Phase Space Reconstruction Using C-C Method

The C-C method is a widely used technique in nonlinear time series analysis for phase space reconstruction, primarily employed to determine optimal time delay and embedding dimension parameters. This method calculates correlation integrals to identify the best time delay parameter, enabling effective phase space reconstruction that serves as foundation for subsequent chaotic characteristic analysis.

### Core Principles Correlation Integral Computation: The core concept involves calculating correlation integrals of time series by statistically analyzing distance distributions between point pairs in phase space. Correlation integrals reflect system correlations at different time scales, helping determine optimal time delays. In code implementation, this typically involves nested loops to compute pairwise distances using Euclidean or Manhattan distance metrics. Time Delay Selection: By evaluating correlation integral variations under different delays, the method identifies the first minimum point on the correlation integral curve, where the corresponding delay time represents the optimal choice. Algorithm implementation requires systematic scanning through delay parameters while storing integral values for extremum detection. Noise Resistance: The C-C method demonstrates certain robustness against noise, making it suitable for analyzing noisy time series in practical engineering applications through statistical averaging techniques.

### Key Programming Steps Data Preprocessing: Normalize input time series to eliminate dimensional influences using standardization or min-max scaling functions. Correlation Integral Calculation: Iterate through different time delays and embedding dimensions to compute corresponding correlation integral values, often implemented with matrix operations for efficiency. Minimum Point Identification: Analyze correlation integral curve trends to determine optimal time delay using peak detection algorithms or differential methods. Visualization Analysis: Plot correlation integral versus delay time curves for intuitive parameter selection, typically using plotting libraries like matplotlib in Python.

### Extended Applications Chaotic Time Series Prediction: After phase space reconstruction, the data can train neural networks or support vector machines for prediction tasks. Fault Diagnosis: Analyze phase space characteristics of mechanical vibration signals to identify equipment abnormal states through pattern recognition algorithms. Financial Data Analysis: Study nonlinear dynamic behaviors of stock prices or exchange rate fluctuations using dimensionality reduction techniques.

The C-C method provides reliable phase space reconstruction for nonlinear time series analysis, applicable to complex system studies across physics, biology, economics, and numerous other fields.