# Combining cost functions#

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## Introduction#

In ruptures, change point detection procedures can only use a single cost function. The choice of this cost function is crucial as it is related to the type of change to find. For instance, CostL2 can detect shifts in the mean, CostNormal can detect shifts in the mean and the covariance structure, CostAR can detect shifts in the auto-regressive structure, etc.

However, in many settings, several types of changes exist within a signal and a single cost function is not able to detect all changes simultaneously. To cope with this issue, a procedure to combine cost functions is needed. In this example, a technique inspired by [Katser2021] is presented. In a nutshell, different costs are combined to yield an aggregated cost function which is sensitive to several types of changes. The aggregated cost can then be used with the window search method to create a change point detection algorithm.

For simplicity, we focus on mean-shifts that occur in different dimensions of a multivariate signal. Each considered cost function can only detect mean-shift in a single dimension. The user can choose between two aggregations procedures that can:

• either detect shifts that occur simultaneously on all dimensions (intersection of experts),
• or detect shifts that occur on any dimension (union of experts).

Here, only CostL2 is considered for all dimensions, but all other costs could be used. The focus is then on the way the costs are combined (see intersection and union). For simplicity, the number of changes is assumed to be known by the user.

## Setup#

First, we make the necessary imports and define a few utility functions.

import matplotlib.pyplot as plt
import numpy as np
import ruptures as rpt
from ruptures.base import BaseCost

WINDOW_SIZE = 200

def minmax_scale(array: np.ndarray) -&gt; np.ndarray:
"""Scale each dimension to the [0, 1] range."""
return (array - np.min(array, axis=0)) / (
np.max(array, axis=0) - np.min(array, axis=0) + 1e-8
)

def fig_ax(figsize=(10, 3)):
return plt.subplots(figsize=figsize)


The merging procedure of cost functions is applied on the following 2D toy signal, where each dimension contains three mean-shifts. Note that only one mean-shift is shared by both dimensions and all other changes occur at different indexes.

# fmt: off
signal_no_noise = np.array([[0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], 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# fmt: on
bkps_on_dim_0 = [329, 656, 1642, 2000]
bkps_on_dim_1 = [656, 972, 1291, 2000]
bkps_all_dims = [329, 656, 972, 1291, 1642, 2000]

signal_with_noise = signal_no_noise + np.random.normal(size=signal_no_noise.shape)

fig, axes = rpt.display(signal_no_noise, bkps_all_dims)
_ = axes[0].set_title("Toy signal (no noise)")

fig, axes = rpt.display(signal_with_noise, bkps_all_dims)
_ = axes[0].set_title("Toy signal (with noise)")


## Detection with a single cost function#

Consider the following cost function (derived from CostL2) that can only detect mean-shift on one dimension at a time.

class CostL2OnSingleDim(BaseCost):
"""This cost function detects mean-shift on a single dimension of a multivariate signal."""

# The 2 following attributes must be specified for compatibility.
model = "CostL2OnSingleDim"
min_size = 1

def __init__(self, dim):
super().__init__()
self.dim = dim

def fit(self, signal):
"""Set the internal parameter."""
self.signal = signal[:, self.dim].reshape(-1, 1)
return self

def error(self, start, end) -&gt; float:
"""Return the approximation cost on the segment [start:end].

Args:
start (int): start of the segment
end (int): end of the segment

Returns:
segment cost

Raises:
NotEnoughPoints: when the segment is too short (less than min_size samples).
"""
if end - start &lt; self.min_size:
raise rpt.exceptions.NotEnoughPoints
if end - start == 1:
return 0.0
return self.signal[start:end].var(axis=0).sum() * (end - start)


Combining this cost function with the window search method yields the following change-points. Note that the true changes are shown with the alternating colors and the estimated changes are shown with the vertical dashed lines.

# Detect mean-shift on the first dimension
cost_function = CostL2OnSingleDim(dim=0)
algo_on_dim_0 = rpt.Window(width=WINDOW_SIZE, custom_cost=cost_function, jump=1).fit(
signal_with_noise
)
bkps_pred = algo_on_dim_0.predict(
n_bkps=len(bkps_on_dim_0) - 1
)  # the number of changes is known
# display signal and changes
fig, axes = rpt.display(signal_with_noise, bkps_on_dim_0, bkps_pred)
_ = axes[0].set_title(
(
f"""Detection of mean-shifts using only Dimension 0:\n"""
f"""true changes: {bkps_on_dim_0[:-1]}, detected changes: {bkps_pred[:-1]}."""
)
)

# Detect mean-shift on the second dimension
cost_function = CostL2OnSingleDim(dim=1)
algo_on_dim_1 = rpt.Window(width=WINDOW_SIZE, custom_cost=cost_function, jump=1).fit(
signal_with_noise
)
bkps_pred = algo_on_dim_1.predict(
n_bkps=len(bkps_on_dim_1) - 1
)  # the number of changes is known
# display signal and changes
fig, axes = rpt.display(signal_with_noise, bkps_on_dim_1, bkps_pred)
_ = axes[0].set_title(
(
f"""Detection of mean-shifts using only Dimension 1:\n"""
f"""true changes: {bkps_on_dim_1[:-1]}, detected changes: {bkps_pred[:-1]}."""
)
)


Observe that, depending on the considered cost function (CostL2OnSingleDim(dim=0) or CostL2OnSingleDim(dim=1)), the detected changes are not the same even though the input signal is the same.

Displaying the scores of the cost functions further illustrates the difference between the two costs. Recall that change-points correspond to high scores.

for dim, algo in enumerate([algo_on_dim_0, algo_on_dim_1]):
fig, ax = fig_ax()
ax.plot(np.r_[np.zeros(WINDOW_SIZE // 2), algo.score, np.zeros(WINDOW_SIZE // 2)])
ax.set_xmargin(0)
ax.set_title(f"Score for CostL2OnSingleDim(dim={dim})")


The question now for a user is how to combine the two cost functions to either detect all changes or only the changes that occur accros all dimensions simultaneously.

## Detection with an aggregated cost function#

### Intersection#

A first way of aggregating the scores is to consider each score along a dimension as being an expert. The idea is to take the intersection of the experts so that the predicted change points are the change points where all the experts are confident. This method is useful when the user is interested in collecting the change points that occur on all dimensions simultaneously. To that end, the scores are scaled to the [0, 1] range then the pointwise minimum is taken.

In the following cell, the intersection procedure correctly predicts the only common change point between the two dimensions. The aggregated score shows clearly the only change point to predict.

# concatenate scores
score_arr = np.c_[algo_on_dim_0.score, algo_on_dim_1.score]
# intersection aggregation
algo_expert_intersection = rpt.Window(width=WINDOW_SIZE, jump=1).fit(signal_with_noise)
algo_expert_intersection.score = (minmax_scale(score_arr)).min(
axis=1
)  # scaling + pointwise min
# only one change point is shared by both dimensions
bkps_intersection_predicted = algo_expert_intersection.predict(n_bkps=1)
# display the aggregated score
fig, ax = fig_ax()
ax.plot(
np.r_[
np.zeros(WINDOW_SIZE // 2),
algo_expert_intersection.score,
np.zeros(WINDOW_SIZE // 2),
]
)
ax.set_xmargin(0)
_ = ax.set_title(
(
"""Aggregated score (intersection of experts)\n"""
f"""true change: {bkps_on_dim_1[0]}, detected change: {bkps_intersection_predicted[0]}"""
)
)


### Union#

Another way of aggregating the scores is to take the union of the experts so that the predicted change points are the change points where at least one expert is confident. This method is useful when the user is interested in collecting all the change points that are present along all dimensions. To that end, the scores are scaled to the [0, 1] range then the pointwise maximum is taken.

In the following cell, the union procedure correctly predicts all the change points. The aggregated score shows 5 clear peaks from which to take the change points.

# union aggregation
algo_expert_union = rpt.Window(width=WINDOW_SIZE, jump=1).fit(signal_with_noise)
algo_expert_union.score = minmax_scale(score_arr).max(axis=1)  # scaling + pointwise max
# 5 change points are present
bkps_union_predicted = algo_expert_union.predict(n_bkps=5)
# display the aggregated score
fig, ax = fig_ax()
ax.plot(
np.r_[
np.zeros(WINDOW_SIZE // 2), algo_expert_union.score, np.zeros(WINDOW_SIZE // 2)
]
)
ax.set_xmargin(0)
_ = ax.set_title(
(
"""Aggregated score (intersection of experts)\n"""
f"""true change: {bkps_all_dims[:-1]}, detected change: {bkps_union_predicted[:-1]}"""
)
)


## Conclusion#

This example shows two ways to aggregate cost functions for a 2D toy signal which contains mean-shifts that do not always co-occur:

Those aggregation procedures can be applied to any set of cost functions and in a wide range of settings.

## Authors#

This example notebook has been authored by Théo VINCENT and edited by Olivier Boulant and Charles Truong.

## References#

[Katser2021] Katser, I., Kozitsin, V., Lobachev, V., & Maksimov, I. (2021). Unsupervised Offline change point Detection Ensembles. Applied Sciences, 11(9), 4280.