Kernelized mean change¶
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 \(\{\Phi(y_t)\}_t\) [KERACH12][KERGBR+12]. 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).
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")
Code explanation¶
-
class
ruptures.costs.CostRbf[source]¶ Kernel cost function (rbf kernel).
References