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STAT 510 - Applied Time Series Analysis
•
ANGEL
•
Department of Statistics
•
Eberly College of Science
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Section 1: Introduction and Basics
Basic Models
Submitted by gfj100 on Mon, 03/22/2010 - 14:04
The analysis of data is always linked to a model. We cannot analyze data without a model. What we will
talk about next are the models that are often used to analyze data in the context of time series. The basic
time series model involves a list of random variables:
...
x
-2
,
x
-1
,
x
0
,
x
,
x
1
,
x
2
,
x
3
, .
..
One thing different here is that they are not necessarily independent and not necessarily identically
distributed.
One defining characteristic of time series is that this is a list of observations where the ordering matters.
For instance, if we have the list of pig weights that we have seen in our example earlier, would it matter if
we switched the labels, i.e., the weight for one pig is now associated with a different pig. With
independent identically distributed data labels can be switched. Pig #1 could be pig#2 or vice versa. The
order is arbitrary. With time series data, however, you cannot do this, ordering is very important because
there is dependency and this could change the meaning of the data.
Another name for this type of models is
a stochastic process
. Stochastic processes are random variables
that are indexed by time, (specifically the model above is called a discrete parameter or discrete time
process). So, a typical time series is where
x
t
is a continuous random variable but it is being indexed by
discrete time.
The models may start out at time negative infinity and go on out to infinity, so we have an infinite
sequence in both directions. An example might be the surface temperature at a point on earth, (This is not
infinite but it is a pretty long time!) we may have to consider this this type of model for theoretical
purposes. Often, whatever process you are talking about it has been going on for a long time before you
started looking at. This is not true for every process, but a lot things like weather, the economy, stocks,
etc. have been active for a long time and at some point the researcher starts to look at this. This
assumption simplifies many theoretical calculations.