# Ensemble in stochastic process

I am learning a time series and forecasting course.In the book "The Analysis of Time Series by Chris Chatfield" it says that

We only have single outcome of the process and a single observation on the random variable at time t.Nevertheless we may regard the observed time series as just one example of the infinite set of time series which might have been observed.This infinite set of time series is sometimes called ensemble.Every member of the ensemble is a possible realization of the stochastic process.

I don't understand what it says here.

How is it possible it get an infinite time series if one observation is possible at a given time t?
Can someone please explain the ensemble part.
Also what is meant by

Every member of the ensemble is a possible realization of the stochastic proces

## 1 Answer

This is most easily understood if you have some experience with probability theory. Mathematically speaking, a random variable is a measurable function from some background probability space into another space. It is just a function. Say, we're measuring the height of the next person passing by my window. We could define a stochastic variable from the background space into the real numbers and use it to describe this height. Now we just have a random variable. When somebody actually passes by, we would have a real number. An realization of that random variable. Let's say that realization is 1.9 m. It could have been infinitely many other numbers.

Your situation is the same. You have a background probability space and a function from that space into some other space. This time we call the function a stochastic process (because it is a function into some space indexed by time, for instance), and the realization of the stochastic process is now called an observed time series. Apart from the names, the situation is as above.

So to answer your questions more directly, think of the ensemble as the set of all the (observed) time series you could possibly see as realizations of a stochastic process, the one you actually observe is just one of them. This is why every member of the ensemble is a possible realization of the process.

The background space is a mathematical construction to define random variables. Another way to think of it would be to say that if we could repeat our observation, we would possibly have seen another time series. Hence, the same stochastic process can give rise to multiple realizations (observed time series and members of the ensemble). The observed time series (members of the ensemble) are the numbers you actually observe, while the stochastic process is a mathematical construction, explaining where the numbers came from. We are often interested in gaining knowledge on the stochastic process but we only have the information in the observed time series.

• Thanks for the explanation. So if I understood correctly this is what happens.In a stochastic process we could have different time series if we repeat the process several times.All these different possible time series are the members of the ensemble.At a particular time when we observe the process what we get is the one possible realization.Since any of the members could have been that actually observed time series every member of the ensemble becomes a possible realization of the process.Can these members of the ensemble be of different probability distributions? – clarkson Sep 29 '14 at 18:02
• I agree with your recap. As for the question: only the stochastic process has a probability distribution. The members of the ensemble don't. They're just arrays of numbers, possible values of the stochastic process. Again, think of the random variable describing height. The random variable has a distribution, 1.9 m doesn't. – swmo Sep 29 '14 at 18:07