Use `KernSeg` with model selection as described in JMLR paper

Sylvain Arlot, Alain Celisse and Zaid Harchaoui provide theory and a heuristic for model selection with KernSeg in their paper A Kernel Multiple Change-point Algorithm via Model Selection. See 3.3.2, Theorem 2 and appendix B.3.

The penalty they propose does not scale linearly with the number of change points, so sadly it is incompatible with the current implementation. Furthermore the heuristic they propose requires knowledge of the respective losses for a set of possible numbers of split points, which currently (to my best understanding) cannot be recovered without expensive refits.

It would be great if this could be added.

[deepcharles]

If you are up for it, that would be great. The idea would be to do a didactic example of this heuristics (in the spirit of this example but with simulated data).

You only need to create a Jupyter notebook in docs/examples and we’ll take care of the integration to the docs.

[mlondschien]

Thanks @deepcharles and @oboulant for the input and thank you for working on ruptures.

I believe the following code implements the heuristic defined in the JMLR paper:

import numpy as np
import ruptures as rpt
from scipy.special import betaln
from sklearn.linear_model import LinearRegression

def kernseg(X, n_bkps_max):
    algo = rpt.KernelCPD(kernel="rbf")

    algo.fit(X).predict(n_bkps=n_bkps_max)

    segmentations_values = [[len(X)]] + list(algo.segmentations_dict.values())
    costs = [algo.cost.sum_of_costs(est) for est in segmentations_values]

    # https://stackoverflow.com/questions/21767690/python-log-n-choose-k
    log_nchoosek = [
        -betaln(1 + n_bkps_max - k, 1 + k) - np.log(n_bkps_max + 1)
        for k in range(0, n_bkps_max + 1)
    ]
    X_lm = np.array([log_nchoosek, range(0, n_bkps_max + 1)]).T
    lm = LinearRegression().fit(
        X_lm[int(0.6 * n_bkps_max) :, :], costs[int(0.6 * n_bkps_max) :]
    )
    adjusted_costs = costs - 2 * X_lm.dot(lm.coef_)

    return [0] + segmentations_values[np.argmin(adjusted_costs)]

Let me know if you would like to add this to the existing package. I could then open a PR.