Parallel and memory-free implementation of RE-ESF-based spatial regression for very large samples. This model estimates residual spatial dependence, constant coefficients, and non-spatially varying coefficients (NVC; coefficients varying with respect to explanatory variable value).
besf( y, x = NULL, nvc = FALSE, nvc_sel = TRUE, coords, s_id = NULL,
covmodel="exp", enum = 200, method = "reml", penalty = "bic", nvc_num = 5,
maxiter = 30, bsize = 4000, ncores = NULL )Matrix with columns for the estimated coefficients on x, their standard errors, z-values, and p-values (K x 4). Effective if nvc =FALSE
Matrix of estimated NVCs on x (N x K). Effective if nvc =TRUE
Matrix of standard errors for the NVCs on x (N x K). Effective if nvc =TRUE
Matrix of t-values for the NVCs on x (N x K). Effective if nvc =TRUE
Matrix of p-values for the NVCs on x (N x K). Effective if nvc =TRUE
Vector of estimated variance parameters (2 x 1). The first and the second elements denote the standard deviation and the Moran's I value of the estimated spatially dependent component, respectively. The Moran's I value is scaled to take a value between 0 (no spatial dependence) and 1 (the maximum possible spatial dependence). Based on Griffith (2003), the scaled Moran'I value is interpretable as follows: 0.25-0.50:weak; 0.50-0.70:moderate; 0.70-0.90:strong; 0.90-1.00:marked
Vector whose elements are residual standard error (resid_SE), adjusted conditional R2 (adjR2(cond)), restricted log-likelihood (rlogLik), Akaike information criterion (AIC), and Bayesian information criterion (BIC). When method = "ml", restricted log-likelihood (rlogLik) is replaced with log-likelihood (logLik)
List indicating whether NVC are removed or not during the BIC/AIC minimization. 1 indicates not removed whreas 0 indicates removed
Vector of estimated random coefficients on Moran's eigenvectors (L x 1)
Vector of estimated spatial dependent component (N x 1)
Vector of predicted values (N x 1)
Vector of residuals (N x 1)
List of other outputs, which are internally used
Vector of explained variables (N x 1)
Matrix of explanatory variables (N x K)
If TRUE, NVCs are assumed on x. Otherwise, constant coefficients are assumed. Default is FALSE
If TRUE, type of coefficients (NVC or constant) is selected through a BIC (default) or AIC minimization. If FALSE, NVCs are assumed across x. Alternatively, nvc_sel can be given by column number(s) of x. For example, if nvc_sel = 2, the coefficient on the second explanatory variable in x is NVC and the other coefficients are constants. The Default is TRUE
Matrix of spatial point coordinates (N x 2)
Optional. ID specifying groups modeling spatially dependent process (N x 1). If it is specified, group-level spatial process is estimated. It is useful. e.g., for multilevel modeling (s_id is given by the group ID) and panel data modeling (s_id is given by individual location id). Default is NULL
Type of kernel to model spatial dependence. The currently available options are "exp" for the exponential kernel, "gau" for the Gaussian kernel, and "sph" for the spherical kernel
Number of Moran eigenvectors to be used for spatial process modeling (scalar). Default is 200
Estimation method. Restricted maximum likelihood method ("reml") and maximum likelihood method ("ml") are available. Default is "reml"
Penalty to select type of coefficients (NVC or constant) to stablize the estimates. The current options are "bic" for the Baysian information criterion-type penalty (N x log(K)) and "aic" for the Akaike information criterion (2K) (see Muller et al., 2013). Default is "bic"
Number of basis functions used to model NVC. An intercept and nvc_num natural spline basis functions are used to model each NVC. Default is 5
Maximum number of iterations. Default is 30
Block/badge size. bsize x bsize elements are iteratively processed during the parallelized computation. Default is 4000
Number of cores used for the parallel computation. If ncores = NULL, the number of available cores - 2 is detected and used. Default is NULL
Daisuke Murakami
Griffith, D. A. (2003). Spatial autocorrelation and spatial filtering: gaining understanding through theory and scientific visualization. Springer Science & Business Media.
Murakami, D. and Griffith, D.A. (2015) Random effects specifications in eigenvector spatial filtering: a simulation study. Journal of Geographical Systems, 17 (4), 311-331.
Murakami, D. and Griffith, D.A. (2019) A memory-free spatial additive mixed modeling for big spatial data. Japan Journal of Statistics and Data Science. DOI:10.1007/s42081-019-00063-x.
resf
require(spdep)
data(boston)
y <- boston.c[, "CMEDV" ]
x <- boston.c[,c("CRIM","ZN","INDUS", "CHAS", "NOX","RM", "AGE",
"DIS" ,"RAD", "TAX", "PTRATIO", "B", "LSTAT")]
xgroup <- boston.c[,"TOWN"]
coords <- boston.c[,c("LON", "LAT")]
######## Regression considering spatially dependent residuals
#res <- besf(y = y, x = x, coords=coords)
#res
######## Regression considering spatially dependent residuals and NVC
######## (coefficients or NVC is selected)
#res2 <- besf(y = y, x = x, coords=coords, nvc = TRUE)
######## Regression considering spatially dependent residuals and NVC
######## (all the coefficients are NVCs)
#res3 <- besf(y = y, x = x, coords=coords, nvc = TRUE, nvc_sel=FALSE)
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