BUUHWE: Bottom Up Unbalanced Haar Wavelet Expansion

In Bagidis: BAses GIving DIStances

Description

Function of a time series x returning its unbalanced Haar wavelet expansion, as obtained with a bottom-up algortithm. BUUHWE is an acronym for Bottom-Up Unbalanced Haar Wavelets Expansion.

Usage

BUUHWE(x)
Breakpoints(Dataset.BUUHWE)
Details(Dataset.BUUHWE)

Arguments

x	a numeric vector of length N
Dataset.BUUHWE	a list of outputs of BUUHWE applied to a dataset of curves.

Details

See BAGIDIS-package for an overview about the BAGIDIS methodology and References for details, in particular Timmermans (2012), Chapter 1, Timmermans and von Sachs (2010) or Fryzlewicz (2007).

Value

BUUHWE(x) returns a list with

detail	detail coefficients starting from rank k=0 up to k=N-1.
basis	unbalanced Haar basis vectors ordered by colums. First column is the constant vector of rank k=0. This is a matrix of dimensions N xN.
split.abs	localization index. For consistency with basis matrix and detail vector dimensions, this is a vector of length N but first coefficient is NA. Index are then ordered by increasing rank k.
series	the initial series x.

Breakpoints returns the breakpoints of the Unbalanced Haar wavelet basis, from rank k=1 to rank k=N-1. In case it is applied to a list of BUUHWE provided for M series of length N, it returns the breakpoints for each elements of the list, as a matrix with M columns and N-1 rows.

Details returns the details of the Unbalanced Haar wavelet basis, from rank k=1 to rank k=N-1. In case it is applied to a list of outputs of BUUHWE provided for M series of length N, it returns the details for each elements of the list, as a matrix with M columns and N-1 rows.

Note

This function can be compared to function uh.bu in package unbalhaar of Piotr Fryzlewicz. Nevertheless outputs are differents as uh.bu is intented for denoising. The unbalanced Haar basis expansion is thus not explicitely obtained in uh.bu. Function BUUHWE.plot provides with a representation of the unbalanced Haar basis expansion.

Function BUUHWE_2D is the image equivalent of BUUHWE.

The BUUHWE expansion is the starting step for comparing curves through the BAGIDIS semi-distance. See function semimetric.BAGIDIS.

Breakpoints and details define what is called the signature of the series in the b-d plane. See references for details. A representation of the signature is obtained with BD.plot.

Author(s)

Catherine Timmermans, Institute of Statistics, Biostatistics and Actuarial Sciences, UCLouvain, Belgium.

Contact: [e-mail]

References

The main references are

• Timmermans C., 2012, Bases Giving Distances. A new paradigm for investigating functional data with applications for spectroscopy. PhD thesis, Universite catholique de Louvain. http://hdl.handle.net/2078.1/112451

• Timmermans C. and von Sachs R., 2015, A novel semi-distance for investigating dissimilarities of curves with sharp local patterns, Journal of Statistical Planning and Inference, 160, 35-50. http://hdl.handle.net/2078.1/154928

• Fryzlewicz P. and Timmermans C., 2015, SHAH: Shape Adaptive Haar wavelets for image processing. Journal of Computational and Graphical Statistics. (accepted - published online 27 May 2015) http://stats.lse.ac.uk/fryzlewicz/shah/shah.pdf

• Timmermans C., Delsol L. and von Sachs R., 2013, Using BAGIDIS in nonparametric functional data analysis: predicting from curves with sharp local features, Journal of Multivariate Analysis, 115, p. 421-444. http://hdl.handle.net/2078.1/118369

Other references include

• Girardi M. and Sweldens W., 1997, A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces, J. Fourier Anal. Appl. 3, 457-474

• Fryzlewicz P., 2007, Unbalanced Haar Technique for Non Parametric Function Estimation, Journal of the American Statistical Association, 102, 1318-1327.

• Timmermans C., von Sachs, R. , 2010, BAGIDIS, a new method for statistical analysis of differences between curves with sharp patterns (ISBA Discussion Paper 2010/30). Url : http://hdl.handle.net/2078.1/91090

• Timmermans, C. , Fryzlewicz, P., 2012, SHAH: Shape-Adaptive Haar Wavelet Transform For Images With Application To Classification (ISBA Discussion Paper 2012/15). Url: http://hdl.handle.net/2078.1/110529

See Also

semimetric.BAGIDIS, BUUHWE.plot, BUUHWE_2D,BD.plot.

Examples

x= c(1,7,3,0,-2,6,4,0,2)
BUUHWE(x)
Breakpoints(list(BUUHWE(x)))
y= c(1,7,5,5,-2,1,4,0,2)
Breakpoints(list(BUUHWE(x),BUUHWE(y)))

Bagidis documentation built on May 29, 2017, noon