Gramatrix 📊

Instagram Engagement Pattern Analysis and Prediction using Linear Algebra

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🎯 Overview

Gramatrix is a Linear Algebra-based project that analyzes Instagram engagement data and predicts the performance of new posts using matrix operations.

The project models real-world engagement data as a system of linear equations and applies mathematical concepts to extract meaningful insights and make predictions.

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🧠 Objective

• Identify key factors influencing engagement
• Detect and analyze redundancy in features
• Discover dominant engagement patterns
• Predict engagement of new Instagram posts

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🧮 Mathematical Model

We model the system as:

[ Ax \approx b ]

• A → Feature matrix (followers, hashtags, etc.)
• x → Weights (importance of each feature)
• b → Actual engagement

We compute the optimal weights using Least Squares:

[ x = (A^T A)^{-1} A^T b ]

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🔄 Project Pipeline

1. Load real-world dataset
2. Convert data into matrix form
3. Apply Gaussian Elimination (RREF)
4. Analyze rank and linear independence
5. Detect redundancy using correlation
6. Apply Gram-Schmidt orthogonalization
7. Compute least squares solution
8. Perform projection (Ax ≈ b)
9. Evaluate error (Mean Squared Error)
10. Visualize results
11. Perform eigenvalue analysis

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📊 Results

• Reels (post_type) have the highest impact on engagement
• Hashtags significantly improve engagement
• Previous engagement strongly influences future performance
• Posting time moderately affects reach
• The model shows strong prediction accuracy

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📸 Results Visualization

📊 Model Performance (Actual vs Predicted)

🧮 Feature Importance

📈 Correlation Matrix

🔮 Prediction Example

🔢 Eigenvalue Analysis

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🔮 Prediction

The model predicts engagement for new posts using:

[ \hat{b} = A_{new} \cdot x ]

Example Input:

[6000, 1, 120, 8, 19, 6, 400]

Output:

Predicted Engagement ≈ 928

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📁 Project Structure

gramatrix/
│
├── data/
│   └── instagram_data.csv
│
├── screenshots/
│   ├── graph.png
│   ├── weights.png
│   ├── correlation.png
│   ├── prediction.png
│   └── eigen.png
│
├── gramatrix.ipynb
├── README.md
└── requirements.txt

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⚙️ Requirements

• Python
• NumPy
• Pandas
• Matplotlib
• SymPy

Install dependencies:

pip install numpy pandas matplotlib sympy

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🚀 How to Run

1. Open the notebook in VS Code or Google Colab
2. Upload instagram_data.csv
3. Run all cells
4. View outputs and predictions

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🎤 Key Concepts Used

• Matrix Representation
• Gaussian Elimination (RREF)
• Rank and Linear Independence
• Gram-Schmidt Orthogonalization
• Least Squares Approximation
• Projection onto Subspace
• Eigenvalues and Eigenvectors

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💡 Conclusion

Gramatrix demonstrates how Linear Algebra can be applied to real-world data analysis. It transforms raw Instagram metrics into meaningful insights and enables prediction of engagement using mathematical modeling.

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License

MIT

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Built by Jashruth K A