Gregory Trubetskoy

Notes to self.

Why Is There No Formal Definition of Time Series?

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If you’re reading this, chances are you may have searched for definition of “Time Series”. And, like me, you were probably disappointed by what you’ve found.

The most popular “definition” I come across amongst our fellow programmer folk is that it’s “data points with timestamps”. Or something like that. And you can make charts from it. And that’s about it, alas.

The word time suggests that is has something to do with time. At first it seems reasonable, I bite. The word series is a little more peculiar. A mathematician would argue that a series is a sum of a sequence. Most people though think “series” and “sequence” are the same thing, and that’s fine. But it’s a clue that time series is not a scientific term, because it would have been called time sequence most likely.

Lets get back to the time aspect of it. Why do data points need timestamps? Or do they? Isn’t it the time interval between points that is most essential, rather than the absolute time? And if the data points are spaced equally (which conforms to the most common definiton of time series), then what purpose would any time-related information attached to a data point serve?

To understand this better, picture a time chart. Of anything - temperature, price of bitcoin over a week, whatever. Now think - does the absolute time of every point provide any useful information to you? Does the essential meaning of the chart change depending on whether it shows the price of bitcoin in the year 2016 or 2098 or 10923?

Doesn’t it seem like “time” in “time series” is a bit of a red herring?

Here is another example. Let’s say I decide to travel from San-Francisco to New York taking measurements of elevation above the sea level at every mile. I then plot that sequence on a chart where x-axis is distance traveled and y-axis is elevation. You would agree that this chart is not a “time series” by any stretch, right? But then if I renamed x-axis to “time traveled” (let’s assume I moved at constant speed), the chart wouldn’t change at all, but now it’s okay to call it “time series”?

So it’s no surprise that there is no formal definition of “time series”. In the end a “time series” is just a sequence. There are no timestamps required and there is nothing at all special regarding a dimension being time as opposed to any other unit, which is why there is no mathematical definition of “time series”. Time series is a colloquial term etymological origins of which are not known to me, but it’s not a thing from a scientific perspective, I’m afraid.

Next time you hear “time series” just substitute it with “sequence” and see how much sense that makes. For example a “time series database” is a “sequence database”, i.e. database optimized for sequences. Aren’t all relational databases optimized for sequences?

Something to think about over the holidays…

Edit: Someone brought up the subject of unevenly-spaced time series. All series are evenly spaced given proper resolution. An unevenly-spaced time series with timestamps accurate to 1 millisecond is a sparse evenly-spaced series with a 1 millisecond resolution.