quadrature: Quadrature rules for the 'goffda' package

In goffda: Goodness-of-Fit Tests for Functional Data

Quadrature rules for the goffda package

Description

Quadrature rules for unidimensional and bidimensional functions, as enhancements of int.simpson from the fda.usc-package.

Usage

integral1D(fx, t, int_rule = "trapezoid", equispaced = FALSE,
  verbose = FALSE)

integral2D(fxy, s, t, int_rule = "trapezoid", equispaced_x = FALSE,
  equispaced_y = FALSE, verbose = FALSE)

w_integral1D(t, int_rule = "trapezoid", equispaced = FALSE,
  verbose = FALSE)

Arguments

fx	a vector of length length(t) with the evaluation of a univariate function at t.
int_rule	quadrature rule for approximating the definite unidimensional integral: trapezoidal rule (int_rule = "trapezoid") and extended Simpson rule (int_rule = "Simpson") are available. Defaults to "trapezoid".
equispaced	flag to indicate if X_fdata$data is valued in an equispaced grid or not. Defaults to FALSE.
verbose	flag to show information about the procedures. Defaults to FALSE.
fxy	a matrix of size c(length(s), length(t)) with the evaluation of a bivariate function at the bivariate grid formed by s and t.
s, t	vectors with grid points where functions are valued.
equispaced_x, equispaced_y	flags to indicate if fxy is valued in equispaced grids or not, at each one of dimensions. Both default to FALSE.

Value

• w_integral1D: a vector of length t with the weights required for the quadrature rule int_rule.

• integral1D: a scalar with the approximation of the univariate integral.

• integral2D: a scalar with the approximation of the bivariate integral.

Author(s)

Code iterated by Javier Álvarez-Liébana and Eduardo García-Portugués from the fda.usc-package originals.

References

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery B. P. (1997). Numerical Recipes in Fortran 77: the art of scientific computing (2nd ed).

Examples

## Numerical integral of 1-D functions

# Equispaced grid points
t1 <- seq(0, 1, l = 201)
t2 <- seq(pi / 4, 3 * pi / 2, l = 201)
fx1 <- 2 * (t1^3) - t1^2 + 5 * t1 - 2 # True integral is equal to 2/3
fx2 <- sin(sqrt(t2)) # True integral is equal to 3.673555
int_fx1_trap <- integral1D(fx = fx1, t = t1, int_rule = "trapezoid",
                           equispaced = TRUE)
int_fx1_Simp <- integral1D(fx = fx1, t = t1, int_rule = "Simpson",
                           equispaced = TRUE)

int_fx2_trap <- integral1D(fx = fx2, t = t2, int_rule = "trapezoid",
                           equispaced = TRUE)
int_fx2_Simp <- integral1D(fx = fx2, t = t2, int_rule = "Simpson",
                           equispaced = TRUE)

# Check if the true integrals is approximated properly
abs(int_fx1_trap - 2/3) / (2/3)
abs(int_fx1_Simp - 2/3) / (2/3)
abs(int_fx2_trap - 3.673555) / 3.673555
abs(int_fx2_Simp - 3.673555) / 3.673555

# Non equispaced grid points
t <- c(seq(0, 0.3, l = 50), seq(0.31, 0.6, l = 150),
       seq(0.61, 1, l = 100))
fx <- 2 * (t^3) - t^2 + 5 * t - 2
int_fx_trap <- integral1D(fx = fx, t = t, int_rule = "trapezoid",
                          equispaced = FALSE)
int_fx_Simp <- integral1D(fx = fx, t = t, int_rule = "Simpson",
                          equispaced = FALSE)

# Check if the true integral is approximated properly
abs(int_fx_trap - 2/3) / (2/3)
abs(int_fx_Simp - 2/3) / (2/3)

## Numerical integral of 2-dimensional functions

# Equispaced grid points
s <- seq(0, 2, l = 101)
t <- seq(1, 5, l = 151)
fxy <- outer(s^2, t^3) # True integral is equal to 416
int_fxy_trap <- integral2D(fxy = fxy, s = s, t = t, int_rule = "trapezoid",
                           equispaced_x = TRUE, equispaced_y = TRUE)
int_fxy_Simp <- integral2D(fxy = fxy, s = s, t = t, int_rule = "Simpson",
                           equispaced_x = TRUE, equispaced_y = TRUE)

# Check if the true integral is approximated properly
abs(int_fxy_trap - 416) / 416
abs(int_fxy_Simp - 416) / 416

# Non equispaced grid points
s <- c(seq(0, 0.3, l = 150), seq(0.31, 1.6, l = 100),
       seq(1.61, 2, l = 250))
t <- c(seq(1, 2.6, l = 170), seq(2.61, 4, l = 100),
       seq(4.01, 5, l = 140))
fxy <- outer(s^2, t^3)

int_fxy_trap <- integral2D(fxy = fxy, s = s, t = t, int_rule = "trapezoid",
                           equispaced_x = FALSE, equispaced_y = FALSE)

# Check if the true integral is approximated properly
abs(int_fxy_trap - 416) / 416

goffda documentation built on Oct. 14, 2023, 5:08 p.m.