resf {spmoran}

Gaussian and non-Gaussian spatial regression models

Description

This model estimates regression coefficients, coefficients varying depending on x (non-spatially varying coefficients; 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 dependence. 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).

Usage

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

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_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

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"

nongauss

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

Details

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

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_g 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 deviation 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 deviations of the NVCs on xconst

s_g

Vector of estimated standard deviations of the group effects

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 contains 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

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

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 (PMF) 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

Author(s)

Daisuke Murakami

References

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. (2020) Balancing spatial and non-spatial variations in varying coefficient modeling: a remedy for spurious correlation. Geographical Analysis, DOI: 10.1111/gean.12310.

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.

See Also

meigen, meigen_f, coef_marginal, besf

Examples

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

#####################################################
######## Gaussian spatial regression models #########
#####################################################

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

######## Group-wise random intercepts ###############
#res2 <- resf(y = y, x = x, meig = meig, xgroup = xgroup)

######## Group-wise random intercepts and ###########
######## Group-level spatial dependence   ###########

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

######## Coefficients varying depending on x ########

#res4  <- resf(y = y, x = x, meig = meig, nvc = TRUE)
#res4

#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

#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

#####################################################
###### Non-Gaussian spatial regression models #######
#####################################################

#### Generalized model for continuous data ##############
# - Data distribution is estimated

#ng5    <- nongauss_y( tr_num = 2 )# 2 SAL transformations to Gaussianize y
#res5	  <- resf(y = y, x = x, meig = meig, nongauss = ng5)
#res5              ## tr_num may be selected by comparing BIC (or AIC)

#plot(res5$pdf,type="l") # Estimated probability density function
#res5$skew_kurt          # Skew and kurtosis of the estimated PDF
#res5$pred_quantile[1:2,]# predicted value by quantile
#coef_marginal(res5)     # Estimated marginal effects (dy/dx)


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

#ng6    <- nongauss_y( tr_num = 2, y_nonneg = TRUE )
#res6	  <- resf(y = y, x = x, meig = meig, nongauss = ng6 )
#coef_marginal(res6)

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

#ng7    <- nongauss_y( y_type = "count" )
#res7	  <- resf(y = y, x = x, meig = meig, nongauss = ng7 )


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

#ng8    <- nongauss_y( y_type = "count", tr_num = 2 )
#res8	  <- resf(y = y, x = x, meig = meig, nongauss = ng8 )