# [R] Diff. between time series and stochastic process

Rolf Turner r.turner at auckland.ac.nz
Tue Oct 2 02:11:23 CEST 2007

```On 2/10/2007, at 12:52 AM, Duncan Murdoch wrote:

> On 10/1/2007 8:39 AM, Megh Dal wrote:
>> Hi, Can anyone give me a good explanation about what is the
>> difference between Stochastic Process and Time Series? In my
>> knowledge Time series process is one type of Stochastic process.
>> Am I right? I need a further explanation.
>
> You're partly right.  When thinking in terms of probability theory,
> "time series" usually means a stochastic process with a discrete time
> index.
>
> But "time series" is also used to mean an observed series of
> observations indexed by time, and I think the closest thing for a
> stochastic process would be "a realization of a stochastic
> process".  So
> "time series" is not just a special case of "stochastic process".

I think that more generally the phrase ``stochastic process'' can be
used to mean
``***realization*** of a stochastic process'' just as ``time series''
can be used to mean
both a discrete time stochastic process and a realization thereof.
The dual usage is
sloppy but it is often convenient to be sloppy (or inconvenient to be
more precise).

Actually the same dichotomy occurs in ``ordinary'' (iid) statistics
where people are
often lax about distinguishing between a random sample and a
realization of a
random sample.  (Often one wishes to switch one's point of view back
and forth
and it gets tedious to be precise about the distinction.)

Anyhow the basic answer to the original question is ``Yes; a time
series is a particular
kind of stochastic process.''  It may be helpful to think about 4
types of stochastic processes:

(1) discrete time --- discrete values (e.g. Markov chains)
(2) discrete time --- continuous values (the ``usual'' sort of time
series, although people do study discrete valued

time series, see e.g. the book by MacDonald and Zucchini)
(3) continuous time --- discrete values (e.g. continuous time Markov
chains)
(4) continuous time --- continuous values (e.g. Brownian motion;
what people usually refer to as (vanilla) stochastic processes).

cheers,

Rolf Turner

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