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🛣️ KAMA — Kaufman Adaptive Moving Average

KAMA is a moving average that changes its own smoothing speed depending on how efficiently the price is trending. In a strong trend it hugs the price closely; in a choppy, sideways market it flattens out almost like a long-period SMA.

💡 Financial Meaning

A fixed-period EMA is a compromise: fast enough to follow trends, but noisy in ranging markets — or the opposite. KAMA removes that trade-off by measuring, at every bar, how much of the raw price movement was "useful" directional travel versus wasted back-and-forth noise, and adapting instantly.

🔢 Mathematical Formula

  1. Efficiency Ratio over the lookback window \(N\) — net distance travelled divided by the total path length walked:

    \[ ER_t = \frac{\left| C_t - C_{t-N} \right|}{\sum_{i=0}^{N-1} \left| C_{t-i} - C_{t-i-1} \right|} \]

    \(ER_t \in [0, 1]\): it is \(1\) for a perfectly straight trend and near \(0\) for pure noise.

  2. Adaptive smoothing constant, interpolating between a fast and a slow EMA constant:

    \[ SC_t = \left[ ER_t \cdot (\alpha_{fast} - \alpha_{slow}) + \alpha_{slow} \right]^2 \]

  3. Recurrence, identical in form to the EMA but with a time-varying coefficient:

    \[ KAMA_t = KAMA_{t-1} + SC_t \cdot (C_t - KAMA_{t-1}) \]

⚙️ Parameters

Parameter Key Default Description
Period (\(N\)) period 10 Lookback window for the Efficiency Ratio, in sessions.

Fast/slow constants are not exposed

The classic Kaufman formulation derives \(\alpha_{fast}\) and \(\alpha_{slow}\) from fixed 2-period and 30-period EMA constants (\(\alpha_{fast}=2/3\), \(\alpha_{slow}\approx 0.065\)). LibreFolio's TA-Lib–backed implementation only exposes the Efficiency Ratio lookback (period) — the fast/slow constants are internal library defaults, not a user-configurable parameter.

🎛️ Signal Processing Equivalent — Adaptive-Gain IIR Filter

KAMA is the same first-order IIR recurrence as the EMA, but with a self-tuning gain \(SC_t\) instead of a fixed \(\alpha\). This is precisely the structure of an adaptive filter (e.g. a simplified LMS-style filter): the "signal-to-noise ratio" of the input (\(ER_t\)) continuously re-tunes the pole location \(z = 1 - SC_t\).

flowchart LR
    C["💹 Close price"] --> ER["📐 Efficiency Ratio (window N)"]
    ER --> SC["⚙️ Adaptive smoothing constant SC"]
    C --> REC["🔁 KAMA(t) = KAMA(t-1) + SC·(C(t) − KAMA(t-1))"]
    SC --> REC
    REC --> OUT["🛣️ KAMA(t)"]
    OUT -.->|"feedback"| REC

Trending vs ranging pole

When \(ER_t \to 1\) (clean trend), \(SC_t \to \alpha_{fast}^2 \approx 0.44\) — a very reactive pole close to the origin. When \(ER_t \to 0\) (pure noise), \(SC_t \to \alpha_{slow}^2 \approx 0.004\) — an extremely sluggish pole near the unit circle, close to a long SMA.

KAMA description (StockCharts)