Pruned Exact Linear Time (PELT)

Description

An R6 class implementing the PELT algorithm for offline change-point detection.

Details

PELT (Pruned Exact Linear Time) is an efficient algorithm for change point detection that prunes the search space to achieve optimal segmentation in linear time under certain conditions.

PELT requires a R6 object of class costFunc, which can be created via costFunc$new(). Currently, the following cost functions are supported:

• "L1" and "L2" for (independent) piecewise Gaussian process with constant variance

• "SIGMA": for (independent) piecewise Gaussian process with varying variance

• "VAR": for piecewise Gaussian vector-regressive process with constant noise variance

• "LinearL2": for piecewise linear regression process with constant noise variance

See ⁠$eval()⁠ method for more details on computation of cost.

Some examples are provided below. See the GitHub README for detailed basic usage!

Methods

$new()

Initialises a PELT object.

$describe()

Describes the PELT object.

$fit()

Constructs a PELT module in ⁠C++⁠.

$eval()

Evaluates the cost of a segment.

$predict()

Performs PELT given a linear penalty value.

$plot()

Plots change-point segmentation in ggplot style.

$clone()

Clones the R6 object.

Active bindings

Methods

Public methods

• PELT$new()

• PELT$describe()

• PELT$fit()

• PELT$eval()

• PELT$predict()

• PELT$plot()

• PELT$clone()

Method new()

Initialises a PELT object.

Usage

PELT$new(minSize, jump, costFunc)

Arguments

Returns

Invisibly returns NULL.

Method describe()

Describes a PELT object.

Usage

PELT$describe(printConfig = FALSE)

Arguments

Returns

Invisibly returns a list storing at least the following fields:

minSize

Minimum allowed segment length.

jump

Search grid step size.

costFunc

The costFun object.

fitted

Whether or not ⁠$fit()⁠ has been run.

tsMat

Time series matrix.

covariates

Covariate matrix (if exists).

n

Number of observations.

p

Number of features.

Method fit()

Constructs a ⁠C++⁠ module for PELT.

Usage

PELT$fit(tsMat = NULL, covariates = NULL)

Arguments

Details

This method constructs a ⁠C++⁠ PELT module and sets private$.fitted to TRUE, enabling the use of ⁠$predict()⁠ and ⁠$eval()⁠.

Returns

Invisibly returns NULL.

Method eval()

Evaluate the cost of the segment (a,b]

Usage

PELT$eval(a, b)

Arguments

Details

The segment cost is evaluated as follows:

• L1 cost function:

  c_{L_1}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \tilde{y}_{(a+1)...b} \|_1

  where \tilde{y}_{(a+1)...b} is the coordinate-wise median of the segment. If a \ge b - 1, return 0.

• L2 cost function:

  c_{L_2}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \bar{y}_{(a+1)...b} \|_2^2

  where \bar{y}_{(a+1)...b} is the empirical mean of the segment. If a \ge b - 1, return 0.

• SIGMA cost function:

  c_{\sum}(y_{(a+1)...b}) := (b - a)\log \det \hat{\Sigma}_{(a+1)...b}

  where \hat{\Sigma}_{(a+1)...b} is the empirical covariance matrix of the segment without Bessel's correction. Here, if addSmallDiag = TRUE, a small bias epsilon is added to the diagonal of estimated covariance matrices to improve numerical stability.
  
  By default, addSmallDiag = TRUE and epsilon = 1e-6. In case addSmallDiag = TRUE, if the computed determinant of covariance matrix is either 0 (singular) or smaller than p*log(epsilon) - the lower bound, return (b - a)*p*log(epsilon), otherwise, output an error message.

• VAR(r) cost function:

  c_{\mathrm{VAR}}(y_{(a+1)...b}) := \sum_{t = a+r+1}^{b} \left\| y_t - \sum_{j=1}^r \hat A_j y_{t-j} \right\|_2^2

  where \hat A_j are the estimated VAR coefficients, commonly estimated via the OLS criterion. If system is singular, a-b < p*r+1 (i.e., not enough observations), or a \ge n-p (where n is the time series length), return 0.

"LinearL2" for piecewise linear regression process with constant noise variance

c_{\text{LinearL2}}(y_{(a+1):b}) := \sum_{t=a+1}^b \| y_t - X_t \hat{\beta} \|_2^2

where \hat{\beta} are OLS estimates on segment (a+1):b. If segment is shorter than the minimum number of points needed for OLS, return 0.

Returns

The segment cost.

Method predict()

Performs PELT given a linear penalty value.

Usage

PELT$predict(pen = 0)

Arguments

Details

The PELT algorithm detects multiple change-points by finding the set of break-points that globally minimises a penalised cost function. PELT uses dynamic programming combined with a pruning rule to reduce the number of candidate change-points, achieving efficient computation.

Let [c_1, \dots, c_k, c_{k+1}] denote the set of segment end-points with c_1 < c_2 < \dots < c_k < c_{k+1} = n, where k is the number of detected change-points and n is the total number of data points. Let c_{(c_i, c_{i+1}]} be the cost of segment (c_i, c_{i+1}]. The total penalised cost is

\text{TotalCost} = \sum_{i=1}^{k+1} c_{(c_i, c_{i+1}]} + \lambda \cdot k,

where \lambda is a linear penalty applied per change-point. PELT finds the set of endpoints that minimises this cost exactly.

The pruning step eliminates candidate change-points that cannot lead to an optimal solution, allowing PELT to run in linear time with respect to the number of data points.

Temporary segment end-points are saved to private$.tmpEndPoints after ⁠$predict()⁠, enabling users to call ⁠$plot()⁠ without specifying endpoints manually.

Returns

An integer vector of regime end-points. By design, the last element is the number of observations.

Method plot()

Plots change-point segmentation

Usage

PELT$plot(
  d = 1L,
  endPts,
  dimNames,
  main,
  xlab,
  tsWidth = 0.25,
  tsCol = "#5B9BD5",
  bgCol = c("#A3C4F3", "#FBB1BD"),
  bgAlpha = 0.5,
  ncol = 1L
)

Arguments

Details

Plots change-point segmentation results. Based on ggplot2. Multiple plots can easily be horizontally and vertically stacked using patchwork's operators / and |, respectively.

Returns

An object of classes gg and ggplot.

Method clone()

The objects of this class are cloneable with this method.

Usage

PELT$clone(deep = FALSE)

Arguments

Author(s)

Minh Long Nguyen edelweiss611428@gmail.com
Toby Dylan Hocking toby.hocking@r-project.org
Charles Truong ctruong@ens-paris-saclay.fr

References

Truong, C., Oudre, L., & Vayatis, N. (2020). Selective review of offline change point detection methods. Signal Processing, 167, 107299.

Killick, R., Fearnhead, P., & Eckley, I. A. (2012). Optimal detection of change points with a linear computational cost. Journal of the American Statistical Association, 107(500), 1590-1598.

Examples

## L2 example
set.seed(1121)
signals = as.matrix(c(rnorm(100,0,1),
                     rnorm(100,5,1)))
# Default L2 cost function
PELTObj = PELT$new(minSize = 1L, jump = 1L)
PELTObj$fit(signals)
PELTObj$predict(pen = 100)
PELTObj$plot()

## SIGMA example
set.seed(111)
signals = as.matrix(c(rnorm(100,-5,1),
                      rnorm(100,-5,10),
                      rnorm(100,-5,1)))
# L2 cost function
PELTObj = PELT$new(minSize = 1L, jump = 1L)
PELTObj$fit(signals)
# We choose pen = 50.
PELTObj$predict(pen = 50)
PELTObj$plot()

# The standard L2 cost function is not suitable.
# Use the SIGMA cost function.
PELTObj$costFunc = costFunc$new(costFunc = "SIGMA")
PELTObj$predict(pen = 50)
PELTObj$plot()

Slicing Window (Window)

Description

An R6 class implementing slicing window for offline change-point detection.

Details

Slicing window is a scalable, linear-time change-point detection algorithm that selects breakpoints based on local gains computed over sliding windows.

Currently supports the following cost functions:

• "L1" and "L2" for (independent) piecewise Gaussian process with constant variance

• "SIGMA": for (independent) piecewise Gaussian process with varying variance

• "VAR": for piecewise Gaussian vector-regressive process with constant noise variance

• "LinearL2": for piecewise linear regression process with constant noise variance

Window requires a R6 object of class costFunc, which can be created via costFunc$new(). Currently, the following cost functions are supported:

• "L1" and "L2" for (independent) piecewise Gaussian process with constant variance

• "SIGMA": for (independent) piecewise Gaussian process with varying variance

• "VAR": for piecewise Gaussian vector-regressive process with constant noise variance

• "LinearL2": for piecewise linear regression process with constant noise variance

See ⁠$eval()⁠ method for more details on computation of cost.

Some examples are provided below. See the GitHub README for detailed basic usage!

Methods

$new()

Initialises a Window object.

$describe()

Describes the Window object.

$fit()

Constructs a Window module in ⁠C++⁠.

$eval()

Evaluates the cost of a segment.

$predict()

Performs Window given a linear penalty value.

$plot()

Plots change-point segmentation in ggplot style.

$clone()

Clones the R6 object.

Active bindings

Methods

Public methods

• Window$new()

• Window$describe()

• Window$fit()

• Window$eval()

• Window$predict()

• Window$plot()

• Window$clone()

Method new()

Initialises a Window object.

Usage

Window$new(minSize, jump, radius, costFunc)

Arguments

Returns

Invisibly returns NULL.

Method describe()

Describes a Window object.

Usage

Window$describe(printConfig = FALSE)

Arguments

Returns

Invisibly returns a list storing at least the following fields:

minSize

Minimum allowed segment length.

jump

Search grid step size.

radius

Radius of each sliding window.

costFunc

The costFun object.

fitted

Whether or not ⁠$fit()⁠ has been run.

tsMat

Time series matrix.

covariates

Covariate matrix (if exists).

n

Number of observations.

p

Number of features.

Method fit()

Constructs a ⁠C++⁠ module for Window.

Usage

Window$fit(tsMat = NULL, covariates = NULL)

Arguments

Details

This method constructs a ⁠C++⁠ Window module and sets private$.fitted to TRUE, enabling the use of ⁠$predict()⁠ and ⁠$eval()⁠. Some precomputations are performed to allow ⁠$predict()⁠ to run in linear time with respect to the number of local change-points (see ⁠$predict()⁠ for more details).

Returns

Invisibly returns NULL.

Method eval()

Evaluate the cost of the segment (a,b]

Usage

Window$eval(a, b)

Arguments

Details

The segment cost is evaluated as follows:

• L1 cost function:

  c_{L_1}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \tilde{y}_{(a+1)...b} \|_1

  where \tilde{y}_{(a+1)...b} is the coordinate-wise median of the segment. If a \ge b - 1, return 0.

• L2 cost function:

  c_{L_2}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \bar{y}_{(a+1)...b} \|_2^2

  where \bar{y}_{(a+1)...b} is the empirical mean of the segment. If a \ge b - 1, return 0.

• SIGMA cost function:

  c_{\sum}(y_{(a+1)...b}) := (b - a)\log \det \hat{\Sigma}_{(a+1)...b}

  where \hat{\Sigma}_{(a+1)...b} is the empirical covariance matrix of the segment without Bessel's correction. Here, if addSmallDiag = TRUE, a small bias epsilon is added to the diagonal of estimated covariance matrices to improve numerical stability.
  
  By default, addSmallDiag = TRUE and epsilon = 1e-6. In case addSmallDiag = TRUE, if the computed determinant of covariance matrix is either 0 (singular) or smaller than p*log(epsilon) - the lower bound, return (b - a)*p*log(epsilon), otherwise, output an error message.

• VAR(r) cost function:

  c_{\mathrm{VAR}}(y_{(a+1)...b}) := \sum_{t = a+r+1}^{b} \left\| y_t - \sum_{j=1}^r \hat A_j y_{t-j} \right\|_2^2

  where \hat A_j are the estimated VAR coefficients, commonly estimated via the OLS criterion. If system is singular, a-b < p*r+1 (i.e., not enough observations), or a \ge n-p (where n is the time series length), return 0.

• "LinearL2" for piecewise linear regression process with constant noise variance

  c_{\text{LinearL2}}(y_{(a+1):b}) := \sum_{t=a+1}^b \| y_t - X_t \hat{\beta} \|_2^2

  where \hat{\beta} are OLS estimates on segment (a+1):b. If segment is shorter than the minimum number of points needed for OLS, return 0.

Returns

The segment cost.

Method predict()

Performs Window given a linear penalty value.

Usage

Window$predict(pen = 0)

Arguments

Details

The algorithm scans the data with a fixed-size window to detect candidate local change-points (lcps) if the gains of its k_\text{thresh} neighbors to the left and right are all smaller than its gain, where k_\text{thresh} is defined as

k_\text{thresh} = \max \left( \frac{\max(2\text{radius}, 2 \cdot \text{minSize})}{2 \cdot \text{jump}}, 1 \right)

After candidate local change-points and computing the local gains, the algorithm selects the "optimal" set of break-points given the linear penalty threshold. Let G_i denote the local gain for candidate change-point i, for i = 1, \dots, n_\text{lcps}. The local gains are ordered such that G_1 \ge G_2 \ge \dots \ge G_{n_\text{lcps}}. Note that it is possible that no local change-points are detected, for example if the window size is too large.

The total cost for the selected k change-points is then calculated as

\text{TotalCost} = - \sum_{i=1}^{k} G_i + \lambda \cdot k,

where \lambda is a linear penalty applied per change-point. We then optimise over k to minimise the penalised cost function.

This approach allows detecting multiple change-points in a time series while controlling model complexity through the linear penalty threshold.

In our implementation, scanning the data to detect candidate local change-points and computing their corresponding local gains is already performed in ⁠$fit()⁠. Therefore, ⁠$predict()⁠ runs in linear time with respect to the number of local change-points.

Temporary segment end-points are saved to private$.tmpEndPoints after ⁠$predict()⁠, enabling users to call ⁠$plot()⁠ without specifying endpoints manually.

Returns

An integer vector of regime end-points. By design, the last element is the number of observations.

Method plot()

Plots change-point segmentation

Usage

Window$plot(
  d = 1L,
  endPts,
  dimNames,
  main,
  xlab,
  tsWidth = 0.25,
  tsCol = "#5B9BD5",
  bgCol = c("#A3C4F3", "#FBB1BD"),
  bgAlpha = 0.5,
  ncol = 1L
)

Arguments

Details

Plots change-point segmentation results. Based on ggplot2. Multiple plots can easily be horizontally and vertically stacked using patchwork's operators / and |, respectively.

Returns

An object of classes gg and ggplot.

Method clone()

The objects of this class are cloneable with this method.

Usage

Window$clone(deep = FALSE)

Arguments

Author(s)

Minh Long Nguyen edelweiss611428@gmail.com
Toby Dylan Hocking toby.hocking@r-project.org
Charles Truong ctruong@ens-paris-saclay.fr

References

Truong, C., Oudre, L., & Vayatis, N. (2020). Selective review of offline change point detection methods. Signal Processing, 167, 107299.

Examples

## L2 example
set.seed(1121)
signals = as.matrix(c(rnorm(100,0,1),
                     rnorm(100,5,1)))
# Default L2 cost function
WindowObj = Window$new(minSize = 1L, jump = 1L)
WindowObj$fit(signals)
WindowObj$predict(pen = 100)
WindowObj$plot()

## SIGMA example
set.seed(111)
signals = as.matrix(c(rnorm(100,-5,1),
                      rnorm(100,-5,10),
                      rnorm(100,-5,1)))
# L2 cost function
WindowObj = Window$new(minSize = 1L, jump = 1L)
WindowObj$fit(signals)
# We choose pen = 50.
WindowObj$predict(pen = 50)
WindowObj$plot()

# The standard L2 cost function is not suitable.
# Use the SIGMA cost function.
WindowObj$costFunc = costFunc$new(costFunc = "SIGMA")
WindowObj$predict(pen = 50)
WindowObj$plot()

Binary Segmentation (binSeg)

Description

An R6 class implementing binary segmentation for offline change-point detection.

Details

Binary segmentation is a classic algorithm for change-point detection that recursively splits the data at locations that minimise the cost function.

binSeg requires a R6 object of class costFunc, which can be created via costFunc$new(). Currently, the following cost functions are supported:

• "L1" and "L2" for (independent) piecewise Gaussian process with constant variance

• "SIGMA": for (independent) piecewise Gaussian process with varying variance

• "VAR": for piecewise Gaussian vector-regressive process with constant noise variance

• "LinearL2": for piecewise linear regression process with constant noise variance

See ⁠$eval()⁠ method for more details on computation of cost.

Some examples are provided below. See the GitHub README for detailed basic usage!

Methods

$new()

Initialises a binSeg object.

$describe()

Describes the binSeg object.

$fit()

Constructs a binSeg module in ⁠C++⁠.

$eval()

Evaluates the cost of a segment.

$predict()

Performs binSeg given a linear penalty value.

$plot()

Plots change-point segmentation in ggplot style.

$clone()

Clones the R6 object.

Active bindings

Methods

Public methods

• binSeg$new()

• binSeg$describe()

• binSeg$fit()

• binSeg$eval()

• binSeg$predict()

• binSeg$plot()

• binSeg$clone()

Method new()

Initialises a binSeg object.

Usage

binSeg$new(minSize, jump, costFunc)

Arguments

Returns

Invisibly returns NULL.

Method describe()

Describes a binSeg object.

Usage

binSeg$describe(printConfig = FALSE)

Arguments

Returns

Invisibly returns a list storing at least the following fields:

minSize

Minimum allowed segment length.

jump

Search grid step size.

costFunc

The costFun object.

fitted

Whether or not ⁠$fit()⁠ has been run.

tsMat

Time series matrix.

covariates

Covariate matrix (if exists).

n

Number of observations.

p

Number of features.

Method fit()

Constructs a ⁠C++⁠ module for binary segmentation.

Usage

binSeg$fit(tsMat = NULL, covariates = NULL)

Arguments

Details

This method constructs a ⁠C++⁠ binSeg module and sets private$.fitted to TRUE, enabling the use of ⁠$predict()⁠ and ⁠$eval()⁠. Some precomputations are performed to allow ⁠$predict()⁠ to run in linear time with respect to the number of data points (see ⁠$predict()⁠ for more details).

Returns

Invisibly returns NULL.

Method eval()

Evaluate the cost of the segment (a,b]

Usage

binSeg$eval(a, b)

Arguments

Details

The segment cost is evaluated as follows:

• L1 cost function:

  c_{L_1}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \tilde{y}_{(a+1)...b} \|_1

  where \tilde{y}_{(a+1)...b} is the coordinate-wise median of the segment. If a \ge b - 1, return 0.

• L2 cost function:

  c_{L_2}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \bar{y}_{(a+1)...b} \|_2^2

  where \bar{y}_{(a+1)...b} is the empirical mean of the segment. If a \ge b - 1, return 0.

• SIGMA cost function:

  c_{\sum}(y_{(a+1)...b}) := (b - a)\log \det \hat{\Sigma}_{(a+1)...b}

  where \hat{\Sigma}_{(a+1)...b} is the empirical covariance matrix of the segment without Bessel's correction. Here, if addSmallDiag = TRUE, a small bias epsilon is added to the diagonal of estimated covariance matrices to improve numerical stability.
  
  By default, addSmallDiag = TRUE and epsilon = 1e-6. In case addSmallDiag = TRUE, if the computed determinant of covariance matrix is either 0 (singular) or smaller than p*log(epsilon) - the lower bound, return (b - a)*p*log(epsilon), otherwise, output an error message.

• VAR(r) cost function:

  c_{\mathrm{VAR}}(y_{(a+1)...b}) := \sum_{t = a+r+1}^{b} \left\| y_t - \sum_{j=1}^r \hat A_j y_{t-j} \right\|_2^2

  where \hat A_j are the estimated VAR coefficients, commonly estimated via the OLS criterion. If system is singular, a-b < p*r+1 (i.e., not enough observations), or a \ge n-p (where n is the time series length), return 0.

• "LinearL2" for piecewise linear regression process with constant noise variance

  c_{\text{LinearL2}}(y_{(a+1):b}) := \sum_{t=a+1}^b \| y_t - X_t \hat{\beta} \|_2^2

  where \hat{\beta} are OLS estimates on segment (a+1):b. If segment is shorter than the minimum number of points needed for OLS, return 0.

Returns

The segment cost.

Method predict()

Performs binSeg given a linear penalty value.

Usage

binSeg$predict(pen = 0)

Arguments

Details

The algorithm recursively partitions a time series to detect multiple change-points. At each step, the algorithm identifies the segment that, if split, would result in the greatest reduction in total cost. This process continues until no further splits are possible (e.g., each segment is of minimal length or each breakpoint corresponds to a single data point).

Then, the algorithm selects the "optimal" set of break-points given the linear penalty threshold. Let [c_1, \dots, c_k, c_{k+1}] denote the set of segment end-points with c_1 < c_2 < \dots < c_k < c_{k+1} = n, where k is the number of detected change-points and n is the total number of data points. and k is the number of change-points. Let c_{(c_i, c_{i+1}]} be the cost of segment (c_i, c_{i+1}]. The total penalised cost is then

\text{TotalCost} = \sum_{i=1}^{k+1} c_{(c_i, c_{i+1}]} + \lambda \cdot k,

where \lambda is a linear penalty applied per change-point. We then optimise over k to minimise the penalised cost function.

This approach allows detecting multiple change-points in a time series while controlling model complexity through the linear penalty threshold.

In our implementation, the recursive step is carried out during ⁠$fit()⁠. Therefore, ⁠$predict()⁠ runs in linear time with respect to the number of data points.

Temporary segment end-points are saved to private$.tmpEndPoints after ⁠$predict()⁠, enabling users to call ⁠$plot()⁠ without specifying endpoints manually.

Returns

An integer vector of regime end-points. By design, the last element is the number of observations.

Method plot()

Plots change-point segmentation

Usage

binSeg$plot(
  d = 1L,
  endPts,
  dimNames,
  main,
  xlab,
  tsWidth = 0.25,
  tsCol = "#5B9BD5",
  bgCol = c("#A3C4F3", "#FBB1BD"),
  bgAlpha = 0.5,
  ncol = 1L
)

Arguments

Details

Plots change-point segmentation results. Based on ggplot2. Multiple plots can easily be horizontally and vertically stacked using patchwork's operators / and |, respectively.

Returns

An object of classes gg and ggplot.

Method clone()

The objects of this class are cloneable with this method.

Usage

binSeg$clone(deep = FALSE)

Arguments

Author(s)

Minh Long Nguyen edelweiss611428@gmail.com
Toby Dylan Hocking toby.hocking@r-project.org
Charles Truong ctruong@ens-paris-saclay.fr

References

Truong, C., Oudre, L., & Vayatis, N. (2020). Selective review of offline change point detection methods. Signal Processing, 167, 107299.

Hocking, T. D. (2024). Finite Sample Complexity Analysis of Binary Segmentation. arXiv preprint arXiv:2410.08654.

Examples

## L2 example
set.seed(1121)
signals = as.matrix(c(rnorm(100,0,1),
                     rnorm(100,5,1)))
# Default L2 cost function
binSegObj = binSeg$new(minSize = 1L, jump = 1L)
binSegObj$fit(signals)
binSegObj$predict(pen = 100)
binSegObj$plot()

## SIGMA example
set.seed(111)
signals = as.matrix(c(rnorm(100,-5,1),
                      rnorm(100,-5,10),
                      rnorm(100,-5,1)))
# L2 cost function
binSegObj = binSeg$new(minSize = 1L, jump = 1L)
binSegObj$fit(signals)
# We choose pen = 50.
binSegObj$predict(pen = 50)
binSegObj$plot()

# The standard L2 cost function is not suitable.
# Use the SIGMA cost function.
binSegObj$costFunc = costFunc$new(costFunc = "SIGMA")
binSegObj$predict(pen = 50)
binSegObj$plot()

costFunc class

Description

An R6 class specifying a cost function

Details

Creates an instance of costFunc R6 class, used in initialisation of change-point detection modules. Currently supports the following cost functions:

• L1 cost function:

  c_{L_1}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \tilde{y}_{(a+1)...b} \|_1

  where \tilde{y}_{(a+1)...b} is the coordinate-wise median of the segment. If a \ge b - 1, return 0.

• L2 cost function:

  c_{L_2}(y_{(a+1)...b}) := \sum_{t = a+1}^{b} \| y_t - \bar{y}_{(a+1)...b} \|_2^2

  where \bar{y}_{(a+1)...b} is the empirical mean of the segment. If a \ge b - 1, return 0.

• SIGMA cost function:

  c_{\sum}(y_{(a+1)...b}) := (b - a)\log \det \hat{\Sigma}_{(a+1)...b}

  where \hat{\Sigma}_{(a+1)...b} is the empirical covariance matrix of the segment without Bessel's correction. Here, if addSmallDiag = TRUE, a small bias epsilon is added to the diagonal of estimated covariance matrices to improve numerical stability.
  
  By default, addSmallDiag = TRUE and epsilon = 1e-6. In case addSmallDiag = TRUE, if the computed determinant of covariance matrix is either 0 (singular) or smaller than p*log(epsilon) - the lower bound, return (b - a)*p*log(epsilon), otherwise, output an error message.

• VAR(r) cost function:

  c_{\mathrm{VAR}}(y_{(a+1)...b}) := \sum_{t = a+r+1}^{b} \left\| y_t - \sum_{j=1}^r \hat A_j y_{t-j} \right\|_2^2

  where \hat A_j are the estimated VAR coefficients, commonly estimated via the OLS criterion. If system is singular, a-b < p*r+1 (i.e., not enough observations), or a \ge n-p (where n is the time series length), return 0.

• "LinearL2" for piecewise linear regression process with constant noise variance

  c_{\text{LinearL2}}(y_{(a+1):b}) := \sum_{t=a+1}^b \| y_t - X_t \hat{\beta} \|_2^2

  where \hat{\beta} are OLS estimates on segment (a+1):b. If segment is shorter than the minimum number of points needed for OLS, return 0.

If active binding ⁠$costFunc⁠ is modified (via assignment operator), the default parameters will be used.

Methods

$new()

Initialises a costFunc object.

$pass()

Describes the costFunc object.

$clone()

Clones the costFunc object.

Active bindings

Methods

Public methods

• costFunc$new()

• costFunc$pass()

• costFunc$clone()

Method new()

Initialises a costFunc object.

Usage

costFunc$new(costFunc, ...)

Arguments

Method pass()

Returns a list of configuration parameters to initialise detection modules.

Usage

costFunc$pass()

Method clone()

The objects of this class are cloneable with this method.

Usage

costFunc$clone(deep = FALSE)

Arguments

Author(s)

Minh Long Nguyen edelweiss611428@gmail.com

Examples

## L2 costFunc (default)
costFuncObj = costFunc$new()
costFuncObj$pass()
## SIGMA costFunc
costFuncObj = costFunc$new(costFunc = "SIGMA")
costFuncObj$pass()
# Modify active bindings
costFuncObj$epsilon = 10^-5
costFuncObj$pass()
costFuncObj$costFunc = "VAR"
costFuncObj$pass()