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So in this lecture, we will be looking at the concepts behind Facebook profit.

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Yes, we know it's a Time series modeling and forecasting tool, but we'd like to have a bit of detail

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about how it works.

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Now, if you're not really concerned with how it works, then feel free to skip ahead to the code preparation.

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However, BE warns that not knowing how it works has led to silly use cases of this library, for instance,

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predicting stocks.

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In fact, we will go through such an example later in this section to show you that this will actually

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perform poorly.

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So that's one downside of only looking at the API without any knowledge of how the model works, you

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may end up doing things that don't make sense.

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However, what I think you will find is that the principles of profit are actually quite simple and

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intuitive.

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We're not going to go over any mathematical details in depth, but just a high level overview.

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In fact, Facebook's on paper also does not go into these details.

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So it's one of those things where if you really want to know the low level details, you just have to

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check the code.

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It is open source, so it's available for you to do that.

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Otherwise, I think the high level principles are good enough for most of us.

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Please note that the paper has been linked in extra reading.

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It's called forecasting at scale.

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So check that out if you want to read the primary source material.

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OK, so as you know, the Facebook profit package is optimized for Business Time series, this means

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that a lot of its features are geared towards complexities that businesses have to deal with.

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One example is seasonality at multiple scales.

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For example, you may have weekly seasonality where you have less customers on Sundays, but you also

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may have yearly seasonality based around holidays such as Christmas, New Year's, Valentine's Day and

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so forth.

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Another example is events which are cyclical but not exactly seasonal with the predefined frequency.

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Some examples are Thanksgiving, Black Friday and Easter.

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You may also have events which are based on the lunar calendar, such as the Lunar New Year.

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This would be difficult to model with the other tools we've learned about, but profit makes this kind

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of thing easy.

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Another feature that prophet can model is changes in trend, this may happen if, for example, you

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create a new website design or perhaps you released some new products.

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Yet another issue that Facebook profit can easily deal with is outliers and missing data.

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We'll see how these can be dealt with quite elegantly.

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OK, so the one and only equation you really need to know in order to understand how Facebook profit

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works is this it says that our model of the Time series is simply the addition of three components,

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along with an unpredictable error term.

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The three components are the trend that the seasonality and the holidays.

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So although some seasonality could be due to holidays, you can see that these are modeled separately.

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So in this case, Gifty represents the trend, which represents the non periodic changes in the series.

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S of T represents periodic changes which may happen at different scales.

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For example, weekly or yearly HFT represents holidays which could be irregular and could occur over

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more than one day.

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Note that like other many time series models, Epsilon T is assumed to come from a normal distribution.

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One interesting fact, which is not totally obvious at this point, is that this model is not auto regressive,

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but rather the paper states that time itself is the only agressor.

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We'll see exactly what this means when we examine each of the components in more detail.

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One consequence of this is that missing data is immaterial.

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No prediction depends on any other value, but only some equation based on time itself.

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Again, if this is not clear, you'll see what we mean by this in the next few slides.

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OK, so let's start by discussing the trend G of T. So essentially there are two ways that profit models

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trends.

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The first method is piecewise linear.

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This automatically presumes that the model will also figure out the boundaries of each linear piece,

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something we call change point detection.

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The second method is logistic growth or decay.

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So an example of this is covid-19.

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Of course, there is a limit to the number of people who can be infected since there is a limited population.

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Another example is sales of a product.

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The sales of a product would most likely be limited by the size of the market.

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The most basic form of logistic growth is shown by the equation here, which you might recognize as

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the sigmoid from deep learning.

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The value of C is the maximum and in profits terminology, this is called the carrying capacity.

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Now, the actual equations representing piecewise linear trend and logistic trend are a bit more complex,

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but they follow the same usual forms.

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Note that for both of these equations, you can see that they are not auto regressive, unlike Arima

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and the other models we've studied.

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Instead, you can see that the time t is the only input argument.

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So for the linear case, we can see that it has the usual form slope multiplied by input plus intercept

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for the logistic case.

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If we squint really hard, we can see the same form we saw earlier.

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It just allows for some adjustment and the carrying capacity, rate of change and the offset.

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These aren't too important for our use case, but they may be useful if you want to explain the model

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to your clients.

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So have a look at the paper and read about these other parameters.

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If your clients are interested in those details.

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So for the seasonal component, we have this interesting equation which should look familiar if you're

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an engineer or a physicist, basically this is just a four year series.

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If you're not an engineer or a physicist, that's OK, too.

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You can see that it's just the sum of science and cosigns, which, of course, produce a periodic time

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series.

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What's important to note about this is it's the sum of multiple signs and cosigns, and each term in

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the sum has a different period thanks to the argument to buy A..

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Over P in this case, Big P is like the overall period.

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So it's three sixty five point two five four yearly data and seven for weekly data.

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When the time variable is scaled in days, the parameters to be optimized are the A's and B's, which

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are found by optimizing some lost function as per usual.

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OK, so the final component is the holiday component, which is essentially represented as a one hot

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vector, thus the influence of any single holiday is governed by a single parameter, which is additive.

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Note that the paper mentions that the model also accounts for days surrounding the holiday, but it

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doesn't specify exactly how.

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OK, so one interesting aspect of the profit model is that it is Bayesian.

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This is not obvious from the way it's been presented so far, but all of the model parameters have priors.

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Behind the scenes, profit is built using STEM, which is a library for Bazian Machine Learning, although

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that sounds complex and allows for some human in the loop optimization.

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So in other words, you can include your client in the modeling process.

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For example, if you find that the change points for the trends are too sensitive, you can try different

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settings for the priors governing the change points to fit the expectations of your client.