Robust Principal Component Analysis

class dtaianomaly.anomaly_detection.RobustPrincipalComponentAnalysis(window_size: str | int, stride: int = 1, max_iter: int = 1000, **kwargs)[source]

Anomaly detection based on Robust Principal Component Analysis (Robust PCA).

Assume that the data matrix is a superposition of a low-rank component and a s parse component. Robust PCA [Candes2011robust] will solve this decomposition as a convex optimization problem. The superposition offers a principeled manner to robust PCA, since the methodology can recover the principal components (first component) of a data matrix even though a positive fraction of the entries are arbitrarly corrupted or anomalous (second component).

Warning

During testing, we found that there are some deviations in the predicted decision scores, depending on if the method was run on windows or linux. The difference in the absolute value is of around the order of 2%, but the general trend of the anomaly scores remains consistent. The only randomness in this implementation of Robust PCA is the PCA solver of scikit-learn, but even setting a random state did not resolve the issue.

Notes

In most existing implementations, Robust PCA only takes one observation at a time into account (i.e., does not look at windows). However, Robust PCA can not be applied to a single variable, which is the case for univariate data. Therefore, we added a parameter window_size to apply Robust PCA in windows of a univariate time series, to make it applicable. Common behavior on multivariate time series can be obtained by setting window_size = 1.

Parameters:

  • window_size (int or str) – The window size to use for extracting sliding windows from the time series. This value will be passed to compute_window_size().

  • stride (int, default=1) – The stride, i.e., the step size for extracting sliding windows from the time series.

  • max_iter (int, default=1000) – The maximum number of iterations allowed to optimize the low rank approximation.

  • kwargs (dict) – Additional parameters to be passed PCA of Sklearn.

window_size_

The effectively used window size for this anomaly detector

Type:

int

pca_

The PCA-object used to project the data in a lower dimension.

Type:

PCA

Examples

>>> from dtaianomaly.anomaly_detection import RobustPrincipalComponentAnalysis
>>> from dtaianomaly.data import demonstration_time_series
>>> x, y = demonstration_time_series()
>>> rpca = RobustPrincipalComponentAnalysis(2).fit(x)
>>> rpca.decision_function(x)
array([1.28436687, 1.29156655, 1.33793287, ..., 1.35563558, 1.25948662,
       1.2923824 ])

References

[Candes2011robust]

Emmanuel J. Candès, Xiaodong Li, Yi Ma, and John Wright. 2011. Robust principal component analysis? J. ACM 58, 3, Article 11 (June 2011), 37 pages. doi: 10.1145/1970392.1970395

check_is_fitted() → None

Check whether this anomaly detector is fitted or not.

Raises:

NotFittedError – If this detector is not fitted yet.

decision_function(X: ndarray) → array

Abstract method, compute anomaly scores.

Parameters:

X (array-like of shape (n_samples, n_attributes)) – Input time series.

Returns:

decision_scores – The computed anomaly scores.

Return type:

array-like of shape (n_samples)

fit(X: ndarray, y: ndarray | None = None, **kwargs) → BaseDetector

Abstract method, fit this detector to the given data.

Parameters:

  • X (array-like of shape (n_samples, n_attributes)) – Input time series.

  • y (array-like, default=None) – Ground-truth information.

Returns:

self – Returns the instance itself.

Return type:

BaseDetector

is_fitted() → bool

Return whether this anomaly detector is fitted.

Returns:

is_fitted – True if and only if this detector is fitted, and can be used for detecting anomalies.

Return type:

bool

predict_confidence(X: ndarray, X_train: ndarray = None, contamination: float = 0.05, decision_scores_given: bool = False)

Predict the confidence of the anomaly scores on the test given test data.

This method implements ExCeeD [perini2020quantifying] (Example-wise Confidence of anomaly Detectors) to estimate the confidence. ExCeed transforms the predicted decision scores to probability estimates using a Bayesian approach, which enables to assign a confidence score to each prediction which captures the uncertainty of the anomaly detector in that prediction.

Parameters:

  • X (array-like of shape (n_samples, n_attributes)) – The test time series for which the confidence of anomaly scores should be predicted.

  • X_train (array-like of shape (n_samples_train, n_attributes), default=None) – The training time series, which can be used as reference. If X_train=None, the test set is used as reference set.

  • contamination (float, default=0.05) – The (estimated) contamination rate for the data, i.e., the expected percentage of anomalies.

  • decision_scores_given (bool, default=False) – Whether the given X and X_train represent time series data or decision scores. If decision_scores_given=False (default), then the given arrays are interpreted as time series. Otherwise, they are interpreted as decision scores, as computed by decision_function().

Returns:

confidence – The confidence of this anomaly detector in each prediction in the given test time series.

Return type:

array-like of shape (n_samples)

References

[perini2020quantifying]

Perini, L., Vercruyssen, V., Davis, J. Quantifying the Confidence of Anomaly Detectors in Their Example-Wise Predictions. In: Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2020. Springer, Cham, doi: 10.1007/978-3-030-67664-3_14.

predict_proba(X: ndarray) → ndarray

Predict anomaly probabilities

Estimate the probability of a sample of X being anomalous, based on the anomaly scores obtained from decision_function by rescaling them to the range of [0, 1] via min-max scaling.

Parameters:

X (array-like of shape (n_samples, n_attributes)) – Input time series.

Returns:

anomaly_scores – 1D array with the same length as X, with values in the interval [0, 1], in which a higher value implies that the instance is more likely to be anomalous.

Return type:

array-like of shape (n_samples)

Raises:

  • ValueError – If scores is not a valid array.

  • ValueError – If the prediction scores from ‘decision_function’ are constant, but not in the interval [0, 1], because these values can not unambiguously be transformed to an anomaly probability.

save(path: str | Path) → None

Save detector to disk as a pickle file with extension .dtai. If the given path consists of multiple subdirectories, then the not existing subdirectories are created.

Parameters:

path (str or Path) – Location where to store the detector.