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Gaussian process change (CostNormal)#

Description#

This cost function detects changes in the mean and covariance matrix of a sequence of multivariate Gaussian random variables. Formally, for a signal \(\{y_t\}_t\) on an interval \(I\), $$ c(y_{I}) = |I| \log\det(\widehat{\Sigma}_I + \epsilon\text{Id}) $$ where \(\widehat{\Sigma}_I\) is the empirical covariance matrix of the sub-signal \(\{y_t\}_{t\in I}\) and \(\epsilon>0\) is a small constant added to cope with badly conditioned covariance matrices (new in version 1.1.5, see Issue 196). It is robust to strongly dependant processes; for more information, see [Lavielle1999] (univariate case) and [Lavielle2006] (multivariate case).

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 CostNormal instance and print the cost of the sub-signal signal[50:150].

c = rpt.costs.CostNormal().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 CostNormal instance (through the argument custom_cost) or set model="normal".

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

To set the small diagonal bias to 0 (default behaviour in versions 1.1.4 and before), simply do the following (change Dynp by the search method you need). 

c = rpt.costs.CostNormal(add_small_diag=False)
algo = rpt.Dynp(custom_cost=c)
# or, equivalently,
algo = rpt.Dynp(model="normal", params={"add_small_diag": False})

References#

[Lavielle1999] Lavielle, M. (1999). Detection of multiples changes in a sequence of dependant variables. Stochastic Processes and Their Applications, 83(1), 79–102.

[Lavielle2006] Lavielle, M., & Teyssière, G. (2006). Detection of multiple change-points in multivariate time series. Lithuanian Mathematical Journal, 46(3).