Z-Score

The Z-Score detrending is a standard score that measures the number of standard deviations a data point is from the mean. It is calculated as:

\[Z = \frac{X - \mu}{\sigma}\]

where \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

from delaynet.detrending_methods import z_score

# Apply Z-Score detrending
detrended_ts = z_score(time_series)

# Apply Z-Score detrending with periodicity
detrended_ts = z_score(time_series, periodicity=24, max_periods=7)

The Z-Score detrending in delaynet can take into account the periodicity of the time series. This is useful for data with seasonal patterns, such as daily or weekly cycles. The periodicity parameter specifies the number of time steps in one period, and the max_periods parameter limits the number of periods to consider before and after the current value.

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import delaynet as dn
from numpy.random import default_rng
import matplotlib.pyplot as plt

# Generate fMRI data for a single node
fmri_data = dn.preparation.data_generator.gen_fmri(
    ts_len=1000,               # Length of time series
    downsampling_factor=2,     # Downsampling factor
    time_resolution=0.2,       # Time resolution
    coupling_strength=2.0,     # Coupling strength
    noise_initial_sd=1.0,      # Standard deviation of initial noise
    noise_final_sd=0.1,        # Standard deviation of final noise
    rng=default_rng(25968)     # Random number generator seed
)[:, 0]

plt.figure(figsize=(10, 5), dpi=300)
plt.plot(fmri_data, label='Original FMRI Data')
plt.title('Original and Z-Score Detrended FMRI Data')
plt.plot(dn.detrending_methods.z_score(fmri_data, periodicity=40), label='Z-Score Detrended FMRI Data with Periodicity=40')
plt.plot(dn.detrending_methods.z_score(fmri_data, periodicity=40, max_periods=3), label='Periodicity=40 + max periods=3')
plt.legend()
plt.ylim(-5, 5)
plt.grid()
plt.show()

detrending_methods.z_score(periodicity: int = 1, max_periods: int = -1) → ndarray

Z-Score (ZS) detrending.

The Z-Score (ZS) detrending is a standard score that measures the number of standard deviations a data point is from the mean. It is calculated as:

\[Z = \frac{X - \mu}{\sigma}\]

where \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

The Z-Score detrending is applied to a time series by computing the mean and standard deviation of a sub time series around each data point. The sub time series is defined by the periodicity and the maximum number of periods to consider before and after the current value. Mean and standard deviation are computed for each sub time series and used to detrend the data point. The current value is removed from the sub time series to avoid bias.

When periodicity > 1, the function compares each point with other points at the same phase of the cycle (same position within each period).

When periodicity == 1:

• If max_periods == -1, a simple Z-score is applied to the entire time series.

• If max_periods != -1, a moving window approach is used, where each point is compared with max_periods points before and after it.

For a valid Z-Score, \(2\times\texttt{periodicity}+1 \leq \texttt{len(ts)}\) needs to be satisfied. Also, \(\texttt{max_periods} \times \texttt{periodicity} \geq\texttt{len(ts)}\) results in including all periods.

Parameters:

• ts (numpy.ndarray) – Time series to detrend.

• periodicity (int > 0) – Periodicity of the time series - reoccurrence of the same pattern.

• max_periods (int >= -1) – Maximum number of periods to consider before and after the current value. If -1, all periods are considered.

Returns:

Detrended time series.

Return type:

numpy.ndarray

Raises:

• ValueError – If the periodicity is not a positive integer.

• ValueError – If the max_periods is not a positive integer or -1.

• ValueError – If \(2\times\texttt{periodicity}+1 \leq \texttt{len(ts)}\) is not satisfied.