# Why are “time series” called such?

Why are “time series” called such?

Series means sum of a sequence.

• Why is it time Series, not time sequence?

• Is time the independent variable?

• Given any sequence $s = s_0, s_1, s_2, \ldots,$ one can associate another sequence $a$ (of successive differences) $a_0=s0, a_1=s_1-s_0, \ldots, a_n=s_n-s_{n-1},\ldots$ for which $s$ is the series of partial sums of $a$. Thus there is no mathematical distinction whatsoever between "sequence" and "series" except for the connotation that one has the intention to analyze a series into its differences. For your answer, then, look at any account of time series models and note the prominent use of the difference operator. – whuber Jan 29 '15 at 18:37
• Time series is a collection of individual transactions/events that occur in a specific bucket of time.If 66 individuals enter a building between the hours of 1:00 pm and 2:00 pm we ascribe the count(66) to that period.Similarly for all other periods/buckets of equal length of 60 minutes.One doesn't start the process of analyzing first or second or any particular difference.If the statistical characteristics of the time series suggest some form of differencing is needed to make the series stationary then one filters the original time series by creating a proxy via the order of differencing. – IrishStat Jan 29 '15 at 19:04
• Why is it World Series and not World Sequence? – The Laconic Dec 31 '18 at 19:37

Why is it "Time Series", not "Time Sequence"?

This inconsistency bugged me too the first time I saw it! But note that outside mathematics, people often use "series" to refer to what mathematicians might call a sequence.

For example, the Oxford English dictionary online gives the main definition of "series" as a "number of events, objects, or people of a similar or related kind coming one after another". This is what is happening in a time series: you have a sequence of observations coming one after the other. This is equivalent to the usage of the word in such phrases as "TV series" (one episode after another), "series circuit" (the current flows through each component successively), the World Series (a sequence of baseball games one after the other) and so on.

The etymology of "series" comes from the early 17th century, "from Latin, literally 'row, chain', from serere 'join, connect'", which is quite instructive. It didn't originally have the meaning of summation, but I can't find separate citations that establish when the word "series" was first used for the sum of the terms in a sequence. In fact it's quite common, particularly in older mathematics textbooks, to see the word "series" used where you might prefer "sequence", and "sum of series" where you might prefer "series". I don't know when this terminology was standardised in its present form. Here's an extract on arithmetic and geometric progressions from Daboll's Schoolmaster's assistant, improved and enlarged being a plain practical system of arithmetic: adapted to the United States - Nathan Daboll's 1814 update to his 1799 original Daboll's schoolmaster's assistant: being a plain, practical system of arithmetic, adapted to the United States, which was one of the most popular mathematics education books in the US throughout much of the 19th century.

The whole of Daboll's Schoolmaster's Assistant is available at archive.org and makes fascinating reading; it is the mathematics textbook that Herman Melville refers to in Moby-Dick (1851) and according to The Historical Roots of Elementary Mathematics by Bunt, Jones and Bedient (Dover Books, 1988) was predominant in American schools until 1850. At some point I may check some later standard texts; I do not think the hard distinction between "sequence" and "series" in mathematics arose until rather later.

Is time the independent variable?

This is basically the right idea: for instance when you plot a time series, we normally show the observations on the vertical axis while the horizontal axis represents time elapsed. And certainly it's true you wouldn't regard time as a dependent variable, since that would make no sense from a causation point of view. Your observations depend on time, and not vice versa.

But note that "time" is usually referred to by an index number to signify the position of the observation ($X_1, X_2, X_3, ...$) rather than by a particular year/date/time - we don't generally see things like $X_\text{1 Jan 1998}, X_\text{2 Jan 1998}, X_\text{3 Jan 1998},...$. Also the time series $X_1, X_2, X_3, ...$ is considered univariate, meaning "one variable". This is in contrast to performing a bivariate ("two variable") regression analysis of your observed values, $X$, against time, $t$. There you would consider your data set as built out of two variables $X_1, X_2, X_3, ...$ against $t_1, t_2, t_3, ...$. In a time series, time is generally represented just by the index number (position in the sequence), not a separate variable in its own right.

• Just to add an example for the last point - in the multi-level model with observations nested in time, time is treated as a variable (and data are in a long format); in the latent growth curve model, the influence of time is reflected in how you constrain the factor loadings (and data are in a wide format). – D L Dahly Jan 29 '15 at 12:36
• @DLDahly Thanks that's useful. I've tried to stray on the side of caution by putting that "time is generally represented just by the index number" which I think it true in most elementary situations that students are likely to come across. Any practical data set almost certainly would include the actual time of course, as a variable in its own right, but it just might not be used in the analysis. Is that a fair enough characterisation, do you think? – Silverfish Jan 29 '15 at 12:42
• I have not understood that "if we observe data over time in Time Series" , doesn't it imply "we are determining the strength of the relationship between dependent variable and time. That is , how does dependent variable change with respect to time" ? So that,time is a independent variable. – user 31466 Jan 30 '15 at 14:12
• or more specifically, i have not understood why can't we do a simple linear regression here ? – user 31466 Jan 30 '15 at 14:16

"Series" is:

a group or a number of related or similar things (http://dictionary.reference.com/browse/series)

a number of things or events that are arranged or happen one after the other (http://www.merriam-webster.com/dictionary/series)

A number of objects or events arranged or coming one after the other in succession (http://www.thefreedictionary.com/series)

A number of events, objects, or people of a similar or related kind coming one after another (http://www.oxforddictionaries.com/definition/english/series)

Time series is a sequence of values "coming one after another". Series does not have to be a sum like in mathematics.

• Dictionaries are usually not good references for technical terms, especially not mathematical or statistical ones. Unless the dictionary is specific to a field, it must focus on common usage rather than technical usage. – whuber Jan 30 '15 at 16:22
• @whuber yes, I agree, but here the name seems to be related more to the non-technical meaning of the phrase and not to the one used in modern mathematics (cf. Silverfish's answer) – Tim Jan 30 '15 at 16:40

The accepted answer is informative (I upvoted it myself), but it assumes that the "series" term in Time Series is really a misnomer and should be "sequence" instead. For the first few decades in the development of time series analysis, 1920s and 1930s, time series was synonymous with ARMA time series. An MA time series is indeed a sum of a sequence of white noise innovations. An AR time series, if covariance stationary, is also a sum of a sequence of white noise innovations. It may well be that the name "series" in time series was very appropriately assigned to time series, when these were synonymous with ARMA time series, but as other types of time series were discovered, which did not have a similar representation as a sum, nobody went back to revise the term because it's been used for decades and it may have just stuck in the statistics community. (https://www.statistics.su.se/english/research/time-series-analysis/a-brief-history-of-time-series-analysis-1.259451)

• An interesting point! You're certainly right that AR and MA were important early techniques in the development of time series analysis. But I don't think it is correct, though, that it was established usage at that time for "series" to be used in the sense of "sum of a sequence" rather than a sequence itself. I did quite a lot of background reading for my answer to this question, especially of economic/finance/accounting literature from around this period, and they commonly referred to their chronologically-ordered data as coming in "series". – Silverfish Dec 31 '18 at 17:25