dimensionality reduction methods on fluxnet series and further processing

file contents

• toolbox.jl basic functionality
• struct_blockdata.jl object structures for dimensionality reduction
• iterate_blockdata.jl parallelized dimensionality reduction computations
• 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