---
title: 'IDfN: Individual Deviation from Normality'
url: https://www.emergentmind.com/topics/individual-deviation-from-normality-idfn
type: topic
---

# IDfN: Individual Deviation from Normality

Individual Deviation from Normality (IDfN) is used in recent literature as a domain-specific measure of how a single unit departs from an explicitly modeled notion of “normal” behavior. The unit may be a subject-specific effect in a hierarchical Bayesian model, an unlabeled instance in anomaly detection, a sample drawn from a finite-\(N\) equilibrium law, a laboratory measurement conditioned on patient history, an entity-time pair in streaming multivariate data, or a projection direction in a multivariate spatial process. The literature therefore suggests that IDfN is not a single canonical statistic, but a family of structurally related quantities defined relative to the normality concept adopted in each field [2112.05503] [2305.12958] [2510.26821] [2605.18701] [2509.18310] [2008.10957].

## 1. Conceptual scope and terminological variation

A recurrent source of ambiguity is that “normality” has different meanings across the cited works. In behavioral individual-differences modeling, normality refers to the additive-scale hierarchical specification of subject effects around a group effect. In AD-MERCS, it refers to local density-supported patterns in low-dimensional subspaces. In the finite-\(N\) statistical-mechanics formulation, it refers to the Gaussian equilibrium law recovered only in the thermodynamic limit. In NORMA, it refers to a patient-specific but population-anchored forecast distribution for the next biomarker value. In online change-point detection, it refers to low reconstruction error under an autoencoder trained on normal windows. In multivariate spatial testing, it refers to Gaussianity of all one-dimensional projections under the union-intersection principle [2112.05503] [2305.12958] [2510.26821] [2605.18701] [2509.18310] [2008.10957].

| Setting | Unit of analysis | IDfN quantity |
|---|---|---|
| Behavioral tasks | Subject \(i\) | Deviation structure of \(\delta_i\) around \(\nu\) |
| AD-MERCS | Instance \(x_i\) | Final anomaly score \(\delta_i \in [0,1]\) |
| Finite-\(N\) equilibrium | Sample from \(p_N\) | Short-tailed deviation parameterized by \(q_N\) |
| NORMA | Observation \(x_t\) | \(d_t = (x_t-\mu_t)/\sigma_t\) |
| Quickest CPD | Entity-time pair \((t,p)\) | \(\mathrm{IDfN}_{t,p} = \frac{1}{F}\sum_{s=1}^F(\hat x_{s,t,p}-x_{s,t,p})^2\) |
| Spatial MVN testing | Direction \(a_i\) | \(\mathrm{IDfN}(a_i)=T_{JB,i}\) |

This variation is substantive rather than merely terminological. A plausible implication is that IDfN is best treated as a comparative framework for per-unit departure scores, not as a universally fixed estimator. What is shared across the formulations is the presence of a reference model of normality and a mapping from an individual unit to a scalar or structured deviation.

## 2. Hierarchical Bayesian constraint models in behavioral individual differences

In the setting of Haaf and Rouder’s hierarchical Bayesian mixed-effects models, IDfN refers to the way individual-level effects are allowed to deviate around a common group effect, and how that deviation is constrained or left unconstrained [2112.05503]. In the simplest two-condition case, with response time \(y_{ij}\) for subject \(i\) on trial \(j\) and condition indicator \(x_j \in \{0,1\}\), the within-subject model is
\[
y_{ij}\mid \mu,\{\alpha_i\},\{\delta_i\},\sigma^2
\sim
\mathcal{N}\bigl(\mu+\alpha_i+x_j\delta_i,\sigma^2\bigr),
\]
where \(\mu\) is the grand-mean intercept, \(\alpha_i\) is the subject-specific random intercept, \(\delta_i\) is the subject-specific random effect, and \(\sigma^2\) is the residual variance.

The individual-difference structure is determined by the prior on \(\delta_i\). Haaf and Rouder consider four models. In the unconstrained model \(\mathcal{M}_u\),
\[
\delta_i \mid \nu,\eta^2 \sim \mathcal{N}(\nu,\eta^2),
\]
with a \(g\)-prior reparameterization in which \(\eta^2 = g_\delta \sigma^2\), \(g_\delta \sim \text{Scale-inv-}\chi^2(r_\delta^2)\), \(\nu \sim \mathcal{N}(0,g_\nu \sigma^2)\), and \(g_\nu \sim \text{Scale-inv-}\chi^2(r_\nu^2)\). The positive-effects model \(\mathcal{M}_+\) truncates that normal prior below at \(0\). The common-effect model \(\mathcal{M}_1\) sets \(\delta_i=\nu\) for all \(i\). The null model \(\mathcal{M}_0\) sets \(\delta_i=0\) for all \(i\). Intercepts are typically given \(\alpha_i \sim \mathcal{N}(0,\omega^2)\) with its own weakly informative hyperprior. The only difference across \(\{\mathcal{M}_u,\mathcal{M}_+,\mathcal{M}_1,\mathcal{M}_0\}\) is the way \(\delta_i\) is constrained or left free on the additive scale.

A common criticism is that the observed data are assumed to be drawn from a normal distribution even though response-time distributions are well known to be non-normal. Faulkenberry examined a shifted-lognormal alternative by fixing a small shift \(\delta_0\), defining \(z_{ij}=\log(y_{ij}-\delta_0)\), and retaining the same hierarchical priors on \(\{\alpha_i\}\), \(\{\delta_i\}\), \(\nu\), \(g_\nu\), and \(g_\delta\). Under this formulation, \(\nu\) is interpreted multiplicatively on the original response-time scale, with \(\exp(\nu)\) as the corresponding factor.

Model comparison proceeds through Bayes factors,
\[
\mathrm{BF}_{ab}=\frac{p(\mathbf y \mid M_a)}{p(\mathbf y \mid M_b)}.
\]
For \(\mathcal{M}_u\), \(\mathcal{M}_1\), and \(\mathcal{M}_0\), Haaf and Rouder use the closed-form results of Rouder et al. (2012) in the BayesFactor R package. For \(\mathcal{M}_+\) versus \(\mathcal{M}_u\), the encompassing-prior method compares the posterior proportion of draws satisfying \(\delta_i>0\) to the corresponding prior proportion.

Faulkenberry’s two numerical-cognition case studies show that the overall pattern of inference is essentially unchanged under the shifted-lognormal alternative [2112.05503]. In the size-congruity dataset (\(N=53\), 19 499 RTs), the normal model yielded posterior \(\nu \approx 60\) ms, individual \(\delta_i\) shrunk into \(\approx[9,116]\) ms from an observed range \([-15,142]\) ms, \(\mathrm{BF}_{+u}=7.19\), and overwhelming preference for \(\mathcal{M}_u\) over \(\mathcal{M}_1\) (\(10^{11}\!:\!1\)) and \(\mathcal{M}_0\) (\(10^{156}\!:\!1\)). The lognormal model gave \(\nu \approx 0.15\), corresponding to \(\exp(0.15)\approx 1.16\) or approximately \(86\) ms, with the same shrinkage pattern and \(\mathrm{BF}_{+u}=6.21\). In the unit-decade compatibility dataset (\(N=53\), 11 600 RTs), the normal model gave posterior \(\nu \approx 43\) ms, individual \(\delta_i \in [15,93]\) ms from observed \([-5,138]\) ms, and \(\mathrm{BF}_{+u}=4.17\); the lognormal model gave \(\nu \approx 0.08\), \(\exp(0.08)\approx 1.09\) or approximately \(57\) ms, and \(\mathrm{BF}_{+u}=8.40\). In both datasets, the positive-effects model best predicted the data.

The resulting controversy is interpretive rather than decisional. Under the normal response-time model, \(\delta_i\) is an additive difference in milliseconds; under the log model, it becomes multiplicative on the original scale. Faulkenberry therefore recommends the normal-response-time formulation as a pragmatic approach for modeling individual differences in behavioral tasks, because it preserves an additive interpretation without materially affecting inferences about IDfN [2112.05503].

## 3. AD-MERCS and instance-level deviation in unsupervised anomaly detection

AD-MERCS models both normality and abnormality in unsupervised anomaly detection by identifying low-dimensional subspaces in which patterns exist and conditions that characterize instances that deviate from those patterns [2305.12958]. The method builds on MERCS, “multi-directional ensembles of regression and classification trees.” Given an unlabeled dataset \(\mathcal{D}=\{x_1,\dots,x_N\}\subset \mathbb{R}^M\), one decision tree \(T_j\) is grown for each attribute \(A_j\) as target, using standard top-down splitting such as CART to maximize information gain or variance reduction. Each tree identifies a low-dimensional subspace \(S_j\), and each leaf \(c_{j,\ell}\) defines a rectangular cell in \(\mathbb{R}^{|S_j|}\) within which the data are assumed to share the same functional pattern for \(A_j\).

Instead of residuals \(\lvert A_j(x)-\hat A_j(x)\rvert\), AD-MERCS fits a local Gaussian kernel density estimate within each leaf:
\[
\kappa_{j,\ell}(y)
=
\frac{1}{n_j h_j}
\sum_{y_i \in c_{j,\ell}}
K\!\Bigl(\frac{y-y_i}{h_j}\Bigr),
\]
with bandwidth \(h_j\) chosen by the method of Botev et al. (2010). Because raw densities are incomparable across leaves and unbounded above, they are squashed into a likelihood score \(\omega_{j,\ell}(y)\in[0,1]\) using a threshold \(\tau_{j,\ell}\) chosen so that exactly \(\rho\%\) of the training densities in \(c_{j,\ell}\) fall below \(\tau_{j,\ell}\); in practice \(\rho \approx 5\%\). If a leaf is noisier than some ancestor node, the ancestor’s density estimate is reused so that scoring never relies on a weaker pattern than what was available higher in the tree.

The initial instance-level IDfN is formed from per-tree local anomaly probabilities
\[
p_{i,j}=1-\omega_{j,\ell}\bigl(x_i^{(j)}\bigr)
\]
and combined through a noisy-OR model:
\[
\delta_i^{(0)}
=
1-\prod_{j=1}^M \bigl(1-\gamma_\delta p_{i,j}\bigr),
\]
where \(0<\gamma_\delta\le 1\) is an inhibition parameter. AD-MERCS then iteratively refines the score by introducing context scores \(\lambda_{j,\ell}\in[0,1]\) for anomalous contexts. Instance scores are recomputed by blending normal-pattern anomaly and context anomaly, and context scores are recomputed by a noisy-AND over the instances they contain. After convergence, the final \(\delta_i\) is the IDfN score of \(x_i\).

The paper states several properties of this construction [2305.12958]. By construction, each \(\delta_i \in [0,1]\). The noisy-OR/AND updates guarantee monotonic non-decrease of \(\delta\) throughout iterations. The split-selection criterion ensures that each tree isolates subspaces of maximal predictive strength for its target. No asymptotic bounds are given, but empirically the method converges in few iterations, typically \(3\)–\(5\).

The examples clarify why density-based scoring differs from mean-residual scoring. In a bimodal leaf with target values \(\{1.4,1.5,1.6\}\cup\{3.4,3.5,3.6\}\), a point at \(y=2.5\) has zero residual to the mean \(\mu=2.5\) and would raise no alarm under residual-based scoring, but \(\kappa(2.5)=0\), hence \(\omega(2.5)=0\), \(p=1\), and the point is maximally anomalous. In the Zoo dataset, the animal “scorpion” is assigned high scores in multiple contexts, including “(has_spine = false)” predicting “has_tail,” yielding the explanation “Among animals with no backbone, having a tail is extremely unlikely,” and another context in which “does not lay eggs” predicts “has_teeth.” The final \(\delta(\text{scorpion})\) is near \(1\), with two concrete subspace-context explanations. This makes IDfN not only a scalar anomaly score but also an explanatory device via low-dimensional deviations from learned normality [2305.12958].

## 4. Finite-\(N\) equilibrium distributions and detectable non-Gaussianity

In the finite-\(N\) statistical-mechanics formulation, IDfN is the systematic short-tailed deviation from Gaussian equilibrium induced by finite particle number \(N\) [2510.26821]. For a one-dimensional ideal-gas-like system with total kinetic energy \(mU/2\), maximizing Havrda–Charvát (Tsallis) entropy yields a compact-support \(q\)-Gaussian density
\[
p_N(x)
=
\frac{
\Gamma\!\Bigl(\frac{2-q_N}{1-q_N}+\tfrac12\Bigr)
}{
\Gamma\!\bigl(\tfrac{2-q_N}{1-q_N}\bigr)\,(\pi U)^{1/2}
}
\Bigl[1-\tfrac{x^2}{U}\Bigr]_{+}^{\frac{1}{1-q_N}},
\]
with
\[
q_N=\frac{N-5}{N-3}.
\]
Taking \(U=N\), the support is \(\lvert x\rvert \le \sqrt{U}\). In the thermodynamic limit \(N\to\infty\), \(q_N\to 1\) and \(p_N(x)\) converges to the Maxwell-Boltzmann Gaussian law. For finite \(N\), the support is strictly compact and the tails are shorter than Gaussian.

This formulation makes the departure from Gaussianity fully parameterized by \(q_N=(N-5)/(N-3)\). As \(N\) decreases toward the minimum \(N>3\), \(q_N\) falls below \(1\) and the support shrinks; as \(N\) grows, all finite-\(N\) effects vanish smoothly. A plausible implication is that \(N\) serves as a physically interpretable non-Gaussianity index in settings where the system is literally or notionally finite.

Shim studies how five standard normality tests respond to this IDfN: Kolmogorov-Smirnov, Anderson-Darling, Cramér-von Mises, Jarque-Bera, and Shapiro-Wilk [2510.26821]. The Monte Carlo design uses system sizes \(N\in\{5,6,\dots,20\}\), sample sizes \(n\in\{10,20,30,\dots,100,500,1000\}\) for \(N\le 19\), and for \(N=20\) extends to \(\{2000,3000,4000,5000,10^4,2\times10^4,5\times10^4,10^5\}\), with \(100\) independent draws per \((N,n)\). KS and AD calibrated under the correct custom null \(p_N\) maintain nominal size \(\alpha=0.05\). Against the Gaussian null, all five statistics tend to grow as \(N\) becomes smaller, but the moment-based tests JB and SW detect short-tailed compact-support deviations far more efficiently than the ECDF-based tests KS, AD, and CvM.

The power map is highly uneven [2510.26821]. For strongly non-Gaussian cases \(N\le 9\), JB and SW achieve nearly \(100\%\) power even for \(n=500\), whereas KS, AD, and CvM plateau around \(30\)–\(50\%\). For \(N\ge 14\), ECDF tests rarely exceed \(20\)–\(30\%\) power at \(n\le 1000\), while JB and SW still exceed \(60\)–\(80\%\). At the near-Gaussian case \(N=20\), ECDF-based tests remain below \(30\%\) power up to \(n\approx 5000\), reaching approximately \(100\%\) only at \(n\ge 5\times 10^4\); JB and SW exceed \(95\%\) power by \(n\approx 5000\) and hit \(100\%\) by \(n=10^4\).

The practical guidance follows directly. When one suspects mild, short-tailed deviation from normality due solely to finite-size constraints, Jarque-Bera or Shapiro-Wilk are preferred. As a rule of thumb, for \(N\le 9\), \(n\approx 500\) suffices with JB or SW for \(>90\%\) power at the \(5\%\) level; for \(10\le N\le 14\), \(n\approx 1000\)–2000 suffices; for \(N\approx 20\), \(n\approx 3000\)–5000 is needed for JB or SW, whereas \(n>5\times 10^4\) would be required for KS, AD, or CvM [2510.26821].

## 5. Personalized laboratory interpretation and NORMA

In laboratory medicine, IDfN is formalized as a personalized deviation score derived from a conditional forecast distribution for the next biomarker value [2605.18701]. NORMA models blood biomarkers from longitudinal histories using a conditional transformer that learns
\[
p(x_{t+1}\mid H_t,P),
\]
where \(x_t\in\mathbb{R}^d\) is the vector of laboratory values at time \(t\), \(H_t=\{(x_1,t_1),\dots,(x_t,t_t)\}\) is the patient’s history, and \(P\) denotes population-level “healthy” prior information such as central \(95\%\) intervals \(\mathrm{Pop\_RI}\). The prediction head is either Gaussian, \(\mathcal{N}(\mu(H_t,P),\sigma^2(H_t,P))\), or quantile-based with five quantiles \(\{q_{0.025},q_{0.25},q_{0.50},q_{0.75},q_{0.975}\}(H_t,P)\).

The input representation combines static covariates, laboratory history, and a “normal” query token. The context token is
\[
z_c=e_g(g)+e_a(a)+e_c(c),
\]
the history token is
\[
z_i^{\mathrm{hist}}=f_v(x_i)+e_s(s_i)+f_\tau(\Delta t_i),
\]
with \(s_i\in\{\text{low},\text{normal},\text{high}\}\) and \(\Delta t_i=t_i-t_{i-1}\), and the query token is
\[
z_{\mathrm{query}}=e_s(s_{T+1}=\text{normal})+f_h(h).
\]
These tokens are processed by transformer decoder layers with multi-head self-attention and feed-forward blocks. The training objective is either the Gaussian negative log-likelihood
\[
L_{\mathrm{Gauss}}=\tfrac12\bigl[\log \sigma^2 + (x_{t+1}-\mu)^2/\sigma^2\bigr]
\]
or the quantile pinball loss
\[
L_\tau(x,q_\tau)=\tau \max(0,x-q_\tau)+(1-\tau)\max(0,q_\tau-x).
\]

From the learned distribution, NORMA extracts either \(\mu_t=\mu(H_t,P)\) and \(\sigma_t=\sigma(H_t,P)\), or \(\mu_t=q_{0.50}(H_t,P)\) and
\[
\sigma_t \approx \frac{q_{0.975}-q_{0.025}}{2\cdot 1.96}.
\]
The IDfN score is then the personalized z-score
\[
d_t=\frac{x_t-\mu_t}{\sigma_t},
\]
and the personalized \(95\%\) reference interval is
\[
[\mu_t-1.96\,\sigma_t,\ \mu_t+1.96\,\sigma_t],
\]
or directly \([q_{0.025},q_{0.975}]\) for the quantile head.

The main methodological issue in this literature is over-personalization. The paper states that purely personalized intervals can overfit to sparse data, inflate false-positive rates, and include unrecognized or subclinical disease; purely personalized intervals routinely overfit, classifying up to \(68\%\) of measurements as abnormal, without corresponding associations with adverse clinical outcomes [2605.18701]. Population intervals, by contrast, ignore stable intra-patient variability. NORMA balances these two extremes by conditioning on both \(H_t\) and \(P\), and by using a “normal” query token at each step. The paper presents a Bayesian-shrinkage analogy,
\[
\mu_B = \frac{\sigma_P^2 \bar x_H + \sigma_H^2 \mu_P}{\sigma_P^2 + \sigma_H^2},
\qquad
\sigma_B^2 = \frac{\sigma_P^2 \sigma_H^2}{\sigma_P^2 + \sigma_H^2},
\]
but states that NORMA learns a non-parametric, time-aware version of this anchoring.

The empirical results span nearly \(2\) billion longitudinal laboratory measurements from over \(1.6\) million individuals across North America, the Middle East, and East Asia [2605.18701]. In CHS, abnormal flags were \(29.6\%\) for \(\mathrm{Pop\_RI}\), \(46.8\%\) for \(\mathrm{Per\_RI}\), and \(39.1\%\) for \(\mathrm{NORMA\_RI}\); among \(\mathrm{Pop\_RI}\)-normal measurements, \(\mathrm{Per\_RI}\) reclassified \(27/100\) and NORMA \(12/100\); median lead time for \(\mathrm{NORMA\_RI}\) versus \(\mathrm{Pop\_RI}\) was \(8.7\) months (IQR \(1.7\)–\(33.8\)). In eICU, abnormal flags were \(50.2\%\), \(68.1\%\), and \(55.8\%\), with reclassification \(37/100\) for \(\mathrm{Per\_RI}\) and \(9/100\) for NORMA, and lead time \(34.8\) h (\(23.1\)–\(72.6\)). In INSPIRE, abnormal flags were \(35.1\%\), \(57.0\%\), and \(42.5\%\), with reclassification \(34/100\) and \(14/100\), and lead time \(23.7\) h (\(11.4\)–\(54.7\)). Among \(\mathrm{Pop\_RI}\)-normal cases in eICU, the positive predictive value per \(100\) reclassified for in-hospital mortality was \(10\) for \(\mathrm{Per\_RI}\) versus \(13\) for NORMA, for acute kidney injury \(15\) versus \(16\), and for prolonged length of stay \(23\) versus \(27\). The clinical interpretation is that IDfN in this setting is a calibrated patient-specific abnormality score anchored to a model of normal variation rather than to either raw population intervals or unconstrained personal baselines.

## 6. Online multi-entity change-point detection

In online change-point detection from multi-entity, multivariate time series, IDfN is defined as a reconstruction-error-based per-entity deviation under an autoencoder trained on normal behavior [2509.18310]. Each sensor channel is first z-score normalized using training-set means \(\mu_s\) and standard deviations \(\sigma_s\), and for entity \(p\) at time \(t\) a sliding window of length \(w\) is formed,
\[
W_p^{(t)}=[X_{t-w+1,p},\dots,X_{t,p}] \in \mathbb{R}^{F\times w}.
\]
A “Simple Autoencoder” then reconstructs the flattened window. The IDfN at time \(t\) for entity \(p\) is the mean squared error
\[
\mathrm{IDfN}_{t,p}
=
\frac{1}{F}\sum_{s=1}^F \bigl(\hat x_{s,t,p}-x_{s,t,p}\bigr)^2.
\]

The architecture is a stack of fully connected dense layers with ReLU activations, a symmetric decoder, a final linear output layer, and dropout of \(0.1\) after each hidden layer. Training is done on normal data only with loss
\[
L=\frac{1}{B}\sum_{\text{batch}} \lVert X-\hat X\rVert^2,
\]
using Adam with initial learning rate \(1\times 10^{-4}\), batch size \(512\), up to \(20\) epochs with early stopping after \(5\) epochs without validation improvement, and learning-rate reduction on plateau by a factor of \(0.5\) down to \(1\times 10^{-6}\). The recommended default window length is \(w=50\).

Per-entity IDfNs are aggregated into system-wide anomaly scores (SWAS) in three ways [2509.18310]. The mean score is
\[
S_t^\mu=\frac{1}{P}\sum_{p=1}^P \mathrm{IDfN}_{t,p},
\]
the variance score is
\[
S_t^{\sigma^2}=\frac{1}{P}\sum_{p=1}^P \bigl(\mathrm{IDfN}_{t,p}-S_t^\mu\bigr)^2,
\]
and the distributional score uses a Gaussian-kernel KDE of the current IDfN distribution compared with a reference KDE over all training IDfNs, with SWAS
\[
S_t^{WD}=WD(\hat f_t,\hat f_{\mathrm{train}}),
\]
the Wasserstein-1 distance. These statistics are converted to deviation scores \(f^\mu(t)\), \(f^{\sigma^2}(t)\), or \(f^{WD}(t)\), accumulated by CUSUM,
\[
C_t=C_{t-1}+f^\alpha(t),
\]
and then tested by an adaptive sequential density-based thresholding rule after the log transform \(\tilde C_i=\ln(1+C_i)\). A change is declared at time \(t\) if the KDE density estimate at \(\tilde C_t\) falls below a threshold \(\delta\), taken as the \(5\)th percentile of the training CUSUM-KDE densities, after a burn-in of \(2\cdot s\) samples.

The validation datasets include auto-regressive series with \(10\) entities and \(F=1\), coupled Chen chaotic oscillators with \(P=4\), \(F=3\), and \(T\approx 1900\), and two Unity crowd-simulation settings using upper-leg acceleration streams with \(F=3\): train-station evacuation with approximately \(20\) agents and a change around \(24\) s, and bidirectional corridor collision with approximately \(30\) agents and a change around \(23\) s [2509.18310]. The paper reports that IDfN trajectories show a clear rise at the ground-truth event time, KDE heatmaps visualize a shifting density of reconstruction errors with a sudden mode change at the incident, aggregated SWAS plots spike near the true change, and CUSUM curves cross adaptive thresholds in tight proximity to the ground-truth change point. In this formulation, IDfN is explicitly local—defined per entity and per time step—but is designed to support system-level inference after aggregation.

## 7. Directional IDfN in multivariate spatial normality testing

In multivariate spatial statistics, IDfN is defined as a directional departure from Gaussianity after accounting for spatial dependence [2008.10957]. The overall null is
\[
H_0: X(s)\sim N_p(\mu,\Sigma)
\quad \text{for } s\in\{s_1,\dots,s_n\}.
\]
By the union-intersection principle, \(H_0\) holds if and only if every projection
\[
Y_a(s)=a^\top X(s)
\]
is Gaussian for every unit vector \(a\in\mathbb{R}^p\). In practice, the continuum of directions is replaced by a finite set \(a_1,\dots,a_K\), such as the coordinate axes, the eigenvectors of an estimate of \(\Sigma\), or random draws from the unit sphere.

For a fixed direction \(a_i\), projected residuals are standardized as
\[
Z_j = a_i^\top(X(s_j)-\bar X)/\hat\sigma_{a_i},
\]
and the sample skewness and kurtosis are
\[
S_{n,i}=\frac1n\sum_{j=1}^n Z_j^3,
\qquad
K_{n,i}=\frac1n\sum_{j=1}^n Z_j^4-3.
\]
Under spatial dependence, the asymptotic variances of \(\sqrt{n}S_{n,i}\) and \(\sqrt{n}K_{n,i}\) depend on the covariance structure of the projected process. Horváth et al. (2020) propose kernel-smoothed estimators
\[
\hat\phi_{S,i}^2
=
6\sum_{h\in\mathbb{Z}^d}
\Bigl[\prod_{\ell=1}^d k(h_\ell/b_\ell)\Bigr]\,
\hat C_i(h)^3,
\]
\[
\hat\phi_{K,i}^2
=
24\sum_{h\in\mathbb{Z}^d}
\Bigl[\prod_{\ell=1}^d k(h_\ell/b_\ell)\Bigr]\,
\hat C_i(h)^4,
\]
where \(\hat C_i(h)\) is the sample autocovariance at lag \(h\), \(k(\cdot)\) is a univariate kernel such as Bartlett, and \(b\) is a bandwidth vector.

The directional Jarque-Bera-type statistic is then
\[
T_{JB,i}
=
\frac{S_{n,i}^2}{\hat\phi_{S,i}^2}
+
\frac{K_{n,i}^2}{\hat\phi_{K,i}^2}.
\]
Under \(H_0(a_i)\), the standardized skewness and kurtosis are jointly Gaussian and asymptotically independent, so \(T_{JB,i}\Rightarrow \chi^2_2\). At nominal level \(\alpha\), the critical value is \(c=\chi^2_{2,1-\alpha}\); for \(\alpha=5\%\), \(\chi^2_{2,0.95}=5.9915\). The global decision rule rejects multivariate normality when
\[
\max_{i=1,\dots,K} T_{JB,i} > c,
\]
or equivalently when any directional \(p\)-value is at most \(\alpha\), possibly with Bonferroni correction or Benjamini-Hochberg control.

In this framework, the directional score itself is interpreted as IDfN:
\[
\mathrm{IDfN}(a_i)=T_{JB,i}.
\]
A large \(T_{JB,i}\) means that, along direction \(a_i\), the data exhibit unusually large skewness or excess kurtosis after accounting for spatial dependence. This provides both a global multivariate normality test and a directional diagnostic. The method thereby makes it possible to locate which linear combinations of variables are most non-Gaussian, rank them by IDfN, and map out a “normality contour” on the unit sphere in \(\mathbb{R}^p\) [2008.10957].

Across these formulations, IDfN functions less as a single theory than as a recurring analytic pattern: choose a reference notion of normality, define an individual unit relative to that reference, and quantify the unit’s departure in a way that is appropriate to the data-generating assumptions of the field. The main controversies concern which normality model is substantively appropriate and how much interpretability is lost when one moves from additive, physically transparent, or clinically transparent scales to transformed or highly adaptive representations. The cited literature consistently treats those choices as consequential for interpretation, but not always for the qualitative ordering of evidence or detection performance [2112.05503] [2305.12958] [2510.26821] [2605.18701] [2509.18310] [2008.10957].

Source: https://www.emergentmind.com/topics/individual-deviation-from-normality-idfn