Fourier State-Space Decomposition for Financial Time-Series Analysis

Abstract

Financial time-series are notoriously complex, and conventional models often fail to capture their multi-layered dynamics. We introduce a novel framework for decomposing financial signals by modeling them as manifestations of underlying wave-like phenomena. This approach operates on the hypothesis that an observable price series is a projection from a higher-dimensional state space containing components visible only in specific analytical domains. Our methodology utilizes wavelet transforms to construct a time-localized composite signal from price data and employs the Fourier transform to reveal latent, frequency-dependent components. We then leverage a genetic algorithm to discover the transformation function that maps these latent time- and frequency-domain representations back to the original price series. This fusion of signal processing and evolutionary computation provides a powerful tool for automated feature engineering and offers a new lens for interpreting market dynamics, with significant implications for prediction, anomaly detection, and the development of more robust theoretical models.

Keywords: State-Space Models, Fourier Analysis, Wavelet Transform, Genetic Algorithms, Financial Time-Series, Econophysics

1. Introduction

The behavior of financial markets is one of the most challenging systems to model, characterized by non-stationarity, non-linearity, and high levels of noise. While classical theories like the random walk hypothesis have provided foundational insights, they are insufficient for capturing the intricate, structured patterns that often appear to govern market movements. This paper departs from classical assumptions and explores the premise that financial time-series are fundamentally wave-like in nature. We posit that the observable price is a lower-dimensional projection of a more complex, underlying state. The central challenge, which this work addresses, is to define and identify this latent state representation and the transformation that connects it to the observable market behavior.

Our primary contribution is a novel methodology for the deconstruction and analysis of financial data. The approach is founded on the hypothesis that crucial information about the market's state is encoded across multiple analytical domains simultaneously. We propose a process that involves several stages of transformation to reveal this hidden architecture:

Wave Parameterization: We first translate standard open-high-low-close (OHLC) data into a set of instantaneous wave parameters—amplitude, frequency, and phase—to represent the market's micro-movements.

Time-Localized State Representation: A Haar wavelet transform is applied over a rolling window of these parameters to synthesize a single, composite signal. This technique produces a signal that captures localized time-frequency characteristics, which is critical for analyzing transient market events.

Frequency Domain Decomposition: The Fast Fourier Transform (FFT) is then applied to this composite wave, transforming it into the frequency domain to expose periodic components that are latent or hidden in the time-domain view.

Transformation Discovery: Finally, we employ a genetic algorithm to probe the relationship between these representations. The algorithm's objective is to discover an optimal linear transformation that maps the time-domain, inverse-transformed, and frequency-domain signals to the original close price.

By successfully identifying this transformation, we not only validate the hypothesis of a multi-domain latent state but also create a rich set of features for downstream applications. This framework moves beyond simple price prediction and offers a method for gaining a deeper theoretical understanding of the forces driving market dynamics.

2. Theoretical Background

Traditional econometric models, such as ARIMA and GARCH, have proven effective for modeling certain characteristics of financial time-series, like auto-correlation and volatility clustering. However, they are often constrained by assumptions of linearity and stationarity and may not fully capture the complex, multi-scale nature of market dynamics.

Our approach draws inspiration from the field of signal processing, which has developed powerful techniques for analyzing complex waveforms.

Fourier Analysis provides a mathematical framework for decomposing any signal into a sum of simple sine and cosine waves. It is a powerful tool for identifying global, periodic components within a dataset but lacks temporal resolution; it can identify what frequencies are present but not when they occur.

Wavelet Analysis addresses this limitation by using basis functions (wavelets) that are localized in both time and frequency. This allows for a multi-resolution analysis that can detect transient phenomena, such as sudden volatility spikes or shifts in market structure, which are of paramount importance in financial analysis.

By combining these techniques with evolutionary computation, we create a system that does not rely on pre-defined models but instead discovers the underlying structure directly from the data.

3. Methodology

Our methodology is a multi-stage pipeline designed to transform raw price data into a structured, multi-domain state representation and then discover the function that maps this state back to the observable price.

3.1. Wave Parameterization

Given a time-series of OHLC data, we first derive three instantaneous wave parameters for each time step t.

Amplitude (At​): The absolute change in the closing price, representing the energy of the price movement. At​=∣Pclose,t​−Pclose,t−1​∣

Frequency (ft​): We use the reciprocal of the True Range (TR) as a proxy for frequency. The TR is a measure of volatility that accounts for price gaps. A high TR (high volatility) implies a low-frequency, high-energy movement, while a low TR implies a high-frequency, low-energy oscillation. TRt​=max(Phigh,t​−Plow,t​,∣Phigh,t​−Pclose,t−1​∣,∣Plow,t​−Pclose,t−1​∣) ft​=TRt​+ϵ1​ where ϵ is a small constant to prevent division by zero.

Phase (ϕt​): The directional angle of the price movement, calculated using the arctan2 function to preserve quadrant information. ϕt​=arctan2(ΔPclose,t​,Δt)

3.2. Time-Localized State Synthesis via Wavelet Transform

To synthesize these three parameters into a single representative signal, we use a rolling-window wavelet transform. The parameters (At​,ft​,ϕt​) are first normalized to a common scale. Then, for each time step t, we take a window of size N ending at t. This window of 3N data points is flattened into a one-dimensional segment. We then apply a one-dimensional Discrete Wavelet Transform (DWT) using the Haar basis function. The combinedwave (Swave​) at time t is defined as the mean of the absolute values of the approximation coefficients (cA) produced by the DWT. This process yields a signal that is sensitive to local changes in the underlying wave dynamics.

3.3. Frequency Domain Decomposition via Fourier Transform

To uncover global periodicities latent within the combinedwave signal, we apply the Fast Fourier Transform (FFT):

F(ω)=F(Swave​)

This transforms the signal from the time domain to the frequency domain, yielding a series of complex numbers. We retain the real (Freal​) and imaginary (Fimag​) parts as separate feature vectors. We also compute the Inverse FFT (Fifft​), which reconstructs the original combined wave and serves as a component in the final transformation.

3.4. Transformation Discovery via Evolutionary Computation

The final and most critical step is to find the transformation function that maps our derived state components back to the original closing price. We model this as an optimization problem solved by a Genetic Algorithm (GA).

Individual (Chromosome): Each candidate solution is an individual represented by a vector of four floating-point numbers: [c1​,c2​,c3​,c4​].

Transformation Function: The coefficients of an individual are used in a linear combination of our derived signals to produce a reconstructed price, P^close​: P^close​=c1​Swave​+c2​Freal​+c3​Fimag​+c4​Fifft​

Fitness Function: The fitness of an individual is determined by how accurately its transformation reconstructs the actual closing price. We seek to minimize the Mean Squared Error (MSE): Fitness=M1​t=1∑M​(P^close,t​−Pclose,t​)2 where M is the number of time steps.

Evolutionary Process: The GA evolves a population of individuals over a set of generations using standard operators: tournament selection, blend crossover (CXBlend), and Gaussian mutation. This process efficiently searches the vast solution space for the vector of coefficients that minimizes the MSE.

4. Experimental Results and Analysis

The proposed framework was applied to a sample of high-frequency EUR/USD minute-level data for illustrative purposes. The GA was configured with a population of 100 individuals and run for 50 generations.

The results demonstrate that the best_transform signal, reconstructed using the optimal coefficients found by the GA, effectively tracks the general trend and structure of the original close price. Visual analysis shows a strong correlation between the two, suggesting the derived latent variables contain significant information about the price. The combined_wave signal acts as a more volatile, responsive indicator that often leads or amplifies movements in the close price, highlighting its potential as a short-term feature.

A correlation analysis confirms these visual findings, typically revealing a strong positive correlation between close and best_transform. The correlation between close and the individual latent signals (combined_wave, fftreal, etc.) is often weaker, which validates our hypothesis: no single component tells the whole story. It is the specific, weighted combination discovered by the GA that successfully models the original signal.

5. Applications and Implications

The successful decomposition and reconstruction of a financial signal using this framework has several profound implications and enables a new class of applications.

Enhanced Feature Engineering: The combined_wave and best_transform signals can be used as highly informative, low-latency features for downstream machine learning models aimed at prediction or classification. They encapsulate complex dynamics that would be difficult to engineer manually.

Advanced Anomaly Detection: The reconstruction error (MSE) from our model can serve as a powerful anomaly score. A sudden spike in the error indicates that the market's behavior is deviating from its previously established internal structure, signaling a potential regime shift, black swan event, or data quality issue.

Market Regime Identification: The optimal transformation coefficients are not static; they represent the market's internal dynamics at a given time. By running this analysis over different time periods, one can track the evolution of these coefficients. Significant changes could provide a quantitative method for identifying shifts between different market regimes (e.g., risk-on vs. risk-off, trending vs. mean-reverting).

Theoretical Modeling: This framework provides a new tool for testing hypotheses about market structure. It allows researchers to move beyond purely statistical descriptions and begin to probe the underlying components that may drive price formation.

6. Future Work

This paper presents a foundational methodology, and several avenues for future research are immediately apparent.

Exploration of Non-Linear Transformations: The current implementation uses a linear combination of features. The next stage of research will involve using techniques like Genetic Programming to allow the GA to evolve the entire mathematical structure of the transformation function, enabling the discovery of non-linear relationships.

Dynamic Coefficient Analysis: A comprehensive study of how the optimal coefficients evolve over time across various market conditions and assets would yield deeper insights into the nature of market regimes and their transitions.

State-Space Expansion: The framework can be extended to include multiple correlated assets. This would allow for the modeling of a shared latent state space, potentially uncovering how information and risk propagate across the financial system.

7. Conclusion

We have presented a novel framework that successfully decomposes a financial time-series into a multi-domain set of latent components and uses an evolutionary algorithm to discover the linear transformation that reconstructs the original signal. This approach validates the hypothesis that observable market behavior is a projection of a more complex, hidden state and provides a powerful, data-driven method for exploring that state's structure. By fusing advanced signal processing with evolutionary computation, we open new pathways for feature engineering, anomaly detection, and the development of a more fundamental, causal understanding of complex systems. The work represents a significant step toward augmenting human analysis with AI systems capable of perceiving and modeling the deep architecture of financial markets.