wavelet_decomposition {mrf}

Redundant Haar Wavelet Decomposition

Description

This function decomposes a time series in its wavelet and smooth coefficients using the redundant Haar wavelet transform.

Usage

wavelet_decomposition(UnivariateData, Aggregation = c(2, 4, 8, 16, 32),
Threshold="hard", Lambda=0.05)

Arguments

UnivariateData

[1:n] Numerical vector with n time series values

Aggregation

[1:Scales] Numerical vector of length 'Scales' carrying numbers whose index is associated with the wavelet level. The numbers indicate the number of values used for aggregation from the original time series.

Threshold

Character indicating if Thresholding is done on the wavelet decomposition or not. Default: Threshold="hard". Possible entries: Threshold="hard" for hard thresholding. Threshold="soft" for soft thresholding. Any other input indicates no thresholding.

Lambda

Numeric value indicating the threshold for computing a hard or soft threshold on the wavelet decomposition.

Details

The resulting wavelet and smooth coefficients are stored in so called wavelet and smooth part levels. The smooth part level is created from the original times series by aggregation (average). This makes the times series in some sense smoother, hence the naming. Each individual smooth part level can be created from the original time series by aggregating over different number of values. The different smooth part levels are ordered, so that the number of values used for aggregation are ascending. A dyadic scheme is recommended (increasing sequences of the power of two). The dyadic scheme for 5 levels would require agg_per_lvl = c(2, 4, 8, 16, 32). So the first smooth part level would be the average of two points of the original time series, the second smooth part level would be the average of four points, and so on. This averaging is applied asymmetrical. That means, that the result of the average of a sequence of points is obtained for the last point in time of that sequence. So each smooth part level starts with a certain offset, since no average can be obtained for the first particular points in time. The wavelet levels are the differences between the original time series and the smooth levels. The first wavelet level is the difference of the original time series and the first smooth part level. The second wavelet level is the difference of the first and second smooth part level and so on.

Value

List of

UnivariateData

[1:n] Numerical vector with n time series values.

WaveletCoefficients

[Scales, n] Matrix with 'Scales' many wavelet scales row-wise with n columns corresponding to the time domain of a time series.

SmoothCoefficients

[Scales, n] Matrix with 'Scales' many smooth approximation scales row-wise with n columns corresponding to the time domain of a time series.

Scales

Number of wavelet levels.

Author(s)

Quirin Stier

References

Aussem, A., Campbell, J., and Murtagh, F. Waveletbased Feature Extraction and Decomposition Strategies for Financial Forecasting. International Journal of Computational Intelligence in Finance, 6,5-12, 1998.

Renaud, O., Starck, J.-L., and Murtagh, F. Prediction based on a Multiscale De- composition. International Journal of Wavelets, Multiresolution and Information Processing, 1(2):217-232. doi:10.1142/S0219691303000153, 2003.

Murtagh, F., Starck, J.-L., and Renaud, O. On Neuro-Wavelet Modeling. Decision Support Systems, 37(4):475-484. doi:10.1016/S0167-9236(03)00092-7, 2004.

Renaud, O., Starck, J.-L., and Murtagh, F. Wavelet-based combined Signal Filter- ing and Prediction. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35(6):1241-1251. doi:10.1109/TSMCB.2005.850182, 2005.

Examples

data(AirPassengers)
plot(AirPassengers, type = "l", col = "black")
UnivariateData = as.vector(array(AirPassengers))
dec_res = wavelet_decomposition(UnivariateData, Aggregation = c(2,4))
plot(dec_res$SmoothCoefficients[2,4:length(dec_res$SmoothCoefficients[2,])],
type = "l", col = "blue")
lines(array(AirPassengers)[4:length(dec_res$SmoothCoefficients[2,])],
col = "black")