Timeseries Engineering

A series whose mean moves from 0 to 1.5 at t = 60. The single best split is found at exactly 60, removing 42% of the squared error. On a series where nothing happens, the method still returns a “best” split (at t = 13) — but it removes only 1.8%. Whether a change is real depends on a penalty, not on the method finding one.

• A changepoint method will always find changes if you let it: the penalty, or the number of changepoints allowed, decides how many, so tune it on data with known breaks.
• A detected shift in the mean can also be a change in variance or a single outlier, depending on what the method assumes.
• An online detector can only flag a break some steps after it happened, once enough evidence has arrived.

rng = np.random.default_rng(9)
def best_split(y):                                # the single split that most reduces squared error
    total = ((y - y.mean()) ** 2).sum()
    gains = [total - ((y[:k] - y[:k].mean()) ** 2).sum() - ((y[k:] - y[k:].mean()) ** 2).sum() for k in range(5, len(y) - 5)]
    k = int(np.argmax(gains)); return k + 5, gains[k] / total

shift = np.r_[rng.normal(0, 1, 60), rng.normal(1.5, 1, 40)]   # the mean moves at t = 60
flat = rng.normal(0, 1, 100)                                   # nothing happens
found, found_flat = best_split(shift), best_split(flat)

Assumes import numpy as np and import pandas as pd. Each snippet simulates or makes up its own series, as the comments say.

# Python — changepoint detection with ruptures
import ruptures as rpt

algo = rpt.Pelt(model="rbf").fit(signal)
breakpoints = algo.predict(pen=10)
rpt.display(signal, breakpoints)