spmoran (version 0.2.1)

resf: Spatial regression for Gaussian and non-Gaussian continuous data

Description

This model estimates residual spatial dependence, constant coefficients, non-spatially varying coefficients (NVC; coefficients varying depending on x), and group effects. The random-effects eigenvector spatial filtering (RE-ESF), which is a low rank Gaussian process approach, is used for the spatial dependence modeling. Compositionally-warping function and/or the Box-Cox transformation function is available for non-Gaussian continuous data (see Murakami et al., 2020 for further detail).

Usage

resf( y, x = NULL, xgroup = NULL, weight = NULL, nvc = FALSE, nvc_sel = TRUE,
      nvc_num = 5, meig, method = "reml", penalty = "bic",
      tr_num = 0, tr_nonneg = FALSE )

Arguments

y

Vector of explained variables (N x 1)

x

Matrix of explanatory variables (N x K). Default is NULL

xgroup

Matrix of group IDs. The IDs may be group numbers or group names (N x K_group). Default is NULL

weight

Vector of weights for samples (N x 1). When non-NULL, the adjusted R-squared value is evaluated for weighted explained variables. Default is NULL

nvc

If TRUE, non-spatiallly varying coefficients (NVCs; coefficients varying with respect to explanatory variable value) are asumed. If FALSE, constant coefficients are assumed. Default is FALSE

nvc_sel

If TRUE, type of each coefficient (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 is NVC and the other coefficients are constants. Default is TRUE

nvc_num

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

meig

Moran eigenvectors and eigenvalues. Output from meigen or meigen_f

method

Estimation method. Restricted maximum likelihood method ("reml") and maximum likelihood method ("ml") are available. Default is "reml"

penalty

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). Default is "bic"

tr_num

Number of the SAL transformations ( SinhArcsinh and Affine, where the use of "L" stems from the "Linear") used to transform non-Gaussian explained variables to Gaussian variables. Default is 0

tr_nonneg

If TRUE, the Box-Cox transfromation used to transform positive non-Gaussian explained variables to Gaussian variables. If tr_num > 0 and tr_nonneg == TRUE, the Box-Cox transformation is applied first. Then, th SAL transformation is applied tr_num times. Default is FALSE

Value

b

Matrix with columns for the estimated constant coefficients on x, their standard errors, t-values, and p-values (K x 4)

b_g

List of K_group matrices with columns for the estimated group effects, their standard errors, and t-values

c_vc

Matrix of estimated NVCs on x (N x K). Effective if nvc = TRUE

cse_vc

Matrix of standard errors for the NVCs on x (N x K). Effective if nvc = TRUE

ct_vc

Matrix of t-values for the NVCs on x (N x K). Effective if nvc = TRUE

cp_vc

Matrix of p-values for the NVCs on x (N x K). Effective if nvc = TRUE

s

Vector of estimated variance parameters (2 x 1). The first and the second elements are the standard error and the Moran's I value of the estimated spatially dependent process, 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

s_c

Vector of standard errors of the NVCs on xconst

s_g

Vector of estimated standard errors of the group effects

e

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)

vc

List indicating whether NVC are removed or not during the BIC/AIC minimization. 1 indicates not removed whreas 0 indicates removed

r

Vector of estimated random coefficients on Moran's eigenvectors (L x 1)

sf

Vector of estimated spatial dependent component (N x 1)

pred

Vector of predicted values (N x 1). If tr_num > 0 or tr_nonneg == TRUE (i.e., y is trnsformed), another column including the predicted values in the transformed/normalized scale (pred_trans) is inserted into the second column

tr_par

List of the estimated parameters in the tr_num SAL transformations

tr_bpar

The estimated parameter in the Box-Cox transformation

tr_y

Vector of the transformed explaied variables

resid

Vector of residuals (N x 1)

other

List of other outputs, which are internally used

Details

If tr_num >0, the resf function iterates the SAL transformation tr_num times to transform the explained variables to Gaussian variables. The SAL transformation is defined as SAL(y)=a+b*sinh(c*arcsinh(y)-d) where a,b,c,d are parameters. Based on Rois and Tober (2019), an iteration of the SAL transformation approximates a wide variety of non-Gaussian distributions without explicitly assuming data distribution. As a result, our spatial regression approach is applicable to a wide variety of non-Gaussian continuous data too. For non-negative explained variables, a Box-Cox transformation is available prior to the SAL transformations by specifying tr_nonneg >0. tr_num and tr_nonneg can be selected by comparing the BIC (or AIC) value across models. This compositionally-warped spatial regression approach is detailed in Murakami et al. (2020).

References

Murakami, D., Kajita, M., Kajita, S. and Matsui, T. (2020) Compositionally-warped additive mixed modeling for a wide variety of non-Gaussian data, Arxiv.

Rios, G. and Tobar, F. (2019) Compositionally-warped Gaussian processes. Neural Networks, 118, 235-246.

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.

Griffith, D. A. (2003). Spatial autocorrelation and spatial filtering: gaining understanding through theory and scientific visualization. Springer Science & Business Media.

See Also

meigen, meigen_f, besf

Examples

# NOT RUN {
require(spdep);require(Matrix)
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")]
meig 	<- meigen(coords=coords)
# meig<- meigen_f(coords=coords)  ## for large samples

######## Regression considering residual spatially dependence
res	  <- resf(y = y, x = x, meig = meig)
res
plot_s(res)    ## spatially dependent component (intercept)

######## Compositionally-warped spatial regression (2 SAL transformations)
res2	  <- resf(y = y, x = x, meig = meig, tr_num = 2)
res2               ## tr_num and tr_nonneg can be selected by comparing BIC (or AIC)
coef_marginal(res2)## marginal effects of x. The median might be useful as a summary statistic

######## Compositionally-warped spatial regression (2 SAL trans. + Box-Cox trans.)
res3	  <- resf(y = y, x = x, meig = meig, tr_num = 2, tr_nonneg=TRUE)
res3            ## tr_num and tr_nonneg can be selected by comparing BIC (or AIC)
coef_marginal(res3)

######## Regression considering residual spatially dependence and NVC
######## (constant coefficients or NVC is selected)
#res4 <- resf(y = y, x = x, meig = meig, nvc = TRUE)
#res4          ## Note: Coefficients on 5,6,and 13-th covariates
               ## are estimated non-spatially varying (NVC) depending on x

#plot_n(res4,5) ## 1D plot of the 5-th NVC
#plot_n(res4,6) ## 1D plot of the 6-th NVC
#plot_n(res4,13)## 1D plot of the 13-th NVC

#plot_s(res4)   ## spatially dependent component (intercept)
#plot_s(res4,5) ## spatial plot of the 5-th NVC
#plot_s(res4,6) ## spatial plot of the 6-th NVC
#plot_s(res4,13)## spatial plot of the 13-th NVC

######## Compositionally-warped spatial regression with NVC (2 SAL trans. + Box-Cox trans.)
######## (constant coefficients or NVC is selected)
#res5 <- resf(y = y, x = x, meig = meig, nvc = TRUE, tr_num = 2, tr_nonneg=TRUE)

######## Regression considering residual spatially dependence and NVC
######## (all the coefficients are NVCs)
#res6 <- resf(y = y, x = x, meig = meig, nvc = TRUE, nvc_sel=FALSE)

######## Regression considering residual spatially dependence and group effects
#res7 <- resf(y = y, x = x, meig = meig, xgroup = xgroup)

######## Regression considering group-level spatially dependence and group effects
#meig_g<- meigen(coords=coords, s_id = xgroup)
#res8 <- resf(y = y, x = x, meig = meig_g, xgroup = xgroup)

# }