest", "non-forest")),
cos_ta = cos(ta_),
log_sl = log(sl_)) Now all pieces are there to fit a SSF. We will use fit_clogit, which is a wrapper around survival::clogit.
m0 <- ssf1 %>% fit_clogit(case_ ~ forest + strata(step_id_))
m1 <- ssf1 %>% fit_clogit(case_ ~ forest + forest:cos_ta + forest:log_sl + log_sl * cos_ta + strata(step_id_))
m2 <- ssf1 %>% fit_clogit(case_ ~ forest + forest:cos_ta + forest:log_sl + log_sl + cos_ta + strata(step_id_))
summary(m0)## Call:
## coxph(formula = Surv(rep(1, 12144L), case_) ~ forest + strata(step_id_),
## data = data, method = "exact")
##
## n= 12144, number of events= 759
##
## coef exp(coef) se(coef) z Pr(>|z|)
## forestnon-forest -0.5200 0.5945 0.1089 -4.776 1.79e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## forestnon-forest 0.5945 1.682 0.4803 0.7359
##
## Concordance= 0.529 (se = 0.007 )
## Likelihood ratio test= 22.06 on 1 df, p=3e-06
## Wald test = 22.81 on 1 df, p=2e-06
## Score (logrank) test = 22.9 on 1 df, p=2e-06summary(m1)## Call:
## coxph(formula = Surv(rep(1, 12144L), case_) ~ forest + forest:cos_ta +
## forest:log_sl + log_sl * cos_ta + strata(step_id_), data = data,
## method = "exact")
##
## n= 12144, number of events= 759
##
## coef exp(coef) se(coef) z Pr(>|z|)
## forestnon-forest 0.77414 2.16873 0.33061 2.342 0.019202 *
## log_sl 0.18971 1.20890 0.04993 3.800 0.000145 ***
## cos_ta -0.26631 0.76620 0.20831 -1.278 0.201102
## forestnon-forest:cos_ta -0.25404 0.77566 0.11761 -2.160 0.030767 *
## forestnon-forest:log_sl -0.24688 0.78124 0.05839 -4.228 2.36e-05 ***
## log_sl:cos_ta 0.02454 1.02484 0.03500 0.701 0.483242
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## forestnon-forest 2.1687 0.4611 1.1345 4.1459
## log_sl 1.2089 0.8272 1.0962 1.3332
## cos_ta 0.7662 1.3051 0.5094 1.1525
## forestnon-forest:cos_ta 0.7757 1.2892 0.6160 0.9767
## forestnon-forest:log_sl 0.7812 1.2800 0.6968 0.8760
## log_sl:cos_ta 1.0248 0.9758 0.9569 1.0976
##
## Concordance= 0.601 (se = 0.013 )
## Likelihood ratio test= 83.97 on 6 df, p=5e-16
## Wald test = 82.65 on 6 df, p=1e-15
## Score (logrank) test = 83.99 on 6 df, p=5e-16summary(m2)## Call:
## coxph(formula = Surv(rep(1, 12144L), case_) ~ forest + forest:cos_ta +
## forest:log_sl + log_sl + cos_ta + strata(step_id_), data = data,
## method = "exact")
##
## n= 12144, number of events= 759
##
## coef exp(coef) se(coef) z Pr(>|z|)
## forestnon-forest 0.78755 2.19801 0.32932 2.391 0.016783 *
## log_sl 0.18953 1.20868 0.04984 3.803 0.000143 ***
## cos_ta -0.13722 0.87178 0.09740 -1.409 0.158885
## forestnon-forest:cos_ta -0.25885 0.77194 0.11740 -2.205 0.027471 *
## forestnon-forest:log_sl -0.24921 0.77941 0.05819 -4.283 1.85e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## forestnon-forest 2.1980 0.4550 1.1527 4.1913
## log_sl 1.2087 0.8273 1.0962 1.3327
## cos_ta 0.8718 1.1471 0.7203 1.0551
## forestnon-forest:cos_ta 0.7719 1.2954 0.6133 0.9717
## forestnon-forest:log_sl 0.7794 1.2830 0.6954 0.8736
##
## Concordance= 0.598 (se = 0.013 )
## Likelihood ratio test= 83.47 on 5 df, p=<2e-16
## Wald test = 81.76 on 5 df, p=4e-16
## Score (logrank) test = 83.22 on 5 df, p=<2e-16#AIC(m0$model)
#AIC(m1$model)
#AIC(m2$model)To be done.
All steps described above, could easily be wrapped into one piped workflow:
m1 <- deer %>%
steps_by_burst() %>% random_steps(n = 15) %>%
extract_covariates(sh_forest) %>%
mutate(forest = factor(sh.forest, levels = 1:2, labels = c("forest", "non-forest")),
cos_ta = cos(ta_),
log_sl = log(sl_)) %>%
fit_clogit(case_ ~ forest + forest:cos_ta + forest:sl_ + sl_ * cos_ta + strata(step_id_))summary(m1)## Call:
## coxph(formula = Surv(rep(1, 12144L), case_) ~ forest + forest:cos_ta +
## forest:sl_ + sl_ * cos_ta + strata(step_id_), data = data,
## method = "exact")
##
## n= 12144, number of events= 759
##
## coef exp(coef) se(coef) z Pr(>|z|)
## forestnon-forest -0.1209311 0.8860950 0.1400205 -0.864 0.38777
## sl_ 0.0006510 1.0006512 0.0001568 4.151 3.31e-05
## cos_ta -0.2893298 0.7487652 0.1095607 -2.641 0.00827
## forestnon-forest:cos_ta -0.2178123 0.8042764 0.1181950 -1.843 0.06536
## forestnon-forest:sl_ -0.0009040 0.9990964 0.0002001 -4.517 6.27e-06
## sl_:cos_ta 0.0002409 1.0002410 0.0001317 1.829 0.06737
##
## forestnon-forest
## sl_ ***
## cos_ta **
## forestnon-forest:cos_ta .
## forestnon-forest:sl_ ***
## sl_:cos_ta .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## forestnon-forest 0.8861 1.1285 0.6734 1.1659
## sl_ 1.0007 0.9993 1.0003 1.0010
## cos_ta 0.7488 1.3355 0.6041 0.9281
## forestnon-forest:cos_ta 0.8043 1.2434 0.6380 1.0139
## forestnon-forest:sl_ 0.9991 1.0009 0.9987 0.9995
## sl_:cos_ta 1.0002 0.9998 1.0000 1.0005
##
## Concordance= 0.603 (se = 0.013 )
## Likelihood ratio test= 98.26 on 6 df, p=<2e-16
## Wald test = 97.25 on 6 df, p=<2e-16
## Score (logrank) test = 111.3 on 6 df, p=<2e-16devtools::session_info()## ─ Session info ──────────────────────────────────────────────────────────
## setting value
## version R version 3.6.1 (2019-07-05)
## os Ubuntu 18.04.3 LTS
## system x86_64, linux-gnu
## ui X11
## language en_US:en
## collate C
## ctype en_US.UTF-8
## tz Europe/Berlin
## date 2019-09-19
##
## ─ Packages ──────────────────────────────────────────────────────────────
## package * version date lib source
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