Least absolute deviation

Description

This cost function detects changes in the median of a signal. Overall, it is a robust estimator of a shift in the central point (mean, median, mode) of a distribution [C1Bai95]. Formally, for a signal \(\{y_t\}_t\) on an interval \(I\),

\[c(y_{I}) = \sum_{t\in I} \|y_t - \bar{y}\|_1\]

where \(\bar{y}\) is the componentwise median of \(\{y_t\}_{t\in I}\).

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 CostL1 instance and print the cost of the sub-signal signal[50:150].

c = rpt.costs.CostL1().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 CostL1 instance (through the argument 'custom_cost') or set model="l1".

c = rpt.costs.CostL1(); algo = rpt.Dynp(custom_cost=c)
# is equivalent to
algo = rpt.Dynp(model="l1")

Code explanation

class ruptures.costs.CostL1

Least absolute deviation.

__init__()

Initialize self. See help(type(self)) for accurate signature.

error(start, end)

Return the approximation cost on the segment [start:end].

Parameters

• start (int) – start of the segment

• end (int) – end of the segment

Returns

segment cost

Return type

float

Raises

NotEnoughPoints – when the segment is too short (less than 'min_size' samples).

fit(signal)

Set parameters of the instance.

Parameters

signal (array) – signal. Shape (n_samples,) or (n_samples, n_features)

Returns

self

References

C1Bai95

J. Bai. Least absolute deviation of a shift. Econometric Theory, 11:403–436, 1995.