## S3 method for class 'fbps'
predict(object, newdata, ...)

##########################
#### True function   #####
##########################
n1 <- 60
n2 <- 80
x <- (1: n1)/n1-1/2/n1
z <- (1: n2)/n2-1/2/n2
MY <- array(0,c(length(x),length(z)))
sigx <- .3
sigz <- .4
for(i in 1: length(x))
  for(j in 1: length(z))
 {
    #MY[i,j] <- .75/(pi*sigx*sigz) *exp(-(x[i]-.2)^2/sigx^2-(z[j]-.3)^2/sigz^2)
    #MY[i,j] <- MY[i,j] + .45/(pi*sigx*sigz) *exp(-(x[i]-.7)^2/sigx^2-(z[j]-.8)^2/sigz^2)
    MY[i,j] = sin(2*pi*(x[i]-.5)^3)*cos(4*pi*z[j])
  }

##########################
#### Observed data   #####
##########################
sigma <- 1
Y <- MY + sigma*rnorm(n1*n2,0,1)

##########################
####   Estimation    #####
##########################
est <- fbps(Y,list(x=x,z=z))
mse <- mean((est$Yhat-MY)^2)
cat("mse of fbps is",mse,"\n")
cat("The smoothing parameters are:",est$lambda,"\n")

########################################################################
########## Compare the estimated surface with the true surface #########
########################################################################

par(mfrow=c(1,2))
persp(x,z,MY,zlab="f(x,z)",zlim=c(-1,2.5), phi=30,theta=45,expand=0.8,r=4,
      col="blue",main="True surface")
persp(x,z,est$Yhat,zlab="f(x,z)",zlim=c(-1,2.5),phi=30,theta=45,
      expand=0.8,r=4,col="red",main="Estimated surface")

##########################
####   prediction    #####
##########################

# 1. make prediction with predict.fbps() for all pairs of x and z given in the original data
#    ( it's expected to have same results as Yhat obtianed using fbps() above )
newdata <- list(x= rep(x, length(z)), z = rep(z, each=length(x)))
pred1 <- predict(est, newdata=newdata)$fitted.values
pred1.mat <- matrix(pred1, nrow=length(x))
par(mfrow=c(1,2))
image(pred1.mat); image(est$Yhat)
all.equal(as.numeric(pred1.mat), as.numeric(est$Yhat))

# 2. predict for pairs of first 10 x values and first 5 z values
newdata <- list(x= rep(x[1:10], 5), z = rep(z[1:5], each=10))
pred2 <- predict(est, newdata=newdata)$fitted.values
pred2.mat <- matrix(pred2, nrow=10)
par(mfrow=c(1,2))
image(pred2.mat); image(est$Yhat[1:10,1:5])
all.equal(as.numeric(pred2.mat), as.numeric(est$Yhat[1:10,1:5]))
# 3. predict for one pair 
newdata <- list(x=x[5], z=z[3])
pred3 <- predict(est, newdata=newdata)$fitted.values
all.equal(as.numeric(pred3), as.numeric(est$Yhat[5,3]))