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OK, so in this lecture, we are going to continue our investigation into applying machine learning

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to the airline passengers time series.

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So this script is going to have the exact same structure as the previous script.

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We're going to go through all the same steps in the exact same order.

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The only difference, no pun intended, is that in this case, we are going to make the time series

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stationary by taking the first difference.

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The implication of this is that when we want to compute our forecast, we have to undo this different

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saying.

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Now, for some people, it might help to look at some math and for others it might help to just look

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at the code.

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So we're going to do both.

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For me personally, I found just going straight to the code, more intuitive.

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I didn't write down this math until I realized I had to make this lecture, but I decided to explain

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the math, since it really helps to explain why we use those functions in the code.

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OK, so suppose we've taken the first difference of our data will say Delta Y of T is just Y of T,

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minus Y of T minus one.

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Then our different data set is just Delta one, Delta Y two and so forth, this is the data we will

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train on because of this.

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Our model predictions will also be for this Difference Time series.

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So Delta Y have one, Delta Y had two and so forth.

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Now, let's suppose we would like to make a train prediction for the original Time series, for instance,

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I want to know why hat t my model does not predict why hat t directly, but I do have Delta.

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Why hat t?

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I know that Delta why hat t is a prediction for Y of T minus Y of T minus one.

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In this way, I can conceptually move Y of T minus one to the other side and say that my prediction

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for Y of T is Y of T minus one plus Delta Y had.

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So this is how I would make a forecast for an N sample prediction.

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This is also how to do a one step forecast.

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This makes sense because if today is T minus one that I know Y of T minus one, I do not know why of

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T which is why I want to make a forecast of this value.

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So I use my model to predict Delta Y of T, and then I add that to the current wave T minus one in order

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to predict tomorrow's Y of T.

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The next question to consider is how do we forecast the multiple times steps into the future beyond

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the in sample data?

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Well, let's begin with the fact that we know how to make multistep forecasts, either using the incremental

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or the multi output method so we know how to obtain a Delta Y hat of big C plus one.

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Delta Y had a big C plus two and so forth.

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So we have all these deltas, but we need to go back to the original undifferentiated series.

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Let's start with the first prediction at time.

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That Big T plus one.

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In this case, we can follow the same formula as before.

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We just add the prediction for the next Delta to the last known value.

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But how do we get the forecast for big C plus two?

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Well, if we knew why a big T plus one, then we could just follow the usual formula.

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But since we do not, then we do the next best thing, which is to plug in our prediction.

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Why had of Big C plus one.

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However, notice that we can just plug in the previous expression for we had a big C plus one in terms

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of the last known value Y of Big T.

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And of course, we can repeat this pattern for the forecast at time at Big T plus three in this case,

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it's why Big T plus the next three deltas.

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OK, so I hope you get the idea.

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The forecast is just the cumulative some of the deltas added to the last known value in the Time series.