modified_z_score_anomaly_detector

Statistical modified Z-score anomaly detector for continuous datasets. It is a statistical anomaly-detection method based on the modified Z-score defined by Iglewicz and Hoaglin (1993): for each known continuous attribute value x, the detector computes 0.6745 * (x - median) / mad, where median is the learned sample median and mad is the learned median absolute deviation, and then aggregates the per-attribute modified Z-scores using the selected learn-time score_mode/1 option.

The library implements the anomaly_detector_protocol defined in the anomaly_detection_protocols library. It learns a detector from a continuous dataset, computes anomaly scores for new instances, predicts normal or anomaly, and exports learned detectors as clauses or files.

Datasets are represented as objects implementing the anomaly_dataset_protocol protocol from the anomaly_detection_protocols library. See the anomaly_detection_protocols/test_datasets directory for examples.

API documentation

Open the ../../apis/library_index.html#modified-z-score link in a web browser.

Loading

To load this library, load the loader.lgt file:

| ?- logtalk_load(modified_z_score_anomaly_detector(loader)).

Testing

To test this library predicates, load the tester.lgt file:

| ?- logtalk_load(modified_z_score_anomaly_detector(tester)).

Features

  • Statistical method: implements robust anomaly detection based on the modified Z-score described by Iglewicz and Hoaglin (1993), using the sample median and median absolute deviation of each continuous attribute to measure how far new observations deviate from the training data distribution.

  • Classical per-attribute modified Z-score: for each known attribute value x, the library computes the standard robust score 0.6745 * (x - median) / mad, where median is the learned sample median for that attribute and mad is the learned median absolute deviation.

  • Continuous features only: accepts datasets whose declared attributes are all continuous.

  • Robust statistics: reuses the statistics library sample object to compute per-attribute medians and the median of absolute deviations.

  • Baseline training selection: supports learn-time baseline_class_values(ClassValues) and baseline_selection_policy(Policy) options. The default baseline class values are [normal]. The default reject policy throws an error if non-baseline examples are present, while filter removes them before fitting.

  • Missing-value tolerant: ignores missing values when fitting attribute statistics. During scoring, queries must provide at least one known value. In the default score_mode(root_mean_square), the raw score is normalized by the number of known values so that scores remain comparable across different missing-value patterns.

  • Configurable scoring semantics: supports both dense multivariate deviation scoring using score_mode(root_mean_square) and sparse anomaly detection using score_mode(any_feature_extreme). The default root-mean-square mode reuses the numberlist library Euclidean norm predicate as part of the computation. The score_mode/1 option only controls how the per-attribute modified Z-scores are aggregated into a single raw anomaly score.

  • Bounded scoring: maps the raw multivariate modified Z-score to [0.0, 1.0) using Score = Raw / (1 + Raw).

  • Default threshold: the default anomaly_threshold(0.7777777777777778) corresponds to the classical raw modified Z-score cutoff 3.5 recommended by Iglewicz and Hoaglin (1993), while remaining overrideable in learn/3 and predict/4.

  • Learn-time score mode: score_mode/1 is recorded in the learned detector and reused for subsequent scoring and prediction. Passing a score_mode/1 option to predict/4 does not override the learned mode.

  • All-missing queries rejected: scoring and prediction throw a domain_error(non_empty_known_values, AttributeNames) exception when every declared feature is missing in the query.

  • Featureless datasets rejected: datasets must declare at least one continuous feature; otherwise learn/2-3 throws a domain_error(non_empty_features, Dataset) exception.

  • Detector export: learned detectors can be exported as predicate clauses.

  • Explicit validation and diagnostics: supports the shared check_anomaly_detector/1, valid_anomaly_detector/1, diagnostics/2, diagnostic/2, and anomaly_detector_options/2 predicates.

Options

The following options are supported by the public API:

  • anomaly_threshold(Threshold): Threshold for predict/3-4 (default: 0.7777777777777778)

  • baseline_class_values(ClassValues): Learn-time class labels that are admissible for baseline fitting (default: [normal])

  • baseline_selection_policy(Policy): Learn-time handling of examples whose class is not listed in baseline_class_values/1. Supported values are filter and reject (default: reject)

  • score_mode(Mode): Learn-time score aggregation mode for learn/3. Supported values are root_mean_square and any_feature_extreme (default: root_mean_square). If passed to predict/4, it is ignored and the value stored in the learned detector is used.

Detector representation

The learned detector is represented by default as:

modified_z_score_detector(TrainingDataset, Encoders, Diagnostics)

Where:

  • TrainingDataset: training dataset object identifier

  • Encoders: list of modified_zscore(Attribute, Median, Scale) records

  • Diagnostics: learned metadata terms including model/1, training_dataset/1, attribute_names/1, feature_count/1, example_count/1, and options/1

When exported using export_to_clauses/4 or export_to_file/4, this detector term is serialized directly as the single argument of the generated predicate clause so that the exported model can be loaded and reused as-is.

Notes

Scoring has three stages. First, the detector computes one classical per-attribute modified Z-score for each known attribute value using 0.6745 * (x - median) / mad. Second, those per-attribute modified Z-scores are aggregated into a single raw anomaly score according to the learned score_mode/1 option. Third, the raw score is mapped to the interval [0.0, 1.0) using Score = Raw / (1 + Raw).

The score_mode/1 option does not change the classical per-attribute formula. It only changes the aggregation step. With score_mode(root_mean_square), the raw score is the root mean square of the per-attribute modified Z-scores. With score_mode(any_feature_extreme), the raw score is the maximum absolute per-attribute modified Z-score.

The baseline_class_values/1 option declares which dataset class labels are admissible for fitting the baseline medians and median absolute deviations. The baseline_selection_policy/1 option then controls what happens when other labels are present in the training data. The default reject policy raises a domain_error(baseline_only_training_data, Dataset) exception when any non-baseline example is found. The filter policy removes non-baseline examples before fitting.

Attributes with zero observed median absolute deviation are assigned a fallback scale of 1.0. This keeps the detector well-defined for singleton datasets or constant columns while still yielding zero score for matching values and positive scores for deviating values.

The root-mean-square aggregation keeps the default threshold stable as the number of observed dimensions grows and avoids penalizing partially observed queries solely for having fewer known attributes.

Use score_mode(any_feature_extreme) when a single extreme feature should be sufficient to flag an anomaly in high-dimensional data.