wtmmTree {wmtsa} | R Documentation |
This function first finds the modulus maxima locations (in time and in scale) of the continuous wavelet transform input. The set of modulus maxima are then segmented into branches, where each branch represents a collection of WTMM that correspond to the same ridge in the WTMM time-scale plane. A coarse-to-fine scale strategy is used to identify the members of each branch as follows: (i) a single WTMM at the coarsest scale is selected as the start of a given branch, (ii) the closest neighboring WTMM in time at the next finest scale is then added to the branch, (iii) step ii is repeated until the smallest scale is reached or an apparent break occurs in the branch across scale, and (iv) steps i-iii are repeated until all WTMM have been accounted. Branches are allowed to wrap off one end of the time-scale plane in time and back onto the other. A branch is not grown unless the nearest neighbor candidate at the next finest scale is close in time to the last recorded branch member, where "close" is defined as being less than the current scale of the neighbor candidate. This means that the window in time for admissible neighbor WTMM candidates (at the next finest scale) shrinks proportionally with scale.
wtmmTree(x, bridge.gaps=FALSE, strength.min=0., n.octave.min=2, tolerance=0.0, border=FALSE)
x |
an object of class wavCWT (as produced by the wavCWT function). |
border |
a logical flag. If border is TRUE WTMM branches which fall off one end
of the CWT branch in time and return on the other are allowed. Otherwise,
these border branches are pruned from the returned list. Default: FALSE . |
bridge.gaps |
a logical flag. Gaps encountered in the search for
additional branch members may be bridged by setting the
bridge.gaps boolean to TRUE . Default: FALSE . |
n.octave.min |
a pruning factor for excluding non-persistent branches. If a WTMM branch does not span this number of octaves, it is excluded from the tree. Default: 2. |
strength.min |
a pruning factor for excluding weak branches. A given WTMM
is excluded from the current branch if it less than or equal
to the product of the maximum WTMM at the current scale and
strength.min . This parameter must be in the range [0,1].
If set to zero, all WTMM are allowed. If set to unity,
only the maximum WTMM at the current scale is allowed.
See the DETAILS section for more details.
Default: 0. |
tolerance |
a tolerance vector used to find moduls maxima in the given CWT. This vector must be as long as there are scales in the CWT such that the jth element defines the tolerance to use in finding modulus maxima at the jth scale of the CWT. Default: 0. |
A point in the CWT W(t,j) is defined
as a maximum if |W(t-1,j)| + tol < |W(t,j)| and |W(t+1,j)| + tol < |W(t,j)|
where tol
is a (scale-dependent) tolerance specified by the user.
The search algorithm is also adpated to identify plateaus in the data,
and will select the the middle of the plateau as a maximum location
when encountered. The data |W(t,j)| is first scaled so that its
maximum value is 1.0, so the tolerances should be adjusted accordingly.
Since the CWT coefficients are in effect a result band-pass filtering operations,
the large scale coefficients form a smoother curve than do the small
scale coefficients. Thus, the tolerance vector allows the user to specify
scale-dependent tolerances, helping to weed out undesirable local maxima.
It is recommended that the tolerance be set proportional to the scale,
e.g., tolerance=C / sqrt(scale)
where C is a constant
0 < C < 1.
an object of class WaveletWTMMTree
.
x[2:5]
. To extract branches which terminate near times 0.47, 0.3, and 1.4, use the syntax
x[time=c(0.47, 0.3, 1.4)]
. To extract all branches which terminate between times 1.2 and 1.5, use the syntax
x[range=c(1.2, 1.5)]
.x
is an output of the wtmmTree
function):
fit=TRUE
, a subset of branches (limited to four) are fit with
various linear regression models on a log(|WTMM|) versus log(scale) basis. The models are specified by the
optional models
argument. This scheme illustrates the process by which
exponents are estimated using the WTMM branches. For example, to see the regressions over chains 10 through 13,
issue plot(x[10:13], fit=TRUE)
. Default: FALSE
.fit=TRUE
. Default: c("lm", "lmsreg", "ltsreg")
.TRUE
, the branch number is placed at the head of each branch. Default: TRUE
.TRUE
, all of the (non-pruned and unbranched) WTMM are
plotted in the time-scale plane. Default: FALSE
.par
function. Default: "o"
.
J.F. Muzy, E. Bacry, and A. Arneodo., ``The multifractal formalism revisited with wavelets.", International Journal of Bifurcation and Chaos, 4, 245–302 (1994).
## 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]) ## plot an illustration of the Holder exponent ## estimation process. select branches between ## times 100 and 200 (only the first four found ## will be fitted) plot(x.tree[range=c(100, 200)], fit=TRUE)