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

(second_difference)=
# Second-Order Differentiation

The second difference detrending calculates the difference of differences, which is a common method for detrending time series.
It works by computing the difference between consecutive first differences, effectively _removing both constant offsets and linear trends_ from the data.
Just as acceleration represents the change in velocity, the second difference captures the change in the rate of change of the time series.

```python
from delaynet.detrending_methods import second_difference

# Apply second difference detrending
detrended_ts = second_difference(time_series)
```

Note that the length of the time series is reduced by 2 after applying this detrending.

```{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]

plt.figure(figsize=(10, 4), dpi=300)
plt.plot(fmri_data, label='Original FMRI Data')
plt.title('Original and Second Difference Detrended FMRI Data')
plt.plot(dn.detrending_methods.second_difference(fmri_data), label='Second Difference Detrended Data')
plt.legend()
plt.grid()
plt.show()
```

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