# dwt: Discrete Wavelet Transform In wavelets: Functions for Computing Wavelet Filters, Wavelet Transforms and Multiresolution Analyses

## Description

Computes the discrete wavelet transform coefficients for a univariate or multivariate time series.

## Usage

 `1` ```dwt(X, filter="la8", n.levels, boundary="periodic", fast=TRUE) ```

## Arguments

 `X` A univariate or multivariate time series. Numeric vectors, matrices and data frames are also accepted. `filter` Either a `wt.filter` object, a character string indicating which wavelet filter to use in the decomposition, or a numeric vector of wavelet coefficients (not scaling coefficients). See `help(wt.filter)` for acceptable filter names. `n.levels` An integer specifying the level of the decomposition. By default this is the value J such that the length of X is at least as great as the length of the level J wavelet filter, but less than the length of the level J+1 wavelet filter. Thus, j <= log((N-1)/(L-1)+1), where N is the length of X. `boundary` A character string indicating which boundary method to use. `boundary = "periodic"` and `boundary = "reflection"` are the only supported methods at this time. `fast` A logical flag which, if true, indicates that the pyramid algorithm is computed with an internal C function. Otherwise, only R code is used in all computations.

## Details

The discrete wavelet transform is computed via the pyramid algorithm, using pseudocode written by Percival and Walden (2000), pp. 100-101. When `boundary="periodic"` the resulting wavelet and scaling coefficients are computed without making changes to the original series - the pyramid algorithm treats `X` as if it is circular. However, when `boundary="reflection"` a call is made to `extend.series`, resulting in a new series which is reflected to twice the length of the original series. The wavelet and scaling coefficients are then computed by using a periodic boundary condition on the reflected sereis, resulting in twice as many wavelet and scaling coefficients at each level.

## Value

Returns an object of class `dwt`, which is an S4 object with slots

 `W` A list with element i comprised of a matrix containing the ith level wavelet coefficients. `V` A list with element i comprised of a matrix containing the ith level scaling coefficients. `filter` A `wt.filter` object containing information for the filter used in the decomposition. See `help(wt.filter)` for details. `level` An integer value representing the level of wavelet decomposition. `n.boundary` A numeric vector indicating the number of boundary coefficients at each level of the decomposition. `boundary` A character string indicating the boundary method used in the decomposition. Valid values are "periodic" or "reflection". `series` The original time series, `X`, in matrix format. `class.X` A character string indicating the class of the input series. Possible values are `"ts"`, `"mts"`, `"numeric"`, `"matrix"`, or `"data.frame"`. `attr.X` A list containing the attributes information of the original time series, `X`. This is useful if `X` is an object of class `ts` or `mts` and it is desired to retain relevant time information. If the original time series, `X`, is a matrix or has no attributes, then `attr.X` is an empty list. `aligned` A logical value indicating whether the wavelet and scaling coefficients have been phase shifted so as to be aligned with relevant time information from the original series. The value of this slot is initially FALSE and can only be changed to TRUE via the `align` function, with the `dwt` object as input. `coe` A logical value indicating whether the center of energy method was used in phase alignement of the wavelet and scaling coefficients. By default, this value is FALSE (and will always be FALSE when `aligned` is FALSE) and will be set to true if the `dwt` object is phase shifted via the `align` function and center of energy method.

## Author(s)

Eric Aldrich. ealdrich@gmail.com.

## References

Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.

`modwt`, `wt.filter`.

## Examples

 ```1 2 3 4 5 6 7``` ```# obtain the two series listed in Percival and Walden (2000), page 42 X1 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,.7,.9,0,.3) X2 <- c(.2,-.4,-.6,-.5,-.8,-.4,-.9,0,-.2,.1,-.1,.1,-.7,.9,0,.3) # combine them and compute DWT newX <- cbind(X1,X2) wt <- dwt(newX, n.levels=3, boundary="reflection", fast=FALSE) ```

### Example output

```
```

wavelets documentation built on March 26, 2020, 6:50 p.m.