Is a time series the same as a stochastic process?

Question

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

Answer 1

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).

Answer 2

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.

Answer 3

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.

Answer 4

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.

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

Answer 5

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.

Answer 6

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.

To your question, stochastic processes are about random fields, so the answer is yes.