# Change detection with a Mahalanobis-type metric (CostMl)#

## 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 [Xing2003, Truong2019] or the inverse of the empirical covariance matrix (yielding the Mahalanobis distance).

## Usage#

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})

## References#

[Xing2003] Xing, E. P., Jordan, M. I., & Russell, S. J. (2003). Distance metric learning, with application to clustering with side-Information. Advances in Neural Information Processing Systems (NIPS), 521–528.

[Truong2019] Truong, C., Oudre, L., & Vayatis, N. (2019). Supervised kernel change point detection with partial annotations. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 1–5.