OrthoQuant

OrthoQuant is a high-performance Python framework for Symmetric Factor Orthogonalization and Cross-Sectional Risk Neutralization in quantitative alpha research.

Designed for quantitative researchers, portfolio managers, and statistical arbitrage teams to eliminate multicollinearity across alpha signals while mathematically minimizing signal distortion.

📖 Full Mathematical Derivations: Complete proofs, lemmas, and spectral analyses are documented in docs/mathematical_formulation.md.

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Mathematical Notation & Definitions

Throughout this framework, all mathematical objects are strictly defined as follows:

Symbol	Mathematical Domain	Description
$N \in \mathbb{N}$	Dimension	Number of assets in the cross-sectional universe ($N \ge K$)
$K \in \mathbb{N}$	Dimension	Number of candidate alpha factors
$M \in \mathbb{N}$	Dimension	Number of common risk exposures ($M < N$)
$\mathbf{F} \in \mathbb{R}^{N \times K}$	Matrix	Factor matrix, $\mathbf{F} = [\mathbf{f}_1, \dots, \mathbf{f}_K]$, where $\mathbf{f}_k \in \mathbb{R}^N$ is factor $k$
$\mathbf{X} \in \mathbb{R}^{N \times M}$	Matrix	Risk exposure matrix (e.g., sector dummy one-hots, Market Beta, Size)
$\mathbf{I}_K \in \mathbb{R}^{K \times K}$	Matrix	Identity matrix of size $K \times K$
$\mathcal{V}_K(\mathbb{R}^N)$	Stiefel Manifold	Set of $N \times K$ orthonormal matrices: $\{ \mathbf{Q} \in \mathbb{R}^{N \times K} : \mathbf{Q}^T \mathbf{Q} = \mathbf{I}_K \}$
$|\cdot|_F$	Matrix Norm	Frobenius norm: $\lVert \mathbf{A} \rVert_F = \sqrt{\mathrm{Tr}(\mathbf{A}^T \mathbf{A})} = \sqrt{\sum_{i,j} a_{ij}^2}$
$\mathrm{Tr}(\cdot)$	Operator	Matrix trace: $\mathrm{Tr}(\mathbf{A}) = \sum_i a_{ii}$
$\kappa_2(\cdot)$	Metric	Matrix condition number: $\kappa_2(\mathbf{A}) = \sigma_{\max}(\mathbf{A}) / \sigma_{\min}(\mathbf{A})$

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Mathematical Formulation & Theorems

1. Lowdin Symmetric Orthogonalization

Problem Statement

Given candidate alpha signals $\mathbf{F} \in \mathbb{R}^{N \times K}$ with $\mathrm{rank}(\mathbf{F}) = K$, we seek an orthonormal factor matrix $\widetilde{\mathbf{F}} \in \mathcal{V}_K(\mathbb{R}^N)$ that minimizes the total Frobenius distortion from the original signals:

$$ \min_{\widetilde{\mathbf{F}} \in \mathbb{R}^{N \times K}} \lVert \mathbf{F} - \widetilde{\mathbf{F}} \rVert_F^2 \quad \text{subject to} \quad \widetilde{\mathbf{F}}^T \widetilde{\mathbf{F}} = \mathbf{I}_K \qquad \text{(1)} $$

Theorem 1 (Optimality & Polar Decomposition)

The unique global minimizer to (1) is:

$$ \widetilde{\mathbf{F}} = \mathbf{F} (\mathbf{F}^T \mathbf{F})^{-1/2} = \mathbf{U} \mathbf{V}^T \qquad \text{(2)} $$

where $\mathbf{F} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T$ is the thin Singular Value Decomposition (SVD) of $\mathbf{F}$ ($\mathbf{U} \in \mathbb{R}^{N \times K}$, $\mathbf{\Sigma} = \mathrm{diag}(\sigma_1, \dots, \sigma_K) \succ 0$, $\mathbf{V} \in \mathbb{R}^{K \times K}$).

Proof Sketch: Expanding $\lVert \mathbf{F} - \widetilde{\mathbf{F}} \rVert_F^2 = \mathrm{Tr}(\mathbf{F}^T \mathbf{F}) - 2 \mathrm{Tr}(\widetilde{\mathbf{F}}^T \mathbf{F}) + \mathrm{Tr}(\mathbf{I}_K)$, minimizing distance is equivalent to maximizing trace overlap $\max_{\widetilde{\mathbf{F}}^T \widetilde{\mathbf{F}} = \mathbf{I}} \mathrm{Tr}(\widetilde{\mathbf{F}}^T \mathbf{F})$. Substituting SVD $\mathbf{F} = \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T$, we have $\mathrm{Tr}(\widetilde{\mathbf{F}}^T \mathbf{U} \mathbf{\Sigma} \mathbf{V}^T) = \mathrm{Tr}(\mathbf{Z}^T \mathbf{\Sigma})$ where $\mathbf{Z} \triangleq \mathbf{U}^T \widetilde{\mathbf{F}} \mathbf{V}$. Since $\lVert \mathbf{Z} \rVert_2 \le 1$, $\lvert z_{kk} \rvert \le 1$, the upper bound $\sum_k \sigma_k$ is attained if and only if $\mathbf{Z} = \mathbf{I}_K \iff \widetilde{\mathbf{F}} = \mathbf{U} \mathbf{V}^T$. $\blacksquare$ (See docs/mathematical_formulation.md for full proof; cf. Lowdin (1950), Higham (1988)).

Theorem 2 (Permutation Symmetry / Equivariance)

Let $\mathbf{P} \in \mathbb{R}^{K \times K}$ be any permutation matrix ($\mathbf{P}^T \mathbf{P} = \mathbf{I}_K$). If factors are permuted as $\mathbf{F}_{\pi} = \mathbf{F} \mathbf{P}$, then:

$$ \widetilde{\mathbf{F}}_{\pi} = \widetilde{\mathbf{F}} \mathbf{P} \qquad \text{(3)} $$

Proof Sketch: $(\mathbf{F} \mathbf{P})((\mathbf{F} \mathbf{P})^T (\mathbf{F} \mathbf{P}))^{-1/2} = \mathbf{F} \mathbf{P} (\mathbf{P}^T \mathbf{S} \mathbf{P})^{-1/2} = \mathbf{F} \mathbf{P} (\mathbf{P}^T \mathbf{S}^{-1/2} \mathbf{P}) = \mathbf{F} \mathbf{S}^{-1/2} \mathbf{P} = \widetilde{\mathbf{F}} \mathbf{P}$. $\blacksquare$

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2. Frisch–Waugh–Lovell Cross-Sectional Risk Neutralization

Let $\mathbf{X} \in \mathbb{R}^{N \times M}$ be the risk exposure matrix ($M < N$). Regressing factors on risk exposures:

$$ \mathbf{F} = \mathbf{X} \mathbf{B} + \mathbf{\mathcal{E}}, \quad \text{where } \mathbf{B} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{F} $$

The residual factor matrix $\mathbf{F}_{\text{neutral}} = \mathbf{\mathcal{E}}$ is the orthogonal projection onto the null space of $\mathbf{X}^T$:

$$ \mathbf{F}_{\text{neutral}} = \mathbf{M}_X \mathbf{F} = (\mathbf{I}_N - \mathbf{X} (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T) \mathbf{F} \qquad \text{(4)} $$

Theorem 3 (Exact Risk Orthogonality)

$$ \mathbf{X}^T \mathbf{F}_{\text{neutral}} = \mathbf{0}_{M \times K} \qquad \text{(5)} $$

Proof: $\mathbf{X}^T \mathbf{M}_X = \mathbf{X}^T (\mathbf{I}_N - \mathbf{X} (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T) = \mathbf{X}^T - (\mathbf{X}^T \mathbf{X})(\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T = \mathbf{0}$. $\blacksquare$ (Frisch & Waugh 1933; Lovell 1963).

Theorem 4 (Numerical Stability via QR Decomposition)

Let $\mathbf{X} = \mathbf{Q}_X \mathbf{R}_X$ be the thin QR decomposition of $\mathbf{X}$ ($\mathbf{Q}_X^T \mathbf{Q}_X = \mathbf{I}_M$, $\mathbf{R}_X$ invertible). Then:

$$ \mathbf{M}_X \mathbf{F} = \mathbf{F} - \mathbf{Q}_X (\mathbf{Q}_X^T \mathbf{F}) \qquad \text{(6)} $$

Proof: $\mathbf{X} (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T = \mathbf{Q}_X \mathbf{R}_X (\mathbf{R}_X^T \mathbf{Q}_X^T \mathbf{Q}_X \mathbf{R}_X)^{-1} \mathbf{R}_X^T \mathbf{Q}_X^T = \mathbf{Q}_X \mathbf{R}_X (\mathbf{R}_X^T \mathbf{R}_X)^{-1} \mathbf{R}_X^T \mathbf{Q}_X^T = \mathbf{Q}_X \mathbf{Q}_X^T$. $\blacksquare$
This formulation eliminates explicit inversion of $\mathbf{X}^T \mathbf{X}$, reducing the condition number from $\kappa(\mathbf{X})^2$ to $\kappa(\mathbf{X})$.

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3. Collinearity & Performance Metrics

Variance Inflation Factor (VIF)

For standardized factors $\mathbf{Z} \in \mathbb{R}^{N \times K}$ with correlation matrix $\mathbf{C} = \frac{1}{N-1} \mathbf{Z}^T \mathbf{Z}$:

$$ \mathrm{VIF}_k = (\mathbf{C}^{-1})_{kk} = \frac{1}{1 - R_k^2} \qquad \text{(7)} $$

where $R_k^2$ is the coefficient of determination from regressing factor $k$ on all other $K-1$ factors (Belsley et al., 2005).
For orthogonal factors, $\mathbf{C} = \mathbf{I}_K \implies \mathrm{VIF}_k = 1.0$ for all $k$.

Effective Number of Factors (ENF)

Using the spectral entropy of the eigenvalue distribution $p_k = \lambda_k / K$ of $\mathbf{C}$ (Meucci, 2009):

$$ \mathrm{ENF} \triangleq \exp\left(-\sum_{k=1}^K p_k \ln(p_k)\right) \qquad \text{(8)} $$

• $\mathrm{ENF} = 1.0$: Completely collinear factors (rank 1).
• $\mathrm{ENF} = K$: Completely orthogonal factors (maximum spectral entropy).

Theorem 5 (R² Additivity Under Orthogonality)

If factors $\mathbf{Q} \in \mathcal{V}_K(\mathbb{R}^N)$ are orthonormal ($\mathbf{Q}^T \mathbf{Q} = \mathbf{I}_K$), then in the multiple regression $\mathbf{y} = \sum_{k=1}^K \beta_k \mathbf{q}_k + \boldsymbol{\varepsilon}$, the overall $R^2$ decomposes strictly additively:

$$ R^2_{\text{total}} = \sum_{k=1}^K R^2_k \qquad \text{(9)} $$

Proof: Since $\mathbf{q}_i^T \mathbf{q}_j = 0$ for $i \ne j$, cross-terms $\beta_i \beta_j \mathbf{q}_i^T \mathbf{q}_j$ vanish identically from the regression sum of squares $\lVert \sum_k \beta_k \mathbf{q}_k \rVert^2 = \sum_k \beta_k^2 \lVert \mathbf{q}_k \rVert^2$. $\blacksquare$

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Comparison: Lowdin vs Gram-Schmidt vs PCA

Dimension	Lowdin Symmetric	Gram-Schmidt	Principal Component Analysis (PCA)
Orthogonality	$\widetilde{\mathbf{F}}^T \widetilde{\mathbf{F}} = \mathbf{I}$	$\widetilde{\mathbf{F}}^T \widetilde{\mathbf{F}} = \mathbf{I}$	$\widetilde{\mathbf{F}}^T \widetilde{\mathbf{F}} = \mathbf{I}$
Order Symmetry	Yes (Permutation Symmetric)	No (Order-dependent)	No (Sorted by eigenvalue variance)
Interpretability	High (Columns align 1-to-1 with original factors)	Medium (First factor pure, rest mixed)	Low (Synthetic eigenvectors lose factor meaning)
Distortion Metric	Minimum Frobenius Distance ($\min |\mathbf{F} - \widetilde{\mathbf{F}}|_F$)	Sub-optimal	Maximizes variance, not signal fidelity
Recommended Use	Multi-factor quantitative equity models	Hierarchical signal protection (Core + Satellites)	Dimension reduction / Statistical factor models

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Installation

git clone https://github.com/huuan26/OrthoQuant.git
cd OrthoQuant
pip install -e .

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Quickstart

import pandas as pd
from orthoquant import (
    generate_synthetic_factors,
    neutralize_factors,
    lowdin_orthogonalize,
    variance_inflation_factors,
    condition_number,
    information_coefficient,
)

# 1. Generate synthetic market universe (100 assets, 5 collinear factors, 4 sectors)
data = generate_synthetic_factors(n_assets=100, n_factors=5, base_correlation=0.75)
raw_factors = data["factors"]
risk_exposures = data["risk_exposures"]
forward_returns = data["forward_returns"]

print("--- Raw Factors Collinearity ---")
print("Condition Number:", condition_number(raw_factors))
print("Variance Inflation Factors (VIF):\n", variance_inflation_factors(raw_factors))

# 2. Cross-Sectional Risk Neutralization (strip sector & beta exposure)
neutral_factors = neutralize_factors(raw_factors, risk_exposures, standardize=True)

# 3. Lowdin Symmetric Orthogonalization
ortho_factors = lowdin_orthogonalize(neutral_factors, standardize=True)

print("\n--- After Lowdin Orthogonalization ---")
print("Condition Number:", condition_number(ortho_factors))  # Exactly 1.0
print("Variance Inflation Factors (VIF):\n", variance_inflation_factors(ortho_factors))  # Exactly 1.0

# 4. Evaluate Predictive Information Coefficient (Rank IC)
rank_ic = information_coefficient(ortho_factors, forward_returns, method="spearman")
print("\nOrthonormal Factors Rank IC:\n", rank_ic)

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Testing & Verification

Run the comprehensive test suite with coverage:

pytest

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Academic References & Bibliography

1. Lowdin, P.-O. (1950). On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals. The Journal of Chemical Physics, 18(3), 365–375.
2. Frisch, R., & Waugh, F. V. (1933). Partial time regressions as compared with individual trends. Econometrica, 1(4), 387–401.
3. Lovell, M. C. (1963). Seasonal adjustment of economic time series and multiple regression analysis. Journal of the American Statistical Association, 58(304), 993–1010.
4. Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations (4th ed.). Johns Hopkins University Press.
5. Higham, N. J. (1988). Computing a nearest symmetric positive semidefinite matrix. Linear Algebra and its Applications, 103, 103–118.
6. Belsley, D. A., Kuh, E., & Welsch, R. E. (2005). Regression diagnostics: Identifying influential data and sources of collinearity. John Wiley & Sons.
7. Meucci, A. (2009). Managing diversification. Risk, 22(5), 74–79.
8. Kakushadze, Z. (2015). Factor models and alpha streams. The Journal of Investment Strategies, 5(1), 1–45.

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Author & License

• Author: Phan Bùi Hữu Ân (huuan26)
• License: MIT License - see the LICENSE file for details.