Gaussian Curve Fitting

Gaussian curve fitting is a common task in data analysis, widely applied in fields such as electromagnetic wave signal analysis and spectral measurements. The Gaussian curve (also known as the normal distribution curve) exhibits a bell-shaped characteristic, with its mathematical expression containing three key parameters: amplitude (peak height), center position (mean), and width (standard deviation).

Core fitting methodology: Data Preparation: Raw collected data may contain noise and require preprocessing such as smoothing or baseline correction. In practice, this can be implemented using functions like scipy.signal.savgol_filter() for Savitzky-Golay smoothing or numpy.polyfit() for polynomial baseline fitting. Parameter Initialization: Initial parameters are estimated through data observation (e.g., peak position as initial mean value). Code implementation often uses numpy.argmax() to locate the data peak for initial center estimation. Optimization Algorithm: Common methods like least squares or gradient descent iteratively adjust parameters to minimize errors between theoretical curves and actual data. Libraries such as scipy.optimize.curve_fit() implement Levenberg-Marquardt algorithm for nonlinear least squares fitting. Result Evaluation: Fit quality is validated through residual analysis or R-squared values. The coefficient of determination (R²) can be calculated as 1 - (sum_squared_residuals / total_sum_squares).

Extended Applications: Multi-Peak Fitting: Complex signals may contain superimposed Gaussian peaks requiring decomposition via mixture models. The GaussianMixture class from sklearn.mixture enables probabilistic modeling of multiple peaks. Real-Time Fitting: For dynamic monitoring (e.g., laser wavelength drift), continuous fitting can be achieved using sliding window algorithms. This involves maintaining a fixed-size data buffer and performing iterative fitting with libraries like lmfit for parameter tracking.

Gaussian fitting not only extracts signal characteristics but also helps quantify signal-to-noise ratios or detect anomalous data points, making it a fundamental tool for electromagnetic wave researchers.