Skill: wu-uk/hvac-control-scipy-curve-fit (version 0.1.0)

--- name: scipy-curve-fit description: Use scipy.optimize.curve_fit for nonlinear least squares parameter estimation from experimental data. --- # Using scipy.optimize.curve_fit for Parameter Estimation ## Overview `scipy.optimize.curve_fit` is a tool for fitting models to experimental data using nonlinear least squares optimization. ## Basic Usage ```python from scipy.optimize import curve_fit import numpy as np # Define your model function def model(x, param1, param2): return param1 * (1 - np.exp(-x / param2)) # Fit to data popt, pcov = curve_fit(model, x_data, y_data) # popt contains the optimal parameters [param1, param2] # pcov contains the covariance matrix ``` ## Fitting a First-Order Step Response ```python import numpy as np from scipy.optimize import curve_fit # Known values from experiment y_initial = ... # Initial output value u = ... # Input magnitude during step test # Define the step response model def step_response(t, K, tau): """First-order step response with fixed initial value and input.""" return y_initial + K * u * (1 - np.exp(-t / tau)) # Your experimental data t_data = np.array([...]) # Time points y_data = np.array([...]) # Output readings # Perform the fit popt, pcov = curve_fit( step_response, t_data, y_data, p0=[K_guess, tau_guess], # Initial guesses bounds=([K_min, tau_min], [K_max, tau_max]) # Parameter bounds ) K_estimated, tau_estimated = popt ``` ## Setting Initial Guesses (p0) Good initial guesses speed up convergence: ```python # Estimate K from steady-state data K_guess = (y_data[-1] - y_initial) / u # Estimate tau from 63.2% rise time y_63 = y_initial + 0.632 * (y_data[-1] - y_initial) idx_63 = np.argmin(np.abs(y_data - y_63)) tau_guess = t_data[idx_63] p0 = [K_guess, tau_guess] ``` ## Setting Parameter Bounds Bounds prevent physically impossible solutions: ```python bounds = ( [lower_K, lower_tau], # Lower bounds [upper_K, upper_tau] # Upper bounds ) ``` ## Calculating Fit Quality ### R-squared (Coefficient of Determination) ```python # Predicted values from fitted model y_predicted = step_response(t_data, K_estimated, tau_estimated) # Calculate R-squared ss_residuals = np.sum((y_data - y_predicted) ** 2) ss_total = np.sum((y_data - np.mean(y_data)) ** 2) r_squared = 1 - (ss_residuals / ss_total) ``` ### Root Mean Square Error (RMSE) ```python residuals = y_data - y_predicted rmse = np.sqrt(np.mean(residuals ** 2)) ``` ## Complete Example ```python import numpy as np from scipy.optimize import curve_fit def fit_first_order_model(data, y_initial, input_value): """ Fit first-order model to step response data. Returns dict with K, tau, r_squared, fitting_error """ t_data = np.array([d["time"] for d in data]) y_data = np.array([d["output"] for d in data]) def model(t, K, tau): return y_initial + K * input_value * (1 - np.exp(-t / tau)) # Initial guesses K_guess = (y_data[-1] - y_initial) / input_value tau_guess = t_data[len(t_data)//3] # Rough guess # Fit with bounds popt, _ = curve_fit( model, t_data, y_data, p0=[K_guess, tau_guess], bounds=([0, 0], [np.inf, np.inf]) ) K, tau = popt # Calculate quality metrics y_pred = model(t_data, K, tau) ss_res = np.sum((y_data - y_pred) ** 2) ss_tot = np.sum((y_data - np.mean(y_data)) ** 2) r_squared = 1 - (ss_res / ss_tot) fitting_error = np.sqrt(np.mean((y_data - y_pred) ** 2)) return { "K": float(K), "tau": float(tau), "r_squared": float(r_squared), "fitting_error": float(fitting_error) } ``` ## Common Issues 1. **RuntimeError**: Optimal parameters not found - Try better initial guesses - Check that data is valid (no NaN, reasonable range) 2. **Poor fit (low R^2)**: - Data might not be from step response phase - System might not be first-order - Too much noise in measurements 3. **Unrealistic parameters**: - Add bounds to constrain solution - Check units are consistent


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