OK, so in this lecture, we will be summarizing everything we learned in this section, the section
was all about exponential smoothing methods, also known as.
We started this section with a very basic concept, the simple moving average.
While it is indeed simple, we saw that it actually has many applications, including algorithmic trading.
The next step was to consider the exponentially weighted moving average.
This kind of update form the basis for all the models we studied in this section.
The next step was to consider simple exponential smoothing.
Mathematically, you saw that this was pretty much exactly the same as the WME, but simply framed in
a different way.
Instead of looking at it as a way to smooth out or average a time series, we saw it as a predictive
model.
From there, we were able to expand on the model by adding more components.
We added a trend component and then it became a linear trend model.
We added a seasonal component.
And then it became the whole winters' model.
After looking at how to implement whole winters' on the famous airline passengers data.
We then applied it to more data sets, including champagne sales and stock prices.
We also discovered that there are many options to choose from when it comes to the whole winters' model.
We can make both Trenorden seasonality additive or multiplicative.
We can damp the trend.
We can use the box, Coggs transform.
So with all these options, we needed a way to systematically test them, but to avoid overfitting while
doing so, we learn that while techniques like cross-validation work for non time series data, we need
to be careful when our observations have a time dependancy to solve this issue.
We learned about walk forward validation.
Now, please keep in mind that walk forward validation isn't unique to Etes methods.
The reason I've introduced it in this section is simply because it's an appropriate and convenient time
to do so.
Walk forward validation can be used for any of the methods we discuss in this course.
At the same time, it would be pretty tedious to do all that work and every script.
So we're basically not going to.
However, this doesn't mean that you shouldn't use this technique in your own time series work.
So if you're an inquisitive student, you're probably wondering why is this collection of methods called
it's X stands for error trend seasonality?
So why is it error trend seasonality and not level trend seasonality?
The answer is because it's possible to transform the update equations you learned about earlier into
what is called stage space form.
Now, the details aren't really important for this course.
But let's look at one example.
This is for simple exponential smoothing.
Basically, you can see that we've rearranged the equations so that we have an error term.
Epsilon t Epsilon T is considered to be, as usual, normally distributed, i.e. Noize, also known
as a Gaussian white noise.
Epsilon C is essentially the residual of our predictions in Time series analysis.
We sometimes call these innovations.
You can think of L.A. as kind of like a hidden state.
We can't observe it, but it's something we want to estimate.
YFC is the thing we actually observe.
Basically, we can see that the error term Epsilon t is the part of this equation, which is unpredictable.
So this tells us how the error term enters the picture and explains why we use the name error trend
seasonality, essentially the kinds of time series we want to model our time series that can be decompose
into these three components, the error of the trend and the seasonality.