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Now you might be curious how a linear model like linear regression is able to perfectly forecast the

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sine wave which is clearly non-linear.

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For those of you who are interested I'd like to share with you the derivation since I find this pretty

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interesting myself.

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If you are not interested you can consider this optional so don't feel like you have to understand this

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in order to understand the rest of the course.

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It turns out that it's possible to predict a sine wave perfectly using only two pass values of the time

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series.

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We call this an R2 model.

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Notice we do not even need a biased term so let's see how this might be true.

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First note that if you want to write a time series which is a function of time in a mathematical form

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it looks like this.

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So x of T equals the sign of Omega times t and the s the letter inside the sign is Omega and not w the

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lowercase Omega just happens to look a lot like a W.

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Now if you recall we actually did something very similar to this when we created the data we said a

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series is equal to and put out sign of zero point one times and put out a range of two hundred so NPR

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got a range of two hundred it represents all the time points and zero point one was the angular frequency

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surprisingly.

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This is all we need to show that our R2 model can learn a sine wave.

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Let's just plug in the sine wave function into our recurrence relation.

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By the way recurrence relation is just a fancy way of describing the auto regressive model.

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The Fibonacci equation is also an example of a recurrence relation.

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In fact it has the exact same form as our R2 model with weights of 1 for this derivation.

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It's actually more convenient to shift the time steps up by 1.

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So instead of predicting x of T using X T minus 1 and X T minus 2 will predict x of T plus 1 using x

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of T and X T minus 1.

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The next step is to multiply out the terms inside of the sine functions.

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This still may not look like much but let's keep going.

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The key here is to remember all the way back to your high school math where you learned about trigonometric

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identities.

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Now don't worry I've forgotten these as well.

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These are not the type of thing you're using your day to day machine learning work so don't be too sad.

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The important identity is this one.

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It's the sign of a plus b plus the sign of a minus B equals two coast B sign a.

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Let's see why.

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First we can manipulate our sign I'll make a t equation by bringing the W term to the left hand side.

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So now we have sign of a mega T plus Omega minus W2 times sine of omega T minus Omega equals w one time

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sign of omega T.

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Now here's the main trick we have to recognize that both of these equations have the same format.

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If we look inside the sine terms we see that omega T is just a and Omega is just B immediately.

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This helps us figure out the constants w 1 and W2.

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We can already see that W2 must equal minus 1.

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This also means that w 1 is equal to 2 cost B which is equal to 2 cos Omega.

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Thus we prove that we can represent the sine wave as a recurrence relation.