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

Memory-free implementation of SNVC modeling for very large samples. 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), and SNVC (= SVC + NVC). Type of coefficients can be selected through BIC/AIC minimization. By default, it estimates a SVC model.

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

1 2 3 4 |

`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 |

`coords` |
Matrix of spatial point coordinates (N x 2) |

`s_id` |
Optional. ID specifying groups modeling spatially dependent process (N x 1). If it is specified, group-level spatial process is estimated. It is useful for multilevel modeling (s_id is given by the group ID) and panel data modeling (s_id is given by individual location id). 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 |

`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 (SNVC, SVC, 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) (see Muller et al., 2013). Default is "bic" |

`maxiter` |
Maximum number of iterations. Default is 30 |

`covmodel` |
Type of kernel to model spatial dependence. The currently available options are "exp" for the exponential kernel, "gau" for the Gaussian kernel, and "sph" for the spherical kernel |

`enum` |
Number of Moran eigenvectors to be used for spatial process modeling (scalar). Default is 200 |

`bsize` |
Block/badge size. bsize x bsize elements are iteratively processed during the parallelized computation. Default is 4000 |

`cl` |
Number of cores used for the parallel computation. If cl = NULL, the number of available cores is detected. Default is NULL |

`b_vc` |
Matrix of estimated SNVC (= SVC + NVC) on x (N x K) |

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

`z_vc` |
Matrix of z-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), z-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), z-values (N x K), and p-values (N x K), respectively |

`c` |
Matrix with columns for the estimated coefficients on xconst, their standard errors, z-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 |

`cz_vc` |
Matrix of z-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 |

`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 |

`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) |

`resid` |
Vector of residuals (N x 1) |

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

Daisuke Murakami

Muller, S., Scealy, J.L., and Welsh, A.H. (2013) Model selection in linear mixed models. Statistical Science, 28 (2), 136-167.

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., and Griffith, D.A. (2019). Spatially varying coefficient modeling for large datasets: Eliminating N from spatial regressions. Spatial Statistics, 30, 39-64.

Murakami, D. and Griffith, D.A. (2019) A memory-free spatial additive mixed modeling for big spatial data. Japan Journal of Statistics and Data Science. DOI:10.1007/s42081-019-00063-x.

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

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 | ```
require(spdep)
data(boston)
y <- boston.c[, "CMEDV"]
x <- boston.c[,c("ZN", "INDUS" ,"LSTAT")]
xconst <- boston.c[,c("CRIM", "NOX", "CHAS", "AGE", "DIS", "RAD", "TAX", "PTRATIO", "B", "RM" )]
coords <- boston.c[,c("LAT","LON")]
######## SVC model
# res <- besf_vc(y=y,x=x,xconst=xconst,coords=coords)
######## SNVC model
# res2 <- besf_vc(y=y,x=x,xconst=xconst,coords=coords,x_nvc=TRUE)
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")]
############## SVC modeling1 #################
######## (SVC on x; Constant coefficients on xconst)
#res <- besf_vc(y=y,x=x,xconst=xconst,coords=coords, x_sel = FALSE )
#res
#plot_s(res,0) # Spatially varying intercept
#plot_s(res,1) # 1st SVC
#plot_s(res,2) # 2nd SVC
############## SVC modeling2 #################
######## (SVC or constant coefficients on x; Constant coefficients on xconst)
#res2 <- besf_vc(y=y,x=x,xconst=xconst,coords=coords )
############## SVC modeling3 #################
######## - Group-level SVC or constant coefficients on x
######## - Constant coefficients on xconst
#res3 <- besf_vc(y=y,x=x,xconst=xconst,coords=coords, s_id=xgroup)
############## SNVC modeling1 #################
######## - SNVC, SVC, NVC, or constant coefficients on x
######## - Constant coefficients on xconst
#res4 <- besf_vc(y=y,x=x,xconst=xconst,coords=coords, x_nvc =TRUE)
############## SNVC modeling2 #################
######## - SNVC, SVC, NVC, or constant coefficients on x
######## - NVC or Constant coefficients on xconst
#res5 <- besf_vc(y=y,x=x,xconst=xconst,coords=coords, x_nvc =TRUE, xconst_nvc=TRUE)
#plot_s(res5,0) # Spatially varying intercept
#plot_s(res5,1) # 1st SNVC
#plot_s(res5,1,snvc=FALSE)# SVC in the 1st SNVC
#plot_n(res5,1,xtype="x") # NVC in the 1st NVC
#plot_n(res5,6,xtype="xconst")
``` |

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.