5.1 Exponential equation .ipynb

Kernel: Python 3 (ipykernel)

Non-Linear Regression with Python

Non-linear regression is used when data shows a curved or non-linear relationship between input features and output. Unlike linear regression which fits straight lines, non-linear regression can model more complex patterns.

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What is Non-Linear Regression?

A regression technique used to model the relationship between a dependent variable and one or more independent variables when the model equation is non-linear in the parameters.

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Exponential Regression

Equation:

Where:

• ( a ) = initial value

• ( b ) = growth or decay rate

• ( e ) = Euler’s number (~2.718)

Common Use Cases:

• Population growth

• Compound interest

• Radioactive decay

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from sklearn.metrics import r2_score

# Simulated exponential data
np.random.seed(0)
X = np.linspace(0, 4, 50)
y = 2 * np.exp(1.5 * X) + np.random.normal(0, 25, size=X.shape)

# Define exponential function
def exp_func(x, a, b):
    return a * np.exp(b * x)

# Fit model
params, _ = curve_fit(exp_func, X, y, p0=(1, 1))  # p0: initial guess
a, b = params
y_pred = exp_func(X, a, b)

# R² Score
r2 = r2_score(y, y_pred)
print(f"Fitted: a = {a:.2f}, b = {b:.2f}, R² = {r2:.4f}")

# Plot
plt.scatter(X, y, label='Observed')
plt.plot(X, y_pred, color='red', label='Exponential Fit')
plt.legend()
plt.title("Exponential Regression")
plt.grid(True)
plt.show()

Fitted: a = 2.12, b = 1.48, R² = 0.9795