A Hilbert-Huang Transform. An HHT v n a is a Hilbert-Huang transform of an n-item time series of items of type a represented using vector v.

Create using hht or hhtEmd.

A Hilbert Trasnform of a given IMF, given as a "skeleton line".

Compute the Hilbert-Huang transform from a given Empirical Mode Decomposition.

Directly compute the Hilbert-Huang transform of a given time series. Essentially is a composition of hhtEmd and emd. See hhtEmd for a more flexible version.

Invert a Hilbert-Huang transform back to an Empirical Mode Decomposition

Since: 0.1.9.0

Construct a time series correpsonding to its hilbert-huang transform.

Since: 0.1.9.0

Compute the full Hilbert-Huang Transform spectrum. At each timestep is a sparse map of frequency components and their respective magnitudes. Frequencies not in the map are considered to be zero.

Takes a "binning" function to allow you to specify how specific you want your frequencies to be.

See hhtSparseSpectrum for a sparser version, and hhtDenseSpectrum for a denser version.

A sparser vesion of hhtSpectrum. Compute the full Hilbert-Huang Transform spectrum. Returns a sparse matrix representing the power at each time step (the Finite n) and frequency (the k).

Takes a "binning" function to allow you to specify how specific you want your frequencies to be.

Since: 0.1.4.0

A denser version of hhtSpectrum. Compute the full Hilbert-Huang Transform spectrum, returning a dense matrix (as a vector of vectors) representing the power at each time step and each frequency.

Takes a "binning" function that maps a frequency to one of m discrete slots, for accumulation in the dense matrix.

Since: 0.1.4.0

Compute the mean marginal spectrum given a Hilbert-Huang Transform. It is similar to a Fourier Transform; it provides the "total power" over the entire time series for each frequency component, averaged over the length of the time series.

A binning function is accepted to allow you to specify how specific you want your frequencies to be.

Since: 0.1.8.0

Compute the marginal spectrum given a Hilbert-Huang Transform. It provides the "total power" over the entire time series for each frequency component. See meanMarginal for a version that averages over the length of the time series, making it more close in nature to the purpose of a Fourier Transform.

A binning function is accepted to allow you to specify how specific you want your frequencies to be.

Compute the instantaneous energy of the time series at every step via the Hilbert-Huang Transform.

Degree of stationarity, as a function of frequency.

Returns the "expected value" of frequency at each time step, calculated as a weighted average of all contributions at every frequency at that time step.

Since: 0.1.4.0

Returns the dominant frequency (frequency with largest magnitude contribution) at each time step.

Since: 0.1.4.0

Fold and collapse a Hilbert-Huang transform along the frequency axis at each step in time along some monoid.

Since: 0.1.8.0

Options for EMD composition.

Default EMDOpts

Note: If you immediately use this and set eoSifter, then v will be ambiguous. Explicitly set v with type applications to appease GHC

defaultEO @(Data.Vector.Vector)
   { eoSifter = scTimes 100
   }

Boundary conditions for splines.

Default Sifter

defaultSifter = siftStdDev 0.3 siftOr siftTimes 50

R package uses siftTimes 20, Matlab uses no limit

End condition for spline

Real part is original series and imaginary part is hilbert transformed series. Creates a "helical" form of the original series that rotates along the complex plane.

Note that since 0.1.7.0, this uses the same algorithm as the matlab implementation https://www.mathworks.com/help/signal/ref/hilbert.html

Hilbert transformed series. Essentially the same series, but phase-shifted 90 degrees. Is so-named because it is the "imaginary part" of the proper hilbert transform, hilbert.

Note that since 0.1.7.0, this uses the same algorithm as the matlab implementation https://www.mathworks.com/help/signal/ref/hilbert.html

The polar form of hilbert: returns the magnitude and phase of the discrete hilbert transform of a series.

The computation of magnitude is unique, but computing phase gives us some ambiguity. The interpretation of the hilbert transform for instantaneous frequency is that the original series "spirals" around the complex plane as time progresses, like a helix. So, we impose a constraint on the phase to uniquely determine it: \(\phi_{t+1}\) is the minimal valid phase such that \(\phi_{t+1} \geq \phi_{t}\). This enforces the phase to be monotonically increasing at the slowest possible detectable rate.

Since: 0.1.6.0

Given a time series, return a time series of the magnitude of the hilbert transform and the frequency of the hilbert transform, in units of revolutions per tick. Is only expected to taken in proper/legal IMFs.

The frequency will always be between 0 and 1, since we can't determine anything faster given the discretization, and we exclude negative values as physically unmeaningful for an IMF.