Our task is to create a model predicting the hourly water temperature for March and April of
2013 using hourly temperatures data from September 2005 to February 2013. We are looking at the changes in temperature given each hour on any given day of the months in order to get a more accurate prediction of the water temperature. The coefficients of month and hour explain the seasonal and hourly effects on the overall water temperature.
Since we are predicting the water temperature, we need preceding hourly temperatures as references and that’s why we have included “temperature (-1)” in our model. In other words, the temperature of current period depends on the preceding period, and the temperature for the preceding period depends on the one before it, the list goes on and on. Because of this relationship between the temperature from last period and this period, we have to adjust our prediction and correct for this relationship, modeling for dependence over time. By letting this relationship present in our data, our predictions will not be as accurate because the changes in temperature may be caused by some outside factors such as weather, climate change, amount of sun, etc. Thus, we have included the AR(1) term, which improves the accuracy of our model and how well the model accounts for variation from 23% to 99%. Our task is to predict the hourly temperature for March and April of 2013, so we need to take a closer look at the hourly and monthly temperature. Therefore, we expanded our data to look at the hourly changes and monthly changes in water temperature. The reason why we don’t need to breakdown the daily changes is because it is the average daily temperature changes of different months, so they vary from season to season. Estimating the change in temperature for each of the days would not give us any insight on the overall trend of the data or help estimate hourly temperature for that matter.
We excluded the last variable in each of the 12 months and 24 hours is because we want something as a reference. Otherwise we will be falling into a loop trying to figure out what each coefficient represents. The constant term in our model represents the average water temperature of the last month