It depends a lot on what you mean by signal and capacity. Those words mean different things in wireless communications, quantitative finance, machine learning, neuroscience, and cognitive psychology. I'll mention a few methodologies that I find broadly useful across domains, then tailor them if you're thinking of a specific field.
1. Start by defining "signal half-life"
Rather than asking "does the signal decay?", I usually estimate:
- Predictive strength vs. time (AUC, R², correlation, mutual information, Sharpe, etc.)
- Half-life: the lag where performance falls to 50% of its initial value
- Asymptotic value: does the signal go to zero or plateau?
This makes it much easier to compare signals across datasets.
A common model is
[
S(t)=S_0e^{-\lambda t}
]
or, if decay slows over time,
[
S(t)=\alpha t^{-\beta}
]
Power-law decay often fits better than exponential in biological and cognitive systems.
2. Compare competing decay models
Instead of assuming exponential decay, fit several models:
| Model | Good for |
|---|
| Exponential | Constant hazard of information loss |
| Power law | Long tails; many natural systems |
| Weibull | Flexible hazard rates |
| Piecewise | Regime changes |
| Bayesian state-space | Hidden latent signal strength |
Then compare using AIC/BIC or cross-validation rather than visual inspection.
3. Capacity should usually be measured jointly
Capacity isn't just "how much signal remains."
I like separating it into:
- Storage capacity (how much information exists)
- Retrieval capacity (how much is usable)
- Effective capacity (how much improves decisions)
Information-theoretic quantities are often much more stable than raw accuracy:
- Mutual information
- Entropy
- KL divergence
- Fisher information
The Information Bottleneck framework is especially useful if you're trying to quantify how much relevant information survives compression or time.
4. Measure decay as a survival process
Another perspective that transfers well across fields:
Treat each useful signal as "alive" until it no longer exceeds some performance threshold.
Then estimate
- Kaplan-Meier survival curves
- Hazard functions
- Cox proportional hazards models
This is surprisingly useful for:
- alpha factors in finance
- predictive features
- anomaly detectors
- recommendation features
Papers I'd recommend
If you're interested in memory / information retention
Sikström (2002), Forgetting Curves: Implications for Connectionist Models remains one of the strongest papers on why power-law forgetting naturally emerges from heterogeneous exponential processes.
A more recent perspective by Kornell and Bjork argues that observed forgetting curves reflect both memory strength distributions and recall thresholds, cautioning against interpreting empirical curves as direct measures of individual memory decay.
If you're thinking about communications
Biglieri, Proakis & Shamai's review on fading channels is still an excellent reference. It connects statistical channel models directly to information-theoretic capacity and coding strategies.
If you're thinking about machine learning
For continual learning, the survey by De Lange et al. is a comprehensive overview of catastrophic forgetting, stability-plasticity tradeoffs, and evaluation methodologies.
There's also interesting recent work exploring whether neural networks exhibit human-like forgetting curves and how those curves can guide rehearsal strategies, though it's newer and should be viewed as emerging research rather than established consensus.
A methodology I've found especially robust
When evaluating any signal over time:
- Measure the initial signal quality.
- Re-evaluate it at multiple future horizons.
- Fit several decay models (not just exponential).
- Estimate half-life with confidence intervals (bootstrap if possible).
- Quantify residual capacity using mutual information or predictive performance.
- Test whether interventions (refreshing, retraining, repetition, adaptation) change the decay rate rather than just the starting performance.
That workflow tends to be more informative than reporting a single decay constant.
If you meant signal decay and capacity in a specific context—such as wireless systems, quantitative trading signals, feature drift in machine learning, or cognitive memory—I can point you to the most influential papers and domain-specific methodologies.
