This repository demonstrates how polynomial regression works using both a simple example with synthetic data and a real-world example for training and testing a polynomial regression model. It contains examples of generating quadratic data, transforming the data into polynomial features, fitting a linear regression model, and visualizing the results using Python, NumPy, and scikit-learn.
Polynomial Regression is a type of regression analysis where the relationship between the independent variable X and the dependent variable 𝑌 is modeled as an nth-degree polynomial.
In this repository, to better understand polynomial regression, we provide:
A simple example using synthetic data to explain the concept of polynomial regression.
A real-world example where we train and test a polynomial regression model for house price prediction, utilizing the data provided above.
Step 1: Generate X Values
Step 2: Generate Y Values
Step 3: Plot the Data
Step 4: Transform Input Features to Polynomial Features
Step 5: Fit a Linear Regression Model to the Polynomial Features
Step 6: Predict Y Values Using the Fitted Model
Step 7: Plot the Synthetic Data Points and the Fitted Polynomial Curve
Important NOTE:
Using these 7 methods helps you decide whether Polynomial Regression is needed to model your data. Here is a brief explanation of each method and why it's used:
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Visual InspectionScatter Plot: To get an initial sense of the data's behavior./ / Linear Patterns: To determine if a simple linear model can explain the relationship./ / Non-linear Patterns: To identify if a more complex model, like polynomial regression, is needed.
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Residual AnalysisResidual Plot: To check the accuracy of the model's predictions./ / Random Scatter: To confirm if the model's assumptions hold true./ / Patterned Residuals: To detect any systematic errors indicating non-linearity.
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Model ComparisonFit Models: To see which model better captures the data's behavior./ / Performance Metrics: To quantitatively assess model fit and prediction accuracy./ / Improvement: To ensure the chosen model significantly improves prediction quality.
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Cross-ValidationK-Fold CV: To evaluate the model's ability to generalize to unseen data./ / Validation Scores: To ensure model consistency and reliability across different data splits./ / Consistent Improvement: To confirm that the polynomial model is consistently better.
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Hypothesis TestingANOVA: To statistically compare the models./ / Significant Improvement: To validate the necessity of a more complex model.
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Information CriteriaAIC/BIC: To balance model fit and complexity./ / Lower AIC/BIC: To prefer the model that best explains the data with the least complexity.
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Domain KnowledgeUnderstanding Relationships: To align the model with theoretical or empirical expectations.// Practical Implications: To ensure the model is practical and meaningful in the context of the problem domain.//
By addressing these points, you ensure a thorough evaluation of whether Polynomial Regression is needed, balancing complexity, accuracy, and practicality.
We have two sample. in this two sample have two figure of them for show data. we show before training to get the scatter need to use polynomial. we show these figure in below.
first figure:
The figure shows a scatter plot of synthetic data that appears to have a quadratic relationship between the variables. Each data point forms a pattern that is clearly not linear, which suggests that a simple linear model would not be appropriate for capturing the underlying relationship in the data. Instead, a polynomial model would be more suitable.
scoand figure:
The subsequent figures illustrate the relationship between geographical features (latitude and longitude) and house prices. These scatter plots reveal non-linear relationships, evidenced by the curved patterns and clusters of data points. This non-linearity suggests that a polynomial model would be more appropriate to capture the complexities in the data, leading to better performance in predicting house prices.
In result:
These scatter plots highlight the need for a polynomial model, specifically a second-degree polynomial regression, to effectively model the observed data. This approach is crucial for accurately capturing the non-linear relationships and underlying patterns that a linear model would miss. Employing a polynomial regression model ensures better predictive accuracy and a deeper understanding of the true nature of the data.
- I’ve only reviewed the first method in the above section. I encourage you to explore the other methods to gain more insights from the data. This will help you make a more informed decision about which algorithm or model to use.
To get started with this project, ensure you have Python 3 installed. Clone the repository and install the required dependencies:
Clone the repository
Install dependencies:
pip install -r requirements.txt
Development Environments: You can use development environments like VS Code, Jupyter Notebook, or Google Colab to run and edit the project.
For any questions or to get in touch, feel free to contact me via email at 12shim73@gmail.com or on LinkedIn: Shima Soleymanipour.