# The Holder spectrum of a time series

### Description

Using a tree, this function returns time localized exponent estimations for a given time series.

### Usage

1 | ```
holderSpectrum(x, n.scale.min=3, fit=lmsreg)
``` |

### Arguments

`x` |
an object of class |

`fit` |
a linear regression function to use in fitting the resulting data. Default: |

`n.scale.min` |
the minimum number of scales (points) that a given |

### Details

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

- i
Calculate the CWT of the time series.

- ii
Find the modulus maxima of the CWT (WTMM).

- iii
Link the WTMM into separate branches based (mainly) on their position in time to form a WTMM tree.

- iv
For each branch in the tree, perform an exponential fit of the WTMM over an admissible range of scale and as the scale approaches zero. The resulting

*scaling exponent*is an estimate of the local exponent for the time series. The occurrence of the singularity in time is recorded as the location in time where the WTMM converges as the scale nears zero.

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.

### Value

a list containing the estimated exponents, associated times and corresponding branch number.

### References

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).

### See Also

`wavCWT`

, `wavCWTFilters`

, `wavCWTTree`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | ```
## create series with a linear trend and two
## cusps: h(x = 1) = 0.5 and h(x = 15) = 0.3
cusps <- function(x) -0.2 * abs(x-1)^0.5 - 0.5* abs(x-15)^0.3 + 0.00346 * x + 1.34
x <- seq(-5, 20, length=1000)
y <- signalSeries(cusps(x), x)
## calculate CWT using Mexican hat filter
W <- wavCWT(y, wavelet="gaussian2")
## calculate WTMM and extract first two branches
## in tree corresponding to the cusps
W.tree <- wavCWTTree(W)[1:2]
## plot the CWT tree overlaid with a scaled
## version of the time series to illustrate
## alignment of branches with cusps
yshift <- y@data - min(y@data)
yshift <- yshift / max(yshift) * 4 - 4.5
plot(W.tree, xlab="x")
lines(x, yshift, lwd=2)
text(6.5, -1, "f(x) = -0.2|x-1|^0.5 - 0.5|x-15|^0.3 + 0.00346x + 1.34", cex=0.8)
## estimate Holder exponents
holder <- holderSpectrum(W.tree)
print(holder)
``` |

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