Polynomial Fitting (POLYFIT)

// Licensed under the Apache License, Version 2.0
// © mihakralj
//@version=6
indicator("Polynomial Fitting (POLYFIT)", "POLYFIT", overlay=true, precision=8)

//@function Rolling polynomial regression fit of degree d over a lookback window
//@param source Series to fit
//@param period Lookback period (number of data points)
//@param degree Polynomial degree (1=linear, 2=quadratic, 3=cubic, etc.)
//@returns Fitted value at the current bar (polynomial endpoint)
//@description Fits a polynomial P(x) = a_0 + a_1*x + ... + a_d*x^d to the most
//  recent `period` data points using the normal equations (X'X)a = X'y.
//  The x-values are normalized to [0,1] for numerical stability.
//  Solves via Gauss-Jordan elimination with partial pivoting.
//  Output is P(1.0) — the polynomial evaluated at the current bar.
//  Degree 1 = linear regression (endpoint), degree 2 = quadratic fit, etc.
//  Complexity: O(period * degree + degree^3) per bar.
polyfit(series float source, simple int period, simple int degree) =>
    if period < 2
        runtime.error("Period must be at least 2")
    if degree < 1
        runtime.error("Degree must be at least 1")
    int d = math.min(degree, period - 1)
    int m = d + 1

    var array<float> buf = array.new_float(period, na)
    var int head = 0
    var int count = 0
    var float lastValid = na

    float curr = source
    if na(curr) and not na(lastValid)
        curr := lastValid
    if not na(curr)
        lastValid := curr

    if not na(curr)
        array.set(buf, head, curr)
        if count < period
            count += 1
        head := (head + 1) % period

    int n = count
    if n < m + 1
        na
    else
        int start = n < period ? 0 : head
        float nf = float(n)
        float invN = 1.0 / (nf - 1.0)

        int matSize = m * m
        array<float> mat = array.new_float(matSize, 0.0)
        array<float> rhs = array.new_float(m, 0.0)

        for i = 0 to n - 1
            int idx = (start + i) % period
            float y = array.get(buf, idx)
            float x = float(i) * invN

            float xp = 1.0
            for r = 0 to d
                float val_r = array.get(rhs, r)
                array.set(rhs, r, val_r + xp * y)

                float xq = xp
                for c = r to d
                    int pos = r * m + c
                    float val_m = array.get(mat, pos)
                    array.set(mat, pos, val_m + xp * xq)
                    if c != r
                        int pos2 = c * m + r
                        array.set(mat, pos2, val_m + xp * xq)
                    xq *= x
                xp *= x

        for col = 0 to d
            int pivRow = col
            float pivMax = math.abs(array.get(mat, col * m + col))
            for row = col + 1 to d
                float absVal = math.abs(array.get(mat, row * m + col))
                if absVal > pivMax
                    pivMax := absVal
                    pivRow := row
            if pivRow != col
                for k = 0 to d
                    int p1 = col * m + k
                    int p2 = pivRow * m + k
                    float tmp = array.get(mat, p1)
                    array.set(mat, p1, array.get(mat, p2))
                    array.set(mat, p2, tmp)
                float tmpR = array.get(rhs, col)
                array.set(rhs, col, array.get(rhs, pivRow))
                array.set(rhs, pivRow, tmpR)

            float piv = array.get(mat, col * m + col)
            if math.abs(piv) < 1e-30
                break

            float invPiv = 1.0 / piv
            for k = col to d
                int pos = col * m + k
                array.set(mat, pos, array.get(mat, pos) * invPiv)
            array.set(rhs, col, array.get(rhs, col) * invPiv)

            for row = 0 to d
                if row != col
                    float factor = array.get(mat, row * m + col)
                    for k = col to d
                        int p1 = row * m + k
                        int p2 = col * m + k
                        array.set(mat, p1, array.get(mat, p1) - factor * array.get(mat, p2))
                    array.set(rhs, row, array.get(rhs, row) - factor * array.get(rhs, col))

        float result = 0.0
        float xp = 1.0
        for r = 0 to d
            result += array.get(rhs, r) * xp
            xp *= 1.0
        result

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

// Inputs
i_period = input.int(20, "Period", minval=2, tooltip="Number of data points in the fitting window")
i_degree = input.int(2, "Degree", minval=1, maxval=6, tooltip="Polynomial degree: 1=linear, 2=quadratic, 3=cubic")
i_source = input.source(close, "Source")

// Calculation
fit_value = polyfit(i_source, i_period, i_degree)

// Plot
plot(fit_value, "POLYFIT", color=color.yellow, linewidth=2)