# Introduction

mgwrsar package estimates linear and local linear models with spatial autocorrelation of the following forms:

$y=\beta_v(u_i,v_i) X_v+\,\epsilon_i\;\;\;\;\;\;\;\; (GWR)$

$y=\beta_c X_c+\beta_v(u_i,v_i) X_v+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR)$

$y=\lambda Wy+\beta_c X_c+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR-SAR(0,k,0))$

$y=\lambda Wy+\beta_v(u_i,v_i)X_v+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR-SAR(0,0,k))$

$y=\lambda Wy+\beta_c X_c+\beta_v(u_i,v_i)X_v+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR-SAR(0,k_c,k_v))$

$y=\lambda(u_i,v_i) Wy+\beta_c X_c+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR-SAR(1,k,0))$

$y=\lambda(u_i,v_i)Wy+\beta_v(u_i,v_i)X_v+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR-SAR(1,0,k))$

$y=\lambda(u_i,v_i)Wy+\beta_cX_c+\beta_v(u_i,v_i)X_v+\,\epsilon_i\;\;\;\;\;\;\;\; (MGWR-SAR(1,k_c,k_v))$

For more details on the estimation method, see Geniaux, G. and Martinetti, D. (2017). A new method for dealing simultaneously with spatial autocorrelation and spatial heterogeneity in regression models. Regional Science and Urban Economics. (https://doi.org/10.1016/j.regsciurbeco.2017.04.001)

The estimation of the previous model can be done using the MGWRSAR wrapper function which requires a dataframe and a matrix of coordinates. Note that:

• When the model implies spatial autocorrelation, a row normalized spatial weight matrix must be provided.
• 2SLS and Best 2SLS method can be used. When model imply local regressions, a bandwidth and a kernel type must be provided.
• When the model implies mixed local regression, the name of stationary covariates must be provided.
• Optimal bandwidth can be estimated using bandwidths_mgwrsar function.

In addition to the ability of considering spatial autocorrelation in GWR/MGWR like models, MGWRSAR function introduces several useful technics for estimating local regression with space coordinates:

• it uses RCCP and RCCPeigen code that speed up computations and allows parallel computing (doParallel package),
• it allows to drop out variables with not enough local variance in local regression, which allows to consider dummies in GWR/MGWR framework without trouble.
• it allows to drop out local outliers in local regression.
• it allows to consider additional variable for kernel, including time (asymmetric kernel) and categorical variables (see Li and Racine 2010). Experimental version.

# Using mgwrsar package

The MGWRSAR function requires a dataframe and a matrix of coordinates, and eventually a spatial weight matrix if model include spatiale dependence. The package includes a simulated data example which is used for this vignette.

library(mgwrsar)
library(dplyr)
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#>     filter, lag
#> The following objects are masked from 'package:base':
#>
#>     intersect, setdiff, setequal, union
data(mydata)
coords=as.matrix(mydata[,c("x","y")])
## Creating a spatial weight matrix (sparce dgCMatrix) of 8 nearest neighbors with 0 in diagonal
W=kernel_matW(H=4,kernels='rectangle',coord_i=coords,NN=4,adaptive=TRUE,diagnull=TRUE,rowNorm=T)

## Estimation

Estimation of a GWR with a gauss kernel without parallel computation:


### without parallel computing with distance computation for all points
ptm1<-proc.time()
model_GWR0<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.13, Model = 'GWR',control=list(SE=T,get_ts=TRUE))
(proc.time()-ptm1)[3]
#> elapsed
#>   0.617

summary_mgwrsar(model_GWR0)
#> Call:
#> MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,
#>     fixed_vars = NULL, kernels = c("gauss"), H = 0.13, Model = "GWR",
#>     control = list(SE = T, get_ts = TRUE))
#> Model: GWR
#> Kernels function: gauss
#> Kernels type: GD
#> Bandwidth: 0.13
#> Computation time: 0.615
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: NO
#> Use of Target Points: NO
#> Number of data points: 1000
#> Varying parameters: Intercept X1 X2 X3
#>         Intercept       X1       X2      X3
#> Min.     -2.63740  0.16880 -0.37306 -1.7588
#> 1st Qu.  -1.46100  0.38590  1.52729 -1.3070
#> Median   -1.08812  0.51409  1.94806 -1.0194
#> Mean     -0.97597  0.48879  1.94778 -1.0289
#> 3rd Qu.  -0.62307  0.59454  2.58871 -0.7439
#> Max.      1.06100  0.79570  3.37755 -0.3083
#> Effective degrees of freedom: 955.0674
#> AICc: 1560.364
#> Residual sum of squares: 64.0684
#> RMSE: 0.2531174

### with parallel computing with distance computation for all points
ptm1<-proc.time()
model_GWR0_parallel<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.03, Model = 'GWR',control=list(SE=T,ncore=2,doMC=TRUE))
(proc.time()-ptm1)[3]
#> elapsed
#>   0.531

# W<-spdep::mat2listw(W)
#
# gp2mM <- lagsarlm(Y_gwr ~ X1+X2+X3,data=mydata, lW, method="Matrix")
#
# mdo<-s2sls(Y_gwr ~ X1+X2+X3,data=mydata,Produc,W)
#
# mm<-lagmess(Y_gwr ~ X1+X2+X3,data=mydata, lW)

## how to speed up computation using rough kernels (0 weights for nearest neighbors > 300th position)
ptm1<-proc.time()
model_GWR<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.03, Model = 'GWR',control=list(SE=TRUE,NN=300))
(proc.time()-ptm1)[3]
#> elapsed
#>   0.347

summary_mgwrsar(model_GWR)
#> Call:
#> MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,
#>     fixed_vars = NULL, kernels = c("gauss"), H = 0.03, Model = "GWR",
#>     control = list(SE = TRUE, NN = 300))
#> Model: GWR
#> Kernels function: gauss
#> Kernels type: GD
#> Bandwidth: 0.03
#> Computation time: 0.343
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: YES, 300 neighbors / 1000
#> Use of Target Points: NO
#> Number of data points: 1000
#> Varying parameters: Intercept X1 X2 X3
#>         Intercept        X1        X2      X3
#> Min.    -3.793217  0.016931 -0.765588 -2.0329
#> 1st Qu. -1.480176  0.351177  1.095174 -1.2921
#> Median  -0.835614  0.542020  1.631293 -1.0081
#> Mean    -0.830668  0.500057  1.674643 -1.0011
#> 3rd Qu. -0.236283  0.637046  2.453404 -0.7241
#> Max.     2.102565  0.942520  4.035453  0.0457
#> Effective degrees of freedom: 572.4888
#> AICc: 1179.575
#> Residual sum of squares: 1.797923
#> RMSE: 0.04240192

## how to speed up computation using rough kernels ( 0 weights for nearest neighbors > 300th position)
TP=find_TP(formula = 'Y_gwr~X1+X2+X3', data =mydata,coords=coords,K=10,type='residuals')
# only 90 targets points are used
length(TP)
#> [1] 31

## how to speed up computation using Target points
ptm1<-proc.time()
model_GWR<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.03, Model = 'GWR',control=list(SE=TRUE,TP=TP))
(proc.time()-ptm1)[3]
#> elapsed
#>   0.182

## how to speed up computation using rough kernels ( 0 weights for nearest neighbors > 300th position)  and Target points

ptm1<-proc.time()
model_GWR<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.1, Model = 'GWR',control=list(SE=TRUE,NN=300,TP=TP))
(proc.time()-ptm1)[3]
#> elapsed
#>   0.174

## how to speed up computation using adapative kernels
ptm1<-proc.time()
model_GWR<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('bisq'),H=20, Model = 'GWR',control=list(SE=TRUE,adaptive=TRUE))
(proc.time()-ptm1)[3]
#> elapsed
#>   0.222

## how to speed up computation using adapative kernels and Target points
ptm1<-proc.time()
model_GWR<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('bisq'),H=20, Model = 'GWR',control=list(SE=TRUE,adaptive=TRUE,TP=TP))
(proc.time()-ptm1)[3]
#> elapsed
#>    0.17

library(microbenchmark)
res=microbenchmark(MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.1, Model = 'GWR',control=list(SE=TRUE)),MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.1, Model = 'GWR',control=list(SE=TRUE,NN=300,TP=TP)),times=2)
#> Warning in microbenchmark(MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, :
#> less accurate nanosecond times to avoid potential integer overflows
res
#> Unit: milliseconds
#>                                                                                                                                                                                          expr
#>                     MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,      fixed_vars = NULL, kernels = c("gauss"), H = 0.1, Model = "GWR",      control = list(SE = TRUE))
#>  MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,      fixed_vars = NULL, kernels = c("gauss"), H = 0.1, Model = "GWR",      control = list(SE = TRUE, NN = 300, TP = TP))
#>       min       lq     mean   median       uq      max neval cld
#>  616.7437 616.7437 616.8783 616.8783 617.0129 617.0129     2  a
#>  170.9253 170.9253 171.1031 171.1031 171.2810 171.2810     2   b

res=microbenchmark(MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=0.1, Model = 'GWR',control=list(SE=TRUE)),MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('bisq'),H=20, Model = 'GWR',control=list(SE=TRUE,adaptive=T,TP=TP)),times=2)
res
#> Unit: milliseconds
#>                                                                                                                                                                                            expr
#>                       MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,      fixed_vars = NULL, kernels = c("gauss"), H = 0.1, Model = "GWR",      control = list(SE = TRUE))
#>  MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,      fixed_vars = NULL, kernels = c("bisq"), H = 20, Model = "GWR",      control = list(SE = TRUE, adaptive = T, TP = TP))
#>       min       lq     mean   median       uq      max neval cld
#>  478.2233 478.2233 620.7698 620.7698 763.3162 763.3162     2   a
#>  170.0988 170.0988 175.2822 175.2822 180.4655 180.4655     2   a

Summary and plot of mgwrsar object using leaflet:

summary_mgwrsar(model_GWR0)
#> Call:
#> MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords,
#>     fixed_vars = NULL, kernels = c("gauss"), H = 0.13, Model = "GWR",
#>     control = list(SE = T, get_ts = TRUE))
#> Model: GWR
#> Kernels function: gauss
#> Kernels type: GD
#> Bandwidth: 0.13
#> Computation time: 0.615
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: NO
#> Use of Target Points: NO
#> Number of data points: 1000
#> Varying parameters: Intercept X1 X2 X3
#>         Intercept       X1       X2      X3
#> Min.     -2.63740  0.16880 -0.37306 -1.7588
#> 1st Qu.  -1.46100  0.38590  1.52729 -1.3070
#> Median   -1.08812  0.51409  1.94806 -1.0194
#> Mean     -0.97597  0.48879  1.94778 -1.0289
#> 3rd Qu.  -0.62307  0.59454  2.58871 -0.7439
#> Max.      1.06100  0.79570  3.37755 -0.3083
#> Effective degrees of freedom: 955.0674
#> AICc: 1560.364
#> Residual sum of squares: 64.0684
#> RMSE: 0.2531174
plot_mgwrsar(model_GWR0,type='B_coef',var='X2')
plot_mgwrsar(model_GWR0,type='t_coef',var='X2')

Plot of the effects of spatially varying coefficients:

plot_effect(model_GWR0,title='Effects')

Estimation of a GWR with spgwr package:

library(spgwr)
mydataSP=mydata
coordinates(mydataSP)=c('x','y')
ptm1<-proc.time()
model_spgwr <- gwr(Y_gwr~X1+X2+X3, data=mydataSP, bandwidth=0.13,hatmatrix=TRUE)
(proc.time()-ptm1)[3]

head(model_spgwr$SDF@data$gwr.e)
model_spgwr

all(abs(model_GWR0$residuals-model_spgwr$SDF@data$gwr.e)<0.00000000001) [1] -0.1585824 -0.1861144 -0.4450174 -0.1767796 -0.2014330 [6] 0.2670349 Call: gwr(formula = Y_gwr ~ X1 + X2 + X3, data = mydataSP, bandwidth = 0.13, hatmatrix = TRUE) Kernel function: gwr.Gauss Fixed bandwidth: 0.13 Summary of GWR coefficient estimates at data points: Min. 1st Qu. Median 3rd Qu. Max. Global X.Intercept. -2.63740 -1.46100 -1.08812 -0.62307 1.06100 -1.4845 X1 0.16880 0.38590 0.51409 0.59454 0.79570 0.5000 X2 -0.37306 1.52729 1.94806 2.58871 3.37755 2.5481 X3 -1.75876 -1.30704 -1.01941 -0.74386 -0.30828 -1.0762 Number of data points: 1000 Effective number of parameters (residual: 2traceS - traceS'S): 62.85371 Effective degrees of freedom (residual: 2traceS - traceS'S): 937.1463 Sigma (residual: 2traceS - traceS'S): 0.2614678 Effective number of parameters (model: traceS): 44.93259 Effective degrees of freedom (model: traceS): 955.0674 Sigma (model: traceS): 0.2590031 Sigma (ML): 0.2531174 AICc (GWR p. 61, eq 2.33; p. 96, eq. 4.21): 186.462 AIC (GWR p. 96, eq. 4.22): 135.0056 Residual sum of squares: 64.0684 Quasi-global R2: 0.9813492 [1] TRUE Estimation of a MGWR with stationary intercept using an adapative gaussian kernel (based on 20 nearest neighbors):  model_MGWR<-MGWRSAR(formula = 'Y_mgwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars='X3',kernels=c('gauss'),H=20, Model = 'MGWR',control=list(SE=T,adaptive=T)) summary_mgwrsar(model_MGWR) #> Call: #> MGWRSAR(formula = "Y_mgwr~X1+X2+X3", data = mydata, coords = coords, #> fixed_vars = "X3", kernels = c("gauss"), H = 20, Model = "MGWR", #> control = list(SE = T, adaptive = T)) #> Model: MGWR #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 20 #> Computation time: 0.717 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 1000 #> Constant parameters: X3 #> -1.00471 #> Varying parameters: Intercept X1 X2 #> Intercept X1 X2 #> Min. -2.81276 0.11756 -0.4245 #> 1st Qu. -1.52541 0.36076 1.3433 #> Median -0.99545 0.53148 1.7245 #> Mean -0.86301 0.49545 1.7449 #> 3rd Qu. -0.38685 0.61695 2.4882 #> Max. 1.22304 0.84347 3.3954 #> Effective degrees of freedom: 932.5146 #> AICc: 998.574 #> Residual sum of squares: 18.81683 #> RMSE: 0.1371744 Estimation of a mgwrsar(0,kc,kv) model with stationary intercept and stationnary spatial multiplier using an adapative gaussian kernel (based on 20 nearest neighbors): mgwrsar_0_kc_kv<-MGWRSAR(formula = 'Y_mgwrsar_0_kc_kv~X1+X2+X3', data = mydata,coords=coords, fixed_vars='X2',kernels=c('gauss'),H=20, Model = 'MGWRSAR_0_kc_kv',control=list(SE=F,adaptive=T,W=W)) summary_mgwrsar(mgwrsar_0_kc_kv) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_0_kc_kv~X1+X2+X3", data = mydata, #> coords = coords, fixed_vars = "X2", kernels = c("gauss"), #> H = 20, Model = "MGWRSAR_0_kc_kv", control = list(SE = F, #> adaptive = T, W = W)) #> Model: MGWRSAR_0_kc_kv #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 20 #> Computation time: 0.609 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 1000 #> Constant parameters: X2 lambda #> 0.870593 0.4120387 #> Varying parameters: Intercept X1 X3 #> Intercept X1 X3 #> Min. -1.254418 0.029588 -1.2702 #> 1st Qu. -0.444425 0.366633 -1.0810 #> Median -0.033372 0.540500 -1.0026 #> Mean 0.065257 0.498703 -1.0016 #> 3rd Qu. 0.445573 0.627798 -0.9243 #> Max. 2.099437 0.873585 -0.6744 #> Residual sum of squares: 137.9687 #> RMSE: 0.3714414 Estimation of a mgwrsar(0,0,kv) model with stationnary spatial multiplier using an adapative gaussian kernel (based on 20 nearest neighbors): mgwrsar_0_0_kv<-MGWRSAR(formula = 'Y_mgwrsar_0_kc_kv~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=20, Model = 'MGWRSAR_0_0_kv',control=list(SE=F,adaptive=T,W=W)) summary_mgwrsar(mgwrsar_0_0_kv) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_0_kc_kv~X1+X2+X3", data = mydata, #> coords = coords, fixed_vars = NULL, kernels = c("gauss"), #> H = 20, Model = "MGWRSAR_0_0_kv", control = list(SE = F, #> adaptive = T, W = W)) #> Model: MGWRSAR_0_0_kv #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 20 #> Computation time: 0.618 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 1000 #> Constant parameters: lambda #> 0.3676871 #> Varying parameters: Intercept X1 X2 X3 #> Intercept X1 X2 X3 #> Min. -3.763567 0.080428 -2.209824 -1.2308 #> 1st Qu. -1.219268 0.357217 0.706989 -1.0647 #> Median -0.618269 0.536381 1.425772 -0.9997 #> Mean -0.513883 0.499074 1.398492 -1.0014 #> 3rd Qu. -0.017472 0.627690 2.638912 -0.9437 #> Max. 2.108716 0.889871 4.369973 -0.6752 #> Residual sum of squares: 151.7064 #> RMSE: 0.3894951 Estimation of a mgwrsar(1,0,kv) model with all parameter spatially varying using an adapative gaussian kernel (based on 20 nearest neighbors): mgwrsar_1_0_kv<-MGWRSAR(formula = 'Y_mgwrsar_1_0_kv~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss'),H=20, Model = 'MGWRSAR_1_0_kv',control=list(SE=F,adaptive=T,W=W)) summary_mgwrsar(mgwrsar_1_0_kv) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_1_0_kv~X1+X2+X3", data = mydata, #> coords = coords, fixed_vars = NULL, kernels = c("gauss"), #> H = 20, Model = "MGWRSAR_1_0_kv", control = list(SE = F, #> adaptive = T, W = W)) #> Model: MGWRSAR_1_0_kv #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 20 #> Computation time: 0.54 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 1000 #> Varying parameters: Intercept X1 X2 X3 #> Intercept X1 X2 X3 lambda #> Min. -5.55249 0.11207 -1.20584 -1.86027 -0.2375 #> 1st Qu. -1.47283 0.36956 0.64156 -1.24644 0.1831 #> Median -0.75406 0.54369 1.72877 -0.99875 0.3845 #> Mean -0.79631 0.50721 1.75803 -1.00349 0.3568 #> 3rd Qu. -0.15439 0.63505 2.86983 -0.71468 0.5382 #> Max. 2.06110 0.95748 5.14466 -0.27205 0.8808 #> Residual sum of squares: 868.1985 #> RMSE: 0.9317717 Estimation of a mgwrsar(1,kc,kv) model with stationary X1 using an adapative gaussian kernel (based on 20 nearest neighbors): mgwrsar_1_kc_kv<-MGWRSAR(formula = 'Y_mgwrsar_1_kc_kv~X1+X2+X3', data = mydata,coords=coords, fixed_vars='X2',kernels=c('gauss'),H=20, Model = 'MGWRSAR_1_kc_kv',control=list(SE=F,adaptive=T,W=W)) summary_mgwrsar(mgwrsar_1_kc_kv) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_1_kc_kv~X1+X2+X3", data = mydata, #> coords = coords, fixed_vars = "X2", kernels = c("gauss"), #> H = 20, Model = "MGWRSAR_1_kc_kv", control = list(SE = F, #> adaptive = T, W = W)) #> Model: MGWRSAR_1_kc_kv #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 20 #> Computation time: 1.364 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 1000 #> Constant parameters: X2 #> 2.459628 #> Varying parameters: Intercept X1 X3 #> Intercept X1 X3 lambda #> Min. -2.785597 0.064134 -1.373242 -0.6412 #> 1st Qu. -0.783444 0.371202 -1.092831 0.0467 #> Median -0.454995 0.534294 -1.006845 0.3538 #> Mean -0.435127 0.501815 -1.009762 0.3553 #> 3rd Qu. -0.111683 0.633466 -0.933401 0.8027 #> Max. 2.392965 0.910200 -0.617579 0.9900 #> Residual sum of squares: 6519.053 #> RMSE: 2.553244 Estimation of a mgwrsar(1,kc,0) model with stationary X1 using an adapative gaussian kernel (based on 20 nearest neighbors): mgwrsar_1_kc_0<-MGWRSAR(formula = 'Y_mgwrsar_1_kc_kv~X1+X2+X3', data = mydata,coords=coords, fixed_vars=c('Intercept','X1','X2','X3'),kernels=c('gauss'),H=20, Model = 'MGWRSAR_1_kc_0',control=list(SE=F,adaptive=T,W=W)) summary_mgwrsar(mgwrsar_1_kc_0) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_1_kc_kv~X1+X2+X3", data = mydata, #> coords = coords, fixed_vars = c("Intercept", "X1", "X2", #> "X3"), kernels = c("gauss"), H = 20, Model = "MGWRSAR_1_kc_0", #> control = list(SE = F, adaptive = T, W = W)) #> Model: MGWRSAR_1_kc_0 #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 20 #> Computation time: 1.307 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 1000 #> Constant parameters: Intercept X1 X2 X3 #> -0.8205521 0.509378 1.69077 -1.035463 #> Varying parameters: #> lambda #> Min. -0.9137 #> 1st Qu. 0.2291 #> Median 0.6009 #> Mean 0.4822 #> 3rd Qu. 0.9900 #> Max. 0.9995 #> Residual sum of squares: 2736.795 #> RMSE: 1.654326 ## Prediction In this example, we use an in-sample of size 800 for model estimation and an out-sample of size 200 for prediction. Note that for model with spatial autocorrelation you must provide a global weight matrix of size 1000, ordered by in-sample then out-sample locations. For GWR and MGWR_1_0_kv, where all coefficients vary in space, the predictions are carried out using the corresponding model estimate with the out-sample location as target points, so the estimated coefficients are not used: only the call and the data of the estimated model are used. For mixed model that have some constant coefficients (MGWR, MGXWR_0_0_kv, MGXWR_1_kc_kv, MGXWR_1_kc_0), the estimated coefficients are extrapolated using a weighting matrix. By default, the weighting matrix for prediction is based on the expected weights of outsample data if they were had been added to insample data to estimate the corresponding MGWRSAR (see Geniaux 2022 for further detail). The user can also choose to extrapolate the model coefficients using a shepperd kernel with custom number of neighbours or using the same kernel and bandwidth as the estimated model. Prediction of GWR model using sheppard kernel with 8 neighbors: length_out=800 index_in=sample(1:1000,length_out) index_out=(1:1000)[-index_in] model_GWR_insample<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata[index_in,],coords=coords[index_in,], fixed_vars=NULL,kernels=c('gauss'),H=8, Model = 'GWR',control=list(adaptive=T)) summary_mgwrsar(model_GWR_insample) #> Call: #> MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata[index_in, ], #> coords = coords[index_in, ], fixed_vars = NULL, kernels = c("gauss"), #> H = 8, Model = "GWR", control = list(adaptive = T)) #> Model: GWR #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 8 #> Computation time: 0.315 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 800 #> Varying parameters: Intercept X1 X2 X3 #> Intercept X1 X2 X3 #> Min. -3.431046 0.057831 -0.613649 -1.9259 #> 1st Qu. -1.481228 0.347417 1.190307 -1.2686 #> Median -0.888871 0.545087 1.748743 -1.0012 #> Mean -0.822152 0.499287 1.704531 -1.0095 #> 3rd Qu. -0.272541 0.628847 2.489687 -0.7335 #> Max. 1.585348 0.897832 3.579135 -0.1883 #> Residual sum of squares: 5.988363 #> RMSE: 0.08651852 newdata=mydata[index_out,] newdata_coords=coords[index_out,] newdata$Y_mgwrsar_1_0_kv=0

Y_pred=predict_mgwrsar(model_GWR_insample, newdata=newdata, newdata_coords=newdata_coords)
#> [1] 0.4664548 2.3094268 4.5104714 1.8733030 3.9355001 2.9333140
head(mydata$Y_gwr[index_out]) #> [1] 0.3224649 2.0656554 4.5648040 1.9159209 3.8348415 2.9941564 sqrt(mean((mydata$Y_gwr[index_out]-Y_pred)^2)) # RMSE
#> [1] 0.13121

model_MGWR_insample<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata[index_in,],coords=coords[index_in,], fixed_vars='X3',kernels=c('gauss'),H=8, Model = 'MGWR',control=list(adaptive=T))
summary_mgwrsar(model_MGWR_insample)
#> Call:
#> MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata[index_in, ],
#>     coords = coords[index_in, ], fixed_vars = "X3", kernels = c("gauss"),
#>     H = 8, Model = "MGWR", control = list(adaptive = T))
#> Model: MGWR
#> Kernels function: gauss
#> Kernels type: GD
#> Bandwidth: 8
#> Computation time: 0.511
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: NO
#> Use of Target Points: NO
#> Number of data points: 800
#> Constant parameters: X3
#> -1.006481
#> Varying parameters: Intercept X1 X2
#>         Intercept        X1      X2
#> Min.    -3.936929  0.060461 -0.6137
#> 1st Qu. -1.498654  0.346186  1.0902
#> Median  -0.948519  0.541705  1.7289
#> Mean    -0.855135  0.498004  1.7338
#> 3rd Qu. -0.235306  0.644812  2.6282
#> Max.     1.640384  0.897495  3.7508
#> Residual sum of squares: 36.93015
#> RMSE: 0.214855

newdata=mydata[index_out,]
newdata_coords=coords[index_out,]
newdata$Y_mgwrsar_1_0_kv=0 ## Prediction with method_pred='tWtp' Y_pred=predict_mgwrsar(model_MGWR_insample, newdata=newdata, newdata_coords=newdata_coords) #> Warnings: method_pred=='TP' is not implemented for Model in ('MGWR','MGWRSAR_1_kc_0','MGWRSAR_1_kc_kv','MGWRSAR_0_kc_kv','MGWRSAR_0_0_kv'): automatic swith to method_pred='tWtp_model' head(Y_pred) #> [1] 0.4585569 2.7000676 4.7308344 1.7981054 4.0253231 3.0013471 head(mydata$Y_gwr[index_out])
#> [1] 0.3224649 2.0656554 4.5648040 1.9159209 3.8348415 2.9941564
sqrt(mean((mydata$Y_gwr[index_out]-Y_pred)^2)) # RMSE #> [1] 0.2826883 Prediction of MGWRSAR_1_0_kv model using sheppard kernel with 4 neighbors and Best Linear Unbiased Predictor (BLUP) :  length_out=800 index_in=sample(1:1000,length_out) index_out=(1:1000)[-index_in] ### Global Spatial Weight matrix W should be ordered by in_sample (S) then out_sample W_in_out=kernel_matW(H=4,kernels='rectangle',coord_i=rbind(coords[index_in,],coords[index_out,]),NN=4,adaptive=TRUE,diagnull=TRUE,rowNorm=T) W_in=W_in_out[1:length(index_in),1:length(index_in)] W_in=mgwrsar::normW(W_in) newdata=mydata[index_out,] newdata_coords=coords[index_out,] ### SAR Model model_SAR_insample<-MGWRSAR(formula = 'Y_sar~X1+X2+X3', data = mydata[index_in,],coords=coords[index_in,], fixed_vars=NULL,kernels=NULL,H=11, Model = 'SAR',control=list(W=W_in)) model_SAR_insample$RMSE
#> [1] 0.1303956
summary_mgwrsar(model_SAR_insample)
#> Call:
#> MGWRSAR(formula = "Y_sar~X1+X2+X3", data = mydata[index_in, ],
#>     coords = coords[index_in, ], fixed_vars = NULL, kernels = NULL,
#>     H = 11, Model = "SAR", control = list(W = W_in))
#> Model: SAR
#> Method for spatial autocorrelation: 2SLS
#> Kernels function:
#> Kernels type: GD
#> Bandwidth: 11
#> Computation time: 0.001
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel:
#> Use of Target Points: NO
#> Number of data points: 800
#> Constant parameters: Intercept X1 X2 X3 lambda
#> 0.1515706 0.5040549 1.153973 -1.012715 0.3147484
#> Effective degrees of freedom: 795
#> Residual sum of squares: 13.6024
#> RMSE: 0.1303956

## without Best Linear Unbiased Predictor
# prediction YTC
Y_pred=predict_mgwrsar(model_SAR_insample, newdata=newdata, newdata_coords=newdata_coords,W=W_in_out,type='YTC')
#> [1] 2.027912 3.731384 5.910859 2.949735 4.971490 5.277246
RMSE_YTC=sqrt(mean((mydata$Y_sar[index_out]-Y_pred)^2)) RMSE_YTC #> [1] 0.08441952 ## Using Best Linear Unbiased Predictor Y_pred=predict_mgwrsar(model_SAR_insample, newdata=newdata, newdata_coords=newdata_coords,W=W_in_out,type='BPN') head(Y_pred) #> [1] 2.015560 3.680499 5.923186 2.923057 4.940062 5.313581 RMSE_BPN=sqrt(mean((mydata$Y_sar[index_out]-Y_pred)^2))
RMSE_BPN
#> [1] 0.06039921

#### MGWRSAR_1_0_kv
model_MGWRSAR_1_0_kv_insample<-MGWRSAR(formula = 'Y_mgwrsar_1_0_kv~X1+X2+X3', data = mydata[index_in,],coords=coords[index_in,], fixed_vars=NULL,kernels=c('gauss'),H=11, Model = 'MGWRSAR_1_0_kv',control=list(W=W_in,adaptive=TRUE,isgcv=F))
model_MGWRSAR_1_0_kv_insample$RMSE #> [1] 0.708789 summary_mgwrsar(model_MGWRSAR_1_0_kv_insample) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_1_0_kv~X1+X2+X3", data = mydata[index_in, #> ], coords = coords[index_in, ], fixed_vars = NULL, kernels = c("gauss"), #> H = 11, Model = "MGWRSAR_1_0_kv", control = list(W = W_in, #> adaptive = TRUE, isgcv = F)) #> Model: MGWRSAR_1_0_kv #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 11 #> Computation time: 0.472 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 800 #> Varying parameters: Intercept X1 X2 X3 #> Intercept X1 X2 X3 lambda #> Min. -12.479607 0.077377 -1.241909 -1.984773 -0.4086 #> 1st Qu. -1.793509 0.356913 0.973790 -1.258653 0.1056 #> Median -0.874141 0.547879 1.882220 -1.014936 0.2745 #> Mean -1.075548 0.515188 2.245951 -0.997859 0.2801 #> 3rd Qu. -0.145199 0.658945 3.482105 -0.688769 0.4424 #> Max. 2.143243 1.040762 12.859764 -0.202572 0.8068 #> Residual sum of squares: 401.9055 #> RMSE: 0.708789 # prediction with method_pred='tWtp' Y_pred=predict_mgwrsar(model_MGWRSAR_1_0_kv_insample, newdata=newdata, newdata_coords=newdata_coords,W=W_in_out,type='BPN',method_pred='tWtp') head(Y_pred) #> [1] 0.5246521 3.4805124 5.6597370 2.6974158 6.5664039 4.4513093 RMSE_tWtp_BPN=sqrt(mean((mydata$Y_mgwrsar_1_0_kv[index_out]-Y_pred)^2))
RMSE_tWtp_BPN
#> [1] 0.5200496

## Using Best Linear Unbiased Predictor with method_pred='model'
Y_pred=predict_mgwrsar(model_MGWRSAR_1_0_kv_insample, newdata=newdata, newdata_coords=newdata_coords,W=W,type='BPN',method_pred='model' )
#> [1] 0.9431835 2.4151135 5.1652828 2.1911860 6.2417757 4.9202018
RMSE_model_BPN=sqrt(mean((mydata$Y_mgwrsar_1_0_kv[index_out]-Y_pred)^2)) RMSE_model_BPN #> [1] 0.9717156 ## Using Best Linear Unbiased Predictor with method_pred='sheppard' W_out=kernel_matW(H=4,kernels='rectangle',coord_i=coords[index_out,],NN=4,adaptive=TRUE,diagnull=TRUE,rowNorm=T) Y_pred=predict_mgwrsar(model_MGWRSAR_1_0_kv_insample, newdata=newdata, newdata_coords=newdata_coords,W=W,type='BPN',method_pred='sheppard',k_extra=8) head(Y_pred) #> [1] 1.318562 2.380605 5.191412 2.228458 6.211813 4.734835 RMSE_sheppard_BPN=sqrt(mean((mydata$Y_mgwrsar_1_0_kv[index_out]-Y_pred)^2))
RMSE_sheppard_BPN
#> [1] 0.9695924

Prediction of MGWRSAR_1_kc_kv model using sheppard kernel with 4 neighbors and Best Linear Unbiased Predictor (BLUP) :


length_out=800
index_in=sample(1:1000,length_out)
index_out=(1:1000)[-index_in]

### Global Spatial Weight matrix W should be ordered by in_ sample (S) then out_sample

W_in=W[index_in,index_in]
W_in=mgwrsar::normW(W_in)

model_MGWRSAR_1_kc_kv_insample<-MGWRSAR(formula = 'Y_mgwrsar_1_kc_kv~X1+X2+X3', data = mydata[index_in,],coords=coords[index_in,], fixed_vars='X3',kernels=c('gauss'),H=11, Model = 'MGWRSAR_1_kc_kv',control=list(W=W_in,adaptive=TRUE,isgcv=F))
model_MGWRSAR_1_kc_kv_insample$RMSE #> [1] 0.3502106 summary_mgwrsar(model_MGWRSAR_1_kc_kv_insample) #> Call: #> MGWRSAR(formula = "Y_mgwrsar_1_kc_kv~X1+X2+X3", data = mydata[index_in, #> ], coords = coords[index_in, ], fixed_vars = "X3", kernels = c("gauss"), #> H = 11, Model = "MGWRSAR_1_kc_kv", control = list(W = W_in, #> adaptive = TRUE, isgcv = F)) #> Model: MGWRSAR_1_kc_kv #> Method for spatial autocorrelation: 2SLS #> Kernels function: gauss #> Kernels adaptive: YES #> Kernels type: GD #> Bandwidth: 11 #> Computation time: 1.165 #> Use of parallel computing: FALSE ncore = 1 #> Use of rough kernel: NO #> Use of Target Points: NO #> Number of data points: 800 #> Constant parameters: X3 #> -1.042863 #> Varying parameters: Intercept X1 X2 #> Intercept X1 X2 lambda #> Min. -10.72384 0.08488 -0.77331 -0.2800 #> 1st Qu. -1.70402 0.35135 2.04683 -0.0410 #> Median -0.78341 0.53193 3.01252 -0.0047 #> Mean -0.99026 0.50042 3.42218 -0.0116 #> 3rd Qu. 0.06569 0.63682 5.18079 0.0268 #> Max. 1.62136 0.98688 12.43667 0.1318 #> Residual sum of squares: 98.11797 #> RMSE: 0.3502106 ## without Best Linear Unbiased Predictor newdata=mydata[index_out,] newdata_coords=coords[index_out,] newdata$Y_mgwrsar_1_kc_kv=0

Y_pred=predict_mgwrsar(model_MGWRSAR_1_kc_kv_insample, newdata=newdata, newdata_coords=newdata_coords,W=W,type='YTC')
#> Warnings: method_pred=='TP' is not implemented for  Model in ('MGWR','MGWRSAR_1_kc_0','MGWRSAR_1_kc_kv','MGWRSAR_0_kc_kv','MGWRSAR_0_0_kv'): automatic swith to method_pred='tWtp_model'
#> [1] 0.6663112 3.6869125 5.8678272 2.6071807 7.9223965 4.2593302
RMSE_YTC=sqrt(mean((mydata$Y_mgwrsar_1_kc_kv[index_out]-Y_pred)^2)) RMSE_YTC #> [1] 0.4522916 ## Using Best Linear Unbiased Predictor Y_pred=predict_mgwrsar(model_MGWRSAR_1_kc_kv_insample, newdata=newdata, newdata_coords=newdata_coords,W=W,type='BPN')#,method_pred='sheppard') #> Warnings: method_pred=='TP' is not implemented for Model in ('MGWR','MGWRSAR_1_kc_0','MGWRSAR_1_kc_kv','MGWRSAR_0_kc_kv','MGWRSAR_0_0_kv'): automatic swith to method_pred='tWtp_model' head(Y_pred) #> [1] 0.667970 3.849695 5.867610 2.607104 7.938369 4.259173 RMSE_BPN=sqrt(mean((mydata$Y_mgwrsar_1_kc_kv[index_out]-Y_pred)^2))
RMSE_BPN
#> [1] 0.4636561

## General kernel Product functions

The package provides additional functions that allow to estimate locally linear model with other dimensions than space. Using the control variable ‘type’, it is possible to add time in the kernel and a limited set of other variables (continuous or categorical). If several dimensions are involved in the kernel, a general product kernel is used and you need to provide a list of bandwidths and a list of kernel types. For time kernel, it uses asymetric kernel, eventually with a decay. For categorical variable, it uses the propositions of Li and Racine (2010); see also np package. Optimization of bandwidths has to be done by yourself.

Note that when time or other additional variables are used for kernels, then two small bandwidths could lead to empty local subsamples. We recommend to use gauss and gauss_adapt kernels that suffering less this issue.

## space + time kernel
mytime_index=sort(rep(1:500,2))
mytime_index[1:150]
#>   [1]  1  1  2  2  3  3  4  4  5  5  6  6  7  7  8  8  9  9 10 10 11 11 12 12 13
#>  [26] 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25
#>  [51] 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38
#>  [76] 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 48 49 49 50 50
#> [101] 51 51 52 52 53 53 54 54 55 55 56 56 57 57 58 58 59 59 60 60 61 61 62 62 63
#> [126] 63 64 64 65 65 66 66 67 67 68 68 69 69 70 70 71 71 72 72 73 73 74 74 75 75

model_MGWRSART_0_kc_kv<-MGWRSAR(formula = 'Y_mgwrsar_0_kc_kv~X1+X2+X3', data = mydata,coords=coords, fixed_vars=c('Intercept'),kernels=c('gauss','gauss'),H=c(50,50), Model = 'MGWRSAR_0_kc_kv',control=list(Z=mytime_index,W=W,adaptive=c(TRUE,TRUE),Type='GDT'))

summary_mgwrsar(model_MGWRSART_0_kc_kv)
#> Call:
#> MGWRSAR(formula = "Y_mgwrsar_0_kc_kv~X1+X2+X3", data = mydata,
#>     coords = coords, fixed_vars = c("Intercept"), kernels = c("gauss",
#>         "gauss"), H = c(50, 50), Model = "MGWRSAR_0_kc_kv", control = list(Z = mytime_index,
#>         W = W, adaptive = c(TRUE, TRUE), Type = "GDT"))
#> Model: MGWRSAR_0_kc_kv
#> Method for spatial autocorrelation: 2SLS
#> Kernels function: gauss gauss
#> Kernels type: GDT
#> Bandwidth: 50 50
#> Computation time: 0.854
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: NO NO
#> Use of Target Points: NO
#> Number of data points: 1000
#> Constant parameters: Intercept lambda
#> -0.4460577 0.2657842
#> Varying parameters: X1 X2 X3
#>                 X1         X2      X3
#> Min.     0.0034374 -0.8162198 -2.6362
#> 1st Qu.  0.3715497  1.4244343 -1.3558
#> Median   0.4585510  2.1506913 -1.1208
#> Mean     0.4571081  2.0970690 -1.1146
#> 3rd Qu.  0.5556844  2.8302322 -0.8747
#> Max.     0.9022376  5.0097052  0.4869
#> Residual sum of squares: 107.8131
#> RMSE: 0.3283491

### space  + continuous variable kernel
model_GWRX<-MGWRSAR(formula = 'Y_gwr~X1+X2+X3', data = mydata,coords=coords, fixed_vars=NULL,kernels=c('gauss','gauss'),H=c(120,120), Model = 'GWR',control=list(Z=mydata$X2,Type='GDX',adaptive=c(T,T))) summary_mgwrsar(model_GWRX) #> Call: #> MGWRSAR(formula = "Y_gwr~X1+X2+X3", data = mydata, coords = coords, #> fixed_vars = NULL, kernels = c("gauss", "gauss"), H = c(120, #> 120), Model = "GWR", control = list(Z = mydata$X2, Type = "GDX",
#> Model: GWR
#> Kernels function: gauss gauss
#> Kernels type: GDX
#> Bandwidth: 120 120
#> Computation time: 0.613
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: NO NO
#> Use of Target Points: NO
#> Number of data points: 1000
#> Varying parameters: Intercept X1 X2 X3
#>         Intercept       X1       X2      X3
#> Min.     -6.45292  0.15257 -0.84387 -1.8089
#> 1st Qu.  -2.30342  0.32481  1.00002 -1.2329
#> Median   -1.13830  0.45537  1.91297 -0.9922
#> Mean     -1.30274  0.46953  2.06651 -1.0200
#> 3rd Qu.  -0.03019  0.62710  3.05773 -0.7868
#> Max.      1.54070  0.73745  6.44482 -0.2804
#> Residual sum of squares: 202.5684
#> RMSE: 0.4500759

### space + categorical kernel (Li and Racine 2010)
Z=1+as.numeric(mydata$X2>quantile(mydata$X2,0.9))*2+as.numeric(mydata$X2<=quantile(mydata$X2,0.1))
table(Z)
#> Z
#>   1   2   3
#> 800 100 100

model_MGWRSARC_0_kc_kv<-MGWRSAR(formula = 'Y_mgwrsar_0_kc_kv~X1+X3', data = mydata,coords=coords, fixed_vars=c('Intercept'),kernels=c('gauss','li','li','li'),H=c(120,0.1,0.9,0.9), Model = 'MGWRSAR_0_kc_kv',control=list(Z=Z,W=W,Type='GDC',adaptive=c(T,F,F,F)))
summary_mgwrsar(model_MGWRSARC_0_kc_kv)
#> Call:
#> MGWRSAR(formula = "Y_mgwrsar_0_kc_kv~X1+X3", data = mydata, coords = coords,
#>     fixed_vars = c("Intercept"), kernels = c("gauss", "li", "li",
#>         "li"), H = c(120, 0.1, 0.9, 0.9), Model = "MGWRSAR_0_kc_kv",
#>     control = list(Z = Z, W = W, Type = "GDC", adaptive = c(T,
#>         F, F, F)))
#> Model: MGWRSAR_0_kc_kv
#> Method for spatial autocorrelation: 2SLS
#> Kernels function: gauss li li li
#> Kernels adaptive: YES NO NO NO
#> Kernels type: GDC
#> Bandwidth: 120 0.1 0.9 0.9
#> Computation time: 0.617
#> Use of parallel computing: FALSE  ncore = 1
#> Use of rough kernel: NO NO NO NO
#> Use of Target Points: NO
#> Number of data points: 1000
#> Constant parameters: Intercept lambda
#> 1.149469 0.3528613
#> Varying parameters: X1 X3
#>              X1      X3
#> Min.    0.13802 -1.3391
#> 1st Qu. 0.34306 -1.2704
#> Median  0.45967 -1.2161
#> Mean    0.43625 -1.1279
#> 3rd Qu. 0.53071 -1.0092
#> Max.    0.65289 -0.6043
#> Residual sum of squares: 1067.886
#> RMSE: 1.033386