QuanTAlib/lib/channels/apchannel/apchannel.md

APCHANNEL: Adaptive Price Channel

APCHANNEL applies exponential smoothing independently to price highs and lows, creating a dynamic envelope that "remembers" significant extremes while gradually fading their influence over time. Unlike rigid Donchian channels that drop price extremes abruptly when they exit the lookback window (the "cliff effect"), APCHANNEL decays them smoothly through leaky integration. The result is a channel with continuously sloping boundaries that responds to volatility without the discontinuous jumps that plague fixed-window approaches. The algorithm is O(1) per bar with only two state variables and no buffers.

Historical Context

Traditional Price Channels (Donchian, 1960s) define range by the absolute highest high and lowest low over a fixed period. When a major high from n bars ago drops out of the window, the upper boundary can collapse instantaneously, producing discontinuous channel behavior that generates false signals. The Adaptive Price Channel addresses this by borrowing the exponential smoothing concept from signal processing, applying the same "leaky integrator" principle that electrical engineers use for envelope detection in AM radio circuits.

The approach is equivalent to running two independent EMAs: one on the High series and one on the Low series. This connection to EMA theory means the channel inherits well-understood convergence properties. The half-life of influence is \ln(2) / \ln(1/(1-\alpha)) bars, and the channel is considered warm after approximately 3/\alpha bars. The single-parameter design (\alpha) makes APCHANNEL simpler to tune than multi-parameter alternatives.

Architecture & Physics

1. Dual EMA Recursion

The upper and lower bands are independent EMA filters on High and Low:

\text{Upper}_t = \alpha \cdot H_t + (1 - \alpha) \cdot \text{Upper}_{t-1} \text{Lower}_t = \alpha \cdot L_t + (1 - \alpha) \cdot \text{Lower}_{t-1}

Using the FMA pattern with \text{decay} = 1 - \alpha:

\text{Upper}_t = \text{FMA}(\text{decay}, \text{Upper}_{t-1}, \alpha \cdot H_t)

2. Midpoint

\text{Middle}_t = \frac{\text{Upper}_t + \text{Lower}_t}{2}

3. Alpha Semantics

• High \alpha (e.g., 0.8): Short memory. Channel snaps quickly to new extremes, forgets old ones rapidly.
• Low \alpha (e.g., 0.1): Long memory. Significant highs persist as resistance for dozens of bars.
• Period approximation: \alpha \approx 2 / (P + 1) where P is the equivalent EMA period.

4. Complexity

O(1) per bar: 2 FMA operations + 1 addition + 1 division. No buffers, no history. The two bands are independent and can be computed in parallel.

Mathematical Foundation

Parameters

Parameter	Description	Default	Constraint
alpha	Smoothing factor (higher = faster decay)	0.2	(0, 1]

Initialization

On the first bar:

\text{Upper}_0 = H_0, \quad \text{Lower}_0 = L_0

Half-Life

The number of bars for a price extreme's influence to decay by 50%:

t_{1/2} = \frac{\ln 2}{\ln(1 / (1 - \alpha))}

For \alpha = 0.2: t_{1/2} \approx 3.1 bars. For \alpha = 0.05: t_{1/2} \approx 13.5 bars.

Pseudo-code

function APCHANNEL(high, low, alpha):
    validate: 0 < alpha ≤ 1
    decay = 1 - alpha

    // EMA of highs
    if first_bar:
        upper = high
    else:
        upper = decay * upper + alpha * high

    // EMA of lows
    if first_bar:
        lower = low
    else:
        lower = decay * lower + alpha * low

    middle = (upper + lower) / 2

    return [middle, upper, lower]

Output Interpretation

Output	Description
upper	Exponentially smoothed high (resistance)
lower	Exponentially smoothed low (support)
middle	Arithmetic mean of upper and lower

Resources

• Wilder, J.W. New Concepts in Technical Trading Systems. Trend Research, 1978. (EMA smoothing foundations)
• Donchian, R. "Trend Following Methods in Commodity Price Analysis." Commodity Research Bureau, 1960. (Fixed-window channel predecessor)
• Haykin, S. Adaptive Filter Theory. Prentice Hall, 2002. (Leaky integrator / exponential smoothing theory)