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 ZADU's outputs. “Best” gives the target value for the property being measured, not a guarantee that the whole projection is faithful. n is the sample count. Parameters and input restrictions follow the tables.

nh, dsc, ivm, and c_evm evaluate a projection against class labels. ca_tnc, l_tnc, and cadi compare class-related structure between spaces. All seven require a label for each row.

Local measures¶

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

Cluster-level measures¶

Measure and paper	ID	Return keys	Range; best
Steadiness & Cohesiveness Jeon et al. (2021)	`snc`	`steadiness`, `cohesiveness`	[0, 1]; 1
Distance Consistency Sips et al. (2009)	`dsc`	`distance_consistency`	[0, 1]; 1
Internal Validation Measure Silhouette, Calinski-Harabasz, Davies-Bouldin	`ivm`	selected measure name	depends on selection; depends on selection
Clustering + External Validation Adjusted Rand, AMI/NMI, V-measure	`c_evm`	`{clustering}_{measure}`	depends on selection; depends on selection
Label Trustworthiness & Continuity¹ Jeon et al. (2024)	`l_tnc`	`label_trustworthiness`, `label_continuity`	[0, 1]; 1
Class Angular Distortion Index Gunaratne et al. (2026)	`cadi`	`class_angular_distortion_index`	[0, 1]; 0

Global measures¶

Measure and paper	ID	Return keys	Range; best
Stress Kruskal (1964a), (1964b)	`stress`	`stress`	non-negative; 0
Non-Metric Stress Kruskal (1964)	`nm_stress`	`non_metric_stress`	non-negative; 0
Scale-Normalized Stress Smelser et al. (2024)	`sn_stress`	`scale_normalized_stress`	non-negative; 0
Kullback-Leibler Divergence Hinton & Roweis (2002)	`kl_div`	`kl_divergence`	non-negative; 0
Distance-to-Measure Chazal et al. (2011)	`dtm`	`distance_to_measure`	non-negative; 0
Topographic Product Bauer & Pawelzik (1992)	`topo`	`topographic_product`	real-valued; 0
Pearson correlation Pearson (1895)	`pr`	`pearson_r`	[-1, 1]; 1
Spearman rank correlation Spearman (1904)	`srho`	`spearman_rho`	[-1, 1]; 1

Pearson and Spearman use each unique off-diagonal distance once. Their distance vectors must have nonzero variance.

Gap-based regional measure¶

Measure and paper	ID	Return keys	Range; best
Gap Index² Ros et al. (2026)	`gi`	`gap_index`	[0, 1]; 0

Gap Index measures distortion in empty triangular regions of a two-dimensional projection. See the dedicated guide for regional output.

Parameters and input restrictions¶

Neighborhood size¶

tnc, mrre, lcmc, nh, ca_tnc, nd, topo, and proc default to k=20. Set an integer with 1 <= k < n; T&C and class-aware T&C require k < n / 2. Thus their default k=20 needs at least 41 samples.

ZADU returns MRRE as one minus normalized rank error, so larger values are better. LCMC's upper bound is 1 - k / (n - 1), not 1. Topographic Product is signed; closeness to zero, rather than a larger raw value, indicates its target.

Density bandwidth¶

dtm and kl_div use sigma=0.1, which must be positive. Each space's distances are divided by their maximum before applying the kernel exp(-(distance / maximum_distance)**2 / sigma). Choose sigma for the density scale you want to compare, and keep it fixed across projections.

Steadiness and Cohesiveness¶

Parameter	Default	Meaning
`iteration`	`150`	Number of sampled iterations
`walk_num_ratio`	`0.3`	Walk count relative to the number of samples
`alpha`	`0.1`	Positive offset in the similarity-to-distance conversion `1 / (similarity + alpha)`
`k`	omitted	Uses `floor(sqrt(n))`; pass an integer to override
`clustering_strategy`	`"dbscan"`	Cluster extraction strategy; also accepts `"kmeans"` and `"N-means"` for an integer N
`random_state`	`None`	Seed or NumPy generator
`n_jobs`	`1`	Iteration workers; collection execution may reduce this to avoid nested ZADU workers

CADI¶

n_triplets=0 selects 10 * n sampled triplets; a positive integer selects an explicit count. random_seed=None uses a new random generator. Set an integer seed for repeatable samples. At least two classes are required, and one class must contain at least two samples.

Gap Index¶

metric="euclidean" measures original-space triangle edges. See the Gap Index guide for other distance functions, precomputed distances, and regional output.

Degenerate inputs¶

Requirements depend on the measure. T&C can handle duplicate points using stable index ties; that does not make duplicate or constant inputs valid for every other measure. Pearson and Spearman require nonconstant distance values. Stress requires nonzero original distances; scale-normalized stress, non-metric stress, and density measures require nonzero distances in both spaces. Procrustes requires nonzero variance in each original neighborhood; Topographic Product requires positive distances in its ratios. IVM requires between 2 and n - 1 classes.

String-valued options¶

Internal validation¶

ivm accepts silhouette, calinski_harabasz, or davies_bouldin as its measure. Silhouette ranges from −1 to 1 (higher is better), Calinski–Harabasz is non-negative (higher is better), and Davies–Bouldin is non-negative (lower is better).

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. All four validation scores are higher-is-better, with 1 indicating agreement. Adjusted Rand and adjusted mutual information may be negative; normalized mutual information and V-measure range from 0 to 1. DBSCAN's noise label is passed to the selected scorer along with cluster labels.

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, the runner returns (global_scores, local_scores). Both lists follow specification order; unsupported local entries are None. The local dictionaries use the keys below. Each value is an array of length n.

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.

ZADU uses its normalized DSC directly in label T&C; it does not apply the paper's DSC rescaling step. ↩
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. ↩