# 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 $$I$$, $$c(y_{I}) = |I| \log\det\widehat{\Sigma}_I$$ where $$\widehat{\Sigma}_I$$ is the empirical covariance matrix of the sub-signal $$\{y_t\}_{t\in I}$$. It is robust to strongly dependant processes; for more information, see [Lavielle1999] (univariate case) and [Lavielle2006] (multivariate case).

## 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 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")


[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).