// The MIT License (MIT)
// © mihakralj
//@version=6
indicator("Linear Trend Moving Average (LTMA)", "LTMA", overlay=true)

//@function Calculates Linear Trend MA using dual EMA with linear extrapolation
//@param source Series to smooth
//@param period Lookback period (determines alpha = 2/(period+1))
//@returns LTMA value: EMA-based linear trend projection from first bar
//@description LTMA tracks both level and slope using two cascaded EMAs,
//  then extrapolates the linear trend forward. Unlike DEMA (which cancels
//  first-order lag via 2×EMA1 − EMA2), LTMA estimates the instantaneous
//  slope from the EMA difference and projects it forward by the full period:
//    slope = (EMA1 − EMA2) / (decay − 1)
//    LTMA  = EMA1 + slope × period
//  where decay = (1 − alpha). This produces a predictive moving average
//  that follows linear trends with zero steady-state error.
//  Uses §2 exponential warmup compensator on both EMAs (e*=beta,
//  c=1/(1-e)) for valid output from bar 1.
ltma(series float source, simple int period) =>
    if period <= 0
        runtime.error("Period must be greater than 0")
    float alpha = 2.0 / (period + 1)
    float beta  = 1.0 - alpha

    var bool  warmup = true
    var float e      = 1.0
    var float ema1   = 0.0
    var float ema2   = 0.0
    var float result = source

    float src = nz(source)

    ema1 := alpha * (src - ema1) + ema1
    ema2 := alpha * (ema1 - ema2) + ema2

    if warmup
        e *= beta
        float c = 1.0 / (1.0 - e)
        float comp1 = c * ema1
        float comp2 = c * ema2
        float slope = comp1 - comp2
        result := comp1 + slope * period
        warmup := e > 1e-10
    else
        float slope = ema1 - ema2
        result := ema1 + slope * period
    result

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

// Inputs
i_period = input.int(14, "Period", minval=1)
i_source = input.source(close, "Source")

// Calculation
ltma_value = ltma(i_source, period=i_period)

// Plot
plot(ltma_value, "LTMA", color=color.yellow, linewidth=2)