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Types of Uni and Bivariate Splines

Lindesay Scott-Hayward

2024-05-08

Source: vignettes/web/FittingDifferentSplines_MRSea.Rmd
FittingDifferentSplines_MRSea.Rmd

This document shows how to fit both one and two dimensional splines.

There are three different types of 1D spline available:

  • B-spline
  • Cyclic Cubic Spline
  • Natural Cubic Spline

and two different 2D radial basis functions:

  • exponential
  • gaussian

One dimensional Smoothing

Starting point

  • Load some data
  • Specify the variable of interest. In this case we pick month of the year.
  • Fit an initial model. For simplicity we fit an intercept only model.
# load data
data(ns.data.re)
ns.data.re$response<- ns.data.re$birds
varlist=c('MonthOfYear')
initialModel<- glm(response ~ 1, family='quasipoisson',data=ns.data.re)

Default: B-spline

  • This section shows the fitting of a quadratic B-spline. If linear or cubic B-splines are preferred, change degree to 1 or 3 respectively.
  • The default setting is for B-splines. If the splines parameter in the salsa1dlist is not specified then B-splines will be fitted.
  • Here I have shown how to add the splines parameter to the salsa1dlist object.

Note: I have chosen the QBIC for chosing the number and location of knots as it is relatively quick and useful for testing. (Q because we have a Quasi-Poisson model).

# set some input information for SALSA
salsa1dlist<-list(fitnessMeasure = 'QBIC', 
                  minKnots_1d = c(1), 
                  maxKnots_1d = c(3), 
                  startKnots_1d = c(1), 
                  degree = c(2),
                  gaps = c(0),
                  splines = c("bs"))
# run SALSA
salsa1dOutput<-runSALSA1D(initialModel, 
                          salsa1dlist, 
                          varlist = varlist, 
                          datain = ns.data.re,
                          suppress.printout = TRUE)
runPartialPlots(salsa1dOutput$bestModel, data=ns.data.re, 
                varlist.in = varlist, showKnots = TRUE)
#> [1] "Making partial plots"

Cyclic spline

To fit a cyclic spline, the splines parameter is set to “cc”. To change the order of a cyclic spline, you can use the degree parameter (order = degree +1).

Note that even though the minimum knots is set to 1, for a cyclic spline the minimum allowed is 3. This is changed automatically.

#set some input info for SALSA
salsa1dlist<-list(fitnessMeasure = 'QBIC', 
                  minKnots_1d = c(1), 
                  maxKnots_1d = c(3), 
                  startKnots_1d = c(1), 
                  degree = c(2),
                  gaps = c(0),
                  splines = c("cc"))
# run SALSA
salsa1dOutput.cc<-runSALSA1D(initialModel, 
                          salsa1dlist, 
                          varlist = varlist, 
                          datain = ns.data.re,
                          suppress.printout = TRUE)
runPartialPlots(salsa1dOutput.cc$bestModel, data=ns.data.re, 
                varlist.in = varlist, showKnots = TRUE)
#> [1] "Making partial plots"

Natural Cubic spline

To fit a natural spline, the splines parameter is set to “ns”.

Note that degree is not necessary for this spline type but it must still be specified in the salsa1dlist object. You can use NA for this.

#set some input info for SALSA
salsa1dlist<-list(fitnessMeasure = 'QBIC', 
                  minKnots_1d = c(1), 
                  maxKnots_1d = c(3), 
                  startKnots_1d = c(1), 
                  degree = c(NA),
                  gaps = c(0),
                  splines = c("ns"))
# run SALSA
salsa1dOutput.ns<-runSALSA1D(initialModel, 
                          salsa1dlist, 
                          varlist = varlist, 
                          datain = ns.data.re,
                          suppress.printout = TRUE)
runPartialPlots(salsa1dOutput.ns$bestModel, data=ns.data.re, 
                varlist.in = varlist, showKnots = TRUE)
#> [1] "Making partial plots"

Multiple different splines

To fit more than one type of spline to different variables, this is specified in the splines parameter in the same way you would specify the starting knots for example.

varlist=c('MonthOfYear', "x.pos")
#set some input info for SALSA
salsa1dlist<-list(fitnessMeasure = 'QBIC', 
                  minKnots_1d = c(1,1), 
                  maxKnots_1d = c(3,3), 
                  startKnots_1d = c(1,1), 
                  degree = c(2, 2),
                  gaps = c(0, 0),
                  splines = c("cc", "bs"))
# run SALSA
salsa1dOutput.multi<-runSALSA1D(initialModel, 
                          salsa1dlist, 
                          varlist = varlist, 
                          datain = ns.data.re,
                          suppress.printout = TRUE)
runPartialPlots(salsa1dOutput.multi$bestModel, data=ns.data.re, 
                varlist.in = varlist, showKnots = TRUE)
#> [1] "Making partial plots"

Two dimensional Smoothing

Starting point

This is the same starting point for one dimensional splines if it is only a two dimensional smooth you want.

  • Load some data
  • specify the variable of interest. In this case we pick month of the year.
  • Fit an initial model. For simplicity we fit an intercept only model.
# load data
baselinedata <- filter(nysted.analysisdata, impact == 1, season == 1)
initialModel<- glm(response ~ 1, family='quasipoisson',data=baselinedata)
kg <- getKnotgrid(baselinedata[, c("x.pos", "y.pos")], numKnots = 300, plot = TRUE)

# make distance matrices for datatoknots and knottoknots
distMats<-makeDists(baselinedata[, c("x.pos", "y.pos")], kg)
# make prediction distance matrix. 
preddata <- filter(nysted.predictdata, impact == 0, season == 1)
p2k <-makeDists(preddata[, c("x.pos", "y.pos")], kg, knotmat = FALSE)$dataDist

Default: Gaussian Radial Basis

This uses a Gaussian radial basis implemented using LRF.g(). It is the default and does not need to be specified.

# make parameter set for running salsa2D
salsa2dlist<-list(fitnessMeasure = 'QBIC', 
                  knotgrid = na.omit(kg),
                  startKnots=10, 
                  minKnots=2, 
                  maxKnots=20, 
                  gap=0)
salsa2dOutput<-runSALSA2D(initialModel,
                          salsa2dlist, 
                          d2k=distMats$dataDist,
                          k2k=distMats$knotDist,
                          basis = "gaussian", ##
                          suppress.printout = TRUE)
preddata$preds.g <- predict(object = salsa2dOutput$bestModel, 
                            newdata = preddata, g2k = p2k)

ggplot() +
  geom_tile(data=preddata, aes(x.pos, y.pos, fill=preds.g, 
                               height=sqrt(area), width=sqrt(area))) + 
  xlab("Easting (km)") + ylab("Northing (km)") + coord_equal() +
  theme_bw() + ggtitle("Gaussian Basis") +
  scale_fill_distiller(palette = "Spectral",name="Animal Counts")

Exponential Basis

This uses an exponential radial basis implemented using LRF.e(). It is more peaked at the knot than the gaussian and for this reason I have used a larger number of start knots.

# make parameter set for running salsa2D
salsa2dlist<-list(fitnessMeasure = 'QBIC', 
                  knotgrid = na.omit(kg),
                  startKnots=10, ##
                  minKnots=2, 
                  maxKnots=20, 
                  gap=0)
salsa2dOutput.exp<-runSALSA2D(initialModel,
                          salsa2dlist, 
                          d2k=distMats$dataDist,
                          k2k=distMats$knotDist,
                          basis = "exponential", ##
                          suppress.printout = TRUE)
preddata$preds.e <- predict(object = salsa2dOutput.exp$bestModel, 
                            newdata = preddata, g2k = p2k)

ggplot() +
  geom_tile(data=preddata, aes(x.pos, y.pos, fill=preds.e,
                               height=sqrt(area), width=sqrt(area))) + 
  xlab("Easting (km)") + ylab("Northing (km)") + coord_equal() +
  theme_bw() + ggtitle("Exponential Basis") +
  scale_fill_distiller(palette = "Spectral",name="Animal Counts")

Raw data for reference 

ggplot() +
     geom_tile(data=baselinedata, aes(x.pos, y.pos, fill=response, height=sqrt(area), width=sqrt(area))) + 
     xlab("Easting (km)") + ylab("Northing (km)") + coord_equal() +
     theme_bw() + ggtitle("Raw Data") +
     scale_fill_distiller(palette = "Spectral",name="Animal Counts")