phase_comparison.jl further processing and plotting (very messy)
functionality
read measurement series of length N
center measurement series, with observation average and variance
perform delay embedding with fixed delay embedding parameter W to create data matrix
centralize data matrix
chose number k of reduced dimension from k<P, P = N - W +1
SSA computes k (left) singular vectors of data matrix: modes
NLSA first samples the diffusion distance distribution for kernel scale parameter e computation
NLSA computes diffusion kernel of data matrix and performs kd-tree to create diffusion distance matrix
NLSA computes k eigenvectors of diffusion distance matrix: modes
estimate mode amplitude by variance coverage of original data matrix
create reconstructed time series from modes
hilbert transform quaternizes individual mode to create instantanious phase in analytic signal representation: protophase
half protophase zero count gives period length to estimate mode frequency
showcase characteristics based on the variance and frequency of the individual modes
purpose
dimensionality reduction methods can do an additive decomposition of time series
this decomposition is datadriven and orthogonal, which poses the question wether it can separate different timescales in time series
since modes are quasiperiodic individual timescales can be estimated by frequencies
dimensionality reduction suffers from artifacts linked to orthogonality: variance compression & degeneracy
questions
how do linear (SSA, keeps global metric) and nonlinear (NLSA,keeps local metric -- diffusion distance) differ in attributing timescales?
how do these artifacts play out?
how does the embedding length parameter influence the attributed timescales?
awnsers
timeseries with consistent frequency and amplitude are getting identical attributed harmonics of the seasonal cycle. amplitude and frequency modulation: SSA identifies strong harmonic structure, NLSA creates more time-localized modes by frequency modulation
the strong seasonal trend in the signal is always the first identified mode and subsequent detections are constrained orthogonal to it, eg. harmonic -- but: additional information can be amplitude modulated on top of this 'carrier', frequency estimation not waterproof
all oscillatory modes are confined to period lengths in multiples of the embedding length parameter -- similar to boundary condition
About
dimensionality reduction on fluxnet series and how does it separate timescales?