// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Holt Exponential Moving Average (HOLT)", "HOLT", overlay=true)

//@function Calculates Holt EMA using double exponential smoothing (level + trend)
//@param source Series to smooth
//@param period Lookback period (determines alpha = 2/(period+1))
//@param gamma Trend smoothing factor (0..1). Default: same as alpha
//@returns Holt EMA value (level + trend) from first bar
//@description Holt's (1957) double exponential smoothing tracks both level and trend.
//  Level equation:  L_t = alpha * y_t + (1 - alpha) * (L_{t-1} + B_{t-1})
//  Trend equation:  B_t = gamma * (L_t - L_{t-1}) + (1 - gamma) * B_{t-1}
//  Output:          HOLT_t = L_t + B_t   (1-step-ahead forecast)
//  When gamma=0, degenerates to standard EMA (no trend correction).
//  When gamma=alpha, provides balanced level/trend tracking.
holt(series float source, simple int period, simple float gamma=0) =>
    float alpha = 2.0 / (period + 1)
    float g = gamma > 0 ? gamma : alpha
    var float level = na
    var float trend = 0.0
    float result = na

    if na(level)
        // First bar: initialize level to source, trend to 0
        level := source
        trend := 0.0
        result := source
    else
        float prevLevel = level
        // Level: alpha * source + (1 - alpha) * (prevLevel + trend)
        level := alpha * source + (1.0 - alpha) * (prevLevel + trend)
        // Trend: gamma * (level - prevLevel) + (1 - gamma) * trend
        trend := g * (level - prevLevel) + (1.0 - g) * trend
        // Output: level + trend (1-step-ahead forecast)
        result := level + trend
    result

// ---------- Main loop ----------

// Inputs
i_period = input.int(10, "Period", minval=1)
i_gamma  = input.float(0, "Gamma (0 = auto)", minval=0, maxval=1, step=0.01)
i_source = input.source(close, "Source")

// Calculation
holt_value = holt(i_source, period=i_period, gamma=i_gamma)

// Plot
plot(holt_value, "HOLT", color=color.yellow, linewidth=2)