# Fit amplitude coeffcients in the bundle model for expectiles 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. ```r fitampl(y, B, alpha, p, a, pord = 2, lambda) ``` ## 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. ## Returns a vector of estimated B-spline coefficients. ## 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 $0 a_jg(x_i)$ 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. ## Note This is a simplification of the model described in the reference. There is no explict term for the trend. ## Examples ```r # 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) ``` ## 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. ## Author(s) Paul Eilers

fitampl function