2013issue C0322-26
Why exponential smoothing uses two over n plus one
The familiar smoothing constant 2/(n+1) is a lag-matching construction. An n-period exponential average is defined so that its average age equals that of an n-period simple average.
- An exponential average of an unlimited ordered series uses weights alpha times (1-alpha) to the power of each observation age. Those weights lie between 0 and 1 and sum to 1.
- The average age of that exponential scheme is (1-alpha)/alpha. An n-period simple average assigns weight 1/n to each of the n most recent ages, so its average age is (n-1)/2.
- Equating those average ages is the convention that produces the smoothing constant 2/(n+1).
- Charts usually plot an average under the latest observation, but both the simple average and the exponential average lag that observation by their average ages.
Exponential weights and their average age
An exponential average of an unlimited ordered series uses weights alpha times (1-alpha) to the power of each observation age. Those weights lie between 0 and 1, and they sum to 1.
The average age of that exponential weighting scheme equals (1-alpha)/alpha. That average age is the fulcrum that balances the weighted timeline from age 0 to infinity.
Equal weights on a finite window
An n-period simple average assigns weight 1/n to each of the n most recent ages and zero to older ages, so its average age equals (n-1)/2.
The n-period exponential convention
Because every exponential weight is nonzero, an n-period exponential average is only a convention. The natural way to define it is to give it the same time lag as an n-period simple average.
Equating the two average ages, (1-alpha)/alpha = (n-1)/2, is the construction that produces the familiar smoothing constant 2/(n+1).
Where the average belongs on a chart
Charts usually plot an averaged value under the latest observation for convenience, even though the natural location of that average is shifted left by its average age.
Both the simple average and the exponential average of price data lag the most recent observation by their respective average ages.
All readings on this track · 41 readings
- 1984Matching average age across price filters
- 1984Residual-impact construction of exponential smoothers
- 1988Constructing simple and exponential moving averages as specified filters
- 1989Evaluating smoothed free reserves and funds-discount spreads
- 1990Integer construction of moving averages and exponential smoothing
- 1990Constructing incremental moving average filters
- 1991Constructing a double-smoothed stochastic oscillator
- 1991Constructing an exponential average from a smoothing weight
- 1991Constructing a double-smoothed momentum oscillator family
- 1991Competing certainty rules from exponential-average conditions
- 1991Hybrid construction of a mechanical timing system from smoothed signals
- 1991Constructing an exponential smoother from a moving-average length
- 1991Equivalent relative-strength-index constructions from a ratio and from smoothed momentum
- 1991Constructing the true strength index from double-smoothed momentum
- 1991True strength index from signed, double-smoothed momentum
- 1992Constructing exponential averages from simple lookbacks
- 1992Constructing exponential averages by the smoothing constant, startup length and lag
- 1992Nested range smoothing and windowed high-low damping
- 1992A weekly equity stance from the discount-bill spread
- 1993Double-smoothed stochastics momentum index construction
- 1993Constructing the true strength index with double exponential smoothing
- 1994Polarized fractal efficiency with a shared exponential smoother
- 1994Five-interval exponential smoothing of ten-interval signed path-efficiency
- 1994Hand-ranked price-volume series with stacked smoothing and stochastic
- 1995Range expansion filter on multi-horizon averages
- 1995The exponential smoothing constant from averaging lag
- 1997Construct a nested-average rainbow band and oscillator
- 1998Filter stacks that combine a moving average with exponential smoothing
- 2002Smoothed relative-strength construction from wilder-averaging and a price prefilter
- 2003Reverse-engineered RSI price targets
- 2004Piecewise trend quality from moving-average reversals
- 2004Nested smoothing recipes versus unaudited oscillator experiments
- 2008Weight construction beyond the N-day label
- 2010A consensus filter for objective RSI and smoothing construction
- 2012Sentiment-zone oscillator from signed closes
- 2012Classic RSI one-over-N smoothing versus the standard exponential average
- 2013Why exponential smoothing uses two over n plus one
- 2015Constructing ratio indicators with moving-average and exponential-smoothing weights
- 2015Slow relative-strength index from an exponentially smoothed price
- 2017Rebuilding the relative strength index as open-scale buy-sell pressure
- 2017Comparing Wilder and exponential smoothing inside a relative-strength index