`xs`: a time series `n`: segment length

`N = len(xs)` `d = floor(N/n)` `ys = np.zeros(d*n)` `mean_x = mean(xs)` `Xs = cumsum(xs - mean_x)` 1. Initialize a plt plot with three subplots, using `fig, axes = subplot_mosaic("A;B", figsize=(20, 20))` 2. On `axes["A"]` plot `Xs`. 3. Divide `Xs` into consecutive segments of length `n`. For each of the `d` segments,

 fit a linear trend by least squares, 
 plot it on `axes["A"]`, as a thin red line.
 add a black dashed vertical line at the end of the segment.
 Store the detrended values to the corresponding segment in `ys`

4. Plot `ys` on `axes["B"]`

```python import matplotlib.pyplot as plt import numpy as np from sklearn.linear_model import LinearRegression

def perform_DFA(xs, n, axa, axb):

   N = len(xs)
   d = np.floor(N/n).astype(int)
   ys = np.zeros(d*n)
   
   # Compute the cumulative sum Xs
   mean_x = np.mean(xs)
   Xs = np.cumsum(xs - mean_x)

   # Plot Xs on axes["A"]
   axa.plot(Xs, label='Cumulative sum')
   
   # Linear regression model for fitting trends
   model = LinearRegression()

   # Loop over the segments
   for i in range(d):
       # Extract the current segment
       segment = Xs[i*n:(i+1)*n]
       
       # Fit a linear trend by least squares
       x = np.arange(n).reshape(-1, 1)
       y = segment.reshape(-1, 1)
       model.fit(x, y)
       trend = model.predict(x)

       # Plot the trend
       axa.plot(range(i*n, (i+1)*n), trend, 'r-', linewidth=0.5)

       # Add a black dashed vertical line at the end of the segment
       if i < d - 1:
           axa.axvline(x=(i+1)*n, color='k', linestyle='--')

       # Store the detrended values
       ys[i*n:(i+1)*n] = segment - trend.flatten()

   # Plot detrended time series
   axb.plot(ys)
   # axa.set_title("Cumulative sum with trend lines")
   # axb.set_title("Detrended cumulative series")
   
   return np.var(ys)

np.random.seed(42)

1. Generate white noise sequence

N = 2**10 xs = np.random.normal(0, 1, N) plt.rcParams.update({'font.size': 0})

1. Initialize a plot with three subplots

fig, axes = plt.subplots(nrows=10, ncols=1, figsize=(20, 20))

for i in range(5, 10):

   n = 2**i
   v = perform_DFA(xs, n, axes[2*(i-5)], axes[2*(i-5)+1])

plt.show()