Let's talk now about some basic definitions and properties of time series data.

We start from the very first definition.

So what is a time series?

A time series can be described as a set of observations.

And if we want to write it in symbols

it would be a capital S time series is equal

to this set of S1 to S capital T

where S of i is a measurement.

So a state measured at a specific time t i that belongs to a given space.

In this case

it could be a d dimensional real valued observations.

It's worth noticing that typically these observations are dependent.

And in fact, studying the nature of this dependency is of particular interest in time

series processing.

And the analysis of time series is actually concerned with techniques that try to analyze

and model these temporal dependencies between data points

between observations.

We gave many examples of time series in the previous sections

but let me give you some

more.

For example, monthly goods shipped from the factory, if we record them every month,

in a systematic way

we will have time series data over time.

And another example could be weekly

the number of weekly road accidents or the daily rainfall

amounts. These are all examples of time series data where every

observation is taken at a different time.

When we talk about time series

we distinguish between regularly sampled and irregularly

sampled time series.

We say that discrete time series is regularly sampled if the observations are equally spaced

in time.

It means that for any of the two observations, the consecutive two observations that we can

take

one at time t i and one at time t i plus one

the difference between time

so

the delta t is constant.

On the other end

we will talk about irregularly sampled time series when observations are

not necessarily equally spaced

which means if we take the delta t between two consecutive

observations, these are not necessarily constant, they can vary.

And here we have a graphical example where on the top we have a regularly sampled time

series where samples are equally spaced

while on the bottom in blue we have an irregularly

sampled time series where observations are sometimes close to each other

sometimes far

away from each other.

For time series, we also distinguish between univariate and multivariate.