Dynamic Mode Decomposition for Financial Trading Strategies

A data-driven analysis applying Dynamic Mode Decomposition (DMD) to model complex dynamical systems for financial price prediction

---

🧠 Introduction & Motivation

Dynamic Mode Decomposition (DMD) is a powerful, data-driven "equation-free" algorithm that addresses this challenge. It analyzes a time-series of "snapshots" from a system and decomposes its complex spatio-temporal behavior into a set of low-rank, coherent structures called DMD modes. Each mode is associated with a specific oscillation frequency and a growth or decay rate (an eigenvalue). This allows us to build a simple, linear model that captures the dominant evolutionary patterns of the system.

📈 Financial Price Prediction: By treating the stock market as a complex dynamical system, we use DMD to identify dominant "growing modes" to predict price movements and inform trading strategies.

---

📊 2. Dataset Description

• Source: National Stock Exchange (NSE500 Index)

• Scope: Minute-wise stock prices of selected companies across sectors

• Sampling Frequency: 1 minute

• Training / Testing Split: 80/20

Parameter	Description	Example
N	Number of companies	10
M	Number of time snapshots	2000
Δt	Sampling interval	1 min
Observable	Stock price at time t	245.5 ₹

---

⚙️ 3. Methodology

We treat the stock market as a discrete-time dynamical system evolving as:

$$ x_{k+1} = F(x_k) $$

where ( x_k \in \mathbb{R}^n ) is the system state (vector of stock prices at time ( k )) and ( F ) is an unknown nonlinear mapping.
DMD approximates this system by finding a best-fit linear operator ( A ) such that:

$$ x_{k+1} \approx A x_k $$

The goal is to find ( A ) that minimizes the least-squares error between actual and predicted states.

---

🧩 3.1 Matrix Formulation

Snapshots of the system are organized as:

$$ X = [x_1, x_2, \dots, x_{m-1}] $$

$$ X' = [x_2, x_3, \dots, x_m] $$

We seek ( A ) satisfying:

$$ X' = A X $$

The optimal approximation is given by:

$$ A = X' X^{+} $$

where ( X^{+} ) is the Moore–Penrose pseudoinverse of ( X ).

---

🧮 3.2 Dimensionality Reduction via SVD

To mitigate noise and reduce dimensionality, we compute the truncated Singular Value Decomposition (SVD):

$$ X = U \Sigma V^{T} $$

where:

$$ \begin{aligned} U &\in \mathbb{C}^{n \times r} &&\text{— POD modes} \\ \Sigma &\in \mathbb{C}^{r \times r} &&\text{— singular values} \\ V &\in \mathbb{C}^{m \times r} &&\text{— temporal coefficients} \end{aligned} $$

The reduced linear operator becomes:

$$ \tilde{A} = U^{T} X' V \Sigma^{-1} $$

---

🧠 3.3 Eigen Decomposition

We compute eigenvalues and eigenvectors of ( \tilde{A} ):

$$ \tilde{A} W = W \Lambda $$

Then, the exact DMD modes are:

$$ \Phi = X' V \Sigma^{-1} W $$

Each mode ( \phi_k) evolves in time as:

$$ x(t) \approx \sum_{k=1}^{r} b_k \phi_k e^{\omega_k t} $$

where:

$$ \omega_k = \frac{\ln(\lambda_k)}{\Delta t} $$

and ( \lambda_k ) are the DMD eigenvalues.

---

💹 3.4 Interpretation for Financial Prediction

If:

$$ \Re(\lambda_k) > 1 $$

→ Growing mode → increasing stock value

If:

$$ \Re(\lambda_k) < 1 $$

→ Decaying mode → decreasing stock value

---

🧰 3.5 Implementation Overview

Implemented using: PyDMD — Python Dynamic Mode Decomposition library
Dataset: NSE500 stock price time series

🖥️ Graphical User Interface (GUI)

Built using Streamlit with the following features:

• 📈 Stock Price Prediction — Predicts future stock movement using DMD reconstruction.
• 💹 Stock Recommendation — Ranks stocks by growing eigenvalues (positive real part).
• 🎨 Visualization Tools — Interactive plots for:
  • DMD Modes
  • Eigenvalue Spectra
  • Temporal Dynamics

---

📊 4. Results & Analysis

🧮 Metric	📏 Value	📖 Interpretation
MAE	15.5299	Average absolute prediction error
MAPE	1.94%	Excellent short-term forecasting accuracy (less than 2% relative error)

---

🧩 Key Observations

• 🔹 Eigenvalue Analysis: Revealed clear growing and decaying modes in the stock time series.

• 🔹 Accuracy: DMD effectively captured short-term market dynamics with minimal computational cost.

• 🔹 Visualization Insight: Stem plots of DMD modes showed distinct temporal oscillations and trend separations.

  

---

⚡ Insight:
The model demonstrates that Dynamic Mode Decomposition is not only effective for physical systems but also a powerful analytical tool in financial forecasting — offering interpretability, speed, and accuracy.

The growth rate (real part of eigenvalue) represents trend strength, while the complex part encodes cyclic behavior.

This allows ranking stocks by their dominant growing modes, forming the basis for automated trading recommendations.