Quantitative Models and Construction Methods - **Model Name**: Floating Frequency Fourier Transform Model **Construction Idea**: The model aims to overcome the limitations of traditional Fourier analysis by introducing floating frequencies, which adapt to real-world market dynamics such as macroeconomic changes and capital structure shifts[64][65][66] **Construction Process**: 1. **Filtering**: Use Butterworth filter to remove high-frequency noise and retain trend data. The filter is applied to the logarithmic weekly closing prices of the Shanghai Composite Index, with cutoff frequency set at 0.1 and sampling frequency at 1.0[65][67] 2. **Frequency and Amplitude Selection**: Perform discrete fast Fourier transform on filtered data to extract amplitude and phase information. Key frequencies are identified based on amplitude peaks, e.g., 94.1 weeks, 80.5 weeks, and 62.6 weeks[75][78] 3. **Waveform Reconstruction**: Combine selected frequencies and amplitudes to reconstruct the market trend curve. The reconstructed curve includes direct current components and sinusoidal terms with specific frequencies and initial phases[68][78] **Evaluation**: The floating frequency approach improves prediction accuracy by preserving real-world frequency characteristics and reducing extrapolation errors[64][65][66] - **Model Name**: Dual Optimization Floating Frequency Fourier Transform Model **Construction Idea**: Enhance the floating frequency Fourier transform by optimizing both amplitude and frequency to better fit market trends and improve prediction accuracy[85][86][87] **Construction Process**: 1. **Inner Optimization**: Use non-linear least squares to optimize amplitude values (sin and cos terms) for given initial frequency values. Frequencies are combined with the top two amplitudes from Fourier transform results to form a fitting curve[87][88] 2. **Outer Optimization**: Apply particle swarm optimization to determine the best floating frequencies (e.g., 39.02 weeks, 19.17 weeks, and 8.13 weeks). Training data spans 10 years (2014-2024), and the model is validated using 2024-2025 data[87][88][100] **Evaluation**: The dual optimization method significantly improves trend accuracy compared to simple floating frequency models, especially in capturing major market movements[85][86][100] Model Backtesting Results - **Floating Frequency Fourier Transform Model**: - Key frequencies: 94.1 weeks, 80.5 weeks, 62.6 weeks[78][80][81] - Observed discrepancies in certain periods, e.g., 2014-2015 and 2017-2019, indicating limitations in amplitude selection methods[81] - **Dual Optimization Floating Frequency Fourier Transform Model**: - Optimized frequencies: 39.02 weeks, 19.17 weeks, 8.13 weeks[100] - Improved accuracy in major trend predictions, e.g., capturing relative highs in September 2021[100] - Future predictions: Long-term upward trend from 2024 to 2027, with key oscillations in August 2025 and August 2026[102][103] Quantitative Factors and Construction Methods - **Factor Name**: Harmonic Sine Wave Factor **Construction Idea**: Simulate price movements using harmonic sine waves to replicate Elliott Wave patterns[39][42] **Construction Process**: - Formula: $y(n) = 4*sin(0.00125n)+2*sin(0.01n)+1*sin(0.04n)+0.5*sin(0.16n)$ - Variables: $n$ represents time, $y(n)$ represents price index - Frequency structure: Base frequency 0.00125, harmonic frequencies follow $2^k$ relationship (k=-3,0,2,4) - Amplitude decay: Coefficients follow $2^{-m}$ rule (m=-1,0,1,2)[39][42] **Evaluation**: Successfully replicates 5-3 wave patterns described in Elliott Wave theory, providing a mathematical basis for wave analysis[39][42] - **Factor Name**: Pletcher Fibonacci Wave Factor **Construction Idea**: Introduce quasi-periodic characteristics by incorporating non-harmonic frequencies into wave simulation[50][51] **Construction Process**: - Formula: $y(n) = 4*sin(0.00125n)+sin(0.01n)+1/2*sin(0.04n)+1/4*sin(0.17n)+1/8*sin(0.72n)+1/16*sin(0.305n)$ - Frequency relationships: Some frequencies maintain harmonic relationships (e.g., 0.00125, 0.01, 0.04), while others exhibit non-harmonic relationships (e.g., 0.17, 0.72, 0.305)[50][51] **Evaluation**: Captures floating frequency characteristics and quasi-periodic behavior, aligning with real-world market dynamics[50][51] Factor Backtesting Results - **Harmonic Sine Wave Factor**: - Successfully replicates Elliott Wave patterns, including 5-3 wave structures[39][42] - **Pletcher Fibonacci Wave Factor**: - Demonstrates quasi-periodic behavior, reflecting floating frequency characteristics in market data[50][51]