# Estimating/predicting maximum of sum process for modeling number of cars on the road?

I have two series of information about (for example) the amount of cars on a two roads over a 1 mile stretch. I have collected 4 outcomes: Number of cars passing through an intersection at 1, 10, and 60 minutes as well as greatest number of cars in the area in the last hour.

I'd like to model the maximum of the sum of the number of cars on both roads in an hour for a given hour.

Adding an example for clarity (as per comments): If (over a 20 minute period, sampled every minutes), my observations are 2,4,7,4,8 and 5,4,3,2,1, the maximum over both roads is 10 (7 + 3), not 13 (5 + 8), because the queue specific maximums happened at different times.

• You should give an example. It took me some minutes to figure out what you mean. Like for 1,3,4 and 4,3,1 it's not 4+4, but 3+3 at $t=2$. – ziggystar Jan 24 '14 at 14:27

Since you just want an approximation (OK, let's glorify it and call it an "estimate"), one simple approach would be to make a bootstrap estimate. Do this in two stages. First, draw a large number of bootstrap samples from each of the two series and collect the maximums from each sample. Second, draw a large number of samples from each of the set of maxima and add them pairwise to get a sample of maximum totals. Use the mean, median, or mode of this sample as your estimate, with the appropriate (empirical) bootstrap confidence interval.

The burning question we all have for you is: is it reasonable to assume the numbers of cars on either road are independent? If so, you can create a smoothing process for the numbers of cars over time and add the smoothing curves together. Something like a smoothing spline in negative binomial regression model would be sufficiently flexible and involve few assumptions.

If the numbers of cars on either road are not independent, then you will need to merge the road information as a function of time into a 3 column array with a column for time and the corresponding columns for road1 and road2 traffic. You can define a 4th column of roadsum traffic and apply the same smoothing spline methods there.

If you do not have these time series indexed by time to find the corresponding traffic occurrences, then you're out of luck. There's no reason to assume that a peak in traffic on road1 could be indicative of a synchronous or lagged peak in traffic on road2. Those assumptions would just be unreasonable.