---
file_format: mystnb
kernelspec:
  name: python3
---

(delta)=
# Delta Detrending

The delta detrending method performs local mean subtraction. For each value in the time series, it subtracts the mean of a window of values centered around that point. (**CITE**)

```python
from delaynet.detrending_methods import delta

# Apply delta detrending with a window size of 10
detrended_ts = delta(time_series, window_size=10)
```

Mathematically, the delta detrending is defined as:

$$x_t' = x_t - \frac{1}{2w + 1} \sum_{k = t - w}^{t + w} x_k$$

where $x_t$ is the value at time $t$, $w$ is the window size, and $x_t'$ is the detrended value.

```{code-cell}
:tags: [hide-input]
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]

# Plot the original fMRI data and the delta detrended fMRI data
plt.figure(figsize=(10, 4), dpi=300)
plt.plot(fmri_data, label='Original FMRI Data')
plt.title('Original and Delta Detrended FMRI Data')
plt.plot(dn.detrending_methods.delta(fmri_data, window_size=5), label=r'Delta Detrended $5$ Window')
plt.plot(dn.detrending_methods.delta(fmri_data, window_size=15), label=r'Delta Detrended $15$ Window')
plt.plot(dn.detrending_methods.delta(fmri_data, window_size=30), label=r'Delta Detrended $30$ Window')
plt.legend()
plt.grid()
plt.show()
```

(link_to_method)=
```{eval-rst}
.. automethod:: delaynet.detrending_methods.delta
    :noindex:
```