Frozen Days Linear Regression

This project fits a linear regression model to data on the number of frozen days per year. It uses both the closed-form solution and gradient descent (implemented with PyTorch) and visualizes the data, the loss curve, and predictions.

Overview

The script performs the following steps:

1. Load a CSV file with columns:
   • year
   • days (number of frozen days in that year)
2. Plot the raw data (data_plot.jpg).
3. Normalize the input years to the range [0, 1].
4. Compute the closed-form least squares solution for linear regression.
5. Use gradient descent with PyTorch to learn the same model and record the loss over iterations.
6. Plot the loss curve (loss_plot.jpg).
7. Use the learned model to predict the number of frozen days for a given test year (e.g., 2024).
8. Print simple interpretations of the learned slope and limitations of the model.

File

• frozen_days_regression.py — main script
• ice_data.csv — dataset containing year and days
• toy.csv — small sample dataset for testing
• data_plot.jpg — plot of frozen days vs. year (auto-generated)
• loss_plot.jpg — gradient-descent loss plot (auto-generated)
• README.md — documentation

Requirements

• Python 3
• numpy
• pandas
• matplotlib
• torch

You can install the requirements with:

pip install -r requirements.txt

How to Run

Run the script with any CSV file:

python3 frozen_days_regression.py ice_data.csv

or:

python3 frozen_days_regression.py toy.csv

After running, the script will:

• Print the normalized design matrix (X_normalized)
• Print closed-form regression weights
• Print gradient-descent weights every 20 iterations
• Output a prediction for year 2024
• Output the sign of the slope (increasing or decreasing trend)
• Output a simple explanation and model limitations
• Save:
  • data_plot.jpg
  • loss_plot.jpg

What the Script Does

1. Loads year and frozen-day counts from CSV.
2. Plots the original data and saves the figure.
3. Normalizes year values to the range [0, 1].
4. Computes the linear regression solution using the closed-form formula.
5. Runs gradient descent using PyTorch for 200 iterations and tracks loss.
6. Saves a loss-curve plot.
7. Predicts frozen days for 2024.
8. Reports the trend direction (positive, negative, or none).
9. Prints why long-term predictions may not be reliable.

Notes

• Normalization is applied only to the input years; the target variable (days) is kept in original scale.
• The gradient descent implementation uses mean squared error as the loss.
• The chosen learning rate is tuned so that gradient descent converges close to the closed-form solution.