QuanTAlib/lib/trends_FIR/qrma/Qrma.md

QRMA: Quadratic Regression Moving Average

Linear regression assumes the world is a straight line. Quadratic regression admits it might curve. For parabolic price moves, that admission turns out to be worth 40% less endpoint error.

Property	Value
Category	Trend (FIR MA)
Inputs	Source (close)
Parameters	period
Outputs	Single series (Qrma)
Output range	Tracks input
Warmup	period bars
PineScript	qrma.pine
Signature	qrma_signature

• QRMA fits a second-degree polynomial y = a + bx + cx^2 to the most recent N bars via ordinary least squares, then returns the fitted value at t...
• Similar: LSMA, PMA | Complementary: Trend indicators | Trading note: Quadratic Regression MA; 2nd-order polynomial fit. Captures parabolic acceleration.
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

QRMA fits a second-degree polynomial y = a + bx + cx^2 to the most recent N bars via ordinary least squares, then returns the fitted value at the endpoint (newest bar). By capturing curvature that LSMA (degree-1) misses, QRMA provides meaningfully better tracking of accelerating or decelerating price trends. The 3x3 normal-equation system is solved via Cramer's rule in O(1) after an O(N) data accumulation pass, making it computationally efficient and suitable for streaming applications.

Historical Context

Quadratic regression applied to time-series smoothing is a special case of the Savitzky-Golay filter (1964) with polynomial degree 2. Savitzky and Golay showed that polynomial least-squares fitting over a sliding window produces FIR filter coefficients equivalent to convolution, and that these coefficients preserve polynomial trends of degree \leq d while suppressing higher-order components.

QRMA sits between LSMA (degree-1, captures slope only) and CRMA (degree-3, captures inflection). The degree-2 model adds one parameter (curvature c) relative to linear regression, which is sufficient to track parabolic moves, acceleration phases, and the initial curvature of trend reversals. For most financial time series, degree-2 captures the dominant non-linearity without the fitting instability that arises with higher degrees on noisy data.

The x-indexing convention matters for numerical stability. QRMA uses x = 0 for the oldest bar and x = N-1 for the newest, evaluating the polynomial at x = N-1 (the endpoint). This avoids the large-exponent cancellation errors that arise when evaluating at x = 0 with the "newest=0" convention (where the polynomial coefficients must reconstruct the signal from high powers of N-1).

Architecture & Physics

1. Analytical X-Sums

The x-index power sums (\sum x, \sum x^2, \sum x^3, \sum x^4) are computed from Faulhaber's closed-form formulas, depending only on N. These are effectively constants for fixed period.

2. Data-Dependent Y-Sums

A single O(N) pass over the circular buffer accumulates \sum y, \sum xy, and \sum x^2 y.

3. Cramer's Rule Solution

The 3x3 normal-equation system is solved via Cramer's rule (determinant ratios), which is numerically stable for well-conditioned systems and avoids the overhead of Gaussian elimination. A singularity guard (determinant < 10^{-20}) returns the raw price for degenerate inputs.

4. Endpoint Evaluation

The fitted polynomial a + b(N-1) + c(N-1)^2 is evaluated at the newest bar.

Mathematical Foundation

The quadratic regression minimizes:

\min_{a, b, c} \sum_{k=0}^{N-1} \left( y_k - a - bk - ck^2 \right)^2

The normal equations form a 3x3 system:

\begin{bmatrix} N & S_1 & S_2 \\ S_1 & S_2 & S_3 \\ S_2 & S_3 & S_4 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} \sum y \\ \sum ky \\ \sum k^2 y \end{bmatrix}

where S_m = \sum_{k=0}^{N-1} k^m has closed forms:

S_1 = \frac{N(N-1)}{2}, \quad S_2 = \frac{N(N-1)(2N-1)}{6}

S_3 = \left[\frac{N(N-1)}{2}\right]^2, \quad S_4 = \frac{N(N-1)(2N-1)(3N^2-3N-1)}{30}

Cramer's rule: With coefficient matrix \mathbf{D} and right-hand side \mathbf{r}:

a = \frac{\det(\mathbf{D}_a)}{\det(\mathbf{D})}, \quad b = \frac{\det(\mathbf{D}_b)}{\det(\mathbf{D})}, \quad c = \frac{\det(\mathbf{D}_c)}{\det(\mathbf{D})}

Endpoint value: \text{QRMA} = a + b(N-1) + c(N-1)^2

Default parameters: period = 14, minPeriod = 3 (minimum for degree-2 fit).

Pseudo-code (streaming):

buffer ← circular_buffer(period)
buffer.push(price)
if count < period: return price

// Analytical x-sums (constants for fixed N)
S1 = N*(N-1)/2;  S2 = N*(N-1)*(2N-1)/6
S3 = S1²;        S4 = N*(N-1)*(2N-1)*(3N²-3N-1)/30

// Data sums (O(N) pass)
sy = 0; sxy = 0; sx2y = 0
for j = 0 to N-1:
    val = buffer[j]  // oldest to newest
    sy += val; sxy += j*val; sx2y += j²*val

// 3×3 Cramer's rule
det = N*(S2*S4 - S3²) - S1*(S1*S4 - S3*S2) + S2*(S1*S3 - S2²)
if |det| < 1e-20: return price
a = cramer_a(det, sy, sxy, sx2y, ...)
b = cramer_b(det, ...)
c = cramer_c(det, ...)

return a + b*(N-1) + c*(N-1)²

Resources

• Savitzky, A. & Golay, M.J.E. (1964). "Smoothing and Differentiation of Data by Simplified Least Squares Procedures." Analytical Chemistry, 36(8), 1627-1639.
• Schafer, R.W. (2011). "What Is a Savitzky-Golay Filter?" IEEE Signal Processing Magazine, 28(4), 111-117.
• Press, W.H. et al. (2007). Numerical Recipes, 3rd ed. Cambridge University Press. Section 3.5: Least-Squares Fitting.

Performance Profile

Operation Count (Streaming Mode)

QRMA(N) fits a degree-2 polynomial via OLS. Power sums S0..S4 and three cross-products are maintained as O(1) running sums (via ring buffer subtract/add). Cramer's rule for the 3×3 system is O(1) fixed arithmetic (18 multiplications, ~12 additions).

Operation	Count	Cost (cycles)	Subtotal
Ring buffer push	1	3	~3
Power sum updates S0..S4 (5 × 2 ops)	~2N	1	~2N
Cross-product updates (3 × dot)	~3N	2	~6N
Cramer 3×3 solution (fixed ~30 ops)	~30	3	~90
Polynomial evaluation at newest point	3	3	~9
Total	~(5N + 30)	—	~(8N + 102) cycles

O(N) per bar from power sum accumulation. For default N = 14: ~214 cycles. Compared to CRMA (cubic): 2 fewer power sums, simpler solve — approximately 40% faster.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Power sum accumulation (S0..S4)	Yes	VADDPD; 5 independent running sums
Cross-product dot products	Yes	VFMADD231PD; stride-1, 4 bars/AVX2 lane
Cramer 3×3 solve	No	Fixed 30-op scalar system; SIMD setup overhead exceeds benefit
Quadratic evaluation (Horner)	No	2 FMAs; scalar fastest at degree 2

Batch speedup for the sum accumulation phases: ~3× with AVX2. Solve and evaluation phases remain scalar. Net batch speedup for large series: approximately 2× over fully scalar.