Package 'ICSsmoothing' reference manual

Title:	Data Smoothing by Interpolating Cubic Splines
Description:	We construct the explicit form of clamped cubic interpolating spline (both uniform - knots are equidistant and non-uniform - knots are arbitrary). Using this form, we propose a linear regression model suitable for real data smoothing.
Authors:	Juraj Hudak, Viliam Kacala, Csaba Torok, Lubomir Antoni
Maintainer:	Lubomir Antoni <[email protected]>
License:	GPL-2
Version:	1.2.8
Built:	2024-11-06 03:10:58 UTC
Source:	https://github.com/cran/ICSsmoothing

277 measurements of the cross sections for $\pi^{-}p$ collision (nuclear physics).

Description

277 measurements of the cross sections for $\pi^{-}p$ collision (nuclear physics).

Usage

CERN
CERN

Format

A data frame with 277 elements.

Source

https://link.springer.com/article/10.1007/BF02683433

Construct the explicit form of non-uniform clamped interpolating cubic spline (NcICS).

Description

cics_explicit constructs the explicit form of non-uniform clamped interpolating cubic spline (via Hermite cubic spline) for nodes uu, function values yy and exterior-node derivatives d.

Usage

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

Arguments

`uu`	a vector of arbitrary nodes (ordered ascendingly), with magnitude `n+2`, `n` $\ge$ `1`.
`yy`	a vector of function values pertaining to nodes in `uu`.
`d`	a vector of two values of derivative, in the first and the last node of `uu`.
`clrs`	a vector of colours that are used alternately to plot the graph of spline's components.
`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

`spline_coeffs`	matrix, whose `i`-th row contains coefficients of non-uniform ICS's `i`-th component.
`spline_polynomials`	list of NcICS's components string representations.
`B`	`4`-element array of `(n+1)x(n+4)` matrices, whereas element in `i`-th row and `j`-th column of `l`-th matrix contains coefficient by `x^{l-1}` of cubic polynomial that is in `i`-th row and `j`-th column of matrix `B` from spline's explicit form $S=B.\gamma.$
`gamma`	$\gamma=$ vector of spline coefficients - function values and exterior-node derivatives that takes part in the explicit form $S=B.\gamma$ .
`aux_BF`	A basis function of the spline
`aux_tridiag_inverse`	An inverse of the tridiagonal matrix used for spline derivatives construction

Examples

cics_explicit(
 uu = c(1, 2.2, 3, 3.8, 7),
 CERN$y[1:5],
 d=c(0,-2),
 xlab="X axis",
 ylab="Y axis"
)

uu <- c(0, 1, 4, 6);
yy <- c(4, 5, 2, 1.8);
sp <- cics_explicit(uu, yy, c(1,0))
sp$spline_polynomials
### <~~>
### Spline components' coefficients
explicit_spline(sp$B, sp$gamma)
sp$spline_coeffs == .Last.value
cics_explicit(
 uu = c(1, 2.2, 3, 3.8, 7),
 CERN$y[1:5],
 d=c(0,-2),
 xlab="X axis",
 ylab="Y axis"
)

uu <- c(0, 1, 4, 6);
yy <- c(4, 5, 2, 1.8);
sp <- cics_explicit(uu, yy, c(1,0))
sp$spline_polynomials
### <~~>
### Spline components' coefficients
explicit_spline(sp$B, sp$gamma)
sp$spline_coeffs == .Last.value

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
)
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
)
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
)

Construct the explicit form of uniform clamped interpolating cubic spline (UcICS).

Description

cics_unif_explicit constructs the explicit form of uniform clamped interpolating cubic spline (via Hermite cubic spline) for nodes uu, function values yy and exterior-node derivatives d.

Usage

cics_unif_explicit(
  uumin,
  uumax,
  yy,
  d,
  clrs = c("blue", "red"),
  xlab = NULL,
  ylab = NULL,
  title = NULL
)
cics_unif_explicit(
  uumin,
  uumax,
  yy,
  d,
  clrs = c("blue", "red"),
  xlab = NULL,
  ylab = NULL,
  title = NULL
)

Arguments

`uumin`	a starting node.
`uumax`	an ending node.
`yy`	a vector of function values pertaining to nodes in `uu`.
`d`	a vector of two values of derivative, in the first and the last node of `uu`.
`clrs`	a vector (optional parameter) of colours that are used alternately to plot the graph of spline's components.
`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 of spline components

`spline_coeffs`	matrix, whose `i`-th row contains coefficients of uniform ICS's `i`-th component.
`spline_polynomials`	list of UcICS's components string representations.
`B`	`4`-element array of `(n+1)x(n+4)` matrices, whereas element in `i`-th row and `j`-th column of `l`-th matrix contains coefficient by `x^{l-1}` of cubic polynomial that is in `i`-th row and `j`-th column of matrix `B` from spline's explicit form $S=B.\gamma.$
`gamma`	$\gamma=$ vector of spline coefficients - function values and exterior-node derivatives that takes part in the explicit form $S=B.\gamma$ .
`aux_BF`	A basis function of the spline
`aux_tridiag_inverse`	An inverse of the tridiagonal matrix used for spline derivatives construction

Examples

yy <- c(4, 5, 2, 1.8);
sp <- cics_unif_explicit(0, 6, yy, c(2, 0.9))
sp$spline_polynomials
### <~~>
### Spline components' coefficients
explicit_spline(sp$B, sp$gamma)
sp$spline_coeffs == .Last.value
yy <- c(4, 5, 2, 1.8);
sp <- cics_unif_explicit(0, 6, yy, c(2, 0.9))
sp$spline_polynomials
### <~~>
### Spline components' coefficients
explicit_spline(sp$B, sp$gamma)
sp$spline_coeffs == .Last.value

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

Description

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

Usage

cics_unif_explicit_smooth(
  xx,
  yy,
  k,
  clrs = c("blue", "red"),
  d,
  xlab = NULL,
  ylab = NULL,
  title = NULL,
  plotTF = TRUE
)
cics_unif_explicit_smooth(
  xx,
  yy,
  k,
  clrs = c("blue", "red"),
  d,
  xlab = NULL,
  ylab = NULL,
  title = NULL,
  plotTF = TRUE
)

Arguments

`xx`	a vector of data set's `x`-coordinates (that are in increasing order).
`yy`	a vector of datanvidi set's `y`-coordinates.
`k`	a chosen number of components of smoothing UcICS (integer $\ge 2$ ).
`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 computed node. 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.
`plotTF`	a boolean value (optional parameter), if TRUE then plot.

Value

a list with components

`nodes`	vector of equidistant nodes, based on which we construct the smoothing spline.
`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 UcICS.
`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


cp <- cics_unif_explicit_smooth(
xx = CERN$x,
  yy = CERN$y,
  k = 19, #23,
  d = c(1, 0),
  xlab = "X axis",
  ylab = "Y axis"
)
cp <- cics_unif_explicit_smooth(
xx = CERN$x,
  yy = CERN$y,
  k = 19, #23,
  d = c(1, 0),
  xlab = "X axis",
  ylab = "Y axis"
)

The function computes the coefficients of the cubic polynomials as spline components of the clamped interpolating cubic spline of class `C^2` in its explicit form `S=B * gamma`.

Description

The function computes the coefficients of the cubic polynomials as spline components of the clamped interpolating cubic spline of class C^2 in its explicit form S=B * gamma.

Usage

explicit_spline(B, gamma)
explicit_spline(B, gamma)

Arguments

`B`	a `4`-element array of `(n+1)x(n+4)` matrices, whereas element in `i`-th row and `j`-th column of `l`-th matrix contains coefficient by `x^{l-1}` of cubic polynomial that is in `i`-th row and `j`-th column of matrix `B` from spline's explicit form `S=B.gamma`.
`gamma`	a vector of spline coefficients - function values and exterior-node derivatives that takes part in the explicit form $S=B.\gamma$ .

Value

a matrix with four columns, whose i-th row contains the coefficients of the splines's i-th component.

Examples

# See functions cics_explicit, cics_unif_explicit and the vignette.
# See functions cics_explicit, cics_unif_explicit and the vignette.

Forecasting demo using cics_unif_explicit_smooth.

Description

Forecasting demo using cics_unif_explicit_smooth.

Usage

forecast_demo()
forecast_demo()

Value

a forecast result

Examples


# Plots as well as the process of computation of future derivatives and values using extrapolation.
ud <- forecast_demo()

# Plots as well as the process of computation of future derivatives and values using extrapolation.
ud <- forecast_demo()

Construct 4 Hermite basis functions.

Description

hermite_bf_matrix constructs matrix of Hermite basis functions' coefficients on [u,v], that is the matrix of 4 cubic polynomials' coefficients of one-component Hermite cubic spline.

Usage

hermite_bf_matrix(u, v)
hermite_bf_matrix(u, v)

Arguments

`u`	a left border of interval [`u`,`v`].
`v`	a right border of interval [`u`,`v`], `u` $\le$ `v`.

Value

The matrix of 4 Hermite basis functions' coefficients.

Examples

hermite_bf_matrix(0,1)
hermite_bf_matrix(-2,3)
hermite_bf_matrix(0,1)
hermite_bf_matrix(-2,3)

Construct inverse of a general tridiagonal matrix.

Description

tridiag_inv_general constructs inverse of a general tridiagonal matrix T of order n, using Usmani's theorem.

Usage

tridiag_inv_general(T, n)
tridiag_inv_general(T, n)

Arguments

`T`	a tridiagonal matrix.
`n`	an order of given tridiagonal matrix.

Value

The inverse of matrix T.

Examples

tridiag_inv_general(matrix(c(1, 4, 0, -9), 2, 2), 2)
tridiag_inv_general(matrix(c(1, 3, 5, -2, 0, 8, 7, 6, 6), 3, 3), 3)
tridiag_inv_general(matrix(c(1, 4, 0, -9), 2, 2), 2)
tridiag_inv_general(matrix(c(1, 3, 5, -2, 0, 8, 7, 6, 6), 3, 3), 3)

Construct inverse of a tridiagonal matrix T_n(a,b,a).

Description

tridiag_inv_unif_by_sums constructs inverse of a regular tridiagonal matrix T_n(a,b,a) with constant entries by a special algorithm using sums of matrix elements.

Usage

tridiag_inv_unif_by_sums(n, a, b)
tridiag_inv_unif_by_sums(n, a, b)

Arguments

`n`	an order of given tridiagonal matrix.
`a`	a value of tridiagonal matrix elements that are off-diagonal.
`b`	a value of tridiagonal matrix diagonal elements.

Value

The inverse of matrix T_n(a,b,a).

Examples

tridiag_inv_unif_by_sums(5, 1, 4)
tridiag_inv_unif_by_sums(9, 10, -1)
tridiag_inv_unif_by_sums(5, 1, 4)
tridiag_inv_unif_by_sums(9, 10, -1)

Package 'ICSsmoothing'

Help Index

277 measurements of the cross sections for π−p\pi^{-}pπ−p collision (nuclear physics).

Description

Usage

Format

Source

Construct the explicit form of non-uniform clamped interpolating cubic spline (NcICS).

Description

Usage

Arguments

Value

Examples

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

Description

Usage

Arguments

Value

Examples

Construct the explicit form of uniform clamped interpolating cubic spline (UcICS).

Description

Usage

Arguments

Value

Examples

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

Description

Usage

Arguments

Value

Examples

The function computes the coefficients of the cubic polynomials as spline components of the clamped interpolating cubic spline of class C^2 in its explicit form S=B * gamma.

Description

Usage

Arguments

Value

Examples

Forecasting demo using cics_unif_explicit_smooth.

Description

Usage

Value

Examples

Construct 4 Hermite basis functions.

Description

Usage

Arguments

Value

Examples

Construct inverse of a general tridiagonal matrix.

Description

Usage

Arguments

Value

Examples

Construct inverse of a tridiagonal matrix T_n(a,b,a).

Description

Usage

Arguments

Value

Examples

277 measurements of the cross sections for $\pi^{-}p$ collision (nuclear physics).

The function computes the coefficients of the cubic polynomials as spline components of the clamped interpolating cubic spline of class `C^2` in its explicit form `S=B * gamma`.