In this lecture, we are going to discuss another integral part of the ARIMA model, the model.
As you know, M-A stands for Moving Average.
However, it's important not to conflate this with the simple moving average and the exponentially weighted
moving average.
We discussed earlier in this course, as you will see shortly, this model is very different from those.
So let's get right to what the moving average model looks like.
The moving average model is similar to the auto regressive model in that it is a linear function of
something.
But what is it a linear function?
In fact, it is a linear function of past error terms.
This should definitely strike you as odd in pretty much all of machine learning.
We typically create models which depend on input data.
In this moving average model, there is no input data to be seen.
Instead, at the time, series depends only on errors.
Of course, in order to know these errors, we must have compared at the previous predictions with the
previous values of the Time series.
Note that our abbreviated form for a moving average model with order Q is may Q.
This means that the output Y depends on Q passed error terms in addition to the latest error term epsilon
sub t.
Here's one way to think of the moving average model that might help you rationalize its name.
Consider what the expected value of Y of T should be.
As you know, we treat errors as normals with mean zero and variance Sigma Square.
Well, if we take the expected value of y of t, we simply get C, since the expected value of each
of the errors is zero.
Hence we can think of the biased term C as the average value and then each of the errors as fluctuations
that make y of t go up or down around c.
Another way to think about the moving average model is in terms of simulation.
In fact, simulating a moving average process is somewhat easier than an auto regressive process.
Let's suppose that we want to simulate an may to process.
So the output y depends on two past error terms in addition to the current error term.
First, we need to generate Epsilon one and Epsilon two.
These are just samples from some normal with mean zero and variance sigma squared.
Then we generate epsilon three, then we calculate y three according to our formula, which depends
on Epsilon one, two and three, as well as the model weights, which we assume have been given.
Next we generate Epsilon four, then we calculate y four, which depends on Epsilon two, three and
four.
Then we generate Epsilon five and we five, Epsilon six and Y six and so on.
So as you see, this process really amounts to nothing but generating samples from the normal and adding
them together.
Therefore, you would use this model if you think that nature actually behaves this way to generate
the data which you are observing.