Impara Curve Fitting and Least Squares | Optimization and Root Finding

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.


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import 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)


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import 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.

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Sezione 3. Capitolo 3

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              12345678910111213141516171819202122232425
            
import 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)


              123456789101112131415161718192021222324252627282930313233
            
import 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)

Tutto è chiaro?

Grazie per i tuoi commenti!

Sezione 3. Capitolo 3

Curve Fitting and Least Squares

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?

Awesome!

Curve Fitting and Least Squares

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?