`wv`

Overview
This repository is dedicated to the Wavelet Variance
(`wv`

) R package where different tools to perform wavelet
variance analysis are provided (both standard and robust analysis).
Below are instructions and examples on how to install and make use of
the `wv`

package.

*Currently the only implemented wavelet filter in the package is
the Haar wavelet filter.*

The `wv`

package is available on both CRAN and GitHub. The
CRAN version is considered stable while the GitHub version is subject to
modifications/updates which may lead to installation problems or broken
functions. You can install the stable version of the `wv`

package with:

`install.packages("wv")`

For users who are interested in having the latest developments, the
GitHub version is ideal
although more dependencies are required to run a stable version of the
package. Most importantly, users **must** have a (C++)
compiler installed on their machine that is compatible with R
(e.g. Clang). Once you’ve made sure that you have a compatible C++
compiler installed on your computer, run the following code in an R
session and you will be ready to use the devlopment version of
`wv`

.

```
# Install dependencies
install.packages(c("devtools"))
# Install/Update the package from GitHub
::install_github("SMAC-Group/wv")
devtools
# Install the package with Vignettes/User Guides
::install_github("SMAC-Group/wv", build_vignettes = TRUE) devtools
```

Below are some examples of how to make use of some of the main
functions in the `wv`

package. Firstly, we highlight the
functions that perform the wavelet decomposition of a time series (both
the discrete and maximum-overlap discrete wavelet transforms) based on
which the following functions can compute the wavelet variance and its
corresponding confidence intervals for inference. These are particularly
useful, for example, when comparing the wavelet variance of different
time series in order to understand if they share common properties or
not.

The DWT performs a wavelet decomposition by applying the wavelet filter to non-overlapping windows of the time series. Below is an example of how to perform this decomposition on a simulated Gaussian random walk process.

```
# Load packages
library(wv)
library(simts)
#>
#> Attaching package: 'simts'
#> The following object is masked from 'package:wv':
#>
#> unitConversion
# Set seed for reproducibility
set.seed(999)
# Simulate a Gaussian random walk
= 10^3
n = RW(gamma2 = 1)
model = gen_gts(n = n, model = model)
Xt
# Plot the simulated random walk
plot(Xt)
```

Based on the above code, we have simulated a random walk with null expectation unit innovation variance. The functions to compute the DWT, access its output and plot the related wavelet coefficients can be found below.

```
# DWT
= dwt(Xt)
Xt.dwt #> Warning in dwt(Xt): The data has been truncated so that it is divisible by
#> `nlevels` (e.g. 2^*)
# Print the Wavelet Coefficients
summary(Xt.dwt)
#>
#> Results of DWT using haar filter with 8 levels:
#> Displaying only the first 6 coefficients...
#> Level 1 Wavelet Coefficients
#> -0.9281198 0.1909687 -0.4002392 -0.8957566 -0.7926733 0.09473632 ...
#> Level 2 Wavelet Coefficients
#> 0.2739394 -2.795066 0.8319477 0.3625764 -0.08550498 0.108777 ...
#> Level 3 Wavelet Coefficients
#> -2.384147 2.388987 -2.733086 -1.928826 -2.133592 -2.809311 ...
#> Level 4 Wavelet Coefficients
#> -6.282995 -5.729953 -7.616116 1.635971 1.2219 4.605007 ...
#> Level 5 Wavelet Coefficients
#> -10.03819 -6.631445 -0.6948675 -19.01644 12.1667 -5.350667 ...
#> Level 6 Wavelet Coefficients
#> -23.08426 -18.32217 18.40802 -8.680854 -4.335243 6.224648 ...
#> Level 7 Wavelet Coefficients
#> -34.68146 -7.485961 -5.448519 -30.51181 ...
#> Level 8 Wavelet Coefficients
#> -50.81154 1.13322 ...
# Plot of Discrete Wavelet Coefficients
plot(Xt.dwt)
```

Compared to the DWT, the MODWT applies the wavelet filter to overlapping windows of the time series (more specifically it slides the filter by one observation at a time). As for the DWT, the functions to perform the MODWT are below.

```
# MODWT
= modwt(Xt)
Xt.modwt
# Summary of Maximum Overlap Discrete Wavelet Coefficients
summary(Xt.modwt)
#>
#> Results of MODWT using haar filter with 9 levels:
#> Displaying only the first 6 coefficients...
#> Level 1 Wavelet Coefficients
#> -0.6562798 0.397592 0.1350352 -0.1386532 -0.2830119 -0.9393291 ...
#> Level 2 Wavelet Coefficients
#> 0.1369697 0.2645046 -0.2126415 -0.822003 -1.397533 -1.344998 ...
#> Level 3 Wavelet Coefficients
#> -0.8429231 -1.362249 -2.044266 -1.991537 -1.571604 -0.827054 ...
#> Level 4 Wavelet Coefficients
#> -1.570749 -1.175945 -0.7167598 -0.3172241 -0.04720348 0.01135659 ...
#> Level 5 Wavelet Coefficients
#> -1.774519 -1.731545 -1.702113 -1.711879 -1.649612 -1.530804 ...
#> Level 6 Wavelet Coefficients
#> -2.885532 -2.821419 -2.829503 -2.839451 -2.783538 -2.726905 ...
#> Level 7 Wavelet Coefficients
#> -3.065437 -3.069304 -3.063743 -3.03344 -2.997907 -2.955661 ...
#> Level 8 Wavelet Coefficients
#> -3.175721 -3.114701 -3.064177 -3.0097 -2.956361 -2.910441 ...
#> Level 9 Wavelet Coefficients
#> -6.967706 -6.925613 -6.884283 -6.838645 -6.79024 -6.735489 ...
# Plot of Maximum Overlap Discrete Wavelet Coefficients
plot(Xt.modwt, index = "all")
```

If the interest of the user lies solely in the wavelet variance
(issued from either DWT or MODWT), then the package provides functions
that directly compute this quantity and deliver the tools necessary to
analyse it (confidence intervals and plots). Below is a code that
simulates a white noise and a random walk process and directly plots
their respective wavelet variances in a log-log plot applying the
`plot()`

function to the package function
`wvar()`

.

```
# Set seed for reproducibility
set.seed(999)
# Simulate Gaussian White noise
= 10^4
n = gen_gts(n = n, model = WN(sigma2 = 1))
Xt
# Simulate Gaussian Random walk
= gen_gts(n = n, model = RW(gamma2 = 1))
Yt
# Plot WV
par(mfrow = c(1,2), mar = c(4,5,1,1))
plot(wvar(Xt), main = "White noise")
plot(wvar(Yt), main = "Random walk", legend_position = NULL)
```

As indicated in the legends, the light shaded blue area represents the 95% confidence intervals for each scale of estimated wavelet variance. However, there could be many practical settings where the time series can suffer from some “contamination” (e.g. outliers) which can seriously bias the standard estimator of wavelet variance. The code below randomly replaces one percent of the observations from the above simulated random walk with observations from a white noise process with larger variance.

```
# Add contamination
= 0.01
gamma = Yt
Yt2 sample(1:n,round(gamma*n))] = rnorm(round(gamma*n),0,5)
Yt2[par(mfrow = c(1,2), mar = c(4,5,1,1))
robust_eda(Yt, main = "RW without contamination")
robust_eda(Yt2, legend_position = NULL, main = "RW with contamination")
```

It can be seen how the classic and robust wavelet variance estimates agree when there is no contamination (left plot) but they classic estimates are heavily biased (especially at the first more informative scales) when the random walk has only 1% contamination.

When dealing with different time series, it is possible to compare their respective wavelet variances to understand if they have similar behaviour/properties. An example is given below where four different first-order autoregressive processes (with different values of the autoregressive parameters) are simulated and succesively their wavelet variance is computed.

```
# Simulate AR processes
= 10^5
n = gen_gts(n = n, model = AR1(phi = 0.10, sigma2 = 1))
Xt = gen_gts(n = n, model = AR1(phi = 0.35, sigma2 = 1))
Yt = gen_gts(n = n, model = AR1(phi = 0.79, sigma2 = 1))
Zt = gen_gts(n = n, model = AR1(phi = 0.95, sigma2 = 1))
Wt
# Compute WV
= wvar(Xt)
wv_Xt = wvar(Yt)
wv_Yt = wvar(Zt)
wv_Zt = wvar(Wt)
wv_Wt
# Plot results
compare_wvar(wv_Xt, wv_Yt, wv_Zt, wv_Wt)
```

As seen above, the function `compare_wvar()`

allows to
plot different outputs of the `wvar()`

function and it can be
seen how the four time series deliver different wavelet variances (as
the autoregressive parameter approaches zero, the shape of the wavelet
variance plot approaches the behaviour of the wavelet variance of a
white noise process).

In the package, we also add some datasets which are the wavelet
variance computed based on real IMU data. Currently the package includes
datasets `adis_wv`

, `imar_wv`

,
`kvh1750_wv`

, `ln200_wv`

and
`navchip_av`

. We can plot these wavelet variance simply with
the `plot`

function. As an example:

```
data("kvh1750_wv")
plot(kvh1750_wv)
```

Various guides ship with package or are available on https://smac-group.com/ to provide insight into how to use the different methods. At the present time, the following vignettes are available:

- Process to Haar Wavelet Variance (Online)