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So at this point, you now understand how convolution is applied to images?

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Well, we could use this approach for Time series.

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The next step is to look more closely at one's convolution, which handles Time series directly.

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So this lecture will look at one de convolution specifically for use on Time series.

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Now we discuss the mathematical operation in the previous lectures, so we don't really need to discuss

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that again.

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But for completion sake, let's just review what Convolution is doing.

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So suppose we have some multivariate input time, series of shape, TBD, if you had multiple samples,

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this would be invited, but let's ignore that extra dimension for now.

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Now, let's also suppose that we've chosen to have more output features and we are doing convolution

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and same mode, thus our output time series, whatever the shape T by M.

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So if we choose a kind of length of K than our kernel or filter, whatever the shape K by M and the

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actual convolution would be done using the same multiplying sum that you've seen many times.

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Now, the point of this lecture is not to do more math, but instead to more deeply understand the math

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we've already seen.

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So one of the first examples I use to explain convolution is the Gaussian blur.

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I think it's pretty obvious what kind of effect this has when we use it on images.

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But clearly this can also be applied to Time series.

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So if we can evolve a Gaussian with a time series, what we get is a blurred version of that Time series.

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Of course, when we talk about Time series, we don't call it blurring, but we call it smoothing.

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So this is another way to implement something you've seen many times before.

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As you recall, the simple moving average and exponential smoothing work in a similar way.

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So you've seen that in learning what we call convolution is really just what everyone else calls cross

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correlation.

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So if you ever talk to a statistician or an engineer, they will think you are doing convolution backwards.

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That's why another way to think of convolutional neural networks is that they are cross correlation

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neural networks.

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Well, let's recall that this is not the first time that we've seen a correlation in this course, auto

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correlation plays a huge role in Arima.

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We use the auto correlation to help us determine the Arima orders.

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So at this point, we can kind of combine our knowledge of deep learning in Arima to more deeply understand

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what correlation is doing.

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As mentioned earlier, a convolutional filter in deep learning is like a pattern finder.

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By doing convolution with a filter, you will get a spike whenever the filter matches the pattern in

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the signal, whether that's an image or a time series.

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So what is autocorrelation doing?

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Well, it's going to give us a spike whenever the time series matches itself, specifically a lag version

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of itself.

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In other words, it's pattern matching different parts of the time series with each other and that can

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tell you which parts data points are useful for making future predictions.

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And that's because if they match, then they are predictive.

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Another thing you can do is cross correlation with different time series, if you see a spike at a lag

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other than zero, then that means one of these times series is predictive of the other.

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Now, in my view, the most interesting way to relate convolution to Arima is this let's write down

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the equation for an ERP process.

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At this point, I shouldn't have to say anything else.

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After watching the previous lectures, you should immediately recognize what this is.

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In fact, this is nothing but convolution.

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And actually this is real convolution, since we have minus K and not plus K.

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That is to say, an AP model is like a mini CNN.

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It's a CNN with only one layer and identity activation.

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In other words, it's linear nonetheless.

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It still is a CNN.

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So that to me is a very interesting connection to make.

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You can think of auto regressive cnes like an extension of IRP or you can think of IRP as a special

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case of CNN's.

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And again, note that there is so much theory behind Arima that we just don't have time for in this

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course and that most students wouldn't be too keen on learning.

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But to give you some idea, once you realize that this is just convolution, there is a lot of analysis

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that you can do, especially involving the frequency domain and for what it transforms.