# windowed_scale_index: Windowed scale index In wavScalogram: Wavelet Scalogram Tools for Time Series Analysis

## Description

This function computes the windowed scale indices of a signal in the scale interval [s_0,s_1], for a given set of scale parameters s_1 and taking s_0 as the minimum scale (see Benítez et al. 2010).

The windowed scale index of a signal in the scale interval [s_0,s_1] centered at time tc and with time windows radius windowrad is given by the quotient

where WS_{windowrad} is the corresponding windowed scalogram with time windows radius windowrad, s_{max} \in [s_0,s_1] is the smallest scale such that WS_{windowrad}(tc,s)≤ WS_{windowrad}(tc,s_{max}) for all s \in [s_0,s_1], and s_{min} \in [s_{max},2s_1] is the smallest scale such that WS_{windowrad}(tc,s_{min})≤ WS_{windowrad}(tc,s) for all s \in [s_{max},2s_1].

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 windowed_scale_index(signal = NULL, wsc = NULL, wsc_coi = NULL, dt = 1, scales = NULL, powerscales = TRUE, s1 = NULL, windowrad = NULL, delta_t = NULL, wname = c("MORLET", "DOG", "PAUL", "HAAR", "HAAR2"), wparam = NULL, waverad = NULL, border_effects = c("BE", "INNER", "PER", "SYM"), makefigure = TRUE, time_values = NULL, figureperiod = TRUE, plot_wsc = FALSE, xlab = "Time", ylab = NULL, main = "Windowed Scale Index", zlim = NULL) 

## Arguments

 signal A vector containing the signal whose windowed scale indices are wanted. wsc A matrix containing the windowed scalograms from which the windowed scale indices are going to be computed (number of times x number of scales, as it is returned by the windowed_scalogram function). If wsc is not NULL, then signal, windowrad, delta_t, waverad and border_effects are not necessary and they are ignored. wsc_coi A vector of length nrow(wsc) (i.e. number of times) containing the values of the maximum scale at each time from which there are border effects in the windowed scalogram wsc. If wsc is NULL, then wsc_coi is not necessary and it is ignored. dt Numeric. The time step of the signal. scales A vector containing the wavelet scales at wich the windowed scalograms are computed. This can be either a vector with all the scales or, following Torrence and Compo 1998, a vector of 3 elements with the minimum scale, the maximum scale and the number of suboctaves per octave. In the first case, powerscales must be FALSE if the given scales are not power 2 scales. In the second case, powerscales must be TRUE in order to construct power 2 scales using a base 2 logarithmic scale). If scales is NULL, they are automatically constructed. powerscales Logical. It must be TRUE (default) only in these cases: If scales are power 2 scales, i.e. they use a base 2 logarithmic scale. If we want to construct power 2 scales automatically. In this case, scales must be NULL. If we want to construct power 2 scales from scales. In this case, length(scales) must be 3. Otherwise, it must be FALSE. s1 A vector containing the scales s_1. The windowed scale indices are computed in the intervals [s_0,s_1], where s_0 is the minimum scale in scales. If s1 are not power 2 scales, then scales should not be power 2 scales either and hence, powerscales must be FALSE. windowrad Integer. Time radius for the windows, measured in dt. By default, it is set to ceiling(length(signal) / 20). delta_t Integer. Increment of time for the construction of windows central times, measured in dt. By default, it is set to ceiling(length(signal) / 256). wname A string, equal to "MORLET", "DOG", "PAUL", "HAAR" or "HAAR2". The difference between "HAAR" and "HAAR2" is that "HAAR2" is more accurate but slower. wparam The corresponding nondimensional parameter for the wavelet function (Morlet, DoG or Paul). waverad Numeric. The radius of the wavelet used in the computations for the cone of influence. If it is not specified, it is asumed to be √{2} for Morlet and DoG, 1/√{2} for Paul and 0.5 for Haar. border_effects A string, equal to "BE", "INNER", "PER" or "SYM", which indicates how to manage the border effects which arise usually when a convolution is performed on finite-lenght signals. "BE": With border effects, padding time series with zeroes. "INNER": Normalized inner scalogram with security margin adapted for each different scale. "PER": With border effects, using boundary wavelets (periodization of the original time series). "SYM": With border effects, using a symmetric catenation of the original time series. makefigure Logical. If TRUE (default), a figure with the windowed scale indices is plotted. time_values A numerical vector of length length(signal) containing custom time values for the figure. If NULL (default), it will be computed starting at 0. figureperiod Logical. If TRUE (default), periods are used in the figure instead of scales. plot_wsc Logical. If TRUE, it plots the windowed scalograms from which the windowed scale indices are computed. xlab A string giving a custom X axis label. ylab A string giving a custom Y axis label. If NULL (default) the Y label is either "s1" or "Period of s1" depending on the value of figureperiod if length(s1) > 1, or "Windowed Scale Index" if length(s1) == 1. main A string giving a custom main title for the figure. zlim A vector of length 2 with the limits for the z-axis (the color bar).

## Value

A list with the following fields:

• wsi: A matrix of size length(tcentral) x length(s1) containing the values of the corresponding windowed scale indices.

• s0: The scale s_0.

• s1: The vector of scales s_1.

• smax: A matrix of size length(tcentral) x length(s1) containing the scales s_{max}.

• smin: A matrix of size length(tcentral) x length(s1) containing the scales s_{min}.

• wsc: A matrix of size length(tcentral) x length(scales) containing the windowed scalograms from which the windowed scale indices are computed.

• scalog_smax: A matrix of size length(tcentral) x length(s1) containing the values of the corresponding scalograms at scales s_{max}.

• scalog_smin: A matrix of size length(tcentral) x length(s1) containing the values of the corresponding scalograms at scales s_{min}.

• tcentral: The vector of central times used in the computation of wsi.

• windowrad: Radius for the time windows, measured in dt.

• fourierfactor: A factor for converting scales into periods.

• coi_maxscale: A vector of length length(tcentral) containing the values of the maximum scale at each time from which there are border effects.

## References

R. Benítez, V. J. Bolós, M. E. Ramírez. A wavelet-based tool for studying non-periodicity. Comput. Math. Appl. 60 (2010), no. 3, 634-641.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 dt <- 0.1 time <- seq(0, 50, dt) signal <- c(sin(pi * time), sin(pi * time / 2)) # First, we try with default s1 scales (a vector with a wide range of values for s1). wsi_full <- windowed_scale_index(signal = signal, dt = dt, figureperiod = FALSE) # Next, we choose a meaningful s1 value, greater than all relevant scales. wsi <- windowed_scale_index(signal = signal, dt = dt, s1 = 4, figureperiod = FALSE) # Another way, giving the windowed scalograms instead of the signal: wsc <- windowed_scalogram(signal = signal, dt = dt, figureperiod = FALSE, energy_density = FALSE, makefigure = FALSE) wsi_full <- windowed_scale_index(wsc = wsc$wsc, wsc_coi = wsc$coi_maxscale, scales = wsc$scales, time_values = wsc$tcentral, figureperiod = FALSE) wsi <- windowed_scale_index(wsc = wsc$wsc, wsc_coi = wsc$coi_maxscale, scales = wsc$scales, s1 = 4, time_values = wsc$tcentral, figureperiod = FALSE) 

wavScalogram documentation built on July 23, 2021, 9:07 a.m.