Integral of Linear Regression Slope (ILRS)

// The MIT License (MIT)
// © mihakralj
//@version=6
indicator("Integral of Linear Regression Slope (ILRS)", "ILRS", overlay=true)

//@function Computes the Integral of Linear Regression Slope — cumulative sum of the
//          least-squares slope computed over a rolling window. Tracks accumulated
//          trend direction as a price-overlay smoothing filter.
//@param source Series to analyze
//@param period Lookback window for slope calculation (>= 2)
//@returns Cumulative integral of the rolling linear regression slope
//@reference John Ehlers, "Rocket Science for Traders" (Wiley, 2001).
//@reference Concept: ILRS is the discrete integral (running sum) of the LinReg slope,
//           producing a smoother trend follower than LSMA. Equivalent to filtering
//           the first derivative and reconstructing via integration.
//@optimized O(period) per bar for slope via circular buffer accumulation
ilrs(series float source, simple int period) =>
    if period < 2
        runtime.error("Period must be at least 2")

    float price = nz(source)

    // --- Circular buffer for rolling window ---
    var array<float> buffer = array.new_float(period, na)
    var int head = 0
    array.set(buffer, head, price)
    head := (head + 1) % period

    // --- Running integral state ---
    var float integral = na

    int count = math.min(bar_index + 1, period)
    if count < 2
        integral := price
        integral
    else
        // --- Compute linear regression slope over the buffer ---
        // x-indices: 0, 1, ..., n-1 (oldest to newest)
        // Analytical x-sums: ΣX = n(n-1)/2, ΣX² = n(n-1)(2n-1)/6
        float n = count
        float sumX  = 0.5 * (n - 1) * n
        float sumX2 = (n - 1) * n * (2 * n - 1) / 6.0

        // Accumulate y-sums from circular buffer
        int start = count < period ? 0 : head
        float sumY  = 0.0
        float sumXY = 0.0
        for i = 0 to int(n) - 1
            int idx = (start + i) % period
            float val = nz(array.get(buffer, idx))
            sumY  += val
            sumXY += i * val

        float denomX = n * sumX2 - sumX * sumX
        float slope = denomX != 0 ? (n * sumXY - sumX * sumY) / denomX : 0.0

        // --- Integrate: ILRS = ILRS[1] + slope ---
        if na(integral)
            integral := price
        integral := integral + slope
        integral

// ── Inputs ──────────────────────────────────────────────────────────────
src = input.source(close, "Source")
per = input.int(14, "Period", minval=2)

// ── Plot ────────────────────────────────────────────────────────────────
plot(ilrs(src, per), "ILRS", color.new(color.yellow, 0), 2)