Description Usage Arguments Details Value Author(s) References See Also Examples

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. This function is available for modeling Gaussian and non-Gaussian data including continuous and count data (see `nongauss_y`

).

1 2 3 4 |

`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). 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 |

`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 |

`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 |

`tr_nonneg` |
Deprecated. Use the nongauss function |

`tr_num` |
Deprecated. Use the nongauss function |

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

.

`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` |
Error statistics. If 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) while rlogLik is replaced with log-likelihood (logLik) if method = "ml". If y_type="count", it includes deviance explained, Gaussian likelihood approximating the model, (Gaussian) AIC, and BIC |

`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 |

`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 |

`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 |

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

`other` |
List of other outputs, which are internally used |

Daisuke Murakami

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., 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.

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

`meigen`

, `meigen_f`

, `coef_marginal`

, `besf`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | ```
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 )
``` |

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