mirror of
https://github.com/mihakralj/QuanTAlib.git
synced 2026-07-27 17:27:43 +00:00
Miha Kralj
114 lines
4.3 KiB
Plaintext
114 lines
4.3 KiB
Plaintext
// This Pine Script™ code is subject to the terms of the Mozilla Public License 2.0
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// https://mozilla.org/MPL/2.0/
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// © QuanTAlib
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//@version=6
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indicator("Quadratic Regression Moving Average (QRMA)", "QRMA", overlay = true)
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//@function Quadratic Regression Moving Average — fits a second-degree polynomial
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// y = a + b·x + c·x² to the lookback window via ordinary least squares and
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// returns the fitted value at the most recent bar (the endpoint). Captures
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// both linear trends and curvature, providing better tracking of parabolic
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// price movements than linear regression (LSMA). Uses x = 0..N-1 convention
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// where x=0 is the oldest bar and the endpoint is evaluated at x=N-1.
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//@param source Series to smooth
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//@param period Lookback window (>= 3, need at least 3 points for quadratic fit)
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//@returns Quadratic regression endpoint value
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//@reference Savitzky, A. & Golay, M.J.E. (1964). Analytical Chemistry, 36(8), 1627–1639.
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// (Quadratic regression is a special case of Savitzky-Golay degree=2 smoothing.)
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//@optimized O(period) per bar via circular buffer; sums precomputed for normal equations
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qrma(series float source, simple int period) =>
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if period < 3
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runtime.error("Period must be at least 3")
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float price = nz(source)
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// --- Circular buffer ---
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var array<float> buffer = array.new_float(period, 0.0)
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var int head = 0
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array.set(buffer, head, price)
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head := (head + 1) % period
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int count = math.min(bar_index + 1, period)
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if count < period
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price
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else
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// --- Precompute constant sums for x = 0, 1, ..., N-1 ---
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// These only depend on period (N) and could be precomputed once,
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// but Pine Script handles them efficiently inline.
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float n = period
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// Σx = N(N-1)/2
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float sx = n * (n - 1.0) / 2.0
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// Σx² = N(N-1)(2N-1)/6
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float sx2 = n * (n - 1.0) * (2.0 * n - 1.0) / 6.0
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// Σx³ = [N(N-1)/2]²
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float sx3 = sx * sx
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// Σx⁴ = N(N-1)(2N-1)(3N²-3N-1)/30
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float sx4 = n * (n - 1.0) * (2.0 * n - 1.0) * (3.0 * n * n - 3.0 * n - 1.0) / 30.0
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// --- Compute data-dependent sums ---
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// Buffer: head points to next-write = oldest entry
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float sy = 0.0
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float sxy = 0.0
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float sx2y = 0.0
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for j = 0 to period - 1
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int idx = (head + j) % period
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float val = array.get(buffer, idx)
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float x = j // x=0 is oldest, x=N-1 is newest
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sy += val
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sxy += x * val
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sx2y += x * x * val
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// --- Solve 3x3 normal equations via Cramer's rule ---
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// [n Σx Σx²] [a] [Σy ]
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// [Σx Σx² Σx³] [b] = [Σxy ]
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// [Σx² Σx³ Σx⁴] [c] [Σx²y]
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float d00 = n
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float d01 = sx
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float d02 = sx2
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float d10 = sx
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float d11 = sx2
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float d12 = sx3
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float d20 = sx2
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float d21 = sx3
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float d22 = sx4
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// Determinant of coefficient matrix
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float det = d00 * (d11 * d22 - d12 * d21) -
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d01 * (d10 * d22 - d12 * d20) +
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d02 * (d10 * d21 - d11 * d20)
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if math.abs(det) < 1e-20
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price
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else
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// Solve for a, b, c using Cramer's rule
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float det_a = sy * (d11 * d22 - d12 * d21) -
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d01 * (sxy * d22 - sx2y * d21) +
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d02 * (sxy * d21 - sx2y * d11)
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float det_b = d00 * (sxy * d22 - sx2y * d21) -
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sy * (d10 * d22 - d12 * d20) +
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d02 * (d10 * sx2y - sxy * d20)
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float det_c = d00 * (d11 * sx2y - sxy * d21) -
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d01 * (d10 * sx2y - sxy * d20) +
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sy * (d10 * d21 - d11 * d20)
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float a = det_a / det
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float b = det_b / det
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float c = det_c / det
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// Evaluate at x = N-1 (newest bar = endpoint)
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float x_end = n - 1.0
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a + b * x_end + c * x_end * x_end
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// ── Inputs ──
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int p_period = input.int(14, "Period", minval = 3)
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float p_src = input.source(close, "Source")
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// ── Calculation ──
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float out = qrma(p_src, p_period)
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// ── Plot ──
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plot(out, "QRMA", color.yellow, 2)
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