Pattern/Variability Heatmaps Overview

• Pattern/variability heatmaps are quantitative visualization methods that extract structured patterns and their stability within complex, high-dimensional data.
• They utilize advanced metrics and algorithms to quantify subgroup deviations and local spatial structures, enhancing traditional heatmaps.
• These techniques facilitate rigorous analysis in fields such as explainable AI, geospatial analytics, and spatial biology through precise, comparative evaluations.

Pattern/variability heatmaps constitute a broad, technically heterogeneous family of quantitative visualization and analysis methods designed to capture, summarize, and compare patterns and their variability within complex, high-dimensional data. Unlike classical heatmaps that simply display an $n \times p$ data matrix or the output of an unsupervised clustering, modern pattern/variability heatmaps aim to express nuanced relationships—including group-wise heterogeneity, local spatial structure, subgroup effects, or part-based explanation overlap—in a form amenable to rigorous quantitative assessment and comparative evaluation. The following sections present the conceptual taxonomy, formal methodologies, principal algorithms, limitations, and use cases for pattern/variability heatmaps, referencing diverse developments spanning explainable AI, dependence visualization, geospatial analytics, supervised model interpretation, and spatial biology.

1. Conceptual Scope and Formal Definitions

Pattern/variability heatmaps are defined by two core axes: (i) the quantification of pattern—that is, the extraction of structure, relationships, or region-specific saliency within a data representation; (ii) the mapping, summarization, or visualization of variability—the degree to which these patterns are stable, heterogeneous, or shifting across instances, subgroups, or spatial domains.

Unlike single-metric matrix visualizations (e.g., Pearson correlation heatmaps), pattern/variability heatmaps may display (a) a suite of pattern measures (e.g., F₁ overlap, mutual information, scagnostics, dependence deviations), (b) subgroup or per-region variability indicators (e.g., quantiles, groupwise summaries, subgroup deviation glyphs), or (c) alternative cues such as spatial clustering, annotation uncertainty, or multi-source reliabilities. Unification is achieved by the translation of local or subgroup-level statistics into a consistent grid (matrix, image, or spatial map) and the application of colormaps, glyph overlays, or side-plots that reveal both the primary structure and the variability inherent in it (Chinwan et al., 29 Nov 2024).

2. Quantitative Part-Based Heatmap Evaluation (PQAH)

The PQAH framework (Tursun et al., 22 May 2024) introduces a granular, objective approach for evaluating how well heatmaps generated by explainable AI (XAI) methods correspond to human-annotated object parts. The method revolves around the PH metric, a part-level F₁ score:

• Given a binarized heatmap $H \in \{0,1\}^{w \times h}$ (normalized and thresholded at $\theta=0.5$), and ground-truth binary masks $M^p$ for each semantic part $p$, define:
  • $TP_p = \sum (M^p \odot H)$ (true-positives for part $p$)
  • $FN_p = \sum M^p - TP_p$
  • Object-level precision is approximated as $\widehat{\mathrm{Prec}} = TP / (TP + FP)$ using the aggregate object mask $M = \sum_p M^p$
  • Per-part recall: $\mathrm{Recall}_p = TP_p / (TP_p + FN_p)$

The per-part PH score is then $$PH_p = F_{1,p} = \frac{2 \cdot \widehat{\mathrm{Prec}} \cdot \mathrm{Recall}_p}{\widehat{\mathrm{Prec}} + \mathrm{Recall}_p}$$. Analogous formulas are used for the background via $1-H$ and $1-M$.

After computing $PH_p$ across images in a category, dataset-level variability is summarized by quartiles: $PH_{Q1}, PH_{Q2}, PH_{Q3}$, visualized as boxplots or other comparison-friendly graphics. High $PH_{Q2}$ (median) indicates robust part coverage; high interquartile range ($Q3-Q1$) exposes instability or inconsistent attention. Comparisons between heatmap generators, architectures, or training regimes are readily facilitated by this summary.

Limitations

• Requires labor-intensive part segmentation for ground truth.
• Approximates precision at an object, not per-part, level.
• The binarization threshold $\theta$ may need method- or dataset-specific tuning.
• Assumes that high overlap with annotated parts connotes explanatory quality, potentially misaligned with some real-world interpretability criteria.

3. Statistical Dependence and Subgroup Variability

Recent advances in multivariate analysis have yielded heatmap-based visualizations that capture both statistical dependence patterns and their variability across subgroups. In "A Tidy Data Structure and Visualisations for Multiple Variable Correlations and Other Pairwise Scores" (Chinwan et al., 29 Nov 2024), the core innovation is the bullseye heatmap, where each cell representing a variable pair displays:

• An inner circle (“bullseye”) showing the statistic for the whole data (e.g., Pearson correlation, MIC, NMI, canonical correlation).
• An outer doughnut, with wedges corresponding to subgroup-specific values (e.g., per-species, per-race).
• Multiple association metrics (pattern measures) per cell, supporting non-linear, categorical, and distributional relationships.

The associated long-form tidy data structure supports the seamless addition of multiple metrics and groupings, enabling rapid filtering, faceting, and comparative analysis via standard tools (tidyverse), without the rigidity of $p \times p$ numeric matrices.

This framework exposes situations such as Simpson’s paradox (overall association direction reversed from within-group), non-monotonic and non-linear variable relationships, and heterogeneous subgroup effects, all in a unified, visually prominent format.

4. Model-Based and Spatial Patterns: Heatmap Algorithms

a) Model-Driven Heatmaps for XAI and Supervised Models

• In supervised learning, pattern/variability heatmaps can summarize both feature importances and two-way interactions extracted via partial dependence (PD) plots or analogous techniques. "Visualizing Variable Importance and Variable Interaction Effects in Machine Learning Models" (Inglis et al., 2021) employs a matrix layout, where diagonal cells show univariate PD, lower triangle shows 2D PD surfaces, and upper triangle encodes normalized interaction strengths (e.g., Friedman’s $H$-stat).
• For neural network explainability, spatial overlap heatmaps and structured attention graphs convey not just "what region" drives a prediction, but also "how variable" alternative explanations are across input regions, as in I-GOS, iGOS++, and SAGs (Fuxin et al., 2021).

b) Statistical Copula- and Variability-Focused Maps

• Copula-based heatmaps (Erdely et al., 2022) depict local deviations from statistical independence within bivariate distributions, using normalized color-coded tiles that reveal regions of positive/negative dependence, bidirectionality, and local anomalies that are invisible to global correlation metrics.
• The construction involves tiling the pseudo-observation space (i.e., after pseudo-rank transforms), computing empirical copula probabilities per cell, and mapping to a palette that distinguishes independence, positive, and negative quadrant dependence on the $[-1,1]$ scale.

c) Aggregation and Spatial Heterogeneity Analysis

• The Numericized Histogram Score (NHS) algorithm (Nguyen, 2017) provides quantitative spatial heatmaps for biological or spatial data by converting histograms of nearest-neighbor distances into continuous, intensity-scaled scores per object. These can be further normalized against uniform distributions and output as color-encoded aggregation/dispersion heatmaps for spatially heterogeneous systems.

5. Interactive, Clustering, and Multi-Modal Variability Heatmaps

Implementations in R (e.g., heatmaply (Galili et al., 2019), superheat (Barter et al., 2015)) and JavaScript (e.g., hilomap (Liu et al., 2022)) support pattern/variability detection and analysis via:

• Deep support for clustering (hierarchical or partitioning), distance metrics (Euclidean, correlation-based), and seriation to enhance contiguous pattern discovery.
• Integration of interactive features (zoom, panning, tooltips), rich color mapping (divergent palettes, quantile-based scaling), and side-plot overlays (scatterplots, barplots, boxplots) to align auxiliary metrics alongside primary structure.
• Hilomap’s algorithm specifically distinguishes and color-codes low and high value trends in spatial point data by separate accumulation and divergence mapping, which permits the neutralization of overlapping extremes—a property unavailable with standard additive blending heatmaps.

6. Interpretive Strategies, Use Cases, and Caveats

Selection and interpretation of pattern/variability heatmaps must reflect the analytic goals and data structure. For explainable AI, fine-grained overlap with semantic parts (PH metric) or identification of multiple sufficient explanations (SAGs) provides a quantitative basis for comparing, diagnosing, and refining model explanations. In multivariate correlation or dependence analysis, the ability to visualize non-linear effects, subgroup reversals, or local dependence heterogeneity (copulas, bullseye glyphs) surpasses the limitations of scalar heatmaps or scatterplot matrices.

In spatial or biomedical domains, NHS, Bayesian HeatmapBCC, and Gaussian uncertainty heatmaps enable nuanced depiction of aggregation, annotation uncertainty, and information source reliability—each accompanied by built-in mechanisms for quantifying and visualizing the stability or variability of detected patterns.

Limitations commonly arise from the need for ground-truth labels, sensitivity to methodological parameters (e.g., thresholds, bin widths), computational complexity for dense or high-resolution domains, and potential misinterpretation of visualized overlap or variability as causal or definitive.

7. Methodological Comparison Table

Method/Framework	Pattern Metric(s)	Variability Summary	Domain/Scenario
PQAH (Tursun et al., 22 May 2024)	Per-part F₁ overlap (PH)	Quartiles (Q1, Q2, Q3), boxplots	Deep Net XAI, segmentation
Bullseye (Chinwan et al., 29 Nov 2024)	Pearson, MI, CANCOR, etc.	Groupwise ring wedges, multistats	Multivariate association
Copula (Erdely et al., 2022)	Local normalized dependence ($\mathcal{H}$)	Cellwise sign, intensity tiles	Dependence visualization
NHS (Nguyen, 2017)	Weighted-Histogram Score (NHS)	Contrast to uniform, thresholded	Spatial/biological data
heatmaply (Galili et al., 2019)	Cell values + clustering	Interactive, cluster boxplots	High-dim data, vector-valued
hilomap (Liu et al., 2022)	Pointwise diverging sums	Low/high trend overlap	Geospatial, map data
Gaussian Uncertainty (Thaler et al., 2021)	Anisotropic heatmap area	Inter- and intra-observer stats	Landmark localization

Pattern/variability heatmaps represent a rapidly evolving class of quantitative analytical tools, each method grounded in statistical rigor or algorithmic precision, aimed at revealing both the structure and stability of patterns in modern scientific, engineering, and interpretive analytics.