Sine Curve Fitting with Python: Parameter Estimation Using Least Squares Optimization

Fundamentals of Sine Curve Fitting

In data analysis and signal processing, sine curve fitting is a common technical requirement, particularly in economics, physics, and engineering. The general form of a sine function can be expressed as: y = A * sin(ωt + φ) + C, where A represents amplitude, ω is angular frequency, φ denotes phase, and C is the offset. The fitting objective is to accurately estimate these parameters from observed data.

Least Squares Optimization Method

Based on the best answer from the Q&A data, we employ the leastsq function from SciPy for least squares optimization. This method finds optimal parameters by minimizing the sum of squared residuals. Here's the complete implementation:

import numpy as np
from scipy.optimize import leastsq
import matplotlib.pyplot as plt

# Generate simulated data
N = 1000
t = np.linspace(0, 4*np.pi, N)
true_freq = 1.15247
data = 3.0 * np.sin(true_freq * t + 0.001) + 0.5 + np.random.randn(N)

# Initial parameter estimates
guess_mean = np.mean(data)
guess_std = 3 * np.std(data) / (2**0.5) / (2**0.5)
guess_phase = 0
guess_freq = 1
guess_amp = guess_std

# Define optimization function
optimize_func = lambda x: x[0] * np.sin(x[1] * t + x[2]) + x[3] - data

# Perform optimization
initial_guess = [guess_amp, guess_freq, guess_phase, guess_mean]
est_amp, est_freq, est_phase, est_mean = leastsq(optimize_func, initial_guess)[0]

# Generate fitted curve
fine_t = np.arange(0, max(t), 0.1)
data_fit = est_amp * np.sin(est_freq * fine_t + est_phase) + est_mean

# Visualize results
plt.figure(figsize=(10, 6))
plt.plot(t, data, '.', alpha=0.5, label='Original Data (with Noise)')
plt.plot(fine_t, data_fit, 'r-', linewidth=2, label='Fitted Curve')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.legend()
plt.grid(True)
plt.show()

Initial Parameter Estimation Strategy

Successful sine curve fitting heavily relies on reasonable initial parameter estimates:

1. Offset Estimation: Directly compute the data mean: guess_mean = np.mean(data)
2. Amplitude Estimation: Estimate based on data standard deviation: guess_amp = 3 * np.std(data) / (2√2)
3. Frequency Estimation: This is the most challenging aspect. If the approximate frequency range is known, it can be manually specified; otherwise, alternative methods are needed
4. Phase Estimation: Typically start from 0, allowing the optimization algorithm to adjust automatically

Frequency Estimation Challenges and Solutions

As noted in the Q&A data, direct optimization of frequency parameters can lead to fitting failures, especially when initial guesses deviate significantly from true values. As a supplement, we can reference the FFT method from the first answer:

def estimate_frequency_fft(tt, yy):
    """Estimate dominant frequency using FFT"""
    ff = np.fft.fftfreq(len(tt), (tt[1] - tt[0]))
    Fyy = abs(np.fft.fft(yy))
    # Exclude zero-frequency peak (corresponding to offset)
    dominant_idx = np.argmax(Fyy[1:]) + 1
    return abs(ff[dominant_idx])

# Improve frequency estimation in original code
guess_freq = estimate_frequency_fft(t, data - guess_mean)
print(f"FFT Estimated Frequency: {guess_freq:.4f}")
print(f"True Frequency: {true_freq:.4f}")
print(f"Relative Error: {abs(guess_freq - true_freq)/true_freq*100:.2f}%")

Fitting Quality Assessment

Evaluating sine curve fitting quality requires multiple metrics:

# Calculate goodness-of-fit metrics
residuals = data - (est_amp * np.sin(est_freq * t + est_phase) + est_mean)
RSS = np.sum(residuals**2)  # Residual Sum of Squares
TSS = np.sum((data - np.mean(data))**2)  # Total Sum of Squares
R_squared = 1 - (RSS / TSS)

print(f"Amplitude Estimate: {est_amp:.4f}")
print(f"Frequency Estimate: {est_freq:.4f}")
print(f"Phase Estimate: {est_phase:.4f}")
print(f"Offset Estimate: {est_mean:.4f}")
print(f"R² Value: {R_squared:.4f}")

Practical Application Considerations

In practical applications, sine curve fitting requires attention to:

1. Data Preprocessing: Ensure uniformly sampled time series, interpolate if necessary
2. Noise Handling: Filtering may be required in high-noise environments
3. Multiple Frequency Components: More complex models needed for signals with multiple frequencies
4. Boundary Conditions: Account for phase parameter periodicity to avoid 2π integer multiples

Extended Applications

The methods described can be extended to more complex scenarios:

1. Multiple Superimposed Sine Waves: Extend optimization function to handle multiple frequency components
2. Nonlinear Least Squares: Use curve_fit function for more flexible interfaces
3. Real-time Fitting: Implement real-time sine parameter estimation with sliding window techniques

By combining least squares optimization with FFT-based frequency estimation, we establish a robust sine curve fitting framework. This approach maintains computational efficiency while handling real-world data with varying noise levels, providing reliable tools for economic cycle analysis, signal processing, and other applications.