Here is the updated README.md file reflecting the use of the Jupyter Notebook file:

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README.md

Assignment: Causality in Data Mining

This repository contains solutions and explanations for Exercise Sheet 3: Causality from the Data Mining course at the University of Vienna, Winter Semester 2024/25. The assignment explores concepts of causality using statistical methods, with a focus on Granger causality, multivariate causal models, and bivariate additive noise models.

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Assignment Overview

Exercise 3-1: Granger Causal Test (6 points)

The objective is to analyze causal relationships between variables using Granger causality with real-world data on chickens ((Y)) and eggs ((X)) from U.S. egg production (1930–1935).

1. Auto-regression for (X) and (Y) (1.5 points)
   Compute regression coefficients ((\beta_0, \beta_1, \beta_2)) for predicting (Y) based on lagged values of (X) and (Y). Compute (RSS1) (Residual Sum of Squares for the full model).

2. Auto-regression without (Y) (1.5 points)
   Repeat the regression excluding (Y) and compute (RSS2) (Residual Sum of Squares for the reduced model).

3. Statistical Test (1.5 points)
   Use the Granger-Sargent test to determine if (X) Granger-causes (Y) with (\alpha = 0.05).

4. Causal Test for the Reverse Direction (1.5 points)
   Repeat the above steps to test if (Y) Granger-causes (X). Interpret the results to address the "chicken or egg" question.

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Exercise 3-2: Multivariate Granger Model (2 points)

Analyze causal relationships in high-dimensional settings using multivariate Granger models.

1. Consistency of Granger Models (1 point)
   Evaluate the consistency of causal inference with ordinary least squares and adaptive Lasso penalties. Discuss if the method generalizes well in high-dimensional data.

2. HMMLGA Algorithm (1 point)
   Explain the High-order Mixed Model Lasso Granger Algorithm (HMMLGA) and describe its key hyperparameters, including regularization parameter ((\lambda)), lag order ((d)), and stopping criteria.

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Exercise 3-3: Bivariate Causal Models on Non-Temporal Data (2 points)

Explore causal relationships using bivariate additive noise models.

1. Non-identifiability in Linear-Gaussian Cases (1 point)
   Explain why causal relationships between variables (X) and (Y) are non-identifiable when the noise is Gaussian and relationships are linear. Discuss the role of symmetry in the joint distribution.

2. Causal and Anti-Causal Directions (1 point)
   For (X =) age and (Y =) blood pressure, explain:

   • How to change (P(\text{effect}|\text{cause})) without affecting (P(\text{cause})).
   • How to change (P(\text{cause})) without affecting (P(\text{effect}|\text{cause})).
   • Challenges in the anti-causal direction.

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Key Topics Covered

• Granger Causality:
  • Statistical framework for analyzing temporal causal relationships.
  • Tests based on differences in residual sums of squares.
• Multivariate Granger Models:
  • Extensions to handle high-dimensional data with sparse penalties like Lasso.
• Bivariate Additive Noise Models:
  • Challenges of causal inference in non-temporal, symmetric distributions.
• Real-World Applications:
  • Example datasets and domain-specific interpretations (e.g., chickens and eggs, age and blood pressure).

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Repository Structure

├── Exercise_Sheet_3_Solution.ipynb  # Jupyter Notebook containing all solutions
├── README.md                        # Overview of the assignment

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Instructions

1. Clone this repository:

   git clone https://github.com/username/causality-assignment.git
   cd causality-assignment

2. Install dependencies:

   pip install -r requirements.txt

3. Run the solutions: Open the Jupyter Notebook in your preferred environment (e.g., Jupyter Lab or Jupyter Notebook):

   jupyter notebook Exercise_Sheet_3_Solution.ipynb

   • Exercise 3-1: Granger causality tests.
   • Exercise 3-2: Multivariate Granger models.
   • Exercise 3-3: Bivariate additive noise models.

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Acknowledgments

This assignment is part of the Data Mining course at the University of Vienna, guided by RNDr. Katerina Schindlerová, CSc., Privatdoz. and Pranava Mummoju, MSc..