R: Functions to include non-parametric continous covariates and...

pspl_terms {pspatreg}

R Documentation

Functions to include non-parametric continous covariates and spatial or spatio-temporal trends in semiparametric regression models.

Description

The pspl() and pspt() functions allow the inclusion of non-parametric continuous covariates and spatial or spatio-temporal trends in semiparametric regression models. Both type of terms are modelled using P-splines.

pspl(): This function allows the inclusion of terms for non-parametric covariates in semiparametric models. Each non-parametric covariate must be included with its own pspl term in a formula.

pspt(): This function allows the inclusion of a spatial or spatio-temporal trend in the formula of the semiparametric spatial or spatio-temporal models. The trend can be decomposed in an ANOVA functional way including main and interaction effects.

Usage

pspl(
  x,
  xl = min(x) - 0.01 * abs(min(x)),
  xr = max(x) + 0.01 * abs(max(x)),
  nknots = 10,
  bdeg = 3,
  pord = 2,
  decom = 2,
  scale = TRUE
)

pspt(
  sp1,
  sp2,
  time = NULL,
  scale = TRUE,
  ntime = NULL,
  xl_sp1 = min(sp1) - 0.01 * abs(min(sp1)),
  xr_sp1 = max(sp1) + 0.01 * abs(max(sp1)),
  xl_sp2 = min(sp2) - 0.01 * abs(min(sp2)),
  xr_sp2 = max(sp2) + 0.01 * abs(max(sp2)),
  xl_time = min(time) - 0.01 * abs(min(time)),
  xr_time = max(time) + 0.01 * abs(max(time)),
  nknots = c(10, 10, 5),
  bdeg = c(3, 3, 3),
  pord = c(2, 2, 2),
  decom = 2,
  psanova = FALSE,
  nest_sp1 = 1,
  nest_sp2 = 1,
  nest_time = 1,
  f1_main = TRUE,
  f2_main = TRUE,
  ft_main = TRUE,
  f12_int = TRUE,
  f1t_int = TRUE,
  f2t_int = TRUE,
  f12t_int = TRUE
)

Arguments

`x`	Name of the covariate.
`xl`	Minimum of the interval for the continuous covariate.
`xr`	Maximum of the interval for the continuous covariate.
`nknots`	Vector including the number of knots of each coordinate for spline bases. Default = c(10,10,5). The order of the knots in the vector follows the order of the specified spatio-temporal parameters so the first value of the vector is the number of knots for `sp1`, the second value is for `sp2` and the third for `time`. See `Examples`.
`bdeg`	Order of the B-spline bases. Default = c(3,3,3).
`pord`	Order of the penalty for the difference matrices in P-spline. Default = c(2,2,2).
`decom`	Type of decomposition of fixed part when P-spline term is expressed as a mixed model. If `decom = 1` the fixed part is given by `X = BU_n` where B is the B-spline basis matrix and U_n* is the nullspace basis of the penalty matrix. If `decom = 2` the fixed part is given by `X = [1\|x\|...\|x^(pord-1)]` . Default = 2.
`scale`	Logical value to scale the spatial and temporal coordinates before the estimation of semiparametric model. Default = 'TRUE'
`sp1`	Name of the first spatial coordinate.
`sp2`	Name of the second spatial coordinate.
`time`	Name of the temporal coordinate. It must be specified only for spatio-temporal trends when using panel data. Default = 'NULL'.
`ntime`	Number of temporal periods in panel data.
`xl_sp1`	Minimum of the interval for the first spatial coordinate.
`xr_sp1`	Maximum of the interval for the first spatial coordinate.
`xl_sp2`	Minimum of the interval for the second spatial coordinate.
`xr_sp2`	Maximum of the interval for the second spatial coordinate.
`xl_time`	Minimum of the interval for the temporal coordinate.
`xr_time`	Maximum of the interval for the temporal coordinate.
`psanova`	Logical value to choose an ANOVA decomposition of the spatial or spatio-temporal trend. Default = 'FALSE'. If 'TRUE', you must specify the divisors for main, and interaction effects. More in `Examples`.
`nest_sp1`	Vector including the divisor of the knots for main and interaction effects for the first spatial coordinate. It is used for ANOVA decomposition models including nested bases. Default = 1 (no nested bases). The values must be divisors and the resulting value of the division should not be smaller than 4.
`nest_sp2`	Vector including the divisor of the knots for main and interaction effects for the second spatial coordinate. It is used for ANOVA decomposition models including nested bases. Default = 1 (no nested bases). The values must be divisors and the resulting value of the division should not be smaller than 4.
`nest_time`	Vector including the divisor of the knots for main and interaction effects for the temporal coordinate. It is used for ANOVA decomposition models including nested bases. Default = 1 (no nested bases). The values must be divisors and the resulting value of the division should not be smaller than 4.
`f1_main`	Logical value to include main effect for the first spatial coordinate in ANOVA models. Default = 'TRUE'.
`f2_main`	Logical value to include main effect for the second spatial coordinate in ANOVA models. Default = 'TRUE'.
`ft_main`	Logical value to include main effect for the temporal coordinate in ANOVA models. Default = 'TRUE'.
`f12_int`	Logical value to include second-order interaction effect between first and second spatial coordinates in ANOVA models. Default = 'TRUE'.
`f1t_int`	Logical value to include second-order interaction effect between first spatial and temporal coordinates in ANOVA models. Default = 'TRUE'.
`f2t_int`	Logical value to include second-order interaction effect between second spatial and temporal coordinates in ANOVA models. Default = 'TRUE'.
`f12t_int`	Logical value to include third-order interaction effect between first and second spatial coordinates and temporal coordinates in ANOVA models. Default = 'TRUE'.

Value

pspl(): An object of class bs including.

`B`	Matrix including B-spline basis for the covariate
`a`	List including nknots, knots, bdeg, pord and decom.

pspt(): An object of class bs including.

`B`	Matrix including B-spline basis for the covariate
`a`	List including sp1, sp2, time, nknots, bdeg, pord, decom, psanova, nest_sp1, nest_sp2, nest_time, f1_main, f2_main, ft_main, f12_int, f1t_int, f2t_int, and f12t_int.

References

Eilers, P. and Marx, B. (1996). Flexible Smoothing with B-Splines and Penalties. Statistical Science, (11), 89-121.
Eilers, P. and Marx, B. (2021). Practical Smoothing. The Joys of P-Splines. Cambridge University Press.
Fahrmeir, L.; Kneib, T.; Lang, S.; and Marx, B. (2021). Regression. Models, Methods and Applications (2nd Ed.). Springer.
Lee, D. and Durban, M. (2011). P-Spline ANOVA Type Interaction Models for Spatio-Temporal Smoothing. Statistical Modelling, (11), 49-69. <doi:10.1177/1471082X1001100104>
Lee, D. J., Durban, M., and Eilers, P. (2013). Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases. Computational Statistics & Data Analysis, (61), 22-37. <doi:10.1016/j.csda.2012.11.013>
Minguez, R.; Basile, R. and Durban, M. (2020). An Alternative Semiparametric Model for Spatial Panel Data. Statistical Methods and Applications, (29), 669-708. <doi: 10.1007/s10260-019-00492-8>
Wood, S.N. (2017). Generalized Additive Models. An Introduction with R (second edition). CRC Press, Boca Raton.

Examples

library(pspatreg)
###############################################
# Examples using spatial data of Ames Houses.
###############################################
library(spdep)
library(sf)
ames <- AmesHousing::make_ames() # Raw Ames Housing Data
ames_sf <- st_as_sf(ames, coords = c("Longitude", "Latitude"))
ames_sf$Longitude <- ames$Longitude
ames_sf$Latitude <- ames$Latitude
ames_sf$lnSale_Price <- log(ames_sf$Sale_Price)
ames_sf$lnLot_Area <- log(ames_sf$Lot_Area)
ames_sf$lnTotal_Bsmt_SF <- log(ames_sf$Total_Bsmt_SF+1)
ames_sf$lnGr_Liv_Area <- log(ames_sf$Gr_Liv_Area)
ames_sf1 <- ames_sf[(duplicated(ames_sf$Longitude) == FALSE), ]
####  GAM pure with pspatreg
form1 <- lnSale_Price ~ Fireplaces + Garage_Cars +
          pspl(lnLot_Area, nknots = 20) + 
          pspl(lnTotal_Bsmt_SF, nknots = 20) +
          pspl(lnGr_Liv_Area, nknots = 20)    
gampure <- pspatfit(form1, data = ames_sf1)
summary(gampure)
 
########### Constructing the spatial weights matrix
coord_sf1 <- cbind(ames_sf1$Longitude, ames_sf1$Latitude)
k5nb <- knn2nb(knearneigh(coord_sf1, k = 5, 
                          longlat = TRUE, use_kd_tree = FALSE), sym = TRUE)
lw_ames <- nb2listw(k5nb, style = "W", 
                  zero.policy = FALSE)
#####################  GAM + SAR Model
gamsar <- pspatfit(form1, data = ames_sf1, 
                   type = "sar", listw = lw_ames,
                   method = "Chebyshev")
summary(gamsar)
### Models with 2d spatial trend
form2 <- lnSale_Price ~ Fireplaces + Garage_Cars +
          pspl(lnLot_Area, nknots = 20) + 
          pspl(lnTotal_Bsmt_SF, nknots = 20) +
          pspl(lnGr_Liv_Area, nknots = 20) +
          pspt(Longitude, Latitude, 
               nknots = c(10, 10), 
               psanova = FALSE)
#####################  GAM + GEO Model
gamgeo2d <- pspatfit(form2, data = ames_sf1)
summary(gamgeo2d)

gamgeo2dsar <- pspatfit(form2, data = ames_sf1,
                        type = "sar", 
                        listw = lw_ames, 
                        method = "Chebyshev")
summary(gamgeo2dsar)
### Models with psanova 2d spatial trend
form3 <- lnSale_Price ~ Fireplaces + Garage_Cars +
          pspl(lnLot_Area, nknots = 20) + 
          pspl(lnTotal_Bsmt_SF, nknots = 20) +
          pspl(lnGr_Liv_Area, nknots = 20) +
          pspt(Longitude, Latitude, 
               nknots = c(10, 10), 
               psanova = TRUE)
gamgeo2danovasar <- pspatfit(form3, data = ames_sf1,
                        type = "sar", 
                        listw = lw_ames, method = "Chebyshev")
summary(gamgeo2danovasar)
 ###############################################
###################### Examples using a panel data of rate of
###################### unemployment for 103 Italian provinces in 1996-2019.
 ###############################################
## load spatial panel and Wsp_it
## 103 Italian provinces. Period 1996-2019
data(unemp_it, package = "pspatreg")
## Wsp_it is a matrix. Create a neighboord list 
lwsp_it <- spdep::mat2listw(Wsp_it, style = "W")
 ### Spatio-temporal semiparametric ANOVA model 
 ### Interaction terms f12,f1t,f2t and f12t with nested basis
 ### Remark: nest_sp1, nest_sp2 and nest_time must be divisors of nknots
 form4 <- unrate ~ partrate + agri + cons +
                   pspl(serv, nknots = 15) + 
                   pspl(empgrowth, nknots = 20) +
                   pspt(long, lat, year, 
                        nknots = c(18, 18, 8), 
                        psanova = TRUE, 
                        nest_sp1 = c(1, 2, 2), 
                        nest_sp2 = c(1, 2, 2),
                        nest_time = c(1, 2, 2))
 sptanova <- pspatfit(form4, data = unemp_it)
 summary(sptanova)
 

 ################################################  
 ### Interaction terms f1t not included in ANOVA decomposition
 form5 <- unrate ~ partrate + agri + cons +
                   pspl(serv, nknots = 15) + 
                   pspl(empgrowth, nknots=20) +
                   pspt(long, lat, year, 
                        nknots = c(18, 18, 8),
                        psanova = TRUE, 
                        nest_sp1 = c(1, 2, 3), 
                        nest_sp2 = c(1, 2, 3),
                        nest_time = c(1, 2, 2), 
                        f1t_int = FALSE)
 ## Add sar specification and ar1 temporal correlation 
 sptanova2_sar_ar1 <- pspatfit(form5, data = unemp_it, 
                              listw = lwsp_it, 
                              type = "sar",
                              cor = "ar1")
summary(sptanova2_sar_ar1)

[Package pspatreg version 1.1.2 Index]