Measure reference

ZADU exposes 22 measures. Short aliases and full module IDs are accepted by ZADU; the tables below use the short aliases intended for specifications. Ranges describe the implemented definition, and “optimum” indicates the direction of perfect preservation when one exists.

Class-label assumptions

dsc, ivm, c_evm, nh, and ca_tnc assess a projection relative to labels that are assumed to describe meaningful, sufficiently separated original-space structure. Interpret them cautiously when that assumption is not supported.

Local measures

Measure	ID	Parameters	Return keys	Range	Optimum	Primary reference
Trustworthiness & Continuity	tnc	k=20	trustworthiness, continuity	[0.5, 1]	1	Venna & Kaski (2006)
Mean Relative Rank Error	mrre	k=20	mrre_false, mrre_missing	[0, 1]	1	Lee & Verleysen (2009)
Local Continuity Meta-Criteria	lcmc	k=20	lcmc	[-k/(n-1), 1-k/(n-1)]	1-k/(n-1)	Chen & Buja (2009)
Neighborhood Hit	nh	k=20	neighborhood_hit	[0, 1]	1	Paulovich et al. (2008)
Neighbor Dissimilarity	nd	k=20	neighbor_dissimilarity	non-negative	0	Fujiwara et al. (2023)
Class-Aware Trustworthiness & Continuity	ca_tnc	k=20	ca_trustworthiness, ca_continuity	[0.5, 1]	1	Colange et al. (2020)
Procrustes Measure	proc	k=20	procrustes	non-negative	0	Goldberg & Ritov (2009)

Cluster-level measures

Measure	ID	Parameters	Return keys	Range	Optimum	Primary reference
Steadiness & Cohesiveness	snc	iteration=150, walk_num_ratio=0.3, alpha=0.1, k=None, clustering_strategy="dbscan", random_state=None, n_jobs=1	steadiness, cohesiveness	[0, 1]	1	Jeon et al. (2021)
Distance Consistency	dsc	none	distance_consistency	[0, 1]	1	Sips et al. (2009)
Internal Validation Measure	ivm	measure="silhouette"	selected measure name	depends on selection	depends on selection	Silhouette, Calinski-Harabasz, Davies-Bouldin
Clustering + External Validation	c_evm	measure="arand", clustering="kmeans", clustering_args=None	{clustering}_{measure}	depends on selection	depends on selection	Adjusted Rand, AMI/NMI, V-measure
Label Trustworthiness & Continuity1	l_tnc	cvm="dsc"	label_trustworthiness, label_continuity	[0, 1]	1	Jeon et al. (2024)
Class Angular Distortion Index	cadi	n_triplets=0, random_seed=None	class_angular_distortion_index	[0, 1]	0	Gunaratne et al. (2026)

Global measures

Measure	ID	Parameters	Return key	Range	Optimum	Primary reference
Stress	stress	none	stress	non-negative	0	Kruskal (1964a), (1964b)
Non-Metric Stress	nm_stress	none	non_metric_stress	non-negative	0	Kruskal (1964)
Scale-Normalized Stress	sn_stress	none	scale_normalized_stress	non-negative	0	Smelser et al. (2024)
Kullback-Leibler Divergence	kl_div	sigma=0.1	kl_divergence	non-negative	0	Hinton & Roweis (2002)
Distance-to-Measure	dtm	sigma=0.1	distance_to_measure	non-negative	0	Chazal et al. (2011)
Topographic Product	topo	k=20	topographic_product	real-valued	0	Bauer & Pawelzik (1992)
Pearson correlation	pr	none	pearson_r	[-1, 1]	1	Pearson (1895)
Spearman rank correlation	srho	none	spearman_rho	[-1, 1]	1	Spearman (1904)

Pearson and Spearman use each unique off-diagonal distance once. The stress and density families reject all-zero distance inputs because their normalizations are undefined.

Gap-based regional measure

Measure	ID	Parameters	Return key	Range	Optimum	Primary reference
Gap Index2	gi	metric="euclidean"	gap_index	[0, 1]	0	Ros et al. (2026)

Gap Index measures distortion in empty triangular regions of a two-dimensional projection. It does not fit cleanly into the local, cluster, or global families. See the dedicated guide for detailed and regional output.

String-valued options

Internal validation

ivm accepts silhouette, calinski_harabasz, or davies_bouldin as its measure.

Clustering and external validation

c_evm accepts:

• measure: arand, ami, nmi, or vmeasure
• clustering: kmeans or dbscan

With clustering="kmeans", n_clusters defaults to the number of unique labels and random_state defaults to 0. Override them through clustering_args.

Label trustworthiness and continuity

l_tnc accepts dsc or ch_btw as its cvm.

Invalid option strings raise ValueError with the allowed values.

Pointwise return keys

With return_local=True, supported measures additionally return:

ID	Local return keys
tnc	local_trustworthiness, local_continuity
mrre	local_mrre_false, local_mrre_missing
lcmc	local_lcmc
nh	local_neighborhood_hit
ca_tnc	local_ca_trustworthiness, local_ca_continuity
snc	local_steadiness, local_cohesiveness

The primary references for the original 17 ZADU measures follow Table 1 of the ZADU paper. Later additions link to the publications that introduced them.

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1. The implementation does not apply the original paper's DSC rescaling step. The transformation was intended to map DSC into [0, 1], which is unnecessary for ZADU's already normalized DSC score. ↩

2. Introduced by Jaume Ros, Alessio Arleo, and Fernando Paulovich. ZADU's adaptation retains the upstream MIT license and pinned provenance; see the third-party notice. ↩