Creative Fatigue Detection in Digital Advertising

CreativeDynamics Library v0.9.8.1

Definition and mathematical framework

Creative fatigue: advertising effectiveness degradation over time due to repeated audience exposure. The library employs path signature analysis from rough path theory to detect creative performance degradations in digital advertising campaigns.

Mathematical formulation:

For performance metric time series Y = {y_1, …, y_T}, the library detects fatigue by:

1. Computing path signatures S_t for sliding windows of size w
2. Calculating signature distances $d_t = \lVert S_t - S_{t-1} \rVert_2$
3. Identifying change points where d_t > μ_d + k·σ_d
4. Classifying post-change-point trends to identify declining performance

Captures non-linear pattern changes preceding visible performance drops, enabling proactive campaign optimisation.

For consistency with the rest of this documentation, we can also write the thresholding rule as

$$d_t > \mu_d + k\,\sigma_d.$$

Mathematical formalism

This section follows the notation in the accompanying paper (see arXiv-2509.09758v3/main.tex).

From a time series to a path

Let $\{y_t\}_{t=1}^{T}$ be a performance metric time series observed at regular time steps. We embed the time series as a two-dimensional path $X$ in $\mathbb{R}^2$ by

$$X_t := (t, y_t).$$

In practice we work with a normalised, piecewise-linear interpolation over a window $[a,b]$: time is scaled to $[0,1]$ and metric values are min–max scaled to $[0,1]$. This ensures signatures are comparable across windows and metrics.

Signatures and log-signatures

For a path $X$ of bounded variation on $[a,b]$, its (tensor) signature is the sequence of iterated integrals

$$S(X)_{a,b} := \left(1,\; S^1(X)_{a,b},\; S^2(X)_{a,b},\; \dots\right),$$

where, for $k \ge 1$,

$$S^k(X)_{a,b} = \int_{a < t_1 < \cdots < t_k < b} dX_{t_1} \otimes \cdots \otimes dX_{t_k}.$$

We compute a truncated representation up to depth $d$. The implementation uses log-signatures (Lie increments) for numerical stability and efficiency, but conceptually the detector compares adjacent window signatures.

Sliding windows and a signature-distance statistic

Fix a window size $w$. Let $S_t$ denote the (truncated) signature for the window ending at time index $t$. We define the signature-distance statistic

$$d_t := \lVert S_t - S_{t-1} \rVert_2.$$

Large values of $d_t$ indicate a change in the geometry of the recent trajectory (trend, volatility, oscillation), often before the raw metric exhibits a clear mean shift.

Thresholding and change points

Let $\mu_d$ and $\sigma_d$ be the mean and standard deviation of $\{d_t\}$ for a given item/metric. A change point is flagged when

$$d_t > \mu_d + k\,\sigma_d,$$

where $k$ is a sensitivity multiplier.

Dual-metric analysis framework

Dual-metric approach analyses both Click-Through Rate (CTR) and Cost Per Click (CPC) for detailed fatigue detection.

Metric complementarity

CTR (Click-Through Rate):

• Measures audience engagement (clicks/impressions)
• Reflects creative relevance and appeal
• Early indicator of audience saturation

CPC (Cost Per Click):

• Measures cost efficiency (spend/clicks)
• Reflects competitive dynamics and quality score
• Indicates platform algorithm adjustments

Rationale for dual analysis

• Different response patterns: CTR and CPC exhibit distinct temporal dynamics under fatigue
• Detailed detection: Pattern changes may appear in one metric before the other
• Correlation context: Correlation analysis highlights when CTR and CPC move inversely; metrics are not combined
• Strategic insights: Different metrics align with different campaign objectives (engagement vs. efficiency)

Metric interpretation

CTR: Measures audience engagement over time. Declining CTR suggests diminishing interest or relevance.

CPC: Indicates cost efficiency per engagement. Rising CPC signals decreasing efficiency from declining relevance or increasing competition.

Detection algorithm

Four-phase detection process for each metric:

Phase 1: Change point detection

• Sliding window size w (default=7) captures weekly patterns
• Signature depth d=4 balances detail vs. computational cost
• Threshold multiplier k=1.5 provides precision≈0.7, recall≈0.6

Phase 2: Trend classification

• Stable: |slope| < threshold
• Improving: slope > threshold (CTR↑ or CPC↓)
• Declining: slope < -threshold (CTR↓ or CPC↑)

Phase 3: Benchmark establishment

• Identifies longest stable/improving segment
• Computes average performance as benchmark
• Validates benchmark reliability (minimum 3 data points)

Phase 4: Impact quantification

• Measures deviation from benchmark during declining periods
• Quantifies operational impact (engagement_gap_clicks) and financial inefficiency (actual_overspend_gbp)
• Provides correlation risk context; metrics are reported separately and not combined

Performance characteristics

Empirical validation across multiple advertising datasets:

• Early detection: Identifies fatigue 3-5 days before traditional methods
• False positive rate: ~30% (controlled by threshold parameter k)
• Computational efficiency: O(T·d²) complexity enables real-time analysis
• Robustness: Handles missing data and outliers through normalisation

Impact metrics

Translates detected fatigue into separate operational and financial impact metrics:

Financial (actual_overspend_gbp):

$$\mathrm{actual\_overspend\_gbp} = \sum_{t \in T_{\mathrm{decline}}} \max\bigl(0, \mathrm{CPC}_t - \mathrm{CPC}_{\mathrm{bench}}\bigr)\,\mathrm{Clicks}_t.$$

Actual overspend due to increased cost-per-click during fatigue periods.

Operational (engagement_gap_clicks):

$$\mathrm{engagement\_gap\_clicks} = \sum_{t \in T_{\mathrm{decline}}} \max\bigl(0, \mathrm{CTR}_{\mathrm{bench}} - \mathrm{CTR}_t\bigr)\,\mathrm{Impressions}_t.$$

Lost clicks due to decreased engagement rates. A GBP reference value may be shown separately as:

$$\mathrm{engagement\_gap\_gbp\_reference} = \mathrm{engagement\_gap\_clicks}\,\mathrm{CPC}_{\mathrm{bench}}.$$

Correlation risk context (metrics not combined)

Correlation analysis provides context when interpreting CTR and CPC together:

Correlation coefficient:

$$\rho(\mathrm{CPC}, \mathrm{CTR}) = \frac{\mathrm{Cov}(\mathrm{CPC}, \mathrm{CTR})}{\sigma_{\mathrm{CPC}}\,\sigma_{\mathrm{CTR}}}.$$

Risk classification:

• Low Risk (ρ > -0.2): Independent or weak correlation
• Medium Risk (-0.5 < ρ ≤ -0.2): Moderate negative correlation
• High Risk (ρ ≤ -0.5): Strong negative correlation

Operational and financial metrics are reported separately and are not added regardless of risk level.

Detailed methodology and visualisation interpretation: Example application: quantifying impact.