> *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.*
- 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...
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:
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:
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.
// 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.
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 |
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.
