Fractal Dimension (State Space)

Contents

Fractal Dimension (State Space)¶

This notebook covers Fractal Dimension measures that work in state space, including:

Correlation Dimension

# Setup notebook state
from nbutils import setup_notebook; setup_notebook()

from neurodsp.sim import sim_powerlaw, sim_synaptic_current, sim_combined
from neurodsp.utils import set_random_seed

# Import custom project code
from apm.io import APMDB
from apm.run import run_sims, run_sims_load
from apm.methods import correlation_dimension
from apm.methods.settings import CD_PARAMS
from apm.plts.sims import plot_ap_sims, plot_pe_sims, plot_sims_two
from apm.plts.settings import COLORS
from apm.plts.utils import figsaver
from apm.sim.examples import get_examples, check_examples
from apm.sim.settings import N_SIMS2 as N_SIMS
from apm.sim.defs import SIM_ITERS

Settings¶

# Get current sampling rate
fs = SIM_ITERS.fs

# Settings for saving figures
SAVE_FIG = True
FIGPATH = APMDB().figs_path / '23b_fractal_state'

# Create helper function to manage figsaver settings
fsaver = figsaver(SAVE_FIG, FIGPATH)

# Set the random seed
set_random_seed(111)

# Define collection of example signals
examples = get_examples()

Correlation Dimension¶

The correlation dimension is a measure of the dimensionality of the space of a set of points.

Correlation dimension is generally considered to be a type of fractal dimension measure.

# Check correlation dimension measure parameters
print(CD_PARAMS)

{'delay': 4, 'dimension': 20}

Calculate Correlation Dimension on Example Signals¶

# Check the correlation dimension values across some example signals
check_examples(examples, correlation_dimension, CD_PARAMS, 'correlation dimension')

Computed correlation dimension:
  powerlaw       : 	 14.2760
  synaptic       : 	 15.0800
  knee           : 	 10.0361
  oscillation    : 	 1.2896
  burst          : 	 0.6142
  combined       : 	 13.1042
  comb_burst     : 	 13.8552
  comb_peak      : 	 12.2803

Correlation Dimension: Aperiodic Measures¶

# Run simulations calculating correlation dimension across exponents
cd_sims_exp = run_sims(sim_powerlaw, SIM_ITERS['ap_exp'],
                       correlation_dimension, CD_PARAMS, N_SIMS)

# Run simulations calculating correlation dimension across exponents with an oscillation
cd_sims_comb = run_sims(sim_combined, SIM_ITERS['comb_exp'],
                        correlation_dimension, CD_PARAMS, N_SIMS)

# Plot correlation dimension estimates across aperiodic variations
plot_ap_sims(cd_sims_exp, cd_sims_comb, 'Correlation Dimension', **fsaver('cd_ap'))

../_images/23B-Fractal-State_16_0.png

Correlation Dimension: Periodic Variations¶

# Run simulations calculating correlation dimension across oscillation frequency
cd_sims_freq = run_sims(sim_combined, SIM_ITERS['osc_freq'],
                        correlation_dimension, CD_PARAMS, N_SIMS)

# Run simulations calculating correlation dimension across oscillation power
cd_sims_pow = run_sims(sim_combined, SIM_ITERS['osc_pow'],
                       correlation_dimension, CD_PARAMS, N_SIMS)

# Plot correlation dimension estimates across periodic variations
plot_pe_sims(cd_sims_freq, cd_sims_pow, 'Correlation Dimension', **fsaver('cd_pe'))

../_images/23B-Fractal-State_20_0.png

Correlation Dimension: Knee Variations¶

# Run simulations calculating correlation dimension across timescales
cd_sims_tscales = run_sims(sim_synaptic_current, SIM_ITERS['syn_tscales'],
                           correlation_dimension, CD_PARAMS, N_SIMS)

# Run simulations calculating correlation dimension across peak bandwidths (sims from file)
cd_sims_knee = run_sims_load('ap-knee-' + str(fs), correlation_dimension, CD_PARAMS, N_SIMS)

# Plot the estimated correlation dimension across different timescales & knees
plot_sims_two(SIM_ITERS['syn_tscales'].values, cd_sims_tscales, 'Timescale',
              SIM_ITERS['kn_knee'].values, cd_sims_knee, 'Knee Parameter',
              ylabel='Correlation Dimension', color=COLORS['KN'], **fsaver('cd_kn_vars'))

../_images/23B-Fractal-State_24_0.png

Correlation Dimension: Peak Variations¶

# Run simulations calculating correlation dimension across peak bandwidths (sims from file)
cd_sims_bw = run_sims_load('comb-bw-' + str(fs), correlation_dimension, CD_PARAMS, N_SIMS)

# Run simulations calculating correlation dimension across burst probabilities
cd_sims_burst = run_sims(sim_combined, SIM_ITERS['comb_burst'],
                         correlation_dimension, CD_PARAMS, N_SIMS)

# Plot the estimated correlation dimension across periodic variations
plot_sims_two(SIM_ITERS['peak_bw'].values, cd_sims_bw, 'Bandwidth',
              SIM_ITERS['comb_burst'].values, cd_sims_burst, 'Burst Probability',
              ylabel='Correlation Dimension', color=COLORS['BW'], **fsaver('cd_pe_vars'))

../_images/23B-Fractal-State_28_0.png

Conclusions¶

Correlation dimension:

generally decreases with increasing aperiodic exponent
is broadly consistent across variations of oscillation frequency and power

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Complexity Measures