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Kernelized mean change (CostRbf)#

Description#

Given a positive semi-definite kernel \(k(\cdot, \cdot) : \mathbb{R}^d\times \mathbb{R}^d \mapsto \mathbb{R}\) and its associated feature map \(\Phi:\mathbb{R}^d \mapsto \mathcal{H}\) (where \(\mathcal{H}\) is an appropriate Hilbert space), this cost function detects changes in the mean of the embedded signal \(\{\Phi(y_t)\}_t\) [Garreau2018, Arlot2019]. Formally, for a signal \(\{y_t\}_t\) on an interval \(I\),

\[ c(y_{I}) = \sum_{t\in I} \| \Phi(y_t) - \bar{\mu} \|_{\mathcal{H}}^2 \]

where \(\bar{\mu}\) is the empirical mean of the embedded sub-signal \(\{\Phi(y_t)\}_{t\in I}\). Here the kernel is the radial basis function (rbf):

\[ k(x, y) = \exp(-\gamma \| x - y \|^2 ) \]

where \(\| \cdot \|\) is the Euclidean norm and \(\gamma>0\) is the so-called bandwidth parameter and is determined according to median heuristics (i.e. equal to the inverse of median of all pairwise distances).

In a nutshell, this cost function is able to detect changes in the distribution of an iid sequence of random variables. Because it is non-parametric, it is performs reasonably well on a wide range of tasks.

Usage#

Start with the usual imports and create a signal.

import numpy as np
import matplotlib.pylab as plt
import ruptures as rpt

# creation of data
n, dim = 500, 3  # number of samples, dimension
n_bkps, sigma = 3, 5  # number of change points, noise standart deviation
signal, bkps = rpt.pw_constant(n, dim, n_bkps, noise_std=sigma)

Then create a CostRbf instance and print the cost of the sub-signal signal[50:150].

c = rpt.costs.CostRbf().fit(signal)
print(c.error(50, 150))

You can also compute the sum of costs for a given list of change points.

print(c.sum_of_costs(bkps))
print(c.sum_of_costs([10, 100, 200, 250, n]))

In order to use this cost class in a change point detection algorithm (inheriting from BaseEstimator), either pass a CostRbf instance (through the argument custom_cost) or set model="rbf".

c = rpt.costs.CostRbf()
algo = rpt.Dynp(custom_cost=c)
# is equivalent to
algo = rpt.Dynp(model="rbf")

References#

[Garreau2018] Garreau, D., & Arlot, S. (2018). Consistent change-point detection with kernels. Electronic Journal of Statistics, 12(2), 4440–4486.

[Arlot2019] Arlot, S., Celisse, A., & Harchaoui, Z. (2019). A kernel multiple change-point algorithm via model selection. Journal of Machine Learning Research, 20(162), 1–56.