For a time, I once taught a module on time series analysis and forecasting to undergraduates. One aspect of such a module is to show different sorts of time series and how to build reasonably accurate forecasting models for them. As the models became more complex, we looked at real life time series to identify what sorts of models were best for each one. Of course, sooner or later, we had to model seasonality. After dispensing with the absurd series for the sales of Christmas trees. month by month, we looked at more serious examples. As I cycled to work, I brought in two series related to cyclists in the UK. The first showed the monthly distance cycled by people in the UK, month by month, based on surveys of people's travel habits. The second showed the monthly numbers of accidents to cyclists in the UK, taken from hospital admissions. Both showed strong seasonality, with peaks in the summer months. (Many casual cyclists do not use their bicycles in the colder months. I did and still do.) With my tongue in my cheek, I graphed these two series and announced that it was clearly more dangerous to cycle in the summer ... and waited for a response. Sooner or later, someone would point out that the total number did not represent risk. So, what does represent risk? A little more thought, and the answer came back; why not take the ratio of accidents to distance? With appropriate scales, that gave a third time series, and that too had seasonality. It peaked in September-October. Now we moved into the psychology behind such a seasonal variation, and my researches had shown that in those months it was still warm enough for the casual cyclists, but many neglected to use lights.
The BBC radio programme "More of Less" earlier this month included Rob Eastaway, playing a statistical game with children on a bus in London - spot the cyclist. One pupil scored the number of cyclists with helmets, the second counted those without. First one to reach a score of ten wins. Now helmet-wearing is not compulsory in the UK, but many cyclists wear helmets. Rob espected the game to end at about 10:5, but in the end it was 10:9 for helmets.
Tina and I have amused ourselves doing this counting on several rides on the cycle paths by the river Exe recently. Generally we find that about 20 to 25% of cyclists are without helmets on these paths, but there is some seasonal variation, by day of the week and time of day. More riders are helmetless at weekends, when there are leisure cyclists. Early morning cyclists are more inclined to wear helmets than those mid-morning. More riders are helmetless within a 3 mile radius of the city centre - suggesting that those who do a longer commute tend to wear a helmet. But there is another factor to the seasonality which the time series models would have difficulty representing- that of people cycling in groups. Three of four people together will often affect one another's use of helmets - either all with or all without.
I can't think of any serious reason for wanting to forecast the number of cyclists wearing helmets that you would see on a given day and time, but the analysis of past data would be an interetsing exercise for students. Any offers?