smam: Statistical Modeling of Animal Movements

Description

Animal movement models including Moving-Resting Process with Embedded Brownian Motion, Brownian Motion with Measurement Error, Moving-Resting-Handling Process with Embedded Brownian Motion, Moving-Resting Process with Measurement Error, Moving-Moving Process with two Embedded Brownian Motions.

Author(s)

• *maintainer, author* Chaoran Hu huchaoran.stat@gmail.com

• *author* Vladimir Pozdnyakov vladimir.pozdnyakov@uconn.edu

• *author* Jun Yan jun.yan@uconn.edu

Auxiliary for Preparing Discrete Distribution used to approximating Standard Normal Distribution

Description

Auxiliary for preparing discrete distribution used to approximate standard normal. This function generates order statistics of standard normal with same probability assigned. Then, the discrete distribution is standardized to variance one and mean zero.

Usage

approxNormalOrder(m)

approxNormalOrder2(m, width)

Arguments

Details

This function use EnvStats::evNormOrdStats to get the order statisics of standard normal distribution. The same probability is assigned for each order statistics.

Value

A numeric matrix with first column is support of discrete distribution and second column is corresponding p.m.f..

Author(s)

Chaoran Hu

See Also

EnvStats::evNormOrdStats for order statisics of standard normal. fitMRMEapprox for fit MRME with approximated measurement error.

Density for Time Spent in Moving or Resting

Description

Density for time spent in moving or resting in a time interval, unconditional or conditional on the initial state.

Usage

dtm(w, t, lamM, lamR, s0 = NULL)

dtr(w, t, lamM, lamR, s0 = NULL)

Arguments

Details

dtm returns the density for time in moving; dtr returns the density for time in resting.

Value

a vector of the density evaluated at w.

Functions

• dtr(): Density of time spent in resting

References

Yan, J., Chen, Y., Lawrence-Apfel, K., Ortega, I. M., Pozdnyakov, V., Williams, S., and Meyer, T. (2014) A moving-resting process with an embedded Brownian motion for animal movements. Population Ecology. 56(2): 401–415.

Examples

lamM <- 1
lamR <- c(1/2, 1, 2)
lr <- length(lamR)
totalT <- 10
old.par <- par(no.readonly=TRUE)
par(mfrow=c(1, 2), mar=c(2.5, 2.5, 1.1, 0.1), mgp=c(1.5, 0.5, 0), las=1)
curve(dtm(x, totalT, 1, 1/2, "m"), 0, totalT, lty=1, ylim=c(0, 0.34),
      xlab="M(10)", ylab="density")
curve(dtm(x, totalT, 1, 1, "m"), 0, totalT, lty=2, add=TRUE)
curve(dtm(x, totalT, 1, 2, "m"), 0, totalT, lty=3, add=TRUE)
mtext(expression("S(0) = 1"))
legend("topleft", legend = expression(lambda[r] == 1/2, lambda[r] == 1,
       lambda[r] == 2), lty = 1:lr)
curve(dtm(x, totalT, 1, 1/2, "r"), 0, totalT, lty=1, ylim=c(0, 0.34),
      xlab="M(10)", ylab="density")
curve(dtm(x, totalT, 1, 1, "r"), 0, totalT, lty=2, add=TRUE)
curve(dtm(x, totalT, 1, 2, "r"), 0, totalT, lty=3, add=TRUE)
mtext(expression("S(0) = 0"))
legend("topleft", legend = expression(lambda[r] == 1/2, lambda[r] == 1,
      lambda[r] == 2), lty = 1:lr)
par(old.par)

Variance matrix of estimators from moving-resting process with measurement error

Description

'estVarMRME_Godambe' uses Godambe information matrix to obtain variance matrix of estimators from 'fitMRME'. 'estVarMRME_pBootstrap' uses parametric bootstrap to obtain variance matrix of estimators from 'fitMRME'. 'estVarMRMEnaive_Godambe' use Godambe information matrix to obtain variance matrix of estimators from 'fitMRME_naive'. 'estVarMRMEnaive_pBootstrap' uses parametric bootstrap to obtain variance matrix of estimators from 'fitMRME_naive'.

Usage

estVarMRME_Godambe(
  est_theta,
  data,
  nBS,
  numThreads = 1,
  gradMethod = "simple",
  integrControl = integr.control()
)

estVarMRME_pBootstrap(
  est_theta,
  data,
  nBS,
  detailBS = FALSE,
  numThreads = 1,
  integrControl = integr.control()
)

estVarMRMEnaive_Godambe(
  est_theta,
  data,
  nBS,
  numThreads = 1,
  gradMethod = "simple",
  integrControl = integr.control()
)

estVarMRMEnaive_pBootstrap(
  est_theta,
  data,
  nBS,
  detailBS = FALSE,
  numThreads = 1,
  integrControl = integr.control()
)

Arguments

Value

variance-covariance matrix of estimators

Author(s)

Chaoran Hu

Examples

## Not run: 
## time consuming example
tgrid <- seq(0, 10*100, length=100)
set.seed(123)
dat <- rMRME(tgrid, 1, 0.5, 1, 0.01, "m")

estVarMRME_Godambe(c(1, 0.5, 1, 0.01), dat, nBS = 10)
estVarMRME_pBootstrap(c(1, 0.5, 1, 0.01), dat, nBS = 10)
estVarMRMEnaive_Godambe(c(1, 0.5, 1, 0.01), dat, nBS = 10)
estVarMRMEnaive_pBootstrap(c(1, 0.5, 1, 0.01), dat, nBS = 10)

estVarMRME_Godambe(c(1, 0.5, 1, 0.01), dat, nBS = 10, numThreads = 6)
estVarMRME_pBootstrap(c(1, 0.5, 1, 0.01), dat, nBS = 10, numThreads = 6)
estVarMRMEnaive_Godambe(c(1, 0.5, 1, 0.01), dat, nBS = 10, numThreads = 6)
estVarMRMEnaive_pBootstrap(c(1, 0.5, 1, 0.01), dat, nBS = 10, numThreads = 6)
estVarMRMEnaive_pBootstrap(c(1, 0.5, 1, 0.01), dat, nBS = 10, numThreads = 6)

## End(Not run)

Estimate Result of smam Estimators

Description

'estimate' function returns the estimate result of 'smam::fitXXXX' from smam package.

Usage

estimate(x, ...)

## S3 method for class 'smam_mrme'
estimate(x, ...)

## S3 method for class 'smam_mr'
estimate(x, ...)

## S3 method for class 'smam_mm'
estimate(x, ...)

## S3 method for class 'smam_mrh'
estimate(x, ...)

## S3 method for class 'smam_bmme'
estimate(x, ...)

Arguments

Examples

## time consuming example
#tgrid <- seq(0, 100, length=100)
#set.seed(123)
#dat <- rMRME(tgrid, 1, 0.5, 1, 0.01, "m")

## fit whole dataset to the MRME model
#fit <- fitMRME(dat, start=c(1, 0.5, 1, 0.01))
#fit

## get covariance matrix of estimators
#estimate(fit)

GPS data of f109

Description

A dataset of GPS coordinates of a mature female mountain lion living in the Gros Ventre Mountain Range near Jackson, Wyoming. The data were collected by a code-only GPS wildlife tracking collar from 2009 to 2012.

Usage

f109

Format

A data frame with 3919 rows and 4 variables:

date

Date when the GPS coordinates were collected

cumTime

Standardized time when the GPS coordinates were collected (unit: hour)

centerE

Standardized UTM easting (unit: km)

centerN

Standardized UTM northing (unit: km)

GPS data of f109 (raw format)

Description

The original format of 'f109' dataset.

Usage

f109raw

Format

A data frame with 3917 rows and 3 variables:

t

Date and time when the GPS coordinates were collected (unit: year)

dE

Standardized UTM easting (unit: meter)

dN

Standardized UTM northing (unit: meter)

Fit a Brownian Motion with Measurement Error

Description

Given discretely observed animal movement locations, fit a Brownian motion model with measurement errors. Using segment to fit part of observations to the model. A practical application of this feature is seasonal analysis.

Usage

fitBMME(
  data,
  start = NULL,
  segment = NULL,
  method = "Nelder-Mead",
  optim.control = list()
)

fitBmme(data, start = NULL, method = "Nelder-Mead", optim.control = list())

Arguments

Details

The joint density of the increment data is multivariate normal with a sparse (tri-diagonal) covariance matrix. Sparse matrix operation from package Matrix is used for computing efficiency in handling large data.

Value

A list of the following components:

References

Pozdnyakov V., Meyer, TH., Wang, Y., and Yan, J. (2013) On modeling animal movements using Brownian motion with measurement error. Ecology 95(2): p247–253. doi:doi:10.1890/13-0532.1.

See Also

fitMR

Examples

set.seed(123)
tgrid <- seq(0, 500, by = 1)
dat <- rBMME(tgrid, sigma = 1, delta = 0.5)

## using whole dataset to fit BMME
fit <- fitBMME(dat)
fit

## using part of dataset to fit BMME
batch <- c(rep(0, 100), rep(1, 200), rep(0, 50), rep(2, 100), rep(0, 51))
dat.segment <- cbind(dat, batch)
fit.segment <- fitBMME(dat.segment, segment = "batch")
head(dat.segment)
fit.segment

Fit a Moving-Moving Model with 2 Embedded Brownian Motion

Description

Fit a Moving-Moving Model with 2 Embedded Brownian Motion with animal movement data at discretely observation times by maximizing a full likelihood constructed from the marginal density of increment. 'estVarMM' uses parametric bootstrap to obtain variance matrix of estimators from 'fitMM'.

Usage

fitMM(
  data,
  start,
  logtr = FALSE,
  method = "Nelder-Mead",
  optim.control = list(),
  integrControl = integr.control()
)

estVarMM(
  est_theta,
  data,
  nBS,
  detailBS = FALSE,
  numThreads = 1,
  integrControl = integr.control()
)

Arguments

Value

a list of the following components:

References

Yan, J., Chen, Y., Lawrence-Apfel, K., Ortega, I. M., Pozdnyakov, V., Williams, S., and Meyer, T. (2014) A moving-resting process with an embedded Brownian motion for animal movements. Population Ecology. 56(2): 401–415.

Pozdnyakov, V., Elbroch, L., Labarga, A., Meyer, T., and Yan, J. (2017) Discretely observed Brownian motion governed by telegraph process: estimation. Methodology and Computing in Applied Probability. doi:10.1007/s11009-017-9547-6.

Examples

## Not run: 
## time consuming example
tgrid <- seq(0, 100, length=100)
set.seed(123)
dat <- rMM(tgrid, 1, 0.1, 1, 0.1, "m1")

## fit whole dataset to the MR model
fit <- fitMM(dat, start=c(1, 0.1, 1, 0.1))
fit

var <- estVarMM(fit$estimate, dat, nBS = 10, numThreads = 6)
var

## End(Not run)

Fit a Moving-Resting Model with Embedded Brownian Motion

Description

Fit a Moving-Resting Model with Embedded Brownian Motion with animal movement data at discretely observation times by maximizing a composite likelihood constructed from the marginal density of increment. Using segment to fit part of observations to the model. A practical application of this feature is seasonal analysis.

Usage

fitMR(
  data,
  start,
  segment = NULL,
  likelihood = c("full", "composite"),
  logtr = FALSE,
  method = "Nelder-Mead",
  optim.control = list(),
  integrControl = integr.control()
)

fitMovRes(
  data,
  start,
  likelihood = c("full", "composite"),
  logtr = FALSE,
  method = "Nelder-Mead",
  optim.control = list(),
  integrControl = integr.control()
)

Arguments

Value

a list of the following components:

References

Yan, J., Chen, Y., Lawrence-Apfel, K., Ortega, I. M., Pozdnyakov, V., Williams, S., and Meyer, T. (2014) A moving-resting process with an embedded Brownian motion for animal movements. Population Ecology. 56(2): 401–415.

Pozdnyakov, V., Elbroch, L., Labarga, A., Meyer, T., and Yan, J. (2017) Discretely observed Brownian motion governed by telegraph process: estimation. Methodology and Computing in Applied Probability. doi:10.1007/s11009-017-9547-6.

Examples

## Not run: 
## time consuming example
tgrid <- seq(0, 10, length=500)
set.seed(123)
## make it irregularly spaced
tgrid <- sort(sample(tgrid, 30)) # change to 400 for a larger sample
dat <- rMR(tgrid, 1, 2, 25, "m")

## fit whole dataset to the MR model
fit.fl <- fitMR(dat, start=c(2, 2, 20), likelihood = "full")
fit.fl

fit.cl <- fitMR(dat, start=c(2, 2, 20), likelihood = "composite")
fit.cl

## fit part of dataset to the MR model
batch <- c(rep(0, 5), rep(1, 7), rep(0, 4), rep(2, 10), rep(0, 4))
dat.segment <- cbind(dat, batch)
fit.segment <- fitMR(dat.segment, start = c(2, 2, 20), segment = "batch",
                     likelihood = "full")
head(dat.segment)
fit.segment

## End(Not run)

Fit a Moving-Resting-Handling Model with Embedded Brownian Motion

Description

Fit a Moving-Resting-Handling Model with Embedded Brownian Motion with animal movement data at discretely observation times by maximizing a full likelihood. Using segment to fit part of observations to the model. A practical application of this feature is seasonal analysis.

Usage

fitMRH(
  data,
  start,
  segment = NULL,
  numThreads = RcppParallel::defaultNumThreads() * 3/4,
  lower = c(0.001, 0.001, 0.001, 0.001, 0.001),
  upper = c(10, 10, 10, 10, 0.999),
  integrControl = integr.control(),
  print_level = 3
)

Arguments

Value

A list of estimation result with following components:

Author(s)

Chaoran Hu

References

Pozdnyakov, V., Elbroch, L.M., Hu, C. et al. On Estimation for Brownian Motion Governed by Telegraph Process with Multiple Off States. Methodol Comput Appl Probab 22, 1275–1291 (2020). doi:10.1007/s11009-020-09774-1

See Also

rMRH for simulation.

Examples

## Not run: 
## time consuming example
set.seed(06269)
tgrid <- seq(0, 400, by = 8)
dat <- rMRH(tgrid, 4, 0.5, 0.1, 5, 0.8, 'm')
fitMRH(dat, c(4, 0.5, 0.1, 5, 0.8)) ## parallel process
fitMRH(dat, c(4, 0.5, 0.1, 5, 0.8), numThreads = -1) ## serial process

## fit part of dataset to the MRH model
batch <- c(rep(0, 10), rep(1, 7), rep(0, 10), rep(2, 10), rep(0, 14))
dat.segment <- cbind(dat, batch)
fit.segment <- fitMRH(dat.segment, start = c(4, 0.5, 0.1, 5, 0.8), segment = "batch")
head(dat.segment)
fit.segment

## End(Not run)

Fit a Moving-Resting Model with Measurement Error

Description

'fitMRME' fits a Moving-Resting Model with Measurement Error. The measurement error is modeled by Guassian noise. Using segment to fit part of observations to the model. A practical application of this feature is seasonal analysis.

Usage

fitMRME(
  data,
  start,
  segment = NULL,
  lower = c(1e-06, 1e-06, 1e-06, 1e-06),
  upper = c(10, 10, 10, 10),
  print_level = 3,
  integrControl = integr.control()
)

fitMRME_naive(
  data,
  start,
  segment = NULL,
  lower = c(1e-06, 1e-06, 1e-06, 1e-06),
  upper = c(10, 10, 10, 10),
  integrControl = integr.control()
)

fitMRMEapprox(
  data,
  start,
  segment = NULL,
  approx_norm_even = approxNormalOrder(5),
  approx_norm_odd = approxNormalOrder(6),
  method = "Nelder-Mead",
  optim.control = list(),
  integrControl = integr.control()
)

Arguments

Value

a list of the following components:

Author(s)

Chaoran Hu

References

Hu, C., Elbroch, L.M., Meyer, T., Pozdnyakov, V. and Yan, J. (2021), Moving-resting process with measurement error in animal movement modeling. Methods in Ecology and Evolution. doi:10.1111/2041-210X.13694

Examples

## time consuming example
#tgrid <- seq(0, 10*100, length=100)
#set.seed(123)
#dat <- rMRME(tgrid, 1, 0.5, 1, 0.01, "m")

## fit whole dataset to the MRME model
#fit <- fitMRME(dat, start=c(1, 0.5, 1, 0.01))
#fit

## fit whole dataset to the MRME model with naive composite likelihood
#fit.naive <- fitMRME_naive(dat, start=c(1, 0.5, 1, 0.01))
#fit.naive

## fit whole dataset to the MRME model with approximate error
#fit.approx <- fitMRMEapprox(dat, start=c(1, 0.5, 1, 0.01))
#fit.approx

## fit part of dataset to the MR model
#batch <- c(rep(0, 5), rep(1, 17), rep(0, 4), rep(2, 30), rep(0, 4), rep(3, 40))
#dat.segment <- cbind(dat, batch)
#fit.segment <- fitMRME(dat.segment, start = c(1, 0.5, 1, 0.01), segment = "batch")
#fit.segment.approx <- fitMRMEapprox(dat.segment, start = c(1, 0.5, 1, 0.01), segment = "batch")
#head(dat.segment)
#fit.segment

Estimation of states at each time point with Moving-Resting Process

Description

Estimate the state at each time point under the Moving-Resting process with Embedded Brownian Motion with animal movement data at discretely time points. See the difference between fitStateMR and fitViterbiMR in detail part. Using fitPartialViterbiMR to estimate the state within a small piece of time interval.

Usage

fitStateMR(data, theta, cutoff = 0.5, integrControl = integr.control())

fitViterbiMR(data, theta, cutoff = 0.5, integrControl = integr.control())

fitPartialViterbiMR(
  data,
  theta,
  cutoff = 0.5,
  startpoint,
  pathlength,
  integrControl = integr.control()
)

Arguments

Details

fitStateMR estimates the most likely state by maximizing the probability of Pr(S(t = t_k) = s_k | X), where X is the whole data and s_k is the possible sates at t_k (moving, resting).

fitViterbiMR estimates the most likely state path by maximizing Pr(S(t = t_0) = s_0, S(t = t_1) = s_1, ..., S(t = t_n) = s_n | X), where X is the whole data and s_0, s_1, ..., s_n is the possible state path.

fitPartialViterbiMR estimates the most likely state path of a small peice of time interval, by maximizing the probability of Pr(S(t = t_k) = s_k, ..., S(t = t_{k+q-1}) = s_{k+q-1} | X), where k is the start time point and q is the length of interested time interval.

Value

A data.frame contains estimated results, with elements:

• original data be estimated.

• conditional probability of moving, resting (p.m, p.r), which is Pr(S(t = t_k) = s_k | X) for fitStateMR; log-Pr(s_0, ..., s_k | X_k) for fitViterbiMR, where X_k is (X_0, ..., X_k); and log-Pr(s_k, ..., s_{k+q-1}|X) for fitPartialViterbiMR.

• estimated states with 1-moving, 0-resting.

Author(s)

Chaoran Hu

See Also

rMR for simulation. fitMR for estimation of parameters.

Examples

set.seed(06269)
tgrid <- seq(0, 400, by = 8)
dat <- rMR(tgrid, 4, 3.8, 5, 'm')
fitStateMR(dat, c(4, 3.8, 5), cutoff = 0.5)
fitViterbiMR(dat, c(4, 3.8, 5), cutoff = 0.5)
fitPartialViterbiMR(dat, c(4, 3.8, 5), cutoff = 0.5, 20, 10)

Estimation of states at each time point with Moving-Resting-Handling Process

Description

Estimate the state at each time point under the Moving-Resting-Handling process with Embedded Brownian Motion with animal movement data at discretely time points. See the difference between fitStateMRH and fitViterbiMRH in detail part. Using fitPartialViterbiMRH to estimate the state during a small piece of time interval.

Usage

fitStateMRH(data, theta, integrControl = integr.control())

fitViterbiMRH(data, theta, integrControl = integr.control())

fitPartialViterbiMRH(
  data,
  theta,
  startpoint,
  pathlength,
  integrControl = integr.control()
)

Arguments

Details

fitStateMRH estimates the most likely state by maximizing the probability of Pr(S(t = t_k) = s_k | X), where X is the whole data and s_k is the possible sates at t_k (moving, resting or handling).

fitViterbiMRH estimates the most likely state path by maximizing Pr(S(t = t_0) = s_0, S(t = t_1) = s_1, ..., S(t = t_n) = s_n | X), where X is the whole data and s_0, s_1, ..., s_n is the possible state path.

fitPartialViterbiMRH estimates the most likely state path of a small peice of time interval, by maximizing the probability of Pr(S(t = t_k) = s_k, ..., S(t = t_{k+q-1}) = s_{k+q-1} | X), where k is the start time point and q is the length of interested time interval.

Value

A data.frame contains estimated results, with elements:

• original data be estimated.

• conditional probability of moving, resting, handling (p.m, p.r, p.h), which is Pr(S(t = t_k) = s_k | X) for fitStateMRH; log-Pr(s_0, ..., s_k | X_k) for fitViterbiMRH, where X_k is (X_0, ..., X_k); and log-Pr(s_k, ..., s_{k+q-1}|X) for fitPartialViterbiMRH.

• estimated states with 0-moving, 1-resting, 2-handling.

Author(s)

Chaoran Hu

References

Pozdnyakov, V., Elbroch, L.M., Hu, C., Meyer, T., and Yan, J. (2018+) On estimation for Brownian motion governed by telegraph process with multiple off states. <arXiv:1806.00849>

See Also

rMRH for simulation. fitMRH for estimation of parameters.

Examples

## Not run: 
## time consuming example
set.seed(06269)
tgrid <- seq(0, 400, by = 8)
dat <- rMRH(tgrid, 4, 0.5, 0.1, 5, 0.8, 'm')
fitStateMRH(dat, c(4, 0.5, 0.1, 5, 0.8))
fitViterbiMRH(dat, c(4, 0.5, 0.1, 5, 0.8))
fitPartialViterbiMRH(dat, c(4, 0.5, 0.1, 5, 0.8), 20, 10)

## End(Not run)

Auxiliary for Controlling Numerical Integration

Description

Auxiliary function for the numerical integration used in the likelihood and composite likelihood functions. Typically only used internally by 'fitMR' and 'fitMRH'.

Usage

integr.control(
  rel.tol = .Machine$double.eps^0.25,
  abs.tol = rel.tol,
  subdivisions = 100L
)

Arguments

Details

The arguments are the same as integrate, but passed down to the C API of Rdqags used by integrate.

Value

A list with components named as the arguments.

Sampling from a Brownian bridge path give a grid time

Description

A Brownian bridge is a continuous-time stochastic process B(t) whose probability distribution is the conditional probability distribution of a standard Wiener process W(t) subject to the condition (when standardized) that W(T) = 0, so that the process is pinned to the same value at both t = 0 and t = T. The implementation here is a generalized Brownian bridge that allows start point and end point at different locations.

Usage

rBB(time, start_pt, end_pt, sigma)

Arguments

Value

A data.frame whose first column is the time points and second column is coordinate of the locations.

Examples

## Brownian bridge starting from location 0 and ending at location 1
## with sigma 0.1 from time 0 to time 10
plot(rBB(seq(0, 10, length.out = 100), 0, 1, 0.1), type = "l")

Sampling from Brown Motion with Measurement Error

Description

Given the volatility parameters of a Brownian motion and normally distributed measurement errors, generate the process at discretely observed time points of a given dimension.

Usage

rBMME(time, dim = 2, sigma = 1, delta = 1)

rBmme(time, dim = 2, sigma = 1, delta = 1)

Arguments

Value

A data.frame whose first column is the time points and whose other columns are coordinates of the locations.

References

Pozdnyakov V., Meyer, TH., Wang, Y., and Yan, J. (2013) On modeling animal movements using Brownian motion with measurement error. Ecology 95(2): p247–253. doi:doi:10.1890/13-0532.1.

Examples

tgrid <- seq(0, 10, length = 1001)
## make it irregularly spaced
tgrid <- sort(sample(tgrid, 800))
dat <- rBMME(tgrid, 1, 1)
plot(dat[,1], dat[,2], xlab="t", ylab="X(t)", type="l")

Sampling from a Moving-Moving Process with 2 Embedded Brownian Motion

Description

A moving-moving process consists of two states: moving (large) and moving (small). The transition between the two states is modeled by an alternating renewal process, with exponentially distributed duration. An animal moves according to two Brownian motions with different volatility parameters.

Usage

rMM(time, lamM1, lamM2, sigma1, sigma2, s0, dim = 2)

Arguments

Value

A data.frame whose first column is the time points and whose other columns are coordinates of the locations.

References

Yan, J., Chen, Y., Lawrence-Apfel, K., Ortega, I. M., Pozdnyakov, V., Williams, S., and Meyer, T. (2014) A moving-resting process with an embedded Brownian motion for animal movements. Population Ecology. 56(2): 401–415.

Pozdnyakov, V., Elbroch, L., Labarga, A., Meyer, T., and Yan, J. (2017) Discretely observed Brownian motion governed by telegraph process: estimation. Methodology and Computing in Applied Probability. doi:10.1007/s11009-017-9547-6.

Examples

tgrid <- seq(0, 100, length=100)

dat <- rMM(tgrid, 1, 0.1, 1, 0.1, "m1")
plot(dat[,1], dat[,2], xlab="t", ylab="X(t)", type='l')

Sampling from a Moving-Resting Process with Embedded Brownian Motion

Description

A moving-resting process consists of two states: moving and resting. The transition between the two states is modeled by an alternating renewal process, with exponentially distributed duration. An animal stays at the same location while resting, and moves according to a Brownian motion while moving.

Usage

rMR(time, lamM, lamR, sigma, s0, dim = 2, state = FALSE)

rMovRes(time, lamM, lamR, sigma, s0, dim = 2)

rMRME(time, lamM, lamR, sigma, sig_err, s0, dim = 2, state = FALSE)

Arguments

Value

A data.frame whose first column is the time points and whose other columns are coordinates of the locations.

References

Yan, J., Chen, Y., Lawrence-Apfel, K., Ortega, I. M., Pozdnyakov, V., Williams, S., and Meyer, T. (2014) A moving-resting process with an embedded Brownian motion for animal movements. Population Ecology. 56(2): 401–415.

Pozdnyakov, V., Elbroch, L., Labarga, A., Meyer, T., and Yan, J. (2017) Discretely observed Brownian motion governed by telegraph process: estimation. Methodology and Computing in Applied Probability. doi:10.1007/s11009-017-9547-6.

Examples

tgrid <- seq(0, 10, length=1001)
## make it irregularly spaced
tgrid <- sort(sample(tgrid, 800))
dat <- rMR(tgrid, 1, 1, 1, "m")
plot(dat[,1], dat[,2], xlab="t", ylab="X(t)", type='l')

dat2 <- rMR(tgrid, 1, 1, 1, "m", state = TRUE)
head(dat2)

dat3 <- rMRME(tgrid, 1, 1, 1, 0.01, "m", state = TRUE)
head(dat3)
plot(dat3[,1], dat3[,3], xlab="t", ylab="Z(t)=X(t)+GWN(0.01)", type="l")

Sampling from a Moving-Resting bridge

Description

A moving-resting process consists of two states: moving and resting. The transition between the two states is modeled by an alternating renewal process, with exponentially distributed duration. An animal stays at the same location while resting, and moves according to a Brownian motion while moving. A moving-resting bridge is an extension of Brownian bridge that uses moving-resting process to bridge given starting point and ending point.

Usage

rMRB(time, start_pt, end_pt, lamM, lamR, sigma, s0)

Arguments

Value

A data.frame whose first column is the time points and whose other columns are coordinates of the locations.

Examples

tgrid <- seq(0, 10, length=1001)
## make it irregularly spaced
tgrid <- sort(sample(tgrid, 800))

## a 2-dim moving-resting bridge starting from c(0, 0)
## and ending at c(10, -10)
dat <- rMRB(tgrid, start_pt = c(0, 0), end_pt = c(10, -10),
            lamM = 1, lamR = 1, sigma = 1, "m")
par(mfrow = c(2, 1))
plot(dat[,1], dat[,2], xlab="t", ylab="X(t)", type='l')
plot(dat[,1], dat[,3], xlab="t", ylab="X(t)", type='l')
par(mfrow = c(1, 1))

Sampling from a Moving-Resting-Handling Process with Embedded Brownian Motion

Description

A moving-resting-handling process consists of three states: moving, resting and handling. The transition between the three states is modeled by an alternating renewal process, with expenentially distributed duration. An animal stays at the same location while resting and handling (the choice of resting and handling depends on Bernoulli distribution), and moves according to a Brownian motion while moving state. The sequence of states is moving, resting or staying, moving, resting or staying ... or versus

Usage

rMRH(time, lamM, lamR, lamH, sigma, p, s0, dim = 2, state = FALSE)

Arguments

Value

A data.frame whose first column is the time points and whose other columns are coordinates of the locations. If state is TRUE, the second column will be the simulation state.

Author(s)

Chaoran Hu

References

Pozdnyakov, V., Elbroch, L.M., Hu, C., Meyer, T., and Yan, J. (2018+) On estimation for Brownian motion governed by telegraph process with multiple off states. <arXiv:1806.00849>

See Also

fitMRH for fitting model.

Examples

set.seed(06269)
tgrid <- seq(0, 8000, length.out=1001)
dat <- rMRH(time=tgrid, lamM=4, lamR=0.04, lamH=0.2,
            sigma=1000, p=0.5, s0="m", dim=2)
plot(dat$time, dat$X1, type='l')
plot(dat$time, dat$X2, type='l')
plot(dat$X1,   dat$X2, type='l')

set.seed(06269) ## show the usage of state
dat2 <- rMRH(time=tgrid, lamM=4, lamR=0.04, lamH=0.2,
             sigma=1000, p=0.5, s0="m", dim=2, state=TRUE)
head(dat)
head(dat2)

Subsetting data during given season for each year (seasonal analysis toolbox)

Description

Return subsets of data from each year, which is in given time interval between startDate and endDate.

Usage

seasonFilter(data, startDate, endDate)

Arguments

Value

A data.frame with inputted data and additional column 'BATCH' indicates which subset of inputted data is located within given time interval. In column 'BATCH', different integers stands for different segments and 0 stands for outside given time interval.

Author(s)

Chaoran Hu

Transfer raw dataset to the standard dataset (seasonal analysis toolbox)

Description

Transfer the raw location dataset of animal to the standard dataset, which is acceptable in this packages. The raw dataset contains at least four components: 1. t1: data information. 2. dt..hr.: the difference of time between two sample points. 3. e1: the GPS coordinate of east-west. 4. n1: the GPS coordinate of north-south. (These weird variable names are from the original GPS data. We will change them in later version.)

Usage

transfData(data, dateFormat, roundValue = NULL, lengthUnit = "km")

Arguments

Value

A data.frame containing the following components, which is standard format of dataset in this package:

• date: tells us the date of collecting this sample point.

• cumTime: cumulative time line. The collection of this data starts from time 0 in this time line. (Time unit is hours.)

• centerE: the centered east-west GPS coordinate with the center is the starting point (when cumTime[1]).

• centerN: the centered north-south GPS coordinate with the center is the starting point (when cumTime[1]).

Author(s)

Chaoran Hu

See Also

as.Date has parameter format, which is the same as the parameter dateFormat in this function.

Variance-Covariance Matrix of smam Estimators

Description

This function calculates variance covariance matrix for estimators from smam package. Different methods will be used for different 'smam' models.

Usage

## S3 method for class 'smam_mrme'
vcov(
  object,
  nBS = 25,
  detailBS = TRUE,
  numThreads = 5,
  gradMethod = "simple",
  vcovMethod = "pBootstrap",
  integrControl = integr.control(),
  ...
)

## S3 method for class 'smam_mm'
vcov(
  object,
  nBS = 25,
  detailBS = TRUE,
  numThreads = 5,
  integrControl = integr.control(),
  ...
)

## S3 method for class 'smam_mrh'
vcov(object, numThreads = 5, integrControl = integr.control(), ...)

## S3 method for class 'smam_mr'
vcov(object, ...)

## S3 method for class 'smam_bmme'
vcov(object, ...)

Arguments

Examples

## time consuming example
#tgrid <- seq(0, 100, length=100)
#set.seed(123)
#dat <- rMRME(tgrid, 1, 0.5, 1, 0.01, "m")

## fit whole dataset to the MRME model
#fit <- fitMRME(dat, start=c(1, 0.5, 1, 0.01))
#fit

## get covariance matrix of estimators
#vcov(fit)