Space-time models: GTWR, multiscale GWR by backfitting, Gaussian hidden Markov models

GtwrFit: local weighted least squares with space-time kernel weights; distance sqrt(lam d_S^2 + mu d_T^2) (Huang et al. 2010) or lam d_S + (1 - lam) d_T + 2 sqrt(lam (1 - lam) d_S d_T) cos(ksi) (distance = "fotheringham", GWmodel::gtwr, which also uses past_only = TRUE); bisquare, gaussian or exponential kernels. MgwrBackfit: multiscale GWR with fixed per-covariate bandwidths by backfitting from OLS until sqrt(|RSS1 - RSS0| / RSS1) < threshold; centred predictors with intercepts reported on the uncentred scale (as GWmodel::gwr.multiscale with fixed bandwidths). GaussianHmm: Baum-Welch EM with scaled forward-backward recursions for one or several series sharing parameters, then Viterbi paths (0-based states).

Usage

GtwrFit(
  y,
  X,
  coords,
  times,
  bandwidth,
  lam = 1,
  mu = 1,
  kernel = "bisquare",
  distance = "huang",
  ksi = 0,
  past_only = FALSE
)

MgwrBackfit(
  y,
  X,
  coords,
  bandwidths,
  kernel = "bisquare",
  center = TRUE,
  threshold = 1e-10,
  max_iter = 5000
)

GaussianHmm(
  series,
  n_states,
  means = NULL,
  sds = NULL,
  trans = NULL,
  init = NULL,
  max_iter = 1000,
  tol = 1e-12
)

Arguments

y

Response.

X

Regressor matrix (an intercept is added).

coords

Coordinate matrix (rows are observations).

times

Observation times.

bandwidth

Space-time bandwidth.

lam, mu

Space and time distance weights.

kernel

"bisquare", "gaussian" or "exponential".

distance

"huang" or "fotheringham".

ksi

Angle of the Fotheringham distance.

past_only

Give weight only to observations at or before the focal time.

bandwidths

Bandwidths for the intercept and each covariate.

center

Centre the covariates.

threshold

Backfitting convergence threshold.

max_iter

Maximum iterations.

series

Numeric vector, or list of vectors.

n_states

Number of hidden states.

means, sds, trans, init

Optional starting values.

tol

Log-likelihood tolerance.

Value

A list.

References

Huang, B., Wu, B. and Barry, M. (2010). IJGIS 24, 383-401. Fotheringham, A. S., Crespo, R. and Yao, J. (2015). Geographical Analysis 47, 431-452. Fotheringham, A. S., Yang, W. and Kang, W. (2017). Annals AAG 107, 1247-1265. Rabiner, L. R. (1989). Proc. IEEE 77, 257-286.

Examples

GaussianHmm(c(0.1, -0.2, 0, 5.1, 4.9, 5.2, 0.2, -0.1), 2)$states
#> [[1]]
#> [1] 0 0 0 1 1 1 0 0
#>