# Is a time series the same as a stochastic process?

A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"?

• A time series is a stochastic process with a discrete-time observation support. A stochastic process can be observed in continuous time. (It may also be that series are more related with observations and stochastic processes with the random object behind.) Commented Dec 5, 2014 at 18:34
• "Series" imply discrete or finite nature as opposed to potentially continuous nature of the "process". Commented Dec 5, 2014 at 19:22
• A stochastic process need not evolve over time; it could be stationary. To my mind, the difference between stochastic process and time series is one of viewpoint. A stochastic process is a collection of random variables while a time series is a collection of numbers, or a realization or sample path of a stochastic process. With additional assumptions about the process, we might wish to use the histogram of values of numbers the time series as an estimate of the common density (or mass function) of all the random variables comprising the process etc. Commented Dec 5, 2014 at 20:02
• @DilipSarwate, time series can be stationary or not. Commented Dec 5, 2014 at 22:18
• @Aksakal I beg to differ. Suppose the statistician has observed the finite-length time series $$1,0,-1,0,1,0,-1$$ Is this a stationary series? How can you tell that it is (or is not)? Unless we have available several time series (for the same time instants) from which we might be able make inferences about the stochastic process ("Gee, the histograms of values taken on by $X_n$ is pretty much the same regardless of choice of $n$"). But a single sequence of numbers? You cannot say whether the series is stationary or not but you could assume so re the underlying stochastic process model Commented Dec 6, 2014 at 1:23

Because many troubling discrepancies are showing up in comments and answers, let's refer to some authorities.

James Hamilton does not even define a time series, but he is clear about what one is:

... this set of $$T$$ numbers is only one possible outcome of the underlying stochastic process that generated the data. Indeed, even if we were to imagine having observed the process for an infinite period of time, arriving at the sequence $$\{y_t\}_{t=\infty}^\infty = \{\ldots, y_{-1}, y_0, y_1, y_2, \ldots, y_T, y_{T+1}, y_{T+2}, \ldots, \},$$ the infinite sequence $$\{y_t\}_{t=\infty}^\infty$$ would still be viewed as a single realization from a time series process. ...

Imagine a battery of $$I$$ ... computers generating sequences $$\{y_t^{(1)}\}_{t=-\infty}^{\infty},$$ $$\{y_t^{(2)}\}_{t=-\infty}^{\infty}, \ldots,$$ $$\{y_t^{(I)}\}_{t=-\infty}^{\infty}$$, and consider selecting the observation associated with date $$t$$ from each sequence: $$\{y_t^{(1)}, y_t^{(2)}, \ldots, y_t^{(I)}\}.$$ This would be described as a sample of $$I$$ realizations of the random variable $$Y_t$$. ...

(Time Series Analysis, Chapter 3.)

Thus, a "time series process" is a set of random variables $$\{Y_t\}$$ indexed by integers $$t$$.

In Stochastic Differential Equations, Bernt Øksendal provides a standard mathematical definition of a general stochastic process:

Definition 2.1.4. A stochastic process is a parametrized collection of random variables $$\{X_t\}_{t\in T}$$ defined on a probability space $$(\Omega, \mathcal{F}, \mathcal{P})$$ and assuming values in $$\mathbb{R}^n$$.

The parameter space $$T$$ is usually (as in this book) the halfline $$[0,\infty)$$, but it may also be an interval $$[a,b]$$, the non-negative integers, and even subsets of $$\mathbb{R}^n$$ for $$n\ge 1$$.

Putting the two together, we see that a time series process is a stochastic process indexed by integers.

Some people use "time series" to refer to a realization of a time series process (as in the Wikipedia article). We can see in Hamilton's language a reasonable effort to distinguish the process from the realization by his use of "time series process," so that he can use "time series" to refer to realizations (or even data).

• (+1) I think the last paragraph is particularly important (albeit subtle). I did want to add, though, that the idea of a "continuous time series" is sometimes seen. Occasionally the phrase is used simply to indicate that the variable itself is continuous, rather than discrete, but I have also seen it used to indicate that time is being sampled continuously, so "indexed by integers" may not be a universally accepted definition. See e.g. here, inside Time Series: Theory & Methods by Brockwell & Davis. Commented Mar 15, 2016 at 23:59
• @Silverfish I appreciate those comments. Ultimately, though, I find them unconvincing for the simple reason that "series" is universally used in mathematics to refer to a function with a countable domain. "Sampled continuously" cannot be included within that concept. I'm not challenging your observations that some authors may have referred to continuous-time stochastic processes as "series"--I'm only saying that if this is the case, then they are abusing a well established terminology.
– whuber
Commented Mar 16, 2016 at 14:31
• I think there is a degree of the "description versus prescription" debate in this. The idea of a "continuous time series" is definitely minority usage (I wonder if this is field-dependent, my limited understanding is that signal processing people would usually refer to a "continuous time signal" rather than "series") and personally I'm inclined to agree that the word "series" logically is more consistent with discrete sampling. I just wanted to point out the minority usage is not unheard of, even among experts, which may account for some of the confusion generated. Commented Mar 16, 2016 at 15:43
• @Silverfish, thus, for this minority who also consider continuous time series, stochastic process is equal to time series? Commented Oct 14, 2019 at 16:39

### Defining a stochastic process

Let $$(\Omega, \mathcal{F}, P)$$ be a probability space. Let $$S$$ be another measurable space (such as the space of real numbers $$\mathbb{R}$$). Speaking somewhat imprecisely:

• A random variable is a measurable function from $$\Omega$$ to $$S$$.
• A stochastic process is a family of random variables indexed by time $$t$$.
• For any time $$t \in \mathcal{T}$$, $$X_t$$ is a random variable
• For any outcome $$\omega \in \Omega$$, $$X(\omega)$$ is a realization of the stochastic process, a possible path taken by $$X$$ over time.

### Defining a time series

While a stochastic process has a crystal clear, mathematical definition. A time series is a less precise notion, and people use time series to refer to two related but different objects:

1. As WHuber describes, a stochastic process indexed by integers or some regular, incremental unit of time that can in a sense by mapped to integers (eg. monthly data).
2. A variable observed over time (often at regular intervals). This could be the realization of a stochastic process. Sometimes this is referred to as time series data.

# Example: two flips of a tack

Let $$\Omega = \{ \omega_{HH}, \omega_{HT}, \omega_{TH}, \omega_{TT}\}$$. Let $$X_1, X_2$$ be the result of flip 1 and 2 respectively.

$$X_1(\omega) = \left\{ \begin{array}{rr} 1: & \omega \in \{ \omega_{HH}, \omega_{HT}\} \\ 0: & \omega \in \{\omega_{TH}, \omega_{TT} \}\end{array} \right.$$

$$X_2(\omega) = \left\{ \begin{array}{rr} 1: & \omega \in \{ \omega_{HH}, \omega_{TH}\} \\ 0: & \omega \in \{\omega_{HT}, \omega_{TT} \}\end{array} \right.$$

So clearly $$\{X_1, X_2\}$$ is a stochastic process. People may also call it a time series since the indexing is by integers. People may also call the realization of $$X$$, eg. $$X(\omega_{HH}) = (H, H)$$, a time series or time series data.

• This is interesting. But given a time-series, how to distinguish if it is just a sequence of realizations of the same random variable or if it is the realization of a sequence of random variables that in turn are realizations of an underlying random process? Commented May 18, 2022 at 16:41
• @Barzi2001 A sequence of independent random variables following the same distribution (eg. a series of rolls of the same 6-sided die) is a special case of a stochastic process. Commented May 18, 2022 at 17:39

A stochastic process is a set or collection of random variables $\left\{X_t\right\}$ (not necessarily independent), where the index t takes values in a certain set, this set is ordered and corresponds to the moment of time. example random walk. Time series Is the realisation of the stochastic process.

The difference between a stochastic process and a time series is somewhat like the difference between a cat on a keyboard and an answer on Stack Exchange: Cats on keyboards can produce answers, but cats on keyboards are not answers. Furthermore, not every answer is produced by a cat on a keyboard.

A time series can be understood as a collection of time-value–data-point pairs. A stochastic process on the other hand is a mathematical model or a mathematical description of a distribution of time series¹. Some time series are a realisation of stochastic processes (of either kind). Or, from another point of view: I can use a stochastic process as a model to generate a time series.

Furthermore, time series can also be generated in other ways:

• They can be the result of observations and are thus generated by reality. While I can model reality as a stochastic process (I could also say that I regard reality as a stochastic process), reality is not a stochastic process in the same way that the interior of a box is not a set of points (though we often regard the two equivalent in modelling contexts).

• They can be generated by deterministic processes. Now, strictly speaking, we could (and arguably should) define stochastic processes and deterministic processes in a way that the latter are special cases of the former, but we very rarely make use of this and speaking of deterministic processes as special cases of stochastic processes may cause some confusion – you could compare it to calling $x=2$ a system of non-linear equations.

¹ If it is a discrete-time stochastic process. Continuous-time stochastic process are distributions of functions rather than time series.

• It is unclear whether you are making a distinction between a model and a dataset or whether you are trying to make some other point. It is also unclear what you take a stochastic process to be. (All you have said is that it is "not even" a "discrete-time stochastic process.") These uncertainties in your exposition might add to the confusion rather than resolving it.
– whuber
Commented Dec 5, 2014 at 21:51
• @whuber: I edited my answer to clarify some aspects, but I think you also misunderstood some the “not even” sentence. Commented Dec 5, 2014 at 22:04

I appreciate all contributed discussions/comments on the subject of Time series vs Stochastic process. Here is my understanding of the difference: Time series is a phenomenon observed, recorded as a series of numbers that is indexed with the time at observation; it is most likely a series of observations of a real life phenomenon such as stock prices on the New York Stock Exchange. On the other hand, stochastic process is as always understood as a mathematical representation (not a production) of the time-series.

• Stochastic processes are more general than time series. For example Markov chains are stochastic processes that are not time series. Commented Apr 9, 2018 at 6:32
• @Michael Chernick: Aren't Markov Chain consistent with the definitions: "a set of random variables indexed by integers t" and "a stochastic process indexed by integers"? Which parts of these definitions Markov Chains do not satisfy or are you disagreeing with these definitions? Commented Dec 31, 2018 at 16:15

A random variable is a random variable

A vector of random variables is a random vector

A set of random variables is a random field

Like random vector, we need to give the random variables an index to identify that variable. Using different indexing schemas result in different real life applications, for example:

• If we index the random variables in the random field with rows and columns of pixel positions, then this random field can be used to represent images;
• If we index the random variables in the random field with discrete time labels, then the random field can be used to represent time series.