2005issue C101-2
Constructing a fractal-dimension adaptive moving average
A fixed-length moving average locks one smoothness-versus-lag compromise on a nonstationary price series. This construction estimates how space-filling a recent high-low path is, then maps that fractal-dimension to a bounded exponential-smoothing weight so the filter can be audited for when it is designed to hug price and when it is designed to stall.
- A fixed-length moving average forces a single smoothness-versus-lag compromise because price series are nonstationary and can show different effective bandwidths on different intervals.
- A box-count-proxy turns each equally spaced high-low range into a covering count, and a two-scale comparison of the half windows against the full even window yields a fractal-dimension treated as ranging from 1 to 2.
- The alpha-map sets the exponential-smoothing weight to e to the power of -4.6 times (D minus 1), then keeps alpha between 0.01 and 1, so a line-like path is paired with a fast blend and a filled path with a slow blend.
- The filter is a one-pole blend of the current midpoint price with the previous filter value, the lookback must be even, and the dimension series can be plotted separately as a signal.
The fixed-length compromise
A fixed-length moving average forces a single smoothness-versus-lag compromise because price series are nonstationary and can show different effective bandwidths on different intervals.
An adaptive-moving-average is a smoother whose responsiveness changes when a measured market-state statistic changes, rather than staying locked to one lookback. This construction uses a fractal-indicator as that statistic: a chart-scale reading of whether recent price structure is more line-like or more space-filling.
A two-scale covering comparison
Comparing how many covering objects of two sizes fit a pattern yields a fractal-dimension estimate. A straight segment evaluates to 1 and a filled square evaluates to 2.
In this construction, fractal-dimension is that two-scale covering comparison, treated as ranging from 1 for a line-like path to 2 for a filled, congested path.
A box-count-proxy on equally spaced samples
When samples are equally spaced, the covering count for an interval can be estimated as that interval's high-low range divided by the interval length. That ratio is the box-count-proxy. It stands in for literally tiling the path with boxes.
An even window and the dimension estimate
The dimension estimate uses two equal consecutive halves plus the full even-length window. It is the logarithm of the sum of the half-window counts minus the logarithm of the full-window count, divided by the logarithm of 2. The result is treated as ranging from 1 to 2.
The length that defines the full-window count is an explicit even input, illustrated at 16. The lookback must be even so the window can be split in half. The dimension series itself can be plotted separately as a signal.
The alpha-map
That dimension is mapped to an exponential-smoothing weight by setting alpha equal to e to the power of -4.6 times (D minus 1). Under this alpha-map, a dimension of 1 produces alpha of 1 and a dimension of 2 produces alpha of 0.01. The mapped weight is then forced to stay between 0.01 and 1.
The one-pole filter
Exponential-smoothing is a one-pole blend of the latest observation with the previous filter value, controlled by a mixing weight. Here the observation is the current midpoint price, and the mixing weight is the bounded alpha from the alpha-map.
Under this mapping the smoother is designed to track almost raw price when the path is line-like and to flatten when the path is more space-filling, which the construction treats as a congestion regime.
All readings on this track · 7 readings
- 1992Constructing fractal templates from successive index changes
- 1994Constructing polarized fractal efficiency as a path filter
- 2002Long memory, regimes, and the limits of bell-curve models
- 2003Constructing the fractal dimension index
- 2005Constructing a fractal-dimension adaptive moving average
- 2007Constructing a fractal-dimension regime filter
- 2015Constructing fractal swings as support-resistance atoms