QuanTAlib/lib/statistics/polyfit/Polyfit.md

POLYFIT: Polynomial Fitting

Polynomial Fitting computes a rolling polynomial regression of configurable degree over a lookback window, returning the fitted value at the current bar. Degree 1 produces a linear regression endpoint (identical to LSQR), degree 2 produces a quadratic fit that captures curvature, and degree 3 produces a cubic fit that captures inflection points. The implementation solves the normal equations \mathbf{X}^T\mathbf{X}\mathbf{a} = \mathbf{X}^T\mathbf{y} via Gauss-Jordan elimination with partial pivoting, evaluating the resulting polynomial at x = 1 (the current bar position). With O(Nd + d^3) complexity per bar where N is the period and d is the degree, POLYFIT provides a general-purpose curve-fitting tool that subsumes linear regression and extends it to arbitrary polynomial order.

Historical Context

Polynomial regression traces to Adrien-Marie Legendre (1805) and Carl Friedrich Gauss (1809), who independently developed the method of least squares. The normal equations formulation provides the minimum-sum-of-squares solution in closed form, though numerical stability requires careful implementation. Gauss-Jordan elimination with partial pivoting (Jordan, 1873) is the standard approach for small systems like those arising in polynomial fitting with degrees 1-6.

In technical analysis, linear regression (degree 1) is well established via the Linear Regression Channel and LSQR indicators. Higher-degree fits are less common due to overfitting concerns, but degree 2 (quadratic) is useful for detecting acceleration/deceleration in trends, and degree 3 (cubic) can capture reversal patterns. The key insight is that higher degrees track price more closely but also amplify noise; the optimal degree depends on the signal-to-noise ratio and the lookback period.

The x-normalization step (mapping time indices to [0, 1]) is critical for numerical stability: without it, the Vandermonde matrix entries x^d would span many orders of magnitude for typical lookback periods, causing catastrophic cancellation in the normal equations. With normalization, the matrix condition number remains manageable up to degree 6.

Architecture and Physics

The implementation uses a circular buffer to maintain the last period values, with NaN substitution via last-valid-value tracking.

Matrix assembly: Constructs the (d+1) \times (d+1) Gram matrix \mathbf{G} = \mathbf{X}^T\mathbf{X} and right-hand side \mathbf{r} = \mathbf{X}^T\mathbf{y} in a single pass over the data. The Vandermonde basis vectors are [1, x, x^2, \ldots, x^d] where x_i = i/(n-1) is the normalized time position. The matrix is symmetric so only the upper triangle needs explicit computation (mirrored to lower).

Solver: Gauss-Jordan elimination with partial pivoting transforms the augmented matrix [\mathbf{G} | \mathbf{r}] into [\mathbf{I} | \mathbf{a}]. Partial pivoting selects the row with the largest absolute value in the current column to minimize round-off error. Singular or near-singular matrices (pivot < 10^{-30}) abort gracefully.

Evaluation: The polynomial P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_d x^d is evaluated at x = 1.0 (current bar, since time is normalized to [0, 1]). This gives the fitted value at the most recent observation.

Degree clamping: If degree exceeds period - 1, it is automatically reduced to prevent underdetermined systems.

Mathematical Foundation

The polynomial model:

P(x) = \sum_{j=0}^{d} a_j x^j = a_0 + a_1 x + a_2 x^2 + \cdots + a_d x^d

The normal equations for least-squares fitting:

\mathbf{X}^T\mathbf{X}\,\mathbf{a} = \mathbf{X}^T\mathbf{y}

where \mathbf{X} is the n \times (d+1) Vandermonde matrix:

X_{ij} = x_i^j, \quad x_i = \frac{i}{n-1} \in [0, 1]

The Gram matrix elements:

G_{jk} = \sum_{i=0}^{n-1} x_i^{j+k}

The right-hand side:

r_j = \sum_{i=0}^{n-1} x_i^j \cdot y_i

Gauss-Jordan with partial pivoting reduces [\mathbf{G} | \mathbf{r}] to [\mathbf{I} | \mathbf{a}]:

1. For each column c: find the row p in [c, d] with maximum |G_{pc}|
2. Swap rows c and p
3. Scale row c so the pivot becomes 1
4. Subtract multiples of row c from all other rows

Output: \hat{y}_{\text{current}} = P(1.0) = \sum_{j=0}^{d} a_j

Parameter constraints: period \ge 2, degree \ge 1 (clamped to period - 1). Computational complexity: O(nd + d^3).

POLYFIT(source, period, degree):
    d = min(degree, period - 1)
    m = d + 1
    normalize x_i = i / (n-1) for i in [0, n-1]

    // Build normal equations
    G = (m x m) matrix of zeros
    r = m-vector of zeros
    for each (x_i, y_i) in window:
        for j = 0 to d:
            r[j] += x_i^j * y_i
            for k = j to d:
                G[j][k] += x_i^(j+k)
                G[k][j] = G[j][k]    // symmetric

    // Gauss-Jordan with partial pivoting
    for col = 0 to d:
        pivot_row = argmax |G[row][col]| for row in [col, d]
        swap rows col and pivot_row in G and r
        scale row col by 1/G[col][col]
        eliminate col from all other rows

    // Evaluate at x = 1.0 (current bar)
    return sum(r[j] for j = 0 to d)

Resources

• Legendre, A.M. "Nouvelles methodes pour la determination des orbites des cometes." 1805.
• Gauss, C.F. "Theoria Motus Corporum Coelestium." 1809.
• Golub, G. & Van Loan, C. "Matrix Computations." 4th edition, Johns Hopkins University Press, 2013.
• Press, W.H. et al. "Numerical Recipes: The Art of Scientific Computing." 3rd edition, Cambridge University Press, 2007. Chapter 15 (Modeling of Data).
• Draper, N. & Smith, H. "Applied Regression Analysis." 3rd edition, Wiley, 1998.