Empirical Bayes Smoothing Splines with Correlated Errors

Description

Empirical Bayes smoothing splines with correlated errors. The method uses a recursive algorithm for signal extraction with a non-parametric estimation of the correlation matrix of the errors.

Details

Index:

eBsc                  Empirical Bayes smoothing splines with correlated errors
plot.eBsc             Plots fitted curves from the filter
summary.eBsc          Summary information of the error

The function eBsc() is used to fit the model. Using the resulting eBsc object and summary information on the errors can be printed using summary.

Author(s)

Francisco Rosales, Paulo Serra, Tatyana Krivobokova Maintainer: Francisco Rosales <francisco.rosales-marticorena@protonmail.com>

References

Serra, P. and Krivobokova, T. (2015)
Adaptive Empirical Bayesian Smoothing Splines

See Also

stl (package stats), HoltWinters (package stats)

Examples

# simulated data for non-correlated errors
library(eBsc)
n <- 250
sigma <- 0.05
beta <- function(x,p,q){
	gamma(p+q)/(gamma(p)*gamma(q))*x^(p-1)*(1-x)^(q-1)
}
x <- seq(0, 1, length.out = n)
mu <- (6 * beta(x, 30, 17) + 4 * beta(x, 3, 11))/10;
mu <- (mu - min(mu))/(max(mu) - min(mu))
noise <- rnorm(n)
y <- mu + sigma * noise

#q assumed known and equal to 3, and correlation unknown
fit <- eBsc(y, method = "N", q=3)
plot(fit, full = FALSE)

Internal ebssc objects

Description

Internal functions

Details

These functions are not to be called by the user

Set of functions in example RcppArmadillo package

Description

These four functions are created when RcppArmadillo.package.skeleton() is invoked to create a skeleton packages.

Usage

rcpparma_hello_world()
rcpparma_outerproduct(x)
rcpparma_innerproduct(x)
rcpparma_bothproducts(x)

Arguments

Details

These are example functions which should be largely self-explanatory. Their main benefit is to demonstrate how to write a function using the Armadillo C++ classes, and to have to such a function accessible from R.

Value

rcpparma_hello_world() does not return a value, but displays a message to the console.

rcpparma_outerproduct() returns a numeric matrix computed as the outer (vector) product of x.

rcpparma_innerproduct() returns a double computer as the inner (vector) product of x.

rcpparma_bothproducts() returns a list with both the outer and inner products.

Author(s)

Dirk Eddelbuettel

References

See the documentation for Armadillo, and RcppArmadillo, for more details.

Examples

  x <- sqrt(1:4)
  rcpparma_innerproduct(x)
  rcpparma_outerproduct(x)

Computation of the Demmler-Reinsch basis.

Description

The Demmler-Reinsch basis is provided for a given smoothness degree in a uniform grid.

Usage

drbasis(nn,qq)

Arguments

Details

The use of large numbers required by the basis is handled by the package Brobdingnag. The method assumes the grid is equidistant. Missing values are not supported.

Value

A list object containing the following information.

Author(s)

Francisco Rosales

References

Rosales F. (2016).
Empirical Bayesian Smoothing Splines for Signals with Correlated Errors: Methods and Applications

Serra, P. and Krivobokova, T. (2015)
Adaptive Empirical Bayesian Smoothing Splines

Examples

oldpar <- par(no.readonly = TRUE)
#plot elements of the basis
library(eBsc)
n <- 100
Basis <- list()
for(i in 1:6){Basis[[i]] <- drbasis( nn = n, qq = i)}

#eigenvalues
par(mfrow = c(3,2), mar = c(4,2,2,2))
for(i in 1:6){
name <- paste("Eigenvalues (q = ",i,")", sep = "")
plot(Basis[[i]]$eigenvalues,
type = 'l', lwd = 2, xlab = "x", ylab = "", main = name)	
}
par(oldpar) 

#eigenvectors for q = 3
par(mfrow = c(3,2), mar = c(4,2,2,2))
for(i in 1:6){
name <- paste("Eigenvector n. ", i + 3, sep = "")
plot(Basis[[i]]$eigenvectorsQR[, i + 3],
type = 'l', lwd = 2, xlab = "x", ylab = "", main = name)	
}
par(oldpar) 

#example of a smooth function in the Demmler-Reinsch basis
library(eBsc)
n <- 200
Basis <- list()
for(i in 1:6){Basis[[i]] <- drbasis(nn = n, qq = i)}
coef3 <- c(rep(0,3), (pi*(2:(n-2))) ^ (-3.1)) * (cos(2*(1:n)))
A3 <- Basis[[3]]$eigenvectors
mu <- -A3%*%coef3
mu <- (mu - min(mu)) / (max(mu) - min(mu))
plot(mu, xlab = "x", ylab = "mu", type = 'l', lwd = 2)
par(oldpar)

Empirical Bayes Smoothing Splines with Correlated Errors

Description

Empirical Bayes smoothing splines with correlated errors. The method uses a recursive algorithm for signal extraction with a non-parametric estimation of the correlation matrix of the errors.

Usage

eBsc(y, q, method, parallel, R0, zero_range, ARpMAq, trace, tol.lambda, tol.rho, max.iter)

Arguments

Details

The method assumes the data is equidistant.

Value

A list object of class eBsc containing the following information.

Author(s)

Francisco Rosales, Paulo Serra, Tatyana Krivobokova

References

Serra, P. and Krivobokova, T. (2015)
Adaptive Empirical Bayesian Smoothing Splines

See Also

stl (package stats), HoltWinters (package stats)

Examples

library(eBsc)
n <- 250
sigma <- 0.05
beta <- function(x,p,q){
  gamma(p+q)/(gamma(p)*gamma(q))*x^(p-1)*(1-x)^(q-1)
}
x <- seq(0, 1, length.out = n)
mu <- (6 * beta(x, 30, 17) + 4 * beta(x, 3, 11))/10;
mu <- (mu - min(mu))/(max(mu) - min(mu))
noise <- rnorm(n)
y <- mu + sigma * noise

#q assumed known and equal to 3, and correlation unknown
fit <- eBsc(y, method = "N", q=3)
plot(fit, full = FALSE)

Plot fitted components

Description

Plot fitted components and the acf of the errors.

Usage

## S3 method for class 'eBsc'
plot(x,full=FALSE,...)

Arguments

Details

if the eBsc plots the fits and the acf of the errors.

Value

The function returns the selected plots.

Author(s)

Francisco Rosales, Paulo Serra, Tatyana Krivobokova.

References

Serra, P. and Krivobokova, T. (2015)
Adaptive Empirical Bayesian Smoothing Splines

Examples

library(eBsc)
n <- 250
sigma <- 0.05
Basis <- list()
for(i in 1:6) Basis[[i]] <- drbasis(nn = n, qq = i)
coef3 <- c(rep(0,3),(pi*(2:(n-2)))^(-3.1))*(cos(2*(1:n)))
A3 <- Basis[[3]]$eigenvectors
mu <- A3%*%coef3
mu <- (mu-min(mu))/(max(mu)-min(mu))
noise <- rnorm(n)
y <- mu + sigma * noise

#q assumed known and equal to 3, and correlation unknown
fit <- eBsc(y, method = "N", q=3)

#simple plot by  
plot(fit, full = FALSE)

eBsc Summary

Description

Takes an eBsc object produced by eBsc and summarizes the information of the errors.

Usage

## S3 method for class 'eBsc'
summary(object,...)

Arguments

Value

The function gives basic statistics of the error from applying eBsc.

Author(s)

Francisco Rosales, Paulo Serra, Tatyana Krivobokova

References

Serra, P. and Krivobokova, T. (2015)
Adaptive Empirical Bayesian Smoothing Splines

See Also

plot.eBsc (package eBsc),

Examples

# simulated data	
library(eBsc)
n <- 250
sigma <- 0.05

Basis <- list()
for(i in 1:6){Basis[[i]] <- drbasis(nn = n, qq = i)}
coef3 <- c(rep(0,3),(pi*(2:(n-2)))^(-3.1)) * (cos(2*(1:n)))
A3 <- Basis[[3]]$eigenvectors
mu <- - A3%*%coef3 
mu <- (mu - min(mu))/(max(mu) - min(mu))
noise <- rnorm(n)
y <- mu + sigma * noise

# correlation matrix assumed known and equal to the identity
fit <- eBsc(y, method = "N", q=3)

summary(fit)