holderSpectrum {wmtsa} | R Documentation |
Using a WTMM tree, this function returns time localized exponent estimations for a given time series.
holderSpectrum(chains, n.scale.min=3, fit=lmsreg)
chains |
a list containing information regarding the WTMM chains as output by the wtmmTree function.
Each element of the list is itself a list containing the named elements "time" , "scale" , and "wtmm"
corresponding to the location of the current chain in time, in scale, and its corresponding WTMM value, respectively. |
fit |
a linear regression function to use in fitting the resulting data. Default: lmsreg . |
n.scale.min |
the minimum number of scales (points) that a given suitable branch segment must have before being considered as an admissible candidate for exponent estimation. Default: 3. |
Many real-world time series contain sharp dicontinuities (cusps) which can be attributed to rapid changes in the observed system. These cusps are called singularities and their strength can be quantified via localized exponents as follows: Let f(t) be a continuous real-valued function containing a singularity at time t0. The exponent h(t0) is defined as the supremum (least upper bound) of all exponents h which satisfies the condition
|f(t) - Pn(t - t0)| <= C|t - t0|^h(t0)
where Pn(t - t0) is a polynomial of degree n <= h(t0) and C is a constant. The collection of exponents for a given time series denotes the so-called spectrum. Mallat demonstrated that a cusp singularity at time t0 can be estimated via the CWT by noting that the wavelet transform modulus maxima behave as W(a,t0) ~ |a|^h(t0) as the scale approaches 0.
Thus, the strength of cusp singularities in a given time series can be quantified by
In practice, the above technique can be unstable when applied to observational data due to
negative moment divergences and so-called outliers which correspond to the end points
of sample singularities. One must also be very careful in selecting an appropriate scaling
region of a tree branch before fitting the data. We accomplish this by first segmenting a
given tree branch into regions which exhibit approximate linear behavior in the log(scale)-log(WTMM) space,
and subsequently selecting the region corresponding to the smallest scales for exponent estimation.
Furthermore, through the n.scale.min
argument, the user can control the minimum number of scales (points)
that must exist in the isolated scaling region before a
exponent estimation is recorded.
a list containing the estimated exponent
and corresponding time
vectors.
S.G. Mallat, A Wavelet Tour of Signal Processing (2nd Edition), Academic Press, Cambridge, 1999.
S.G. Mallat and W.L. Hwang, ``Singularity detection and processing with wavelets", IEEE Transactions on Information Theory, 38, 617–643 (1992).
S.G. Mallat and S. Zhong, ``Complete signal representation with multiscale edges", IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 710–732 (1992).
J.F. Muzy, E. Bacry, and A. Arneodo, ``The multifractal formalism revisited with wavelets.", International Journal of Bifurcation and Chaos, 4, 245–302 (1994).
wavCWT
,
wavCWTFilters
,
wtmmTree
.
## calculate the CWT of a random walk series and ## the plot the result with an overlay of the ## original time series above the CWT image set.seed(100) x <- cumsum(rnorm(1024)) x.cwt <- wavCWT(x) plot(x.cwt, series=TRUE) ## create a WTMM tree and plot the first 50 ## branches found x.tree <- wtmmTree(x.cwt) plot(x.tree[1:50]) ## estimate and plot the Holder spectrum x.holder <- holderSpectrum(x.tree) old.plt <- splitplot(2,1,1) plot(x, type="l", col="blue", ylab="RANDOM WALK", xlab="TIME") splitplot(2,1,2) plot(x.holder$time, x.holder$exponent, type="s", xlab="TIME",ylab="Holder exponent") par(old.plt)