spmoran (version 0.2.1)

resf_vc: Spatially and non-spatially varying coefficient (SNVC) modeling for Gaussian and non-Gaussian continuous data

Description

The model estimates residual spatial dependence, constant coefficients, spatially varying coefficients (SVCs), non-spatially varying coefficients (NVC; coefficients varying with respect to explanatory variable value), SNVC (= SVC + NVC), and group effects. The random-effects eigenvector spatial filtering (RE-ESF), which is a low rank Gaussian process approach, is used for the spatial process modeling. While the resf_vc function estimates a SVC model by default, the type of coefficients (constant, SVC, NVC, or SNVC) can be selected through a BIC/AIC minimization. In addition, to transform non-Gaussian explained variables to Gaussian variables, the compositionally-warping function and/or the Box-Cox transformation is available (see Murakami et al., 2020).

Note: SNVCs can be mapped just like SVCs. SNVC model is more robust against spurious correlation (multicollinearity) and stable than SVC models (see Murakami and Griffith, 2020).

Usage

resf_vc(y, x, xconst = NULL, xgroup = NULL, weight = NULL,
        x_nvc = FALSE, xconst_nvc = FALSE, x_sel = TRUE, x_nvc_sel = TRUE,
        xconst_nvc_sel = TRUE, nvc_num = 5, meig, method = "reml",
        penalty = "bic", maxiter = 30, tr_num = 0, tr_nonneg = FALSE )

Arguments

y

Vector of explained variables (N x 1)

x

Matrix of explanatory variables with spatially varying coefficients (SVC) (N x K)

xconst

Matrix of explanatory variables with constant coefficients (N x K_c). Default is NULL

xgroup

Matrix of group IDs. The IDs may be group numbers or group names (N x K_g). 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

x_nvc

If TRUE, SNVCs are assumed on x. Otherwise, SVCs are assumed. Default is FALSE

xconst_nvc

If TRUE, NVCs are assumed on xconst. Otherwise, constant coefficients are assumed. Default is FALSE

x_sel

If TRUE, type of coefficient (SVC or constant) on x is selected through a BIC (default) or AIC minimization. If FALSE, SVCs are assumed across x. Alternatively, x_sel can be given by column number(s) of x. For example, if x_sel = 2, the coefficient on the second explanatory variable in x is SVC and the other coefficients are constants. The Default is TRUE

x_nvc_sel

If TRUE, type of coefficient (NVC or constant) on x is selected through the BIC (default) or AIC minimization. If FALSE, NVCs are assumed across x. Alternatively, x_nvc_sel can be given by column number(s) of x. For example, if x_nvc_sel = 2, the coefficient on the second explanatory variable in x is NVC and the other coefficients are constants. The Default is TRUE

xconst_nvc_sel

If TRUE, type of coefficient (NVC or constant) on xconst is selected through the BIC (default) or AIC minimization. If FALSE, NVCs are assumed across xconst. Alternatively, xconst_nvc_sel can be given by column number(s) of xconst. For example, if xconst_nvc_sel = 2, the coefficient on the second explanatory variable in xconst is NVC and the other coefficients are constants.The 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 varying coefficients and 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"

maxiter

Maximum number of iterations. Default is 30

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_vc

Matrix of estimated spatially and non-spatially varying coefficients (SNVC = SVC + NVC) on x (N x K)

bse_vc

Matrix of standard errors for the SNVCs on x (N x k)

t_vc

Matrix of t-values for the SNVCs on x (N x K)

p_vc

Matrix of p-values for the SNVCs on x (N x K)

B_vc_s

List summarizing estimated SVCs (in SNVC) on x. The four elements are the SVCs (N x K), the standard errors (N x K), t-values (N x K), and p-values (N x K), respectively

B_vc_n

List summarizing estimated NVCs (in SNVC) on x. The four elements are the NVCs (N x K), the standard errors (N x K), t-values (N x K), and p-values (N x K), respectively

c

Matrix with columns for the estimated coefficients on xconst, their standard errors, t-values, and p-values (K_c x 4). Effective if xconst_nvc = FALSE

c_vc

Matrix of estimated NVCs on xconst (N x K_c). Effective if xconst_nvc = TRUE

cse_vc

Matrix of standard errors for the NVCs on xconst (N x k_c). Effective if xconst_nvc = TRUE

ct_vc

Matrix of t-values for the NVCs on xconst (N x K_c). Effective if xconst_nvc = TRUE

cp_vc

Matrix of p-values for the NVCs on xconst (N x K_c). Effective if xconst_nvc = TRUE

b_g

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

s

List of variance parameters in the SNVC (SVC + NVC) on x. The first element is a 2 x K matrix summarizing variance parameters for SVC. The (1, k)-th element is the standard error of the k-th SVC, while the (2, k)-th element is the Moran's I value is scaled to take a value between 0 (no spatial dependence) and 1 (strongest 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. The second element of s is the vector of standard errors of the NVCs

s_c

Vector of standard errors of the NVCs on xconst

s_g

Vector of standard errors of the group effects

vc

List indicating whether SVC/NVC are removed or not during the BIC/AIC minimization. 1 indicates not removed (replaced with constant) whreas 0 indicates removed

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)

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. (2020) Balancing spatial and non-spatial variations in varying coefficient modeling: a remedy for spurious correlation. ArXiv.

Murakami, D., Yoshida, T., Seya, H., Griffith, D.A., and Yamagata, Y. (2017) A Moran coefficient-based mixed effects approach to investigate spatially varying relationships. Spatial Statistics, 19, 68-89.

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_vc

Examples

# NOT RUN {
require(spdep)
data(boston)
y	      <- boston.c[, "CMEDV"]
x       <- boston.c[,c("CRIM", "AGE")]
xconst  <- boston.c[,c("ZN","DIS","RAD","NOX",  "TAX","RM", "PTRATIO", "B")]
xgroup  <- boston.c[,"TOWN"]
coords  <- boston.c[,c("LON", "LAT")]
meig 	  <- meigen(coords=coords)
# meig	<- meigen_f(coords=coords)  ## for large samples

############## SVC modeling1 #################
######## - SVC or constant coefficients on x
######## - Constant coefficients on xconst
res	    <- resf_vc(y=y,x=x,xconst=xconst,meig=meig )
res

plot_s(res,0) # Spatially varying intercept
plot_s(res,1) # 1st SVC (Not shown because the SVC is estimated constant)
plot_s(res,2) # 2nd SVC

############## Compositionally-warped SVC modeling #################
######## - SVC or constant coefficients on x
######## - Constant coefficients on xconst
######## - 2 SAL transformations on y

#res2	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, tr_num = 2 )
#res2                   # tr_num and tr_nonneg can be selected by comparing BIC (or AIC)
#coef_marginal_vc(res2) # marginal effects of x.
                        # The median might be useful as a summary statistic

######## - 2 SAL transformations + Box-Cox transformation on y

#res3	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, tr_num = 2, tr_nonneg=TRUE )
#res3
#coef_marginal_vc(res3)

############## SVC modeling2 #################
######## (SVC on x; Constant coefficients on xconst)
#res4	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_sel = FALSE )

############## SVC modeling3 #################
######## - Group-level SVC or constant coefficients on x
######## - Constant coefficients on xconst
######## - Group effects

#meig_g <- meigen(coords, s_id=xgroup)
#res5	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig_g,xgroup=xgroup)

############## SNVC modeling1 #################
######## - SNVC, SVC, NVC, or constant coefficients on x
######## - Constant coefficients on xconst

#res6	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_nvc =TRUE)


############## Compositionally-warped SNVC modeling #################
######## - SNVC, SVC, NVC, or constant coefficients on x
######## - 2 SAL transformations + Box-Cox transformation on y

#res7	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_nvc =TRUE, tr_num = 2, tr_nonneg=TRUE)
#res7       # tr_num and tr_nonneg can be selected by comparing BIC (or AIC)


############## SNVC modeling2 #################
######## - SNVC, SVC, NVC, or constant coefficients on x
######## - NVC or Constant coefficients on xconst

#res8	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_nvc =TRUE, xconst_nvc=TRUE)
#plot_s(res8,0)            # Spatially varying intercept
#plot_s(res8,1)            # Spatial plot of 1st SNVC (SVC + NVC)
#plot_s(res8,1,btype="svc")# Spatial plot of SVC in the SNVC on x[,1]
#plot_s(res8,1,btype="nvc")# Spatial plot of NVC in the SNVC on x[,1]
#plot_n(res8,1)            # 1D plot of NVC in the SNVC on x[,1]

#plot_s(res8,6,xtype="xconst")# Spatial plot of NVC in the SNVC on xconst[,6]
#plot_n(res8,6,xtype="xconst")# 1D plot of NVC on xconst[,6]

# }