remotePARTS (version 1.0.4)

fitCor: Estimate spatial parameters from time series residuals

Description

fitCor() estimates parameter values of a distance-based variance function from the pixel-wise correlations among time series residuals.

Usage

fitCor(
  resids,
  coords,
  distm_FUN = "distm_scaled",
  covar_FUN = "covar_exp",
  start = list(r = 0.1),
  fit.n = 1000,
  index,
  save_mod = TRUE,
  ...
)

Value

fitCor returns a list object of class "remoteCor", which contains these elements:

call

the function call

mod

the nls fit object, if save_mod=TRUE

spcor

a vector of the estimated spatial correlation parameters

max.distance

the maximum distance among the sampled pixels, as calculated by dist_FUN.

logLik

the log-likelihood of the fit

Arguments

resids

a matrix of time series residuals, with rows corresponding to pixels and columns to time points

coords

a numeric coordinate matrix or data frame, with two columns and rows corresponding to each pixel

distm_FUN

a function to calculate a distance matrix from coords

covar_FUN

a function to estimate distance-based covariances

start

a named list of starting parameter values for covar_FUN, passed to nls

fit.n

an integer indicating how many pixels should be used to estimate parameters.

index

an optional index of pixels to use for parameter estimation

save_mod

logical: should the nls model be saved in the output?

...

additional arguments passed to nls.

Details

For accurate results, resids and coords must be paired matrices. Rows of both matrices should correspond to the same pixels.

Distances between sapmled pixels are calculated with the function specified by distm_FUN. This function can be any that takes a coordinate matrix as input and returns a distance matrix between points. Some options provided by remotePARTS are distm_km(), which returns distances in kilometers and distm_scaled(), which returns distances scaled between 0 and 1.

covar_FUN can be any function that takes a vector of distances as its first argument, and at least one parameter as additional arguments. remotePARTS provides three suitable functions: covar_exp, covar_exppow, and covar_taper; but user-defined functions are also possible.

Parameters are estimated with stats::nls() by regressing correlations among time series residuals on a function of distances specified by covar_FUN.

start is used by nls to determine how many parameters need estimating, and starting values for those parameters. As such, it is important that start has named elements for each parameter in covar_FUN.

The fit will be performed for all pixels specified in index, if provided. Otherwise, a random sample of length fit.n is used. If fit.n exceeds the number of pixels, all pixels are used. When random pixels are used, parameter estimates will be different for each call of the function. For reproducible results, we recommend taking a random sample of pixels manually and passing in those values as index.

Caution: Note that a distance matrix, of size \(n \times n\) must be fit to the sampled data where \(n\) is either fit.n or length(index). Take your computer's memory and processing time into consideration when choosing this size.

Parameter estimates are always returned in the same scale of distances calculated by distm_FUN. It is very important that these estimates are re-scaled by users if output of distm_FUN use units different from the desired scale. For example, if the function covar_FUN = function(d, r, a){-(d/r)^a} is used with distm_FUN = "distm_scaled", the estimated range parameter r will be based on a unit-map. Users will likely want to re-scaled it to map units by multiplying r by the maximum distance among points on your map.

If the distm_FUN is on the scale of your map (e.g., "distm_km"), re-scaling is not needed but may be preferable, since it is scaled to the maximum distance among the sampled data rather than the true maximum distance. For example, dividing the range parameter by max.distance and then multiplying it by the true max distance may provide a better range estimate.

Examples

# simulate dummy data
set.seed(19)
time.points = 30 # time series length
map.width = 8 # square map width
coords = expand.grid(x = 1:map.width, y = 1:map.width) # coordinate matrix

## create empty spatiotemporal variables:
X <- matrix(NA, nrow = nrow(coords), ncol = time.points) # response
Z <- matrix(NA, nrow = nrow(coords), ncol = time.points) # predictor

## setup first time point:
Z[, 1] <- .05*coords[,"x"] + .2*coords[,"y"]
X[, 1] <- .5*Z[, 1] + rnorm(nrow(coords), 0, .05) #x at time t

## project through time:
for(t in 2:time.points){
  Z[, t] <- Z[, t-1] + rnorm(map.width^2)
  X[, t] <- .2*X[, t-1] + .1*Z[, t] + .05*t + rnorm(nrow(coords), 0 , .25)
}

AR.map = fitAR_map(X, coords, formula = y ~ Z, X.list = list(Z = Z), resids.only = FALSE)

# using pre-defined covariance function
## exponential covariance
fitCor(AR.map$residuals, coords, covar_FUN = "covar_exp", start = list(range = .1))

## exponential-power covariance
fitCor(AR.map$residuals, coords, covar_FUN = "covar_exppow", start = list(range = .1, shape = .2))

# user-specified covariance function
fitCor(AR.map$residuals, coords, covar_FUN = function(d, r){d^r}, start = list(r = .1))

# un-scaled distances:
fitCor(AR.map$residuals, coords, distm_FUN = "distm_km", start = list(r = 106))

# specify which pixels to use, for reproducibility
fitCor(AR.map$residuals, coords, index = 1:64)$spcor #all
fitCor(AR.map$residuals, coords, index = 1:20)$spcor #first 20
fitCor(AR.map$residuals, coords, index = 21:64)$spcor # last 43
# randomly select pixels
fitCor(AR.map$residuals, coords, fit.n = 20)$spcor #random 20
fitCor(AR.map$residuals, coords, fit.n = 20)$spcor # different random 20