Month: December 2023

Srijana Mada MTH 522 Project3

In my recent analysis, I delved into various data-driven models to predict future temperatures using the historical data of “Temp_Avg.” These models are designed to pick up on patterns in the temperature changes over time, helping us better understand and foresee temperature fluctuations. I considered a range of methods, from simple linear and exponential models to more complex ones involving quadratic trends and seasonal variations. To make these models flexible, I added terms like ‘t,’ ‘t_squared,’ and ‘log(t)’ to the dataset, representing different time-related characteristics.

Let’s take a look at how these models performed:

1. Linear Model: This assumes a straightforward, linear relationship between time and temperature. It did reasonably well, with a Root Mean Square Error (RMSE) of about 14.56, indicating its ability to capture linear trends.
2. Exponential Model: This model, attempting to catch exponential growth or decay in temperature, didn’t perform as well, showing a high RMSE of about 4.37. This suggests it might not be the best fit for our dataset.
3. Quadratic Model: Introducing a squared term to account for curvature in temperature trends, this model did fairly well with an RMSE of around 14.66, capturing more complex patterns than the linear model.
4. Additive Seasonality: Considering seasonal variations, this model performed impressively with an RMSE of approximately 2.59, showing its effectiveness in capturing repeating patterns over months.
5. Additive Seasonality with Linear Trend: Combining a linear trend with seasonal variations, this model achieved an RMSE of about 2.63, indicating reasonably accurate predictions of temperature fluctuations.
6. Additive Seasonality with Quadratic Trend: Introducing a quadratic trend along with seasonality, this model did well with an RMSE of approximately 2.72, capturing more complex temperature patterns.
7. Multiplicative Seasonality: Considering both seasonal and overall multiplicative variations, this model didn’t perform well, showing an extremely high RMSE.
8. Multiplicative Seasonality with Linear Trend: Similar to the multiplicative model, including a linear trend and seasonality, this model also showed an extremely high RMSE, suggesting challenges in accurately predicting temperature variations.
9. Multiplicative Seasonality with Quadratic Trend: Combining a quadratic trend with multiplicative seasonality, this model yielded an extremely high RMSE, indicating difficulties in accurately forecasting temperature patterns.

Conclusion: Among the tested models, those incorporating additive seasonality (both linear and quadratic trends) performed the best, with lower RMSE values. These models effectively captured seasonal variations in temperature, providing more accurate forecasts compared to other methods. The linear and quadratic models without seasonality did reasonably well but were outperformed by the seasonal models. On the other hand, the exponential and multiplicative models showed challenges and might not be the best choices for predicting temperatures in this dataset.

For my next step, I looked into different ways of predicting future temperatures using a method called Exponential Smoothing. This method is great for capturing patterns and trends in historical temperature data. I tested four specific techniques: Simple Exponential Method, Holt’s Method, and two variations of Holt-Winter’s method that consider different seasonal patterns. The goal was to see which one does the best job of predicting temperature changes over time.

Here’s a simple comparison of the results:

1. Simple Exponential Method: This method, which looks at the historical average temperature without considering trends or seasons, had an error of around 24.38%.
2. Holt’s Method: Adding trend information to the historical average temperature increased the error to about 37.79%.
3. Holt-Winter’s with Additive Seasonality and Additive Trend: This model, considering both trend and seasonal variations, had a lower error of around 3.85%. It assumes that both the trend and seasons add up to influence temperature changes.
4. Holt-Winter’s with Multiplicative Seasonality and Additive Trend: Similar to the previous one but assuming that seasonal variations have a multiplying effect, this method had an error of approximately 3.99%.

Conclusion: Among these methods, the one that considers both additive seasonality and additive trend performed the best, with the lowest error. This suggests that this model might be the most suitable for predicting temperatures based on the given historical data

Information Gain in Decision Trees: 

1. Purpose of Decision Trees: Decision trees help make decisions by breaking down a problem into smaller, manageable steps, like a flowchart.

2. Entropy: Entropy is a measure of confusion or disorder. In decision trees, it gauges how mixed up our data is in terms of categories.

3. Information Gain: Information Gain is like a guide for decision trees. It helps decide which question (feature) to ask first to make our dataset less confusing.

4. How it Works: At each step, the tree looks at different questions (features) and picks the one that reduces confusion the most—this is high Information Gain.

5. Goal: The goal is to keep asking the best questions (features) to split our data until we reach clear and tidy groups.