For testing and educational purposes it is always good to work with synthetic data. Syncopy brings its own suite of synthetic data generators, but it is also possible to devise your own synthetic data using standard NumPy.
These functions return a multi-trial
AnalogData object representing multi-channel time series data:
A harmonic with frequency freq.
Plain white noise with unity standard deviation.
Uncoupled multi-channel AR(1) process realizations.
A linear trend on all channels from 0 to y_max in nSamples.
Linear (harmonic) phase evolution plus a Brownian noise term inducing phase diffusion around the deterministic phase velocity (angular frequency).
Simulation of a network of coupled AR(2) processes
With the help of basic arithmetical operations we can combine different synthetic signals to arrive at more complex ones. Let’s look at an example:
import syncopy as spy # set up cfg cfg = spy.StructDict() cfg.nTrials = 40 cfg.samplerate = 500 cfg.nSamples = 500 cfg.nChannels = 5 # start with a simple 60Hz harmonic sdata = spy.synthdata.harmonic(freq=60, cfg=cfg) # add some strong AR(1) process as surrogate 1/f sdata = sdata + 5 * spy.synthdata.red_noise(alpha=0.95, cfg=cfg) # plot all channels for a single trial sdata.singlepanelplot(trials=10) # compute spectrum and plot trial average of 2 channels spec = spy.freqanalysis(sdata, keeptrials=False) spec.singlepanelplot(channel=[0, 2], frequency=[0,100])
A diffusing phase can be modeled by adding white noise \(\xi(t)\) to a fixed angular frequency:
with the instantaneous frequency \(\omega(t)\).
Integration then yields the phase trajectory:
Here \(W(t)\) being the Wiener process, or simply a one dimensional diffusion process. Note that for the trivial case \(\epsilon = 0\), so no noise got added, the phase describes a linear constant motion with the phase velocity \(\omega = 2\pi f\). This is just a harmonic oscillation with frequency \(f\). Finally, by wrapping the phase trajectory into a \(2\pi\) periodic waveform function, we arrive at a time series (or signal). The simplest waveform is just the cosine, so we have:
This is exactly what the
phase_diffusion() function provides.
Phase diffusing models have some interesting properties, let’s have a look at the power spectrum:
import syncopy as spy cfg = spy.StructDict() cfg.nTrials = 250 cfg.nChannels = 2 cfg.samplerate = 500 cfg.nSamples = 2000 # harmonic frequency is 60Hz, phase diffusion strength is 0.01 signals = spy.synthdata.phase_diffusion(freq=60, eps=0.01, cfg=cfg) # add harmonic frequency with 20Hz, there is no phase diffusion signals += spy.synthdata.harmonic(freq=20, cfg=cfg) # freqanalysis without tapering and absolute power cfg_freq = spy.StructDict() cfg_freq.keeptrials = False cfg_freq.foilim = [2, 100] cfg_freq.output = 'abs' cfg_freq.taper = None spec = spy.freqanalysis(signals, cfg=cfg_freq) spec.singlepanelplot(channel=0)
We see a natural (no tapering) spectral broadening for the phase diffusing signal at 60Hz, reflecting the fluctuations in instantaneous frequency.
To create a synthetic timeseries data set follow these steps:
write a function which returns a single trial as a 2d-
ndarraywith desired shape
collect all the trials into a Python
list, for example with a list comprehension or simply a for loop
AnalogDataobject by passing this list holding the trials as
dataand set the desired
In (pseudo-)Python code:
def generate_trial(nSamples, nChannels): trial = .. something fancy .. # These should evaluate to True isinstance(trial, np.ndarray) trial.shape == (nSamples, nChannels) return trial # collect the trials nSamples = 1000 nChannels = 2 nTrials = 100 trls =  for _ in range(nTrials): trial = generate_trial(Samples, nChannels) # manipulate further as needed, e.g. add a constant trial += 3 trls.append(trial) # instantiate syncopy data object my_fancy_data = spy.AnalogData(data=trls, samplerate=my_samplerate)
The same recipe can be used to generally instantiate Syncopy data objects from NumPy arrays.
Syncopy data objects also accept Python generators as
data, allowing to stream
in trial arrays one by one. In effect this allows creating datasets which are larger
than the systems memory. This is also how the build in generators of
syncopy.synthdata (see above) work under the hood.
Let’s create two harmonics and add some white noise to it:
import numpy as np import syncopy as spy def generate_noisy_harmonics(nSamples, nChannels, samplerate): f1, f2 = 20, 50 # the harmonic frequencies in Hz # the sampling times vector tvec = np.arange(nSamples) * 1 / samplerate # define the two harmonics ch1 = np.cos(2 * np.pi * f1 * tvec) ch2 = np.cos(2 * np.pi * f2 * tvec) # concatenate channels to to trial array trial = np.column_stack([ch1, ch2]) # add some white noise trial += 0.5 * np.random.randn(nSamples, nChannels) return trial nTrials = 50 nSamples = 1000 nChannels = 2 samplerate = 500 # in Hz # collect trials trials =  for _ in range(nTrials): trial = generate_noisy_harmonics(nSamples, nChannels, samplerate) trials.append(trial) synth_data = spy.AnalogData(trials, samplerate=samplerate)
Here we first defined the number of trials (
nTrials) and then the number of samples (
nSamples) and channels (
nChannels) per trial. With a sampling rate of 500Hz and 1000 samples this gives us a trial length of two seconds. The function
generate_noisy_harmonics adds a 20Hz harmonic on the 1st channel, a 50Hz harmonic on the 2nd channel and white noise to all channels, Every trial got collected into a Python
list, which at the last line was used to initialize our
synth_data. Note that data instantiated that way always has a default trigger offset of -1 seconds.
Now we can directly run a multi-tapered FFT analysis and plot the power spectra of all 2 channels:
spectrum = spy.freqanalysis(synth_data, foilim=[0,80], tapsmofrq=2, keeptrials=False) spectrum.singlepanelplot()
As constructed, we have two harmonic peaks at the respective frequencies (20Hz and 50Hz) and the white noise floor on all channels.