Normalcy Score (NS): Cross-Domain Metrics

• Normalcy Score is a domain-specific metric that measures how closely observations conform to a predefined normal state using statistical transformations or learned patterns.
• It is applied across fields such as contextual anomaly detection, trajectory modeling, personalized clinical assessment, and citation normalization to capture deviations effectively.
• Variations include probabilistic, rank-based, and threshold-based formulations that address different uncertainties and structural orders in data.

Searching arXiv for recent and directly relevant papers on “Normalcy Score” and closely related usages across domains. Attempting arXiv search for exact phrase and related terms. Across the literature, “Normalcy Score” (NS) denotes a family of quantities that measure how closely an observation, trajectory, or parameter configuration conforms to a reference notion of normality or normalcy. In some works it is an explicit random standardized residual for contextual anomaly detection; in others it is an implicit complement of an anomaly score, a deviation measure from a learned pattern of life, or a score derived from a patient-specific predictive distribution. Related constructions also appear in statistical tests of distributional normality, in normalization-based citation analysis, and in particle-physics discussions of “normalcy” as proximity of a mixing matrix to diagonal structure (Bindini et al., 6 Jul 2025, Shah et al., 18 May 2026, Nakagawa et al., 2012, Conover et al., 2017, Denton, 2020).

1. Conceptual scope and recurring structure

Taken together, these works suggest that NS is not a single canonical statistic but a domain-specific scoring paradigm. The common pattern is a reference model of what counts as “normal,” together with a transformation that maps agreement or deviation into a scalar or distribution. In contextual anomaly detection, the reference is the conditional distribution $p(y \mid x)$; in trajectory modeling it is a learned pattern of life or a self-supervised reconstruction model; in laboratory medicine it is a conditional predictive interval anchored to population “normal” states; in statistical normality testing it is the null distribution under Gaussian sampling; and in neutrino physics it is a set of ordering inequalities on the moduli of mixing-matrix entries (Bindini et al., 6 Jul 2025, Zhang et al., 2024, Shah et al., 18 May 2026, Denton, 2020).

A recurring source of confusion is that the term is often implicit rather than formal. CLAWS, ReeSPOT, GPS-MTM, and NORMA all admit natural NS constructions, but the supplied analyses explicitly note that the original papers do not always introduce a symbol called “Normalcy Score.” By contrast, the contextual anomaly detection framework defines NS directly as a random variable and builds inference around its posterior distribution (Zaheer et al., 2020, Zhang et al., 2024, Garg et al., 28 Sep 2025, Shah et al., 18 May 2026, Bindini et al., 6 Jul 2025).

Another distinction concerns scale. Some NS formulations are signed and unbounded, such as z-like residuals; some lie in $[0,1]$, such as $1-\hat{y}$ or tail-probability mappings; some are binary decisions induced by whether a value lies outside a $95\%$ interval; and some are not scalar in the original source at all, but only become scalar after a secondary construction. This suggests that NS is best treated as a modeling role rather than a fixed formula.

2. Statistical normality, skewness, and sequential rank transforms

One statistical lineage treats NS as a measure of closeness to a normal distribution. The skewness test based on the sample Pearson measure of skewness constructs $spms$ by substituting the sample skewness $\sqrt{b_1}$ and sample kurtosis $b_2$ into Pearson’s population measure of skewness. Under normality, $E[spms]=0$, $$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$, and the null kurtosis is approximated by

$$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$

A Johnson $[0,1]$0 transformation is then used to map the standardized statistic $[0,1]$1 to a transformed variable $[0,1]$2 that is approximately $[0,1]$3. The supplied adaptation turns this into a skewness-based NS via $[0,1]$4, $[0,1]$5, or $[0,1]$6. This construction is explicitly skewness-focused: it is powerful for moderate skew with light tails, but it is not an omnibus normality score for symmetric heavy-tailed departures (Nakagawa et al., 2012).

The same section of the literature includes a rank-based sequential construction that effectively treats a “Sequential Normal Score” as NS. For i.i.d. continuous observations $[0,1]$7, the sequential rank is

$[0,1]$8

with plotting probability

$[0,1]$9

and score

$1-\hat{y}$0

The key result is that $1-\hat{y}$1 are mutually independent and $1-\hat{y}$2 are independent, asymptotically standard normal random variables. This gives a one-pass, nonparametric normalizing transform for sequential analysis, including extensions with known quantiles and batched observations, while preserving compatibility with normal-theory CUSUM and EWMA procedures (Conover et al., 2017).

These two statistical strands emphasize different meanings of “normalcy.” In the skewness-test adaptation, NS quantifies departure from Gaussian symmetry. In the sequential-rank framework, NS is a distribution-free transform whose output behaves as if sampled from $1-\hat{y}$3. The first is a goodness-of-fit score for normality; the second is a normalization device that makes downstream procedures normal-theory compatible.

3. Contextual anomaly detection and weakly supervised anomaly scoring

The most explicit modern definition appears in contextual anomaly detection. Given contextual variables $1-\hat{y}$4 and a real-valued behavioral variable $1-\hat{y}$5, NS is defined using heteroscedastic Gaussian process regression with independent GP priors on a mean function $1-\hat{y}$6 and a log-standard-deviation function $1-\hat{y}$7: $1-\hat{y}$8 The Normalcy Score is then the random standardized residual

$1-\hat{y}$9

With posterior approximations

$95\%$0

the expected score used for ranking is

$95\%$1

and anomaly ranking uses $95\%$2. The framework samples the full NS distribution, estimates a $95\%$3 Highest Density Interval $95\%$4, and uses its width $95\%$5 as a measure of epistemic uncertainty. This explicitly separates aleatoric uncertainty, represented by $95\%$6, from epistemic uncertainty, reflected in the GP posterior variances and the width of the HDI. It also induces an abstention rule: confident anomaly if $95\%$7, confident normal if $95\%$8, uncertain otherwise (Bindini et al., 6 Jul 2025).

Empirically, this uncertainty-aware NS outperforms context-free anomaly detection and compares favorably with contextual baselines such as ROCOD, QCAD, and classic Z-score formulations. On Abalone it reaches ROC AUC $95\%$9, PR AUC $spms$0, and Precision@$spms$1 $spms$2; on Toxicity it reaches ROC AUC $spms$3, PR AUC $spms$4, and Precision@$spms$5 $spms$6; on SynMachine it matches Z-score performance at ROC AUC and PR AUC $spms$7. In the cardiology application, the improvements over QCAD and ROCOD are statistically significant by Delong test with $spms$8 (Bindini et al., 6 Jul 2025).

A weaker but operationally important form appears in weakly supervised video anomaly detection. CLAWS produces a per-segment anomaly score $spms$9 from a small MLP backbone, and the natural implicit normalcy score is

$\sqrt{b_1}$0

The model’s normalcy suppression modules apply a softmax across the temporal dimension for each feature channel, then reweight intermediate features so that normal regions are suppressed and anomalous regions are highlighted. Clustering loss further encourages compact normal representations and separation from anomalous segments. The full method reaches frame-level AUC $\sqrt{b_1}$1 on UCF Crime and $\sqrt{b_1}$2 on ShanghaiTech, and the ablations show that the suppression modules sharply improve separation between anomaly and normal scores (Zaheer et al., 2020).

These two formulations illustrate a central divide. In CAD, NS is explicitly probabilistic, uncertainty-aware, and context-conditioned. In weak supervision, NS is the complement of an anomaly output learned under architectural and loss-based inductive biases. Both are context-sensitive, but only the former formalizes uncertainty in the score itself.

4. Trajectory normalcy in graphs and self-supervised sequence models

Human mobility research uses “normalcy” to denote patterns of life encoded in temporal, spatial, or semantic structure. ReeSPOT models an individual’s daily trajectories with a Reeb graph built over time. Two trajectories are connected at time $\sqrt{b_1}$3 if their positions are within an $\sqrt{b_1}$4-threshold; connected components define bundles; and graph nodes arise from appear, disappear, connect, and disconnect events. Rare location anomalies, rare route visit anomalies, uncommon time visits, and uncommon stay durations all manifest as new or altered nodes and edges. The paper defines a distance between a baseline normal Reeb graph $\sqrt{b_1}$5 and an updated graph $\sqrt{b_1}$6 by summing per-time distances between nodes. The supplied synthesis treats this distance as an anomaly score $\sqrt{b_1}$7, with a natural NS obtained by a decreasing transform such as

$\sqrt{b_1}$8

The original paper demonstrates the qualitative behavior of this distance: rare locations produce the largest day-level anomaly scores, while uncommon times and uncommon stay durations produce smaller but still visible deviations (Zhang et al., 2024).

A more representation-learning-oriented approach appears in GPS-MTM, a foundation model for mobility data that decomposes trajectories into states, given by point-of-interest categories $\sqrt{b_1}$9, and actions, encoded by details $b_2$0. The model uses a bi-directional Transformer encoder with 4 encoder layers, model dimension 256, 4 attention heads, and dropout 0.1, trained by masked modeling to reconstruct missing states and actions. The paper states that it “captures patterns of normalcy in human movement,” but it does not define a scalar NS. The supplied synthesis therefore derives token-level and trajectory-level normalcy from quantities already present in the model: $b_2$1, regression error $b_2$2, and combined negative log-likelihood or reconstruction loss. A natural token-level construction is $b_2$3, where $b_2$4 combines state negative log-likelihood and action error; a natural trajectory-level construction is $b_2$5, where $b_2$6 averages token anomalies over a segment or day (Garg et al., 28 Sep 2025).

These two mobility paradigms emphasize complementary notions of normalcy. ReeSPOT is topological and explicitly interpretable: anomalies are localized by graph events at concrete times and places. GPS-MTM is probabilistic and semantic: normalcy is the consistency of a stop or trajectory with learned state–action regularities. This suggests two broad classes of trajectory NS: graph-distance scores and reconstruction-likelihood scores.

5. Personalized clinical normalcy and blood biomarker interpretation

Clinical laboratory interpretation makes the tension between population normality and individualized normality explicit. NORMA begins from the observation that population reference intervals classify measurements as “low,” “normal,” or “high” against fixed external ranges, while purely personalized intervals may overfit to sparse histories or absorb unrecognized chronic disease into the baseline. In the comparison reported by the paper, purely personalized intervals classify up to $b_2$7 of measurements as abnormal without corresponding associations with adverse clinical outcomes, and abnormality rates are consistently higher for Per$b_2$8 than for Pop$b_2$9 or NORMA$E[spms]=0$0: in CHS the rates are $E[spms]=0$1, $E[spms]=0$2, and $E[spms]=0$3; in eICU they are $E[spms]=0$4, $E[spms]=0$5, and $E[spms]=0$6; and in INSPIRE they are $E[spms]=0$7, $E[spms]=0$8, and $E[spms]=0$9 (Shah et al., 18 May 2026).

NORMA itself is a conditional decoder-only transformer that predicts the next laboratory value from longitudinal history, timing, age, sex, lab code, and a query token specifying a future laboratory state. To generate a personalized reference interval, the query token is conditioned on a future “normal” state relative to population reference intervals, and the model outputs either a Gaussian predictive distribution or fixed quantiles. Under the Gaussian head, the loss is

$$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$0

and the resulting NORMA reference interval is

$$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$1

Under the quantile head, the interval is $$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$2. The paper itself uses a binary abnormality indicator for values outside the interval, but the supplied synthesis derives a natural scalar NS from the same predictive distribution, for example

$$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$3

or a tail-probability score based on the predictive CDF. In this formulation, NS measures deviation from a patient-conditional, population-anchored “future normal” distribution rather than from either a purely population-based interval or a purely personalized setpoint (Shah et al., 18 May 2026).

Outcome validation is central to the interpretation of this biomedical NS. NORMA-derived intervals achieve stronger or comparable prognostic enrichment than population intervals and markedly outperform purely personalized intervals. In CHS, for creatinine, abnormal flags have hazard ratios Pop$$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$4 $$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$5 $$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$6, Per$$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$7 $$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$8 $$\operatorname{Var}(spms) \approx \lambda_2 \sim \frac{3}{2n}$$9, and NORMA$$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$0 $$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$1 $$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$2; for hemoglobin they are Pop$$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$3 $$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$4 $$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$5, Per$$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$6 $$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$7 $$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$8, and NORMA$$\beta_2(spms)=3+20n^{-1}+\left(\frac{48544}{3}\right)n^{-2}+\left(\frac{10386704}{9}\right)n^{-3}+O(n^{-4}).$$9 $[0,1]$00 $[0,1]$01. The model also shows a controlled form of personalization: interval width narrows with added history up to approximately 30 prior measurements and then stabilizes, rather than shrinking indefinitely. This directly motivates a clinical reading of NS as a personalized but population-anchored measure of deviation that is less prone to overcalling than purely individualized intervals (Shah et al., 18 May 2026).

6. Other domain-specific meanings: citation normalization and fermion mixing

A distinct usage appears in scientometrics, where the nearest analogue is the Mean Normalized Citation Score, not an anomaly score but a field-normalized impact indicator. The “new crown indicator” computes

$[0,1]$02

a mean of publication-level citation-to-expectation ratios. The critique is twofold. First, field normalization based on ISI Subject Categories introduces indexer effects and artifacts, especially for journals assigned to multiple categories; second, the mean is “not a proper statistics for measuring differences among skewed distributions.” The proposed remedy is to reinterpret MNCS as the Median Normalized Citation Score and relate it directly to percentile-based evaluation, including the top $[0,1]$03 of highly cited papers. A plausible implication is that any citation-oriented NS should be publication-level normalized and summarized by median or percentiles rather than by the arithmetic mean (Leydesdorff et al., 2010).

Particle physics uses “normalcy” in a still different sense. For fermion mixing, normalcy means that the mixing matrix is “close to diagonal” in the sense that lighter charged leptons predominantly couple to lighter neutrino mass eigenstates, and heavier leptons to heavier ones. The framework introduces six normalcy conditions: $[0,1]$04

$[0,1]$05

$[0,1]$06

$[0,1]$07

$[0,1]$08

$[0,1]$09

The quark mixing matrix satisfies all six, whereas none is known to be fully satisfied for leptons at high significance. Under anarchy, only $[0,1]$10 of Haar-random parameter space satisfies the normalcy region. The paper does not define a scalar NS, but the supplied synthesis proposes two natural constructions: a count-based score equal to the number of satisfied conditions, and a continuous score based on inequality margins. Here normalcy is neither probabilistic typicality nor statistical Gaussianity, but an ordering principle on matrix elements and mass eigenstates (Denton, 2020).

These cases make clear that “Normalcy Score” is not semantically uniform across disciplines. In scientometrics, the central issue is robust normalization under skewed citation distributions. In neutrino physics, it is structured proximity to diagonal mixing. Both are conceptually related to “normalcy,” but neither is reducible to the uncertainty-aware contextual NS of anomaly detection or to the probabilistic normalcy of trajectory and laboratory models.

Normalcy Score is therefore best understood as a cross-domain schema for quantifying agreement with a chosen reference notion of normality. Its formal expression ranges from transformed skewness statistics and sequential rank maps, to heteroscedastic Bayesian residuals, to complements of learned anomaly scores, to graph distances and personalized predictive z-scores. The major methodological questions recur across domains: what defines the reference model, whether uncertainty is represented, how skewness or heterogeneity is handled, and whether the score measures typicality, deviation, normalization, or structural ordering.