Mental Health Detection

Overview

Social media posts contain temporal linguistic signals that static models discard. By embedding each post with MentalRoBERTa (D=768, domain-adapted to mental health language) and storing them as timestamped trajectories in ChronosVector, we recover behavioral and linguistic patterns invisible to snapshot-based analysis:

• Velocity of linguistic change accelerates before crisis episodes
• Posting rhythms encode circadian disruption (a clinical depression marker)
• Topical migration reveals progressive withdrawal from social topics
• Path signatures fingerprint the shape of a user’s linguistic evolution over time

This page walks through each CVX analytical function applied to eRisk (Reddit, depression detection) and CLPsych (Twitter, mental health triage).

---

Data Model

Each social media user maps to a CVX entity. Each post becomes a timestamped vector. The timestamp is the post’s publication time (Unix seconds).

import chronos_vector as cvx
import numpy as np

# Create index with HNSW parameters tuned for 768-dim embeddings
index = cvx.TemporalIndex(m=16, ef_construction=200, ef_search=50)
index.enable_quantization(-1.0, 1.0)  # MentalRoBERTa outputs are normalized

# entity_ids: np.ndarray[uint64] — user IDs
# timestamps: np.ndarray[int64]  — post times (Unix seconds)
# vectors:    np.ndarray[float32, (N, 768)] — MentalRoBERTa embeddings
n = index.bulk_insert(entity_ids, timestamps, vectors, ef_construction=50)
print(f"Ingested {n} posts from {len(np.unique(entity_ids))} users")

Lowering ef_construction to 50 during bulk load speeds up ingestion substantially. The HNSW graph self-organizes: higher levels form coarse semantic regions used later for topical analysis.

---

Temporal Calculus --- Velocity and Change Points

Velocity: rate of linguistic change

velocity() computes the instantaneous rate of change in embedding space via central finite differences. A sudden velocity spike means the user’s language is shifting rapidly, which can signal the onset of a depressive episode or crisis.

traj = index.trajectory(entity_id=user_id)

# Velocity at a specific timestamp
mid_ts = traj[len(traj) // 2][0]
vel = cvx.velocity(traj, timestamp=mid_ts)
magnitude = np.linalg.norm(vel)
print(f"Velocity magnitude at t={mid_ts}: {magnitude:.4f}")

Clinically, sustained high velocity corresponds to rapid topic switching or emotional volatility. A velocity that spikes and does not return to baseline may indicate a regime change.

Change point detection: behavioral regime shifts

detect_changepoints() uses the PELT algorithm to find timestamps where the user’s linguistic behavior shifts significantly. Each change point includes a severity score proportional to the statistical cost reduction.

changepoints = cvx.detect_changepoints(
    entity_id=user_id,
    trajectory=traj,
    penalty=None,         # Auto-calibrate via BIC
    min_segment_len=5     # At least 5 posts per segment
)

for ts, severity in changepoints:
    print(f"  Regime shift at t={ts}, severity={severity:.2f}")

On eRisk data, depression users show significantly more change points than controls, reflecting the episodic nature of depressive states. The severity scores help distinguish minor topic shifts from major behavioral transitions.

---

Stochastic Characterization --- Hurst Exponent

The Hurst exponent H characterizes the long-range dependence of a trajectory via R/S (rescaled range) analysis:

H value	Behavior	Clinical interpretation
H < 0.5	Anti-persistent (mean-reverting)	Erratic topic switching, inability to sustain focus
H = 0.5	Random walk	No temporal memory
H > 0.5	Persistent (trending)	Sustained topical engagement

h = cvx.hurst_exponent(traj)
print(f"Hurst exponent: {h:.3f}")

On CLPsych, depression users show significantly lower Hurst exponents than controls (Cohen’s d = -0.41, p < 0.001). This anti-persistence reflects a clinical pattern: depressed individuals cycle erratically between topics rather than maintaining coherent thematic threads. The effect is consistent across both datasets, making H a robust temporal biomarker.

---

Point Process Analysis --- Posting Patterns

event_features() analyzes the timing of posts as an independent behavioral signal, separate from content. It extracts 11 features from the inter-post interval distribution.

timestamps_user = [ts for ts, _ in traj]
features = cvx.event_features(timestamps_user)

Key features and their clinical relevance:

Feature	Depression signal	Effect size (eRisk)
burstiness	Higher — posting in intense bursts followed by withdrawal	d = 0.28
memory	Lower — less temporal autocorrelation in gaps	d = -0.19
circadian_strength	Lower — disrupted daily rhythm	d = -0.31
gap_cv	Higher — irregular posting intervals	d = 0.32
max_gap	Higher — longer withdrawal periods	d = 0.25

The strongest individual behavioral signal is night posting ratio (posts between 00:00-06:00): depression users post 31% at night vs. 22% for controls (d = 0.534, p < 0.001 on eRisk). Circadian disruption is a well-established clinical marker of major depressive disorder.

---

Semantic Region Migration

CVX discovers semantic regions from the HNSW graph hierarchy. At Level 3, the graph produces 60-97 coarse regions that act as topical clusters. Tracking how a user’s posts distribute across these regions over time reveals topical migration patterns.

Region trajectory

# Discover coarse semantic regions
regions = index.regions(level=3)
print(f"Found {len(regions)} regions at Level 3")

# Track user's region distribution over time (EMA-smoothed)
region_traj = index.region_trajectory(
    entity_id=user_id,
    level=3,
    window_days=14,   # 14-day sliding window
    alpha=0.3         # EMA smoothing factor
)
# region_traj: list of (timestamp, distribution) where distribution sums to ~1.0

Wasserstein drift: geometry-aware topical shift

wasserstein_drift() measures how a user’s topical distribution changes between two time points, respecting the spatial structure of regions. Unlike L2 distance between histograms, Wasserstein accounts for which regions are semantically close vs. distant.

centroids = [centroid for _, centroid, _ in regions]

# Compare early vs. late distributions
early_dist = region_traj[0][1]
late_dist = region_traj[-1][1]

w_drift = cvx.wasserstein_drift(early_dist, late_dist, centroids, n_projections=50)
print(f"Wasserstein drift: {w_drift:.4f}")

Fisher-Rao distance: invariant distributional metric

For pure distributional comparison (independent of region geometry), fisher_rao_distance() provides the unique Riemannian metric on the statistical manifold, invariant under sufficient statistics.

fr_dist = cvx.fisher_rao_distance(
    [float(x) for x in early_dist],
    [float(x) for x in late_dist]
)
print(f"Fisher-Rao distance: {fr_dist:.4f}")  # Range [0, pi]

Depression users show higher Wasserstein drift over time, indicating progressive migration away from social and neutral topics toward isolation-related semantic regions.

---

Topological Analysis

topological_features() applies persistent homology to region centroids, revealing the structure of a user’s topic space rather than just its content.

# Extract centroids for regions the user actually occupies
user_region_ids = [
    i for i, weight in enumerate(late_dist) if weight > 0.01
]
user_centroids = [centroids[i] for i in user_region_ids]

topo = cvx.topological_features(user_centroids, n_radii=20, persistence_threshold=0.1)
print(f"Significant clusters: {topo['n_components']}")
print(f"Max persistence:      {topo['max_persistence']:.3f}")
print(f"Persistence entropy:  {topo['persistence_entropy']:.3f}")

Depression users’ topic spaces tend to show:

• Higher fragmentation (more connected components) — topics are scattered without coherent clustering
• Lower max persistence — no dominant topical cluster, suggesting lack of sustained interest
• Different Betti curves — the rate at which components merge as radius grows differs, reflecting a fundamentally different topological structure of their semantic engagement

---

Path Signatures --- Trajectory Fingerprinting

Path signatures from rough path theory provide a universal nonlinear feature of sequential data. Applied to region trajectories, they capture the ordered, multi-scale shape of how a user’s topical engagement evolves.

# Compute Level-3 path signature on region trajectory
# Applied on region distributions (~80 dims), NOT raw embeddings (768 dims)
sig = cvx.path_signature(region_traj, depth=3, time_augmentation=True)
print(f"Signature dimension: {len(sig)}")

At depth 3, the signature captures:

• Depth 1: Net displacement — where did the user’s topic focus move
• Depth 2: Signed areas — how it moved (rotation, correlation between topic shifts)
• Depth 3: Higher-order interactions — complex temporal patterns

Cohort detection via signature distance

signature_distance() finds users whose linguistic evolution follows similar temporal patterns. This enables at-risk cohort detection: find users whose trajectory shape matches known depression cases.

# Compare two users' trajectory signatures
sig_a = cvx.path_signature(region_traj_a, depth=3, time_augmentation=True)
sig_b = cvx.path_signature(region_traj_b, depth=3, time_augmentation=True)

dist = cvx.signature_distance(sig_a, sig_b)
print(f"Signature distance: {dist:.4f}")
# Lower distance = more similar temporal evolution pattern

Signature distance operates in O(K^2) per comparison, making it practical for screening large user populations against a library of known at-risk trajectory templates.

---

Results Summary

Dataset	Model	AUC	F1	Early Detection
eRisk (Reddit, N=2,285)	Full (Temporal + Region + Behavioral)	0.911	0.458	AUC@10% = 0.849
CLPsych (Twitter, N=674)	Temporal + Behavioral	0.804	0.571	AUC@20% = 0.813

Key takeaways:

• MentalRoBERTa embeddings are the single most impactful choice (static baseline AUC = 0.901 on eRisk)
• Temporal features add signal beyond static aggregation, especially for early detection
• Behavioral posting patterns (night posting, burstiness) are lightweight and independently discriminative
• Region-based features compress 768-dim embeddings to ~80 dims with minimal AUC loss (0.890 vs 0.911)
• Early detection is viable: AUC > 0.84 with only 10% of a user’s post history on eRisk

---

References

1. Couto, M. et al. (2025). Temporal word embeddings for psychological disorder early detection. Journal of Healthcare Informatics Research.
2. Goh, K.-I. & Barabasi, A.-L. (2008). Burstiness and memory in complex systems. EPL (Europhysics Letters), 81(4).
3. Lyons, T. (1998). Differential equations driven by rough signals. Revista Matematica Iberoamericana, 14(2).
4. Rao, C.R. (1945). Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 37.