JOPS (version 0.2.0)

fitampl: Fit amplitude coeffcients in the bundle model for expectiles

Description

There are two functions for fitting the expectile bundle model, one for estimating asymmetry parameters (fitasy), the other for estimating the amplitude function, fitampl, this function. See the details below.

Usage

fitampl(y, B, alpha, p, a, pord = 2, lambda)

Value

a vector of estimated B-spline coefficients.

Arguments

y

a response vector.

B

a proper B-spline basis matrix, see bbase().

alpha

a vector of B-spline coefficients.

p

a vector of asymmetries.

a

a vector of asymmetry parameters.

pord

the order of the difference penalty, default is 2.

lambda

the positive tuning parameter for the penalty.

Author

Paul Eilers

Details

The expectile bundle model determines a set of expectile curves for a point cloud with data vectors x and y, as \(\psi_j{x_i} = a_j g(x_i)\). Here \(a_j\) is the asymmetry parameter corresponding to a given asymmetry \(p_j\). A vector of asymmetries with all \(0 <p_j < 1\) is specified by the user.

The asymmetric least squares objective function is $$\sum_j \sum_i w_{ij}(y_i - \sum_j a_j g_j(x_i))^2.$$ The function \(g(\cdot)\) is called the amplitude. The weights depend on the residuals: $$w_{ij} = p_j$$ if \(y_i > a_jg(x_i)\) and \(w_{ij} = 1- p_j\) otherwise.

The amplitude function is a sum of B-splines with coefficients alpha. There is no direct solution, so alpha and the asymmetry parameters a must be updated alternatingly. See the example.

References

Schnabel, S.K. and Eilers, P.H.C. (2013) A location-scale model for non-crossing expectile curves. Stat 2: 171–183.

Eilers, P.H.C. and Marx, B.D. (2021). Practical Smoothing, The Joys of P-splines. Cambridge University Press.

Examples

# Get the data
data(bone_data)
x = bone_data$age
y = bone_data$spnbmd
m <- length(x)

# Set asymmetry levels
p = c(0.005, 0.01, 0.02, 0.05, 0.2, 0.5, 0.8, 0.9, 0.95, 0.98, 0.99, 0.995)
np <- length(p)

# Set P-spline parameters
x0 <- 5
x1 <- 30
ndx <- 20
bdeg <- 3
pord <- 2

# Compute bases
B <- bbase(x, x0, x1, ndx, bdeg)
xg <- seq(from = min(x), to = max(x), length = 100)
Bg <- clone_base(B, xg)
n <- ncol(B)

lambda = 1
alpha <- rep(1,n)
a = p
for (it in 1:20){
  alpha <- fitampl(y, B, alpha, p, a, pord, lambda)
  alpha <- alpha / sqrt(mean(alpha ^ 2))
  anew <- fitasy(y, B, alpha, p, a)
  da = max(abs(a - anew))
  a = anew
  cat(it, da, '\n')
     if (da < 1e-6) break
}

# Compute bundle on grid
ampl <- Bg %*% alpha
Z <- ampl %*% a

# Plot data and bundle
plot(x, y, pch = 15, cex = 0.7, col = 'grey', xlab = 'Age', ylab = 'Density')
cols = colorspace::rainbow_hcl(np, start = 10, end = 350)
matlines(xg, Z, lty = 1, lwd = 2, col = cols)