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OK, so in this lecture, we will be summarizing everything we learned in this section, the section

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was all about exponential smoothing methods, also known as.

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We started this section with a very basic concept, the simple moving average.

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While it is indeed simple, we saw that it actually has many applications, including algorithmic trading.

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The next step was to consider the exponentially weighted moving average.

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This kind of update form the basis for all the models we studied in this section.

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The next step was to consider simple exponential smoothing.

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Mathematically, you saw that this was pretty much exactly the same as the WME, but simply framed in

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a different way.

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Instead of looking at it as a way to smooth out or average a time series, we saw it as a predictive

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model.

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From there, we were able to expand on the model by adding more components.

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We added a trend component and then it became a linear trend model.

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We added a seasonal component.

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And then it became the whole winters' model.

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After looking at how to implement whole winters' on the famous airline passengers data.

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We then applied it to more data sets, including champagne sales and stock prices.

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We also discovered that there are many options to choose from when it comes to the whole winters' model.

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We can make both Trenorden seasonality additive or multiplicative.

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We can damp the trend.

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We can use the box, Coggs transform.

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So with all these options, we needed a way to systematically test them, but to avoid overfitting while

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doing so, we learn that while techniques like cross-validation work for non time series data, we need

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to be careful when our observations have a time dependancy to solve this issue.

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We learned about walk forward validation.

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Now, please keep in mind that walk forward validation isn't unique to Etes methods.

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The reason I've introduced it in this section is simply because it's an appropriate and convenient time

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to do so.

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Walk forward validation can be used for any of the methods we discuss in this course.

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At the same time, it would be pretty tedious to do all that work and every script.

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So we're basically not going to.

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However, this doesn't mean that you shouldn't use this technique in your own time series work.

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So if you're an inquisitive student, you're probably wondering why is this collection of methods called

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it's X stands for error trend seasonality?

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So why is it error trend seasonality and not level trend seasonality?

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The answer is because it's possible to transform the update equations you learned about earlier into

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what is called stage space form.

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Now, the details aren't really important for this course.

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But let's look at one example.

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This is for simple exponential smoothing.

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Basically, you can see that we've rearranged the equations so that we have an error term.

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Epsilon t Epsilon T is considered to be, as usual, normally distributed, i.e. Noize, also known

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as a Gaussian white noise.

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Epsilon C is essentially the residual of our predictions in Time series analysis.

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We sometimes call these innovations.

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You can think of L.A. as kind of like a hidden state.

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We can't observe it, but it's something we want to estimate.

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YFC is the thing we actually observe.

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Basically, we can see that the error term Epsilon t is the part of this equation, which is unpredictable.

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So this tells us how the error term enters the picture and explains why we use the name error trend

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seasonality, essentially the kinds of time series we want to model our time series that can be decompose

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into these three components, the error of the trend and the seasonality.