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So far, we've gone through quite a few steps from a simple moving average to the exponentially weighted

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moving average to viewing the moving average as a predictive model and finally adding a trend component

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in this lecture, we are going to complete this journey with the full winter's model.

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The whole winters' model extends the whole linear trend model by adding a seasonal component.

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This makes sense, especially for our data set, because there is a clear seasonal pattern.

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In fact, for many time series data sets, there is a clear seasonal pattern.

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Think of something like air conditioner sales.

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Obviously, more people buy air conditioners in the summer than in the winter.

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We have all kinds of sales that are seasonal, such as back to school sales, Boxing Day sales and nowadays

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Black Friday sales, which I'm sure you're all familiar with.

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So how does the whole winter's model work?

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The basic idea is that for the kinds of Time series we would like to model, there are generally three

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components.

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There is the trend component, which we've seen.

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There is the seasonal component, which we are just now seeing, and there is the level component,

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which is sort of the average value around which everything else fluctuates.

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These three combine together to form the full time series signal.

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The question is how do we combine them together, in fact, for the whole winter's model?

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There are two ways of doing this.

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The first method is called the additive method for this model, the seasonal component or the periodic

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component is simply added to the trend and the level.

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So you can imagine a noisy line added to some periodic signal.

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And this periodic signal being periodic does not change over time.

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The second method is called the multiplicative method for this model.

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The seasonal component changes proportional to the level of the Time series.

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This seems like it might be a good fit for our data set, because it seems like the wavy part actually

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increases in amplitude as the level gets larger and larger.

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So let's look at the additive method in component form.

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As usual, I think the best way to understand these models is to pay careful attention to each of these

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equations.

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The forecast equation is sort of the centerpiece of the model because it tells you how each of the components

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combine to give you the full output.

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So what does the forecast equation look like here?

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Well, first, let's recognize that two thirds of this equation we've already seen this is just holds

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linear trend model.

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It says that for each step that H increases the non seasonal part of the forecast increases by B of

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T.

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The new part is the S term, which obviously corresponds to the seasonal component.

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Now, you might find these indices very confusing at first, but we'll go through them slowly so you

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understand what they mean.

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First, we need to define this new variable M, which is used to define the period of the seasonal component.

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Some people refer to this as the frequency, but it's more accurate to think of it as the period.

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Remember, that period refers to the length of time that it takes for a signal to repeat itself.

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For example, the period of the sine function and the cosine function is too high.

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Frequency is actually inversely related to the period.

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In any case, this is one of those variables that must be chosen manually.

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But in real world situations, it probably won't be that hard.

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As an example, if you're looking at sales data over each month, then we might expect the period to

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be 12 since there are certain events that happen every year which affect sales.

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Obviously, this will require some domain knowledge.

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So the data science practitioner, that's you should apply what you know about the data to choose this

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value correctly.

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So now we know what M is.

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We also know what T and since we've seen them before, T is just the last known time for which we have

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real observations and the number of steps ahead that we are forecasting.

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But what is K?

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Let's ignore this strange formula for now.

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Intuitively, K is a number chosen so that we look back the appropriate number of steps in our training

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data to choose the correct seasonal component.

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For example, suppose that our data goes up until December twenty twenty and the frequency of the data

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is monthly.

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Suppose that we would like to make a forecast for March twenty twenty one.

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In this case, the training data is from December twenty twenty and any time before that to get the

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seasonal component for the March twenty twenty one forecast, we would like to use the seasonal component

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for the latest month of March that we know of, which is March twenty twenty.

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We would not want to have, say, the seasonal component from November twenty twenty because that's

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an entirely different month, which includes Black Friday, which definitely includes data we don't

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care about in March.

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On the other hand, note that the level L and the trend B both come from time t, the latest measurement,

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only the seasonal component looks further back in time to grab the appropriate point over the entire

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season.

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If you're mathematically inclined, then we can do an example to make sure that this works, if you

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are not mathematically inclined, then just understand that intuition and that's OK.

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Remember that we're going to be using a library anyway, so let's suppose that we have it.

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T equals thirty six and our data is monthly, so M equals 12.

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Suppose that we want to forecast 15 steps into the future.

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So H is equal to 15.

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So if one refers to January then 12 refers to December as those twenty four and thirty six.

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So our training data is three years from January to December.

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Three years later, we want to predict 15 months into the future, which is March, but it's the March.

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One year after the latest measurement, we know intuitively that the seasonal component should come

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from T equals twenty seven Y, because that's the last known March measurement that we have.

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In other words, the one that's closest to thirty six.

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So let's see if our formula works, we have equal to the floor of each minus one over M plus one, if

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we plug in all of our numbers, we get K equal to the floor of 14 over 12 plus one, which is all equal

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to two.

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Therefore, the time index for S will be thirty six plus fifteen minus twelve times two, which is equal

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to twenty seven.

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And thus we have confirmed that this strange formula does indeed work.

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All right, so now that we understand the forecast equation, the others are pretty simple.

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Next, we have the legal equation.

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This is almost the same as the linear trend model.

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And you should feel comfort about the fact that it's the same old exponential moving average.

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The main difference is that we subtract the seasonal component from Y note that we subtract the previous

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seasonal component at index T minus M.

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This makes sense since in our equations generally, why is considered the forecast.

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So it's always ahead of all the other components.

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Next, we have the trend equation.

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This one is exactly the same as before.

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It's an exponential moving average of the trend as measured by the difference in levels.

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Finally, we have the seasonal equation, this one might seem strange at first, but at least you can

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recognize that this, again, is an exponential moving average in front of Gammer.

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We have the new value and in front of one minus Gamma, we have the old value.

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Again, recognize that the old value comes from one period ago rather than one timestep ago.

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So that's why the previous S is indexed by T minus.

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For the new value, this makes sense when you consider the forecast recall that Y is equal to the level

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plus trend plus seasonal component.

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Therefore the seasonal component is Y minus the level, minus the trend.

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If you think of it like an equation, you're just moving the level in the trend to the other side to

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isolate the seasonal part.

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All right, so now that you understand the additive model, let's move on to the multiplicative model.

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Since you already understand how the additive model works, it's not that much effort to understand

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the multiplicative model as before.

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Let's start with understanding the forecast equation based on the name of the model.

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I think the forecast equation makes intuitive sense instead of adding the seasonal component.

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We now take the linear part and multiply it by the seasonal component.

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The general equation at a high level can be thought of as Y equals L plus B all times s for the level

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update.

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We now have Y divided by S.

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That makes sense because before we had Y minus S.

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For the trend equation, there is no change for the seasonal equation.

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We have Y divided by L plus B that makes sense because before we had a Y minus O plus B.

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All right, so as with our previous lectures, now that we understand the theory, we're going to discuss

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the main parts of the code.

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So first, we'll need a new class, which is simply called exponential smoothing.

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This might be a little confusing since one might think of the Ummah as exponential smoothing.

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But just remember that in this library, we qualify it by saying it's simple exponential smoothing.

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So not simple.

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Exponential smoothing refers to whole winters'.

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Next, we instantiate our model in the constructor, we pass in the data, and we also specify whether

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the trend and seasonal components are additive or multiplicative.

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We also specify the period which in our case will be 12 since our data is monthly.

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Note that in this lecture we discussed two models with a seasonal component was either additive or multiplicative

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in both these cases.

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The trend was additive.

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Stat's models also gives you the option of making the trend a multiplicative if you think that fits

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your data.

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So that might apply when your trend curves rather than going in a straight line.

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The next step is to call the fit function, which does not require any parameters.

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And as usual, this returns.

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A whole winters' results object from which we can then obtain the fitted values as well as call the

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forecast function.