# Coarse-to-fine spatial linear modeling Prediction and regression via coarse-to-fine spatial linear modeling. ```r cf_lm(y, x, coords, x0 = NULL, coords0 = NULL, mod_hv) ``` ## Arguments - `y`: Vector of response variables (N x 1). - `x`: Matrix of covariates (N x K). - `coords`: Matrix of 2-dimensional point coordinates (N x 2). - `x0`: Optional. Matrix of covariates at prediction sites (N0 x K). - `coords0`: Optional. Matrix of 2-dimensional point coordinates at prediction sites (N0 x 2). - `mod_hv`: Output object of the `cf_lm_hv` function. ## Returns A list with the following elements: - **beta**: Regression coefficients, their standard errors, and the lower and upper limits of the 95 percent confidence intervals. - **sd_summary**: Standard deviation of the regression term (xb), spatial process (spatial_scale1, spatial_scale2,...), additional learning, and residuals. - **e_summary**: R-squared and RMSE for validation samples, and residual standard deviation (residual_SD), and root mean squared error for the validation samples (validation_RMSE) - **pred**: Predictive means and standard deviations (sample sites). - **pred0**: Predictive means and standard deviations (prediction sites). - **bands**: Bandwidth values for each scale. The i-th bandwidth is used for the spatial process corresponding to the i-th column of the Z matrix). - **Z**: Predictive mean of the spatial process in each scale (sample sites; list). - **Z_sd**: Predictive standard deviation of the spatial process in each scale (sample sites; list). - **Z0**: Predictive mean of the spatial process in each scale (prediction sites; list). - **Z0_sd**: Predictive standard deviation of the spatial process in each scale (prediction sites; list). - **Other**: Other internal output objects. ## Examples ```r set.seed(123) require(sp); require(sf) data(meuse) data(meuse.grid) ### Data y <- log(meuse[,"zinc"]) coords <- meuse[,c("x","y")] x <- data.frame(dist = meuse[,"dist"], ffreq2 = as.integer(meuse$ffreq == 2), ffreq3 = as.integer(meuse$ffreq == 3)) ### Data at prediction sites coords0 <- meuse.grid[,c("x","y")] x0 <- data.frame(dist = meuse.grid[,"dist"], ffreq2 = as.integer(meuse.grid$ffreq == 2), ffreq3 = as.integer(meuse.grid$ffreq == 3)) ### Holdout validation optimizing the number of spatial scales mod_hv <- cf_lm_hv(y = y, x = x, coords = coords, add_learn = "none") ### Spatial modeling and prediction mod <- cf_lm(y = y, x = x, x0 = x0, coords = coords, coords0 = coords0, mod_hv = mod_hv) mod ### Mapping predictive mean and standard deviations (SD) meuse.grid$pred <- mod$pred0$pred meuse.grid$pred_sd<- mod$pred0$pred_sd meuse.grid_sf <- st_as_sf(meuse.grid, coords = c("x","y")) plot(meuse.grid_sf[,"pred"], pch = 15, cex = 0.5, nbreaks = 20) # Predictive mean plot(meuse.grid_sf[,"pred_sd"], pch = 15, cex = 0.5, nbreaks = 20)# Predictive SD ### Multiscale spatial pattern/feature extraction mod_s1<- sp_scalewise(mod,bw_range=c(1000,Inf)) # Large scale (1000 <= bandwdith) mod_s2<- sp_scalewise(mod,bw_range=c(500,1000)) # Middle scale (500 <= bandwdith <= 1000) mod_s3<- sp_scalewise(mod,bw_range=c(0,500)) # Small scale (bandwdith <= 500) z1 <- mod_s1$pred0$pred # Predictive mean z2 <- mod_s2$pred0$pred z3 <- mod_s3$pred0$pred z1_sd <- mod_s1$pred0$pred_sd # Predictive SD z2_sd <- mod_s2$pred0$pred_sd z3_sd <- mod_s3$pred0$pred_sd meuse.grid_sf3 <- cbind(meuse.grid_sf, z1, z2, z3, z1_sd, z2_sd, z3_sd) plot(meuse.grid_sf3[,c("z1","z2","z3")], pch = 15, cex = 0.5, nbreaks = 20,key.pos=4,axes=TRUE) # Predictive means plot(meuse.grid_sf3[,c("z1_sd","z2_sd","z3_sd")], pch = 15, cex = 0.5, nbreaks = 20,key.pos=4,axes=TRUE) # Predictive SD ``` ## References Murakami, D., Comber, A., Yoshida, T., Tsutsumida, N., Brunsdon, C., & Nakaya, T. (2025). Coarse-to-fine spatial modeling: A scalable, machine-learning-compatible spatial model. *arXiv:2510.00968*. ## See Also `cf_lm_hv`, `sp_scalewise` ## Author(s) Daisuke Murakami

cf_lm function