fitVAR function

VAR model parameters to simulate correlated parent Gaussian random vectors and fields

Compute VAR model parameters to simulate parent Gaussian random vectors with specified spatiotemporal correlation structure using the method described by Biller and Nelson (2003).

fitVAR(
  spacepoints,
  p,
  margdist,
  margarg,
  p0,
  distbounds = c(-Inf, Inf),
  stcsid,
  stcsarg,
  scalefactor = 1,
  anisotropyid = "affine",
  anisotropyarg = list(phi1 = 1, phi2 = 1, phi12 = 0, theta = 0),
  advectionid = "uniform",
  advectionarg = list(u = 0, v = 0)
)

Arguments

• spacepoints: it can be a numeric integer, which is interpreted as the side length m of the square field (m x m), or a matrix (d x 2) of coordinates (e.g. longitude and latitude) of d spatial locations (e.g. d gauge stations)
• p: order of VAR(p) model
• margdist: target marginal distribution of the field
• margarg: list of marginal distribution arguments. Please consult the documentation of the selected marginal distribution indicated in the argument margdist for the list of required parameters
• p0: probability zero
• distbounds: distribution bounds (default set to c(-Inf, Inf))
• stcsid: spatiotemporal correlation structure ID
• stcsarg: list of spatiotemporal correlation structure arguments. Please consult the documentation of the selected spatiotemporal correlation structure indicated in the argument stcsid for the list of required parameters
• scalefactor: factor specifying the distance between the centers of two pixels (default set to 1)
• anisotropyid: spatial anisotropy ID (affine by default, swirl or wave)
• anisotropyarg: list of arguments characterizing the spatial anisotropy according to the syntax of the function anisotropyT. Isotropic fields by default
• advectionid: advection field ID (uniform by default, rotation, spiral, spiralCE, radial, or hyperbolic)
• advectionarg: list of arguments characterizing the advection field according to the syntax of the function advectionF. No advection by default

Details

The fitting algorithm has $O(m*m)^3$ complexity for a $(m*m)$ field or equivalently $O(d^3)$ complexity for a $d$-dimensional vector. Very large values of $(m*m)$ (or $d$) and high order AR correlation structures can be unpractical on standard machines.

Here, we give indicative CPU times for some settings, referring to a Windows 10 Pro x64 laptop with Intel(R) Core(TM) i7-6700HQ CPU @ 2.60GHz, 4-core, 8 logical processors, and 32GB RAM.

: CPU time:

d = 100 or m = 10, p = 1: ~ 0.4s

d = 900 or m = 30, p = 1: ~ 6.0s

d = 900 or m = 30, p = 5: ~ 47.0s

d = 2500 or m = 50, p = 1: ~100.0s

Note

While all the advection types can be applied to isotropic random fields, anisotropic random fields require more care. We suggest combining affine

anysotropy with uniform advection, and swirl anisotropy with rotation or spiral advection with the same rotation center.

Examples

## for multivariate simulation
coord <- cbind(runif(4)*30, runif(4)*30)

fit <- fitVAR(
  spacepoints = coord,
  p = 1,
  margdist ='burrXII',
  margarg = list(scale = 3,
                 shape1 = .9,
                 shape2 = .2),
  p0 = 0.8,
  stcsid = "clayton",
  stcsarg = list(scfid = "weibull",
                 tcfid = "weibull",
                 copulaarg = 2,
                 scfarg = list(scale = 20,
                               shape = 0.7),
                 tcfarg = list(scale = 1.1,
                               shape = 0.8))
)

dim(fit$alpha)
dim(fit$res.cov)

fit$m
fit$margarg
fit$margdist

## for random fields simulation
fit <- fitVAR(
  spacepoints = 10,
  p = 1,
  margdist ='burrXII',
  margarg = list(scale = 3, shape1 = .9, shape2 = .2),
  p0 = 0.8,
  stcsid = "clayton",
  stcsarg = list(scfid = "weibull", tcfid = "weibull",
                 copulaarg = 2,
                 scfarg = list(scale = 20, shape = 0.7),
                 tcfarg = list(scale = 1.1, shape = 0.8))
)

dim(fit$alpha)
dim(fit$res.cov)

fit$m
fit$margarg
fit$margdist

References

Biller, B., Nelson, B.L. (2003). Modeling and generating multivariate time-series input processes using a vector autoregressive technique. ACM Trans. Model. Comput. Simul. 13(3), 211-237, tools:::Rd_expr_doi("10.1145/937332.937333")

Papalexiou, S.M. (2018). Unified theory for stochastic modelling of hydroclimatic processes: Preserving marginal distributions, correlation structures, and intermittency. Advances in Water Resources, 115, 234-252, tools:::Rd_expr_doi("10.1016/j.advwatres.2018.02.013")

Papalexiou, S.M., Serinaldi, F. (2020). Random Fields Simplified: Preserving Marginal Distributions, Correlations, and Intermittency, With Applications From Rainfall to Humidity. Water Resources Research, 56(2), e2019WR026331, tools:::Rd_expr_doi("10.1029/2019WR026331")

Papalexiou, S.M., Serinaldi, F., Porcu, E. (2021). Advancing Space-Time Simulation of Random Fields: From Storms to Cyclones and Beyond. Water Resources Research, 57, e2020WR029466, tools:::Rd_expr_doi("10.1029/2020WR029466")