Exponential Moving Average (EMA) Filter

An Exponential Moving Average (EMA) Filter is an infinite impulse response filter that applies exponentially decreasing weights to past observations, giving more importance to recent data points while never fully discarding older observations.

• Context:
  • It can (typically) track momentum by giving higher weight to recent changes in price or data compared to older data.
  • It can (often) be used in Optimization Algorithms, such as in the Adam Optimizer for deep learning, where it helps track the exponentially weighted average of past gradients during training.
  • ...
  • It can be calculated using the formula: [math]\displaystyle{ EMA_t = \alpha x_t + (1-\alpha) EMA_{t-1} }[/math], where [math]\displaystyle{ \alpha }[/math] is the smoothing factor and [math]\displaystyle{ 0 \lt \alpha \lt 1 }[/math].
  • It can be adjusted by changing the smoothing factor [math]\displaystyle{ \alpha }[/math], which controls how quickly the EMA responds to changes in the data (e.g., a higher [math]\displaystyle{ \alpha }[/math] gives more weight to recent data).
  • It can be integrated into various technical analysis strategies, including Relative Strength Index (RSI), Moving Average Convergence Divergence (MACD), and Bollinger Bands.
  • It can be sensitive to the choice of the initial value, often starting with the first data point or the simple average of the first few observations to seed the EMA.
  • It can be implemented in an iterative process, where each new observation updates the EMA by blending it with the previous EMA value.
  • It can smooth data in signal processing applications, acting as a low-pass filter to remove high-frequency noise and retain slower-moving trends.
  • ...
• Example(s):
  • As used in stock price analysis to identify market trends by smoothing daily price fluctuations, particularly with tools like moving average crossovers.
  • As applied in exponentially weighted moving average (EWMA) Control Charts used for monitoring industrial processes or quality control over time.
  • A 10-day EMA is applied to stock prices to smooth daily price fluctuations and identify underlying trends.
  • An Exponentially Weighted Moving Average control chart used in an industrial quality control process to detect small shifts in process performance.
  • In machine learning, the EMA is used in the Adam Optimizer to compute exponentially weighted averages of the gradients and squared gradients for efficient parameter updates.
  • ...
• Counter-Example(s):
  • A Simple Moving Average (SMA), which assigns equal weight to all observations within a fixed window, unlike the EMA that weights recent data more heavily.
  • A Cumulative Moving Average (CMA), where each new data point is averaged over all previous data points, leading to slower adaptation to new data trends.
• See: Infinite Impulse Response, Exponential Decay, Time-Series Forecasting, Signal Processing, Simple Moving Average (SMA), Optimization Algorithm, Adam Optimizer, Moving Average Convergence Divergence (MACD), Bollinger Bands

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Moving_average Retrieved:2024-9-12.
  • An exponential moving average (EMA), also known as an exponentially weighted moving average (EWMA), is a first-order infinite impulse response filter that applies weighting factors which decrease exponentially. The weighting for each older datum decreases exponentially, never reaching zero. This formulation is according to Hunter (1986).

2024

• (Pagliardini et al., 2024) ⇒ Matteo Pagliardini, Pierre Ablin, and David Grangier. (2024). “The AdEMAMix Optimizer: Better, Faster, Older.” In: arXiv preprint arXiv:2409.03137.
  • NOTE: The paper introduces AdEMAMix, a novel optimization method that combines two Exponential Moving Averages (EMAs) to better leverage both recent and older gradient information.

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