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Build a spline density from a provided grid density

SplineDensity.Rd

Build a spline density from a provided grid density.

The methods plot, print and predict can be used.

Usage

SplineDensity(x, f,
              xmin = min(x), xmax = max(x),
              leftDerivEst = c(0L, 1L), rightDerivEst = c(0L, 1L),
              leftDeriv = NULL, rightDeriv = NULL,
              knots = NULL,
              nKnots = 24L,
              order = 4L,
              plot = TRUE,
              check = FALSE)

Arguments

x

Numeric vector of values at which the density is provided.

f

Numeric vector of density values corresponding to x.

xmin

Left (or lower) end-point of the distribution.

xmax

Right (or upper) end-point.

leftDerivEst

Integer vector giving the order of the derivatives that will be estimated from the finite differences of the points in x and f. These values will be used even if derivatives are provided.

rightDerivEst

Similar to leftDerivEst.

leftDeriv

Vector of known derivatives at left end-point (if any). The given values are for order \(0\) to \(k-2\) in that order where \(k\) is the order. Unknown values are to be given as NA.

rightDeriv

Similar to rightDeriv for the upper end-point.

knots

A numeric vector of knots in ascending order.

nKnots

Number of knots to be used if knots is not provided.

order

The spline order \(k\), e.g. \(k = 1\) for a broken line spline and k = 4 for a cubic spline.

plot

Logical. If TRUE a plot is provided.

check

Logical. When TRUE, some check of the computations are carried over and results are printed.

Value

A list object that can be used for density computations. This object is given an S3 class "SplineDensity". The structure of this list might evolve in the future life of the package.

Details

A spline approximation for a given density is found by a constrained regression. First, a suitable basis of B-splines is built. Then the coefficients are found in order to minimise the distance to the provided density values with constraints arising from boundary conditions and from the normalisation condition. Boundary conditions can be given. By default, the values of the density and of its first order derivative are taken equal to a finite difference estimation from x and f. This works correctly when the grid is fine enough, and when the provided values correspond to those of a continuous function with continuous derivative on the closed interval with end-points xmin and xmax.

References

de Boor, C (2001) A Practical Guide to Splines, revised edition. Springer-Verlag.

Caution

The spline is not warranted to be positive. This will be the case if positive density values are provided in f and if the grid is fine enough.

Author

Yves Deville

See also

rSplineDensity to generate randomly drawn objects (e.g. for tests). predict.SplineDensity for the evaluation at chosen points.

Examples

data(Brest.tide)
SD <- SplineDensity(x = Brest.tide$x, f = Brest.tide$y)
#> leftDeriv = 0 9.381932e-06 NA 
#> rightDeriv = 0 -2.479663e-05 NA 
SD24 <- SplineDensity(x = Brest.tide$x, f = Brest.tide$y, nKnots = 24)
#> leftDeriv = 0 9.381932e-06 NA 
#> rightDeriv = 0 -2.479663e-05 NA 

## approximate a bounded GPD (negative shape) by a spline density
shape <- 2 + rexp(1)
x <- seq(from = 0, to = 1, length.out = 200)
f <- (1 - x )^(shape - 1) * shape
SDGP <- SplineDensity(x = x, f = f)
#> leftDeriv = 3.342225 -7.80186 NA 
#> rightDeriv = 0 -0.00274446 NA 
plot(SDGP)

#> NULL