Temporal Analytics Toolkit

ChronosVector provides 19 analytical functions for extracting insights from temporal vector data. Each function is grounded in a specific mathematical framework and applicable across multiple domains.

Overview

Category	Functions	Mathematical Basis
Differential Calculus	velocity, drift, temporal_features	Finite differences, feature engineering
Stochastic Characterization	hurst_exponent, detect_changepoints	R/S analysis, PELT algorithm
Path Signatures	path_signature, log_signature, signature_distance	Rough path theory (Lyons, 1998)
Trajectory Comparison	frechet_distance	Computational geometry
Distributional Distances	wasserstein_drift, fisher_rao_distance, hellinger_distance	Optimal transport, information geometry
Point Process Analysis	event_features	Temporal point processes
Topological Analysis	topological_features	Persistent homology (TDA)
Anchor Projection	project_to_anchors, anchor_summary	Coordinate system change to reference frame
Prediction	predict	Linear extrapolation / Neural ODE

---

Differential Calculus

cvx.velocity(trajectory, timestamp)

Theory: Instantaneous rate of change $\frac{dv}{dt}$ via central finite differences. Measures how fast an entity is moving through the embedding space.

Output: Velocity vector (same dimensionality as input).

Applications:

• Mental health: Rate of linguistic change — accelerating drift may signal crisis
• Finance: Speed of portfolio evolution — regime transitions show velocity spikes
• Molecular dynamics: Conformational transition speed between states
• MAP-Elites: Rate of archive exploration — stagnation detection

cvx.drift(v1, v2, top_n=5)

Theory: Total displacement between two vectors, decomposed by dimension. Identifies what changed and how much.

Output: (l2_magnitude, cosine_drift, top_dimensions) — the overall magnitude, angular change, and the dimensions contributing most.

cvx.temporal_features(trajectory)

Theory: Fixed-size summary vector combining mean, std, velocity statistics, Hurst exponent, and change point count. Designed for downstream ML classification.

Output: Vector of size $2D + 5$ where $D$ = input dimensionality.

---

Stochastic Characterization

cvx.hurst_exponent(trajectory)

Theory: The Hurst exponent $H \in (0, 1)$ characterizes the long-range dependence of a time series via R/S (rescaled range) analysis:

• $H = 0.5$: random walk (no memory)
• $H > 0.5$: persistent (trending) — past increases predict future increases
• $H < 0.5$: anti-persistent (mean-reverting) — past increases predict future decreases

Applications:

• Mental health: Depression users show anti-persistent topical dynamics ($H < 0.5$, $d = -0.41$ on CLPsych) — erratic topic switching
• Finance: $H > 0.5$ indicates trending markets; $H < 0.5$ indicates mean-reversion opportunities
• Evolutionary optimization: $H \to 0.5$ indicates convergence (search becoming random)

cvx.detect_changepoints(entity_id, trajectory, penalty, min_segment_len)

Theory: PELT (Pruned Exact Linear Time) algorithm for offline change point detection. Minimizes:

$\sum_{i=1}^{K+1} C(y_{t_{i-1}:t_i}) + K \cdot \beta$

where $C$ is the segment cost and $\beta$ is the penalty per change point.

Output: List of (timestamp, severity) pairs marking regime transitions.

Applications:

• Molecular dynamics: Conformational state transitions
• MLOps: Model drift onset detection
• Social media: Behavioral regime changes in users

---

Path Signatures

Based on rough path theory (Lyons, 1998). See Path Signatures for full mathematical treatment.

cvx.path_signature(trajectory, depth=2, time_augmentation=False)

Theory: The truncated path signature is the universal nonlinear feature of sequential data. Any continuous function of a path can be approximated by a linear function of its signature. It captures ordered, multi-scale structure:

• Depth 1 ($K$ features): Net displacement — where did the entity move
• Depth 2 ($K + K^2$ features): Signed areas — how it moved (rotation, correlation, volatility). Distinguishes “right-then-up” from “up-then-right”
• Depth 3 ($K + K^2 + K^3$ features): Higher-order interactions

Key property — Chen’s Identity: $S(\alpha \ast \beta) = S(\alpha) \otimes S(\beta)$. When a new point is appended, the signature updates in $O(K^2)$ instead of full recomputation in $O(N \cdot K^2)$. This makes path signatures native to incremental-insert databases.

Practical note: Applied on region trajectories ($K \sim 80$ at HNSW level 3), not raw embeddings ($D = 768$), keeping the computation tractable.

cvx.log_signature(trajectory, depth=2, time_augmentation=False)

Theory: The log-signature removes redundant symmetric components via the Baker-Campbell-Hausdorff formula. Contains the same information in fewer dimensions: $K + K(K-1)/2$ instead of $K + K^2$ at depth 2.

cvx.signature_distance(sig_a, sig_b)

Theory: $L_2$ distance between path signatures. A fast trajectory similarity measure: $O(K^2)$ per comparison capturing all order-dependent temporal dynamics.

Applications (all domains):

• Molecular dynamics: Find simulations with similar conformational dynamics
• Drug discovery: Identify optimization campaigns that followed similar paths through chemical space
• MAP-Elites: Find solutions with similar evolutionary trajectories
• Mental health: Detect users with similar temporal patterns to known at-risk cases

---

Trajectory Comparison

cvx.frechet_distance(traj_a, traj_b)

Theory: The discrete Fréchet distance measures the maximum minimum distance between corresponding points when both paths are traversed monotonically. Informally: the shortest leash needed if you walk along path A while your dog walks along path B, both moving only forward.

$d_F(A, B) = \inf_{\alpha, \beta} \max_t \| A(\alpha(t)) - B(\beta(t)) \|$

Complexity: $O(n \times m)$ via dynamic programming.

When to use vs. signature distance:

• Signature distance ($O(K^2)$): universal features, very fast — recommended default
• Fréchet distance ($O(nm)$): exact geometric comparison — when the precise geometric shape of the path matters

---

Distributional Distances

These operate on region distributions — histograms over HNSW graph regions that describe an entity’s topical composition at a given time.

cvx.wasserstein_drift(dist_a, dist_b, centroids, n_projections=50)

Theory: The Wasserstein (optimal transport) distance measures the minimum cost of transforming one distribution into another, respecting the geometry of the space. Unlike $L_2$ between histograms, Wasserstein accounts for which regions are close vs far:

$W_1(p, q) = \inf_{\gamma \in \Pi(p,q)} \int \|x - y\| \, d\gamma(x, y)$

Implemented as Sliced Wasserstein (random 1D projections) for computational efficiency.

Key insight: Shifting mass between neighboring regions costs less than between distant ones. Verified: distant shift (1.2) > adjacent shift (0.7) in tests.

Applications:

• MLOps: Detect concept drift that respects feature space geometry
• Drug discovery: Measure how a campaign’s chemical space focus shifted
• Mental health: Track topical migration between related vs. unrelated topics

cvx.fisher_rao_distance(p, q)

Theory: The Fisher-Rao metric is the unique Riemannian metric on the statistical manifold that is invariant under sufficient statistics (Chentsov’s theorem, 1982). For categorical distributions, it has a closed form via the Bhattacharyya angle:

$d_{FR}(p, q) = 2 \arccos\left(\sum_i \sqrt{p_i \cdot q_i}\right)$

Properties:

• Symmetric, metric (satisfies triangle inequality)
• Bounded: $[0, \pi]$ ($0$ = identical, $\pi$ = disjoint support)
• Invariant under reparametrization of the probability space

When to use vs. Wasserstein: Fisher-Rao is the mathematically natural distance between distributions. Wasserstein additionally incorporates the geometry of the support space (region centroids). Use Fisher-Rao for pure distributional comparison; Wasserstein when the spatial structure of regions matters.

cvx.hellinger_distance(p, q)

Theory: Related to Fisher-Rao, bounded in $[0, 1]$: $H(p, q) = \sqrt{\frac{1 - BC(p,q)}{2}}$ where $BC$ is the Bhattacharyya coefficient.

---

Point Process Analysis

cvx.event_features(timestamps)

Theory: Extracts features from the timing of events, independent of vector content. The inter-event intervals themselves encode behavioral patterns modeled by temporal point process theory.

Output: Dictionary with 11 features:

Feature	Formula	Interpretation
burstiness	$B = \frac{\sigma - \mu}{\sigma + \mu}$	$-1$ = perfectly regular, $0$ = Poisson (random), $+1$ = maximally bursty
memory	Autocorrelation at lag 1	$> 0$: short gaps follow short gaps (clustering). $< 0$: alternating
temporal_entropy	$H = -\sum p_i \log p_i$	Higher = more unpredictable timing
intensity_trend	Slope of event rate	Positive = accelerating, negative = decelerating
gap_cv	$\sigma_{gap} / \mu_{gap}$	Coefficient of variation of intervals
circadian_strength	24h Fourier amplitude	$0$ = no daily rhythm, $1$ = strong rhythm
max_gap	Longest silence	Withdrawal period duration

Applications:

• Mental health: Night posting ratio ($d = 0.534$, $p < 0.001$ on eRisk) and burstiness are behavioral biomarkers of depression. Circadian disruption is a known clinical marker.
• Cybersecurity: Bursty network activity patterns indicate automated attacks or data exfiltration
• Evolutionary computation: Accelerating event rate indicates convergence; stagnation shows flattening
• Finance: Trading bursts precede volatility spikes; memory coefficient detects algorithmic trading patterns

References: Goh & Barabási (2008) introduced burstiness/memory. Hawkes (1971) formalized self-exciting processes.

---

Topological Analysis

cvx.topological_features(points, n_radii=20, persistence_threshold=0.1)

Theory: Persistent homology from Topological Data Analysis (TDA) tracks how the topology of a point cloud changes as a filtration radius grows. For dimension 0 (connected components):

• Build a Vietoris-Rips complex: connect points within radius $r$
• Track when components appear (birth) and merge (death)
• Betti number $\beta_0(r)$ = number of connected components at radius $r$

Implemented via Union-Find on the pairwise distance graph (single-linkage equivalent). Applied on region centroids ($K \sim 80$) for tractability.

Output: Dictionary with:

• n_components: significant clusters (persistence $>$ threshold)
• max_persistence: most prominent topological gap
• persistence_entropy: $-\sum \frac{p_i}{P} \log \frac{p_i}{P}$ — uniformity of feature lifetimes
• betti_curve: $\beta_0(r)$ sampled at n_radii points

What topology reveals that geometry doesn’t:

• Increasing $\beta_0$ over time → fragmentation (topic space splitting)
• Decreasing $\beta_0$ → convergence (topics merging)
• High persistence entropy → uniform cluster structure (no dominant gap)
• Low persistence entropy → clear cluster separation (one dominant gap)

Applications:

• Drug discovery: Track how the active chemical space fragments or converges during optimization campaigns
• Molecular dynamics: Detect when the conformational landscape develops new basins (new $\beta_0$)
• MAP-Elites: Monitor archive topology — is the solution space connected or fragmented?
• MLOps: Detect structural changes in embedding space topology (not just distributional shifts)

References: Edelsbrunner & Harer (2010). Zigzag persistence for temporal networks (EPJ Data Science, 2023).

---

Anchor Projection

See RFC-006: Anchor Projection for design rationale and clinical validation.

cvx.project_to_anchors(trajectory, anchors, metric='cosine')

Theory: Projects a trajectory from absolute embedding space $\mathbb{R}^D$ into an anchor-relative coordinate system $\mathbb{R}^K$, where $K$ is the number of anchors. Each output dimension $k$ is the distance (cosine or L2) from the trajectory point to anchor $k$:

$\text{projected}_t[k] = d(\mathbf{x}_t, \mathbf{a}_k), \quad k = 1, \ldots, K$

This is a coordinate system change, not a new analytics paradigm. The output is itself a trajectory, so all existing CVX functions (velocity, hurst_exponent, detect_changepoints, path_signature, etc.) compose with it natively.

Output: $(T, K)$ array — a trajectory in $\mathbb{R}^K$ where each dimension represents distance to the corresponding anchor.

Applications:

• Mental health: Measure drift toward/away from clinical poles (depression, anxiety, neutral). Anchor-relative features improved F1 from 0.600 to 0.781 on eRisk
• Finance: Track portfolio proximity to sector archetypes (tech-heavy, defensive, balanced)
• Drug discovery: Monitor compound evolution relative to known active/toxic/selective reference molecules
• MLOps: Measure model embedding drift relative to known-good and known-bad reference distributions

cvx.anchor_summary(projected)

Theory: Aggregates per-anchor distance dynamics into a fixed-size summary. For each anchor $k$, computes statistics over the projected trajectory’s $k$-th dimension.

Output: Dictionary per anchor with:

Statistic	Formula	Interpretation
mean	$\bar{d}_k = \frac{1}{T}\sum_t d_k(t)$	Average proximity to anchor $k$
min	$\min_t d_k(t)$	Closest approach to anchor $k$
trend	Linear slope of $d_k(t)$	Positive = drifting away, negative = approaching
last	$d_k(T)$	Current distance to anchor $k$

Applications:

• Mental health: Trend toward depression anchor with decreasing min signals progressive deterioration
• Finance: last vs mean reveals whether current positioning is typical or extreme
• MLOps: Rising trend across all anchors indicates the model is entering an out-of-distribution region

---

Prediction

cvx.predict(trajectory, target_timestamp)

Theory: Linear extrapolation from trajectory history. For Neural ODE prediction, use TemporalIndex(model_path="model.pt").predict().

---

Cross-Domain Application Matrix

Function	Mental Health	Finance	Drug Discovery	MAP-Elites	Molecular Dynamics	MLOps
velocity	Crisis speed	Regime transition	Optimization speed	Exploration rate	Transition speed	Drift rate
hurst_exponent	Anti-persistence (d=-0.41)	Trending vs reverting	Convergence	Stagnation	Basin stability	Drift persistence
detect_changepoints	Behavioral regime change	Market regime	Campaign phase	Archive reorganization	Conformational transition	Drift onset
path_signature	Trajectory pattern matching	Path-dependent options	Campaign similarity	Solution lineage	Folding pathway	Training dynamics
frechet_distance	Similar user trajectories	Portfolio path comparison	Campaign comparison	Archive comparison	Trajectory similarity	Model comparison
wasserstein_drift	Topical migration	Sector rotation	Chemical space shift	Niche redistribution	State redistribution	Feature drift
fisher_rao_distance	Distribution shift	Risk profile change	Activity profile change	Diversity metric	State population change	Class balance drift
event_features	Night posting, burstiness	Trading patterns	Experiment cadence	Generation dynamics	Simulation events	Request patterns
topological_features	Topic fragmentation	Market structure	Landscape topology	Archive connectivity	Conformational basins	Embedding structure
project_to_anchors	Drift toward clinical poles	Sector proximity	Reference compound distance	Archetype tracking	State proximity	Distribution drift
anchor_summary	Deterioration trend	Position vs. norm	Campaign summary	Exploration summary	Basin residence	Drift summary