The data explored here examines daily average temperatures collected from January 1, 1955 to August 13, 2010 at weather stations located in Raleigh, McGuire Airforce Base, Fairbanks and New Orleans. These slides will walk you through multiple approaches for deciding how to analyze large data sets containing seasonality and high day-to-day variability.

A good place to start is just plotting the data. Below are plots of the average daily temperatures (in degrees Fahrenheit) for all four locations. What do these plots indicate?

Figure 1: Plots of Average daily temperature

The next slide has an applet that lets you change the years to compare the daily temperature between two years. Can you find two years that would indicate tempuratures are rising? Can you find two years that would indicate the tempuratures are falling? Is one year obviously warmer or cooler than another? Or other observations?Discuss any observations.

Boxplots are one way to summarize data to get a sense of the overall distribution.

The next slide has an applet that displays boxplots. You can choose yearly, monthly or daily. For example, if you choose yearly, this creates a boxplot for each year while if you choose monthly this creates 12 boxplot; one for each month.

Look at all three (yearly, monthly and daily) and discuss your observations.

It can be difficult to see changes over time because of the seasonal effect. One way to eliminate this is to plot the temperature for one day each year. The next slide has an applet that lets you pick date and plots the temperture for that one date from 1955 to 2010. The monthly option plots the average temperature for the month that you selected rather than the particular day.

Pick a day in the winter and one in the summer. What do you notice about the variablity? Does this make sense?

Is there more or less variabilty between the daily and monthly option? Why?

Another way to remove seasonality from data in a series through time is to compare points that should be the same with respect to seasonal effects. For instance, there should be no seasonal effect if you plot only the temperature readings taken on January 1 of each year or only those taken on August 17 - which you just played with. Let’s look for a more holistic approach.

The temperature data has a seasonal component with a period of 365 days. Letting \(T_t\) denote the temperature reading at time \(t\), the following differences remove the seasonal component:

\[\LARGE D_t = T_t - T_{t-365}\]

Does it make sense that the red line is close to 0? If there were a linear trend in temperature, then the differences should be randomly distributed about the average yearly change.

We are searching for a small signal (in this case a temperature change), if any, within data that are very noisy (due to day to day variations). One way to smooth out some of this variability is to use averaging. The next app plots yearly average temperature over time.