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

In rominsal/pspatreg: Spatial and Spatio-Temporal Semiparametric Regression Models with Spatial Lags

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 = 3
)

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 = 3,
  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 = B*U_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.
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'.
scale	Logical value to scale the spatial and temporal coordinates before the estimation of semiparametric model. Default = 'TRUE'
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.

See Also

pspatfit estimate semiparametric spatial or spatio-temporal regression models.

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)
########### Constructing the spatial weights matrix
ames_sf1 <- ames_sf[(duplicated(ames_sf$Longitude) == FALSE), ]
coord_sf1 <- cbind(ames_sf1$Longitude, ames_sf1$Latitude)
ID <- row.names(as(ames_sf1, "sf"))
col_tri_nb <- tri2nb(coord_sf1)
soi_nb <- graph2nb(soi.graph(col_tri_nb, 
                            coord_sf1), 
                   row.names = ID)
lw_ames <- nb2listw(soi_nb, style = "W", 
                    zero.policy = 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)
#########  GAM pure with spatial trend (2d)
 
#####################  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)
 ### 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)                                 
 

rominsal/pspatreg documentation built on July 8, 2022, 9:28 p.m.