Exponential Moving Average

📐 Definition

For sequence $\{x_n\}$, smoothing factor $\alpha \in (0,1]$, and initial $m_0$, define

$$m_n = (1-\alpha) m_{n-1} + \alpha x_n$$

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Domain and Codomain

Applies to real or complex sequences with bounded values. Output follows the same codomain.

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⚙️ Key Properties

Weights decay geometrically: contribution of $x_{n-k}$ is $(1-\alpha)^k \alpha$. Effective window length is $O(1/\alpha)$. Linear and causal.

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As $\alpha \to 1$, $m_n$ tracks the latest sample; as $\alpha \to 0$, smoothing approaches a constant memory with slow adaptation. For constant input $x_n = c$, $m_n \to c$.

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🎯 Special Cases and Limits

• $\alpha=1$ yields $m_n=x_n$ (no smoothing).
• Small $\alpha$ yields slow adaptation with long memory.

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🔗 Related Functions

Welford statistics provide unbiased variance updates; simple mean uses uniform weighting rather than exponential decay.

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Usage in Oakfield

Oakfield uses EMAs for smoothing diagnostics and lock detection:

• Phase coherence EMA: runtime stats track phase_coherence_ema using an exponential smoothing constant (seconds) and a hysteresis lock state.
• General smoothing helper: runtime/sim_smoothing.h provides a canonical a = exp(-dt/tau) form used for stable time-constant smoothing in other subsystems.

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Historical Foundations

📜 Recursive Smoothing

Exponential smoothing is a standard recursive filter in time series analysis, trading unbiasedness for adaptivity and a simple one-step update.

🌍 Modern Perspective

EMA is widely used for online diagnostics and stabilizing noisy signals in simulation and optimization loops.

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📚 References

• Hamilton, Time Series Analysis
• Oppenheim & Schafer, Discrete-Time Signal Processing