spmoran (version 0.2.3)

resf_vc: Gaussian and non-Gaussian spatial regression models with varying coefficients

Description

This model estimates regression coefficients, spatially varying coefficients (SVCs), non-spatially varying coefficients (NVC; coefficients varying with respect to explanatory variable value), SNVC (= SVC + NVC), group effects, and residual spatial dependence. The random-effects eigenvector spatial filtering, which is an approximate Gaussian process approach, is used for modeling the spatial process in coefficients and residuals. 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. The explained variables are transformed to fit the data distribution if nongauss is specified. Thus, this function is available for modeling Gaussian and non-Gaussian continuous data and count data (see nongauss_y).

Note 1: 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).

Note 2: The SVC model can be less accurate for large samples due to a degeneracy/over-smoothing problem (see Murakami et al., 2023). The addlearn_local is useful to mitigate this problem (See the coding example below).

Usage

resf_vc(y, x, xconst = NULL, xgroup = NULL, weight = NULL, offset = 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", miniter = NULL, maxiter = 30, nongauss = NULL )

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 deviation of the k-th SVC, while the (2, k)-th element is the Moran's I value that 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 deviations of the NVCs

s_c

Vector of standard deviations of the NVCs on xconst

s_g

Vector of standard deviations 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

Error statistics. When y_type="continuous", it includes residual standard error (resid_SE), adjusted conditional R2 (adjR2(cond)), restricted log-likelihood (rlogLik), Akaike information criterion (AIC), and Bayesian information criterion (BIC). rlogLik is replaced with log-likelihood (logLik) if method = "ml". resid_SE is replaced with the residual standard error for the transformed y (resid_SE_trans) if nongauss is specified. When y_type="count", the error statistics includes root mean squared error (RMSE), Gaussian likelihood approximating the model, AIC and BIC based on the likelihood, and the proportion of the null deviance explained by the model (deviance explained (%)). deviance explained, which is also used in the mgcv package, corresponds to the adjusted R2 in case of the linear regression

pred

Matrix of predicted values for y (pred) and their standard errors (pred_se) (N x 2). If y is transformed by specifying nongauss_y, the predicted values in the transformed/normalized scale are added as another column named pred_trans

pred_quantile

Matrix of the quantiles for the predicted values (N x 15). It is useful to evaluate uncertainty in the predictive value

tr_par

List of the parameter estimates for the tr_num SAL transformations. The k-th element of the list includes the four parameters for the k-th SAL transformation (see nongauss_y)

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)

pdf

Matrix whose first column consists of evenly spaced values within the value range of y and the second column consists of the estimated value of the probability density function for y if y_type in nongauss_y is "continuous" and probability mass function if y_type = "count". If offset is specified (and y_type = "count"), the PMF given median offset value is evaluated

skew_kurt

Skewness and kurtosis of the estimated probability density/mass function of y

other

List of other outputs, which are internally used

Arguments

y

Vector of explained variables (N x 1)

x

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

xconst

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

xgroup

Matrix of group IDs for modeling group-wise random effects. 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). If non-NULL, the adjusted R-squared value is evaluated for weighted explained variables. Default is NULL

offset

Vector of offset variables (N x 1). Available if y is count (y_type = "count" is specified in the nongauss_y function). Default is NULL

x_nvc

If TRUE, SNVCs are assumed on x (i.e., a non-linear function of x is added on each SVC to robustify the estimate). 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 for model estimation and selection. "bic" for the Baysian information criterion-type penalty (N x log(K)) and "aic" for the Akaike information criterion (2K). Default is "bic"

miniter

Minimum number of iterations. Default is NULL

maxiter

Maximum number of iterations. Default is 30

nongauss

Parameter setup for modeling non-Gaussian continuous and count data. Output from nongauss_y

Author

Daisuke Murakami

Details

This function estimates Gaussian and non-Gaussian spatial model for continuous and count data. For non-Gaussian modeling, see nongauss_y.

References

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.

Murakami, D., Kajita, M., Kajita, S. and Matsui, T. (2021) Compositionally-warped additive mixed modeling for a wide variety of non-Gaussian data. Spatial Statistics, 43, 100520.

Murakami, D., and Griffith, D.A. (2021) Balancing spatial and non-spatial variations in varying coefficient modeling: a remedy for spurious correlation. Geographical Analysis, DOI: 10.1111/gean.12310.

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, coef_marginal, besf_vc, addlearn_local

Examples

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

#####################################################
############## Gaussian SVC models ##################
#####################################################

#### SVC or constant coefficients on x ##############

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

### For large samples (n > 5,000), the following additional learning
### mitigates an degeneracy/over-smoothing problem in SVCs
# res_adj<- addlearn_local(res)
# res_adj
# plot_s(res_adj,0)
# plot_s(res_adj,1)
# plot_s(res_adj,2)

#### SVC on x #######################################

#res2	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_sel = FALSE )

#### Group-level SVC or constant coefficients on x ##
#### Group-wise random intercepts ###################

#meig_g <- meigen(coords, s_id=xgroup)
#res3	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig_g,xgroup=xgroup)


#####################################################
############## Gaussian SNVC models #################
#####################################################

#### SNVC, SVC, NVC, or constant coefficients on x ###

#res4	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_nvc =TRUE)

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

#res5	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, x_nvc =TRUE, xconst_nvc=TRUE)
#plot_s(res5,0)            # Spatially varying intercept
#plot_s(res5,1)            # Spatial plot of the SNVC (SVC + NVC) on x[,1]
#plot_s(res5,1,btype="svc")# Spatial plot of SVC in the SNVC
#plot_s(res5,1,btype="nvc")# Spatial plot of NVC in the SNVC
#plot_n(res5,1)            # 1D plot of the NVC

#plot_s(res5,6,xtype="xconst")# Spatial plot of the NVC on xconst[,6]
#plot_n(res5,6,xtype="xconst")# 1D plot of the NVC on xconst[,6]


#####################################################
############## Non-Gaussian SVC models ##############
#####################################################

#### Generalized model for continuous data ##########
# - Probability distribution is estimated from data

#ng6    <- nongauss_y( tr_num = 2 )# 2 SAL transformations to Gaussianize y
#res6	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, nongauss = ng6 )
#res6                   # tr_num may be selected by comparing BIC (or AIC)

#coef_marginal_vc(res6) # marginal effects from x (dy/dx)
#plot(res6$pdf,type="l") # Estimated probability density function
#res6$skew_kurt          # Skew and kurtosis of the estimated PDF
#res6$pred_quantile[1:2,]# predicted value by quantile


#### Generalized model for non-negative continuous data
# - Probability distribution is estimated from data

#ng7    <- nongauss_y( tr_num = 2, y_nonneg = TRUE )
#res7	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, nongauss = ng7 )
#coef_marginal_vc(res7)

#### Overdispersed Poisson model for count data #####
# - y is assumed as a count data

#ng8    <- nongauss_y( y_type = "count" )
#res8	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, nongauss = ng8 )

#### Generalized model for count data ###############
# - y is assumed as a count data
# - Probability distribution is estimated from data

#ng9    <- nongauss_y( y_type = "count", tr_num = 2 )
#res9	  <- resf_vc(y=y,x=x,xconst=xconst,meig=meig, nongauss = ng9 )