cics_explicit_smooth {ICSsmoothing}

Smooth given data set by k-component non-uniform clamped interpolating spline (NcICS).

Description

cics_explicit_smooth constructs the non-uniform clamped interpolating spline with k components that smoothes given data set {(xx[i],yy[i]), i=1..length(xx)}.

Usage

cics_explicit_smooth(
  xx,
  yy,
  uu,
  clrs = c("blue", "red"),
  d,
  xlab = NULL,
  ylab = NULL,
  title = NULL
)

Arguments

xx

a vector of data set's x-coordinates (that are in increasing order).

yy

a vector of data set's y-coordinates.

uu

a vector of arbitrary nodes, based on which we construct the smoothing spline. uu[1] and uu[length(uu)] must be equal to xx[1] and xx[length(xx)], respectively.

clrs

a vector of colours that are used alternately to plot the graph of spline's components.

d

a vector (optional parameter) that contains two values of derivative, in the first and the last node from uu. If missing, values of derivative are estimated by given linear regression model. If present, their contribution is removed from linear model and only function values are estimated.

xlab

a title (optional parameter) for the x axis.

ylab

a title (optional parameter) for the y axis.

title

a title (optional parameter) for the plot.

Value

a list with components

est_spline_coeffs

4-element array of (k)x(k+3) matrices, whereas element in i-th row and j-th of l-th matrix contains coefficient by x^{l-1} of cubic polynomial, which is in i-th row and j-th column of matrix B from smoothing spline's explicit form

S=B.\gamma.

est_spline_polynomials

list of string representations of smoothing NcICS.

est_gamma

vector of estimated smoothing spline's coefficients (function values and exterior-node derivatives).

aux_BF

A basis function of the spline

aux_tridiag_inverse

An inverse of the tridiagonal matrix used for spline derivatives construction

aux_M

An estimation matrix used to compute est_gamma

Examples

cics_explicit_smooth(
xx = CERN$x,
yy = CERN$y,
d = c(0, 1),
uu = c(1, sort(runif(20,1,277)), 277),
xlab = "X axis",
ylab = "Y axis"
)

yy <- c(1, 2, 3, 4, 3, 2, 2, 3, 5, 6, 7, 6, 5, 5, 4, 3, 2, 1, 0)
xx <- c(1:length(yy))
uu <- c(1,7,10,19)
sp <- cics_explicit_smooth(xx,yy,uu)
### We can change the derivatives at the end nodes:
sp <- cics_explicit_smooth(xx,yy, uu, d=c(3,-7/10))

### CERN:
uu <- c(1, 15, 26, 63, 73, 88, 103, 117, 132, 200, 203, 219, 258, 277)
sp <- cics_explicit_smooth(
  xx = CERN$x,
  yy = CERN$y,
  d = c(1, 0),
  uu
)