Analyze Temperatures and Update Regression

You are given historical daily temperature data for New York City and several nearby towns. Each row contains a date, the NYC temperature, and the temperature for each town on that date. Answer the following: 1. **Volatility analysis** - Determine which town has the largest temperature fluctuation over time. - Clearly define the metric you use for fluctuation. 2. **Similarity analysis** - Determine which town's temperature pattern is most similar to NYC's. - Clearly define the similarity metric you use. 3. **Prediction task** - Use the towns' temperature data to predict NYC temperature. - Train a regression model and evaluate it using mean squared error, or MSE. 4. **Greedy feature selection** - Given a target number `k`, choose `k` towns to use as features. - Start with no selected towns. - At each step, add the town that gives the largest reduction in validation MSE when combined with the already selected towns. - Return the selected towns and the final MSE. 5. **No-intercept linear regression** - Implement simple linear regression without an intercept for a single predictor `x` and target `y`. - First solve the batch case, where all data is available at once. - Then solve the streaming case, where `(x, y)` pairs arrive one at a time and the current slope must be updated without recomputing from scratch over all past data.