Mahalanobis-type metric¶
Description¶
Given a positive semi-definite matrix \(M\in\mathbb{R}^{d\times d}\), this cost function detects changes in the mean of the embedded signal defined by the pseudo-metric
\[\|x-y\|_M^2 = (x-y)^t M (x-y)\]
Formally, for a signal \(\{y_t\}_t\) on an interval \(I\), the cost function is equal to
\[c(y_{I}) = \sum_{t\in I} \|y_t - \bar{\mu}\|_{M}^2\]
where \(\bar{\mu}\) is the empirical mean of the sub-signal \(\{y_t\}_{t\in I}\). The matrix \(M\) can for instance be the result of a similarity learning algorithm [MLXJR03] or the inverse of the empirical covariance matrix (yielding the Mahalanobis distance).
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 CostMl instance and print the cost of the sub-signal signal[50:150].
M = np.eye(dim)
c = rpt.costs.CostMl(metric=M).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 CostMl instance (through the argument 'custom_cost') or set model="mahalanobis".
c = rpt.costs.CostMl(metric=M); algo = rpt.Dynp(custom_cost=c)
# is equivalent to
algo = rpt.Dynp(model="mahalanobis", params={"metric": M})
Code explanation¶
-
class
ruptures.costs.CostMl(metric=None)[source]¶ Mahalanobis-type cost function.
-
__init__(metric=None)[source]¶ Create a new instance.
- Parameters
metric (ndarray, optional) – PSD matrix that defines a Mahalanobis-type pseudo distance. If None, defaults to the Mahalanobis matrix. Shape (n_features, n_features).
- Returns
self
-
References
- MLXJR03
E. P. Xing, M. I. Jordan, and S. J. Russell. Distance metric learning, with application to clustering with side-Information. In Advances in Neural Information Processing Systems 21 (NIPS 2003), 521–528. 2003.