Curve Fitting and Least Squares
Curve fitting is a fundamental tool in data analysis that allows you to model the relationship between variables by fitting a mathematical function to observed data points. By identifying the curve that best represents your data, you can make predictions, understand underlying trends, and quantify relationships. Curve fitting is widely used in scientific research, engineering, and economics for tasks such as calibrating instruments, modeling experimental data, and forecasting.
12345678910111213141516171819202122232425import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt # Generate synthetic data with noise def model_func(x, a, b, c): return a * np.exp(-b * x) + c np.random.seed(0) x_data = np.linspace(0, 4, 50) y_data = model_func(x_data, 2.5, 1.3, 0.5) + 0.2 * np.random.normal(size=x_data.size) # Fit the model to the data popt, pcov = curve_fit(model_func, x_data, y_data) # Plot the data and the fitted curve plt.scatter(x_data, y_data, label="Data") plt.plot(x_data, model_func(x_data, *popt), color="red", label="Fitted Curve") plt.legend() plt.xlabel("x") plt.ylabel("y") plt.title("Curve Fitting with curve_fit") plt.show() print("Fitted parameters:", popt)
123456789101112131415161718192021222324252627282930313233import numpy as np from scipy.optimize import least_squares import matplotlib.pyplot as plt # Generate synthetic data def true_func(x): return 3 * x**2 + 2 * x + 1 np.random.seed(1) x = np.linspace(-3, 3, 30) y = true_func(x) + np.random.normal(scale=2, size=x.size) # Define the residuals function for least_squares def residuals(params, x, y): a, b, c = params return a * x**2 + b * x + c - y # Initial guess for parameters initial_guess = [1, 1, 1] result = least_squares(residuals, initial_guess, args=(x, y)) a_fit, b_fit, c_fit = result.x # Plot the results plt.scatter(x, y, label="Data") plt.plot(x, true_func(x), label="True Function", linestyle="dashed") plt.plot(x, a_fit * x**2 + b_fit * x + c_fit, color="red", label="Least Squares Fit") plt.legend() plt.xlabel("x") plt.ylabel("y") plt.title("Least Squares Fitting with least_squares") plt.show() print("Estimated parameters:", a_fit, b_fit, c_fit)
After fitting a model to your data, you obtain parameter estimates that best describe the underlying relationship according to the chosen function. These fitted parameters can be interpreted in the context of your model; for example, the coefficients in a polynomial or the rate in an exponential decay. The quality of the fit can be assessed by examining residuals (differences between observed and predicted values), visualizing the fit, and computing statistical metrics such as the sum of squared residuals or the coefficient of determination (R²). Understanding the meaning of these parameter estimates and their uncertainties is crucial for drawing reliable conclusions from your analysis.
1. Which function is used for curve fitting in SciPy?
2. What does least squares fitting minimize?
3. How can you assess the quality of a curve fit?
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Curve fitting is a fundamental tool in data analysis that allows you to model the relationship between variables by fitting a mathematical function to observed data points. By identifying the curve that best represents your data, you can make predictions, understand underlying trends, and quantify relationships. Curve fitting is widely used in scientific research, engineering, and economics for tasks such as calibrating instruments, modeling experimental data, and forecasting.
12345678910111213141516171819202122232425import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt # Generate synthetic data with noise def model_func(x, a, b, c): return a * np.exp(-b * x) + c np.random.seed(0) x_data = np.linspace(0, 4, 50) y_data = model_func(x_data, 2.5, 1.3, 0.5) + 0.2 * np.random.normal(size=x_data.size) # Fit the model to the data popt, pcov = curve_fit(model_func, x_data, y_data) # Plot the data and the fitted curve plt.scatter(x_data, y_data, label="Data") plt.plot(x_data, model_func(x_data, *popt), color="red", label="Fitted Curve") plt.legend() plt.xlabel("x") plt.ylabel("y") plt.title("Curve Fitting with curve_fit") plt.show() print("Fitted parameters:", popt)
123456789101112131415161718192021222324252627282930313233import numpy as np from scipy.optimize import least_squares import matplotlib.pyplot as plt # Generate synthetic data def true_func(x): return 3 * x**2 + 2 * x + 1 np.random.seed(1) x = np.linspace(-3, 3, 30) y = true_func(x) + np.random.normal(scale=2, size=x.size) # Define the residuals function for least_squares def residuals(params, x, y): a, b, c = params return a * x**2 + b * x + c - y # Initial guess for parameters initial_guess = [1, 1, 1] result = least_squares(residuals, initial_guess, args=(x, y)) a_fit, b_fit, c_fit = result.x # Plot the results plt.scatter(x, y, label="Data") plt.plot(x, true_func(x), label="True Function", linestyle="dashed") plt.plot(x, a_fit * x**2 + b_fit * x + c_fit, color="red", label="Least Squares Fit") plt.legend() plt.xlabel("x") plt.ylabel("y") plt.title("Least Squares Fitting with least_squares") plt.show() print("Estimated parameters:", a_fit, b_fit, c_fit)
After fitting a model to your data, you obtain parameter estimates that best describe the underlying relationship according to the chosen function. These fitted parameters can be interpreted in the context of your model; for example, the coefficients in a polynomial or the rate in an exponential decay. The quality of the fit can be assessed by examining residuals (differences between observed and predicted values), visualizing the fit, and computing statistical metrics such as the sum of squared residuals or the coefficient of determination (R²). Understanding the meaning of these parameter estimates and their uncertainties is crucial for drawing reliable conclusions from your analysis.
1. Which function is used for curve fitting in SciPy?
2. What does least squares fitting minimize?
3. How can you assess the quality of a curve fit?
Danke für Ihr Feedback!
