Pruned Exact Linear Time (PELT)

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
"LinearSIGMA": for piecewise linear regression process with varying noise covariance
"LinearL1": for piecewise linear regression process under L1 (least absolute deviations) loss

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

Some examples are provided below. See the package website for detailed 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.
$segments(): Returns the cost and parameter estimates of each segment from the latest $predict().
$plot(): Plots change-point segmentation in ggplot style.
$clone(): Clones the R6 object.

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.

Author

Minh Long Nguyen edelweiss611428@gmail.com
Toby Dylan Hocking toby.hocking@r-project.org
Charles Truong ctruong@ens-paris-saclay.fr
Huy Nhat Minh Nguyen sleepysnorlax0115@gmail.com

Active bindings

minSize: Integer. Minimum allowed segment length. Can be accessed or modified via $minSize. Modifying minSize will automatically trigger $fit().
jump: Integer. Search grid step size. Can be accessed or modified via $jump. Modifying jump will automatically trigger $fit().
costFunc: R6 object of class costFunc. Search grid step size. Can be accessed or modified via $costFunc. Modifying costFunc will automatically trigger $fit().
tsMat: Numeric matrix. Input time series matrix of size $n \times p$. Can be accessed or modified via $tsMat. Modifying tsMat will automatically trigger $fit().
covariates: Numeric matrix. Input time series matrix having a similar number of observations as tsMat. Can be accessed or modified via $covariates. Modifying covariates will automatically trigger $fit().

Methods

Method `new()`

Initialises a PELT object.

Usage

PELT$new(minSize, jump, costFunc)

Arguments

minSize: Integer. Minimum allowed segment length. Default: 1L.
jump: Integer. Search grid step size: only positions in {k, 2k, ...} are considered. Default: 1L.
costFunc: A R6 object of class costFunc. Should be created via costFunc$new() to avoid error. Default: costFunc$new("L2").

Returns

Invisibly returns NULL.

Method `describe()`

Describes a PELT object.

Usage

PELT$describe(printConfig = FALSE)

Arguments

printConfig: Logical. Whether to print object configurations. Default: FALSE.

Returns

Invisibly returns a list storing at least the following fields:

minSize: Minimum allowed segment length.
jump: Search grid step size.
costFunc: The costFunc 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

tsMat: Numeric matrix. A time series matrix of size $n \times p$ whose rows are observations ordered in time. If tsMat = NULL, the method will use the previously assigned tsMat (e.g., set via the active binding $tsMat or from a prior $fit(tsMat)). Default: NULL.
covariates: Numeric matrix. A time series matrix having a similar number of observations as tsMat. Required for models involving both dependent and independent variables. If covariates = NULL and no prior covariates were set (i.e., $covariates is still NULL), the model is force-fitted with only an intercept. Default: NULL.

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

a: Integer. Start index of the segment (exclusive). Must satisfy start < end.
b: Integer. End index of the segment (inclusive).

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: $$c_{\Sigma}(y_{(a+1)...b}) := (b - a)\log \det \hat{\Sigma}_{(a+1)...b}$$ 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 covariance matrix is numerically singular (its log-determinant cannot be computed) or its log-determinant is smaller than the lower bound p*log(epsilon), return (b - a)*p*log(epsilon), otherwise, output an error message.
VAR(r) cost function: $$c_{\mathrm{VAR}}(y_{(a+1)...b}) := \sum_{t = \max(a, r)+1}^{b} \left\| y_t - \hat c - \sum_{j=1}^r \hat A_j y_{t-j} \right\|_2^2$$ where $\hat c$ and $\hat A_j$ are the OLS estimates of the intercept and VAR coefficients on the segment. The lagged values $y_{t-j}$ may come from before $a+1$, so only the first $r$ observations of the whole series lack a full set of lags. If the system is singular, an approximate (minimum-norm) least-squares solve is used. If $b-a < p*r+1$ (i.e., not enough observations), or $a > n-r$ (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.

"LinearSIGMA" for piecewise linear regression process with varying noise covariance $$c_{\text{LinearSIGMA}}(y_{(a+1):b}) := (b-a)\log \det \hat\Sigma_{(a+1):b}$$ where $\hat\Sigma_{(a+1):b}$ is the empirical covariance matrix of OLS residuals $y - X\hat{\beta}$ on segment $(a+1):b$, divided by $b-a$ with no degrees-of-freedom correction for the fitted coefficients, and otherwise estimated the same way as in the SIGMA cost function.
"LinearL1" for piecewise linear regression process under L1 (least absolute deviations) loss $$c_{\text{LinearL1}}(y_{(a+1):b}) := \sum_{t=a+1}^b \| y_t - X_t \hat{\beta} \|_1$$ where $\hat{\beta}$ is fit column-by-column via IRLS on segment $(a+1):b$, iterated until convergence (tol) or maxIter. Unlike the other regression costs, this has no $O(1)$-per-segment closed form.

Returns

The segment cost.

Method `predict()`

Performs PELT given a linear penalty value.

Usage

PELT$predict(pen = 0)

Arguments

pen: Numeric. Penalty per change-point. Default: 0.

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 $0 = c_0 < c_1 < c_2 < \dots < c_k < c_{k+1} = n$, where $k$ is the number of detected change-points and $n$ is the total number of data points. Let $c_{(c_{i-1}, c_i]}$ be the cost of segment $(c_{i-1}, c_i]$. The total penalised cost is $$ \text{TotalCost} = \sum_{i=1}^{k+1} c_{(c_{i-1}, c_i]} + \lambda \cdot k, $$ 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 `segments()`

Returns the cost and parameter estimates of each segment from the latest $predict().

Usage

PELT$segments()

Details

Let $0 = c_0 < c_1 < \dots < c_{k+1} = n$ be the end-points from the latest $predict(pen). Segment $i$ is $(c_{i-1}, c_i]$, Cost is $c_{(c_{i-1}, c_i]}$ and Params is its minimiser. The costs therefore sum to the optimal total penalised cost minus $\lambda \cdot k$.

Temporary end-points are cleared by $fit(), so $predict() must be run again after modifying the object via its active bindings.

Returns

A list with one element per segment $(Start, End]$. Each element is a list with:

Start: Start index of the segment (exclusive, 0-based), same convention as $eval().
End: End index of the segment (inclusive).
Cost: The segment cost, as returned by $eval(Start, End).
Params: A named list of the segment parameter estimates, e.g. list(mean = ...) for "L2". See costFactory for the fields returned by each cost function.

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

d: Integer vector. Dimensions to plot. Default: 1L.
endPts: Integer vector. End points. Default: latest temporary changepoints obtained via $predict().
dimNames: Character vector. Feature names matching length of d. Defaults to "X1", "X2", ....
main: Character. Main title. Defaults to "PELT: d = ...".
xlab: Character. X-axis label. Default: "Time".
tsWidth: Numeric. Line width for time series and segments. Default: 0.25.
tsCol: Character. Time series color. Default: "#5B9BD5".
bgCol: Character vector. Segment colors, recycled to length of endPts. Default: c("#A3C4F3", "#FBB1BD").
bgAlpha: Numeric. Background transparency. Default: 0.5.
ncol: Integer. Number of columns in facet layout. Default: 1L.

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

deep: Whether to make a deep clone.

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$predict(pen = 100)
#> [1] 100 200
PELTObj$plot()
#> `endPts` is missing. Proceed to use the temporary `endPts`!
#> `dimNames` is missing. Proceed to use the default `dimNames`! e.g., `paste0('X', d))`.


## 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$predict(pen = 50)
#>  [1] 101 102 104 107 108 109 110 112 113 114 115 116 118 121 122 123 124 125 126
#> [20] 127 130 131 132 133 135 136 137 140 142 143 146 147 151 152 154 155 158 159
#> [39] 161 162 168 169 170 171 174 175 179 184 185 188 189 192 200 300
PELTObj$plot()
#> `endPts` is missing. Proceed to use the temporary `endPts`!
#> `dimNames` is missing. Proceed to use the default `dimNames`! e.g., `paste0('X', d))`.


# The standard L2 cost function is not suitable.
# Use the SIGMA cost function.
PELTObj$costFunc = costFunc$new(costFunc = "SIGMA")
#> `costFunc` has been updated. Re-fitting the model.
PELTObj$predict(pen = 50)
#> [1] 100 200 300
PELTObj$plot()
#> `endPts` is missing. Proceed to use the temporary `endPts`!
#> `dimNames` is missing. Proceed to use the default `dimNames`! e.g., `paste0('X', d))`.

Details

Methods

References

Author

Active bindings

Methods

Public methods

Method new()

Usage

Arguments

Returns

Method describe()

Usage

Arguments

Returns

Method fit()

Usage

Arguments

Details

Returns

Method eval()

Usage

Arguments

Details

Returns

Method predict()

Usage

Arguments

Details

Returns

Method segments()

Usage

Details

Returns

Method plot()

Usage

Arguments

Details

Returns

Method clone()

Usage

Arguments

Examples

Pruned Exact Linear Time (`PELT`)

Method `new()`

Method `describe()`

Method `fit()`

Method `eval()`

Method `predict()`

Method `segments()`

Method `plot()`

Method `clone()`