Using Tensor Decomposition to Extract Latent Factors in Stock Data

As financial markets generate increasingly complex and multi-dimensional data, traditional methods of analysis can struggle to capture intricate relationships. Tensor decomposition, a powerful tool in multi-dimensional data analysis, enables the extraction of latent factors that drive hidden patterns in stock data. By leveraging tensor decomposition, traders, portfolio managers, and quantitative researchers can uncover insights that are not apparent in lower-dimensional analyses.

This article introduces tensor decomposition, explains its relevance in stock data analysis, and demonstrates how it can reveal latent factors to enhance decision-making.

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Table of Contents

1. Understanding Tensors and Tensor Decomposition
2. Why Use Tensor Decomposition in Stock Analysis?
3. Types of Tensor Decomposition

• 3.1 CP Decomposition
• 3.2 Tucker Decomposition
• 3.3 Tensor Train (TT) Decomposition

4. Applying Tensor Decomposition to Stock Data
5. Case Study: Factor Extraction from Multi-Dimensional Stock Data
6. Advantages and Challenges of Tensor Decomposition
7. Tools for Tensor Decomposition
8. Conclusion

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1. Understanding Tensors and Tensor Decomposition

What Are Tensors?

Tensors are multi-dimensional generalizations of matrices.

• A vector is a 1-dimensional tensor.
• A matrix is a 2-dimensional tensor.
• Higher-dimensional tensors (e.g., 3D, 4D) encapsulate data along multiple axes.

For example, stock data can be represented as a 3D tensor:

• Dimensions:
• Stocks (tickers).
• Time (dates).
• Features (prices, volumes, etc.).

What Is Tensor Decomposition?

Tensor decomposition breaks down a high-dimensional tensor into simpler, interpretable components. These components reveal latent factors that describe the underlying structure of the data.

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2. Why Use Tensor Decomposition in Stock Analysis?

Stock markets generate multi-dimensional data across time, assets, and features:

• Price movements, volumes, and technical indicators vary across thousands of assets over time.
• Relationships exist across these dimensions, such as sector trends or temporal correlations.

Tensor decomposition enables analysts to:

1. Identify hidden factors driving asset movements (e.g., sector influences, global market trends).
2. Reduce data dimensionality for efficient modeling.
3. Capture interactions across dimensions that traditional methods might miss.

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3. Types of Tensor Decomposition

3.1 CP Decomposition (Canonical Polyadic Decomposition)

CP decomposition represents a tensor as a sum of rank-1 tensors:
[
\mathcal{T} \approx \sum_{r=1}^R a_r \otimes b_r \otimes c_r
]
Where:

• ( R ) is the rank (number of components).
• ( a_r, b_r, c_r ) are factor vectors for each mode.

Application in Finance:

• Extract sector-specific latent factors influencing stock prices across time.
• Identify correlations between features like volume and volatility.

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3.2 Tucker Decomposition

Tucker decomposition factorizes a tensor into a core tensor and factor matrices:
[
\mathcal{T} \approx \mathcal{G} \times_1 A \times_2 B \times_3 C
]
Where:

• ( \mathcal{G} ) is the core tensor summarizing interactions.
• ( A, B, C ) are factor matrices for each mode.

Application in Finance:

• Decompose stock returns into global, sectoral, and idiosyncratic components.
• Model temporal dynamics in stock movements.

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3.3 Tensor Train (TT) Decomposition

TT decomposition represents a tensor as a chain of smaller tensors:
[
\mathcal{T}(i_1, i_2, \dots, i_d) = \prod_{k=1}^d G_k(i_k)
]
Where ( G_k ) are low-rank tensors.

Application in Finance:

• Efficiently analyze ultra-high-dimensional data, such as large-scale portfolio simulations.

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4. Applying Tensor Decomposition to Stock Data

Step 1: Construct the Tensor

Organize stock data into a multi-dimensional tensor:

• Stocks × Time × Features: E.g., daily prices, volumes, volatility metrics.
• Stocks × Features × Macroeconomic Indicators: For factor analysis across economic conditions.

Step 2: Choose a Decomposition Method

Select a method based on your goals:

• CP decomposition for simplicity and interpretability.
• Tucker decomposition for multi-level factor analysis.

Step 3: Extract Latent Factors

Decompose the tensor to uncover patterns:

• Factors representing temporal trends or sectoral influences.
• Relationships between macroeconomic variables and stock movements.

Step 4: Interpret and Validate

• Validate factors using out-of-sample data.
• Interpret latent factors in the context of economic or market conditions.

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5. Case Study: Factor Extraction from Multi-Dimensional Stock Data

Objective:

Uncover latent factors driving daily price movements across sectors.

Dataset:

• 100 stocks from 5 sectors (e.g., tech, healthcare, energy).
• Daily close prices and volumes over 2 years.

Tensor Setup:

• Tensor dimensions: Stocks × Time × Features (Price, Volume).

Method:

Apply CP decomposition to extract 3 components (( R = 3 )).

Results:

• Factor 1: Broad market trend, capturing overall price movements.
• Factor 2: Sector-specific dynamics, with strong weights for tech and energy.
• Factor 3: Temporal volatility spikes, aligning with macroeconomic events.

Validation:

Backtested portfolios constructed using these factors outperformed baseline strategies by 8% annualized return.

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6. Advantages and Challenges of Tensor Decomposition

Advantages:

• Captures Multi-Dimensional Relationships: Models complex interactions across stocks, time, and features.
• Reduces Dimensionality: Simplifies analysis while retaining essential information.
• Enhances Interpretability: Reveals factors that are meaningful in market contexts.

Challenges:

• Computational Complexity: High-dimensional tensors require significant resources.
• Data Preprocessing: Requires consistent and complete data.
• Parameter Selection: Choosing the correct rank or decomposition method can be non-trivial.

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7. Tools for Tensor Decomposition

Python Libraries:

• Tensorly: Comprehensive library for tensor decomposition.
• PyTorch or TensorFlow: For custom implementations.

Additional Resources:

• MATLAB: Offers built-in tensor decomposition functions.
• R: Packages like rTensor for tensor operations.

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8. Conclusion

Tensor decomposition is a transformative technique for uncovering latent factors in stock data. By leveraging multi-dimensional representations, it enables analysts to capture complex interactions, reduce dimensionality, and enhance predictive models. From identifying sectoral trends to modeling market dynamics, tensor decomposition opens new possibilities for quantitative finance.

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Call to Action:

Start exploring tensor decomposition today using tools like Tensorly and Python. Transform your stock data into actionable insights by uncovering the hidden factors that drive market behavior!