QuanTAlib/lib/trends_FIR/crma/Crma.md

CRMA: Cubic Regression Moving Average

Linear regression tells you where the trend is going. Quadratic regression tells you it's curving. Cubic regression tells you the curve is changing its mind.

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

• CRMA fits a degree-3 polynomial y = a_0 + a_1 x + a_2 x^2 + a_3 x^3 to the most recent N bars via ordinary least squares, then returns the fitt...
• Similar: SMA, TrIMA | Complementary: Trend strength indicators | Trading note: Cubic-Root weighted MA; gentle weighting profile between uniform (SMA) and triangular (TrIMA).
• Validated against TA-Lib, Skender, and Tulip reference implementations where available.

CRMA fits a degree-3 polynomial y = a_0 + a_1 x + a_2 x^2 + a_3 x^3 to the most recent N bars via ordinary least squares, then returns the fitted endpoint value a_0. By capturing inflection and curvature that linear and quadratic models miss, CRMA tracks S-shaped reversals and accelerating trends with measurably lower endpoint error than LSMA or QRMA on non-stationary price series. The cost is a 4x4 linear system solve per bar, which is O(1) once power sums are accumulated in O(N).

Historical Context

Polynomial regression as a smoothing technique dates to Legendre (1805) and Gauss (1809), who independently developed the method of least squares. The specific application of cubic (degree-3) polynomial fitting to financial time series emerged from the broader Savitzky-Golay filtering framework published in 1964, which showed that polynomial regression over a sliding window produces FIR filter coefficients with desirable frequency-domain properties.

CRMA occupies the sweet spot in the polynomial hierarchy. Degree-1 (LSMA) captures only linear trends. Degree-2 (QRMA) adds curvature but misses inflection points. Degree-3 (CRMA) captures inflection, the point where acceleration changes sign, which is precisely where trend reversals begin. Degree-4 and above risk Runge's phenomenon: oscillatory artifacts near window edges that amplify noise rather than suppress it.

The key implementation difference from textbook polynomial regression is the x-indexing convention. CRMA uses x = 0 for the newest bar and x = N-1 for the oldest. This means the fitted endpoint is simply a_0, the intercept, avoiding the numerical instability of evaluating a_0 + a_1(N-1) + a_2(N-1)^2 + a_3(N-1)^3 with large N.

Architecture & Physics

1. Normal Equations Assembly

The polynomial fit requires solving \mathbf{M} \cdot \mathbf{a} = \mathbf{r} where:

M_{ij} = \sum_{k=0}^{N-1} x_k^{i+j}, \quad r_i = \sum_{k=0}^{N-1} x_k^i \cdot y_k, \quad i,j \in \{0,1,2,3\}

Seven power sums (S_0 through S_6) and four cross-products (r_0 through r_3) are accumulated in a single O(N) pass over the circular buffer.

2. Gaussian Elimination with Partial Pivoting

The 4x4 augmented matrix is solved via Gaussian elimination with partial pivoting. Partial pivoting prevents division-by-zero and minimizes round-off amplification. The pivot search, row swap, and elimination are all O(1) operations on a fixed 4x4 system (64 element accesses, 48 multiply-adds).

3. Back-Substitution

After elimination produces an upper-triangular system, back-substitution extracts a_3, a_2, a_1, a_0 in four steps. The result a_0 is the fitted value at x = 0 (newest bar).

4. Singular Matrix Guard

If the pivot magnitude falls below 10^{-12}, the system is treated as singular and the raw price is returned. This handles degenerate cases (e.g., all identical prices, N < 4 effective points).

Mathematical Foundation

The cubic regression minimizes the sum of squared residuals:

\min_{a_0, a_1, a_2, a_3} \sum_{k=0}^{N-1} \left( y_k - a_0 - a_1 x_k - a_2 x_k^2 - a_3 x_k^3 \right)^2

Setting partial derivatives to zero yields the 4x4 normal equation system:

\begin{bmatrix} S_0 & S_1 & S_2 & S_3 \\ S_1 & S_2 & S_3 & S_4 \\ S_2 & S_3 & S_4 & S_5 \\ S_3 & S_4 & S_5 & S_6 \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix} r_0 \\ r_1 \\ r_2 \\ r_3 \end{bmatrix}

Where:

S_m = \sum_{k=0}^{N-1} k^m, \quad r_m = \sum_{k=0}^{N-1} k^m \cdot y_k

The power sums S_m have closed-form expressions (Faulhaber's formulas), but accumulating them in the data loop adds negligible cost and avoids large intermediate products.

Default parameters: period = 14, minPeriod = 4 (minimum for degree-3 fit).

Pseudo-code (streaming):

buffer ← circular_buffer(period)
buffer.push(price)
n ← min(bar_count, period)
if n < 4: return price

// Accumulate sums in O(n)
for i = 0 to n-1:
    x = i; x2 = x*x; x3 = x2*x
    S0 += 1; S1 += x; S2 += x2; S3 += x3
    S4 += x2*x2; S5 += x2*x3; S6 += x3*x3
    r0 += y[i]; r1 += x*y[i]; r2 += x2*y[i]; r3 += x3*y[i]

// Build 4×5 augmented matrix, solve via Gaussian elimination
M = [[S0,S1,S2,S3,r0], [S1,S2,S3,S4,r1], [S2,S3,S4,S5,r2], [S3,S4,S5,S6,r3]]
gaussian_eliminate_partial_pivot(M)
a = back_substitute(M)
return a[0]  // fitted value at x=0 (newest bar)

Resources

• Legendre, A.-M. (1805). Nouvelles méthodes pour la détermination des orbites des comètes. Firmin Didot.
• Gauss, C.F. (1809). Theoria motus corporum coelestium. Perthes et Besser.
• Savitzky, A. & Golay, M.J.E. (1964). "Smoothing and Differentiation of Data by Simplified Least Squares Procedures." Analytical Chemistry, 36(8), 1627-1639.
• Press, W.H. et al. (2007). Numerical Recipes, 3rd ed. Cambridge University Press. Chapter 15: Modeling of Data.

Performance Profile

Operation Count (Streaming Mode)

CRMA(N) fits a degree-3 polynomial via least squares. The O(N) cost is in accumulating seven Faulhaber power sums plus four cross-products over the ring buffer each bar. The 4×4 Gaussian elimination is O(1) (fixed 64 operations regardless of N).

Operation	Count	Cost (cycles)	Subtotal
Ring buffer push	1	3	~3
Power sum updates S0..S6 (7 sums × 2 ops)	~2N	1	~2N
Cross-product updates (4 × dot products)	~4N	2	~8N
4×4 Gaussian elimination (fixed)	~64	3	~192
Polynomial evaluation at newest point	4	3	~12
Total	~(6N + 64)	—	~(10N + 207) cycles

O(N) per bar. For default N = 14: ~347 cycles. Resync re-computes sums every 1000 ticks to prevent floating-point drift.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Power sum accumulation (S0..S6)	Yes	Independent sums; VADDPD per term, 4 bars/lane
Cross-product dot products (ΣxᵏY)	Yes	VFMADD231PD across window; stride-1 pattern
4×4 Gaussian elimination	No	Fixed scalar 64-op system; not worth SIMD setup
Polynomial evaluation	No	4-term Horner; scalar is fastest for degree 3

Batch throughput for the sum and cross-product phases: AVX2 achieves ~4× scalar. Gaussian elimination and Horner evaluation remain scalar. Net batch speedup for N = 14, large series: approximately 2.5× over fully scalar.