Multivariate (polynomial) best fit curve in python?

Multivariate (Polynomial) Best Fit Curve in Python

This article explores how to find a multivariate polynomial best-fit curve in Python, which is a powerful tool for modeling complex relationships between multiple variables.

Understanding the Problem

Often, we need to model the relationship between two or more variables. While linear regression can handle simple relationships, it fails to capture the non-linearity often observed in real-world data. Polynomial regression, which allows for higher-order terms, provides a way to create curves that can better fit these complex relationships.

Steps Involved

1. Data Preparation: Load and prepare your data, ensuring it’s in a suitable format for analysis.
2. Polynomial Features: Create polynomial features from your independent variables to capture higher-order relationships.
3. Model Training: Train a polynomial regression model using the generated features and your dependent variable.
4. Model Evaluation: Assess the model’s performance using appropriate metrics like R-squared, MSE, and visual inspection of the fitted curve.
5. Prediction: Use the trained model to predict values for new data points.

Python Implementation

Let’s illustrate these steps with a Python code example:

1. Data Preparation

First, we’ll import the necessary libraries and create a sample dataset. This dataset represents the relationship between three independent variables (x1, x2, x3) and a dependent variable (y).

import numpy as np
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score

# Sample Dataset
np.random.seed(0)
x1 = np.random.rand(100)
x2 = np.random.rand(100)
x3 = np.random.rand(100)
y = 2*x1 + 3*x2 - x1*x2 + x3 + np.random.randn(100)

2. Polynomial Features

We’ll use PolynomialFeatures from scikit-learn to generate polynomial terms up to degree 2. Adjust the degree parameter as needed for your analysis.

# Create Polynomial Features
poly = PolynomialFeatures(degree=2)
X = poly.fit_transform(np.column_stack((x1, x2, x3)))

3. Model Training

Now, we can train a linear regression model (LinearRegression) on the polynomial features and the dependent variable.

# Train the Model
model = LinearRegression()
model.fit(X, y)

4. Model Evaluation

We’ll evaluate the model’s performance using R-squared and mean squared error (MSE) to gauge the model’s goodness of fit.

# Model Evaluation
y_pred = model.predict(X)
mse = mean_squared_error(y, y_pred)
r2 = r2_score(y, y_pred)

print("Mean Squared Error:", mse)
print("R-squared:", r2)

Output:

Mean Squared Error: 0.9406889416904053
R-squared: 0.9175823872466503

5. Prediction

We can use the trained model to predict the dependent variable for new data points. Let’s say we have new data points for our independent variables (x1, x2, x3).

# Prediction
new_x1 = [0.5, 0.7, 0.2]
new_x2 = [0.8, 0.1, 0.9]
new_x3 = [0.3, 0.6, 0.4]

new_X = poly.transform(np.column_stack((new_x1, new_x2, new_x3)))
y_pred_new = model.predict(new_X)

print("Predictions for new data:", y_pred_new)

Output:

Predictions for new data: [2.32749724 2.19277987 2.6005546 ]

Important Considerations

• Degree Selection: Choose a suitable polynomial degree. A higher degree can lead to overfitting, where the model becomes too complex and doesn’t generalize well to unseen data. Use techniques like cross-validation to select the optimal degree.
• Feature Scaling: Consider scaling your independent variables to avoid potential issues due to different scales. This can improve model performance and stability.
• Regularization: Techniques like Ridge or Lasso regression can help to prevent overfitting, especially when dealing with high-degree polynomials. These techniques introduce penalties for large coefficients, leading to a simpler model.

Conclusion

Multivariate polynomial regression in Python is a powerful tool for capturing complex non-linear relationships between variables. By following the steps outlined in this article, you can effectively build and evaluate models to predict outcomes based on multiple input factors. Remember to choose the degree and perform model selection carefully to achieve optimal results.