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So in this lecture, we will be looking at Gargash more in depth and will describe conceptually how

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you might extend this model to deepen our own networks.

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So as we've gone through this course, you may have noticed one omission, which is that we didn't spend

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much time on discussing how these models are actually trained.

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This is because this is an application focused course rather than a math and programming type.

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Of course, in this lecture we will dive into some of these details, which I believe will help you

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understand further.

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So with any machine learning or statistical model, there are really two steps of learning how they

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work.

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The first step is necessary and it's the one we've covered in this course in any other course I teach.

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That is, how does the model make predictions?

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Or in other words, how do we get from the input to the output?

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Often this simply means learning the equation that tells us how to do this.

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As usual, this involves your typical mathematical operations like adding, multiplying and so forth.

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As you've seen for Gurche, this involves the model parameters, Omega, the Alphas and the bait is

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what we have not yet discussed is how are these parameters actually found.

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This is the second step now for deep learning.

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We've kind of discussed this, but not really.

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We know that we call some function called fit and we know that this does something like gradient descent,

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which looks like you're traveling down a hill.

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That's nice, but that's only about 10 percent of what you really need to know.

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If you want to make something like this yourself.

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That's not to say understanding those things isn't important, but it does say that the journey to mastery

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may be longer than expected.

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Of course, this lecture will take you closer to that point.

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OK, so the central item of interest when we want to fit or train a model is the objective for deep

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learning Arima and many other machine learning models.

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This is based on the log likelihood.

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Basically, our goal is to maximize the likelihood with respect to the model parameters.

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When we find the model parameters that maximize or minimize this objective, we consider the model to

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be trained.

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Now, it's not my goal in this lecture to explain maximum likelihood from scratch, but you can check

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the resources in extra reading dot text if you would like to learn more.

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This lecture is more geared towards those who already understand maximum likelihood estimation.

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So normally when we want to perform regression, we use the squared error objective.

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I know you've all seen this before, so I won't bother to explain this in detail.

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Now, if you've taken my in-depth courses before, then you also know that minimizing the sum of squared

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errors is the same as maximizing the log likelihood when the distribution of errors is normal.

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That is, when you use the squared error objective, you assume that your errors come from a normal

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distribution.

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What you may not have known is that this also requires you to make another assumption.

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This assumption is that the variance of those errors is constant.

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In order to understand why this is the case, we're going to review the steps to prove that minimizing

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the squared error is equivalent to maximizing the likelihood of the normal.

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OK, so in order to maximize the likelihood, we start by writing out the likelihood function, which

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makes use of the normal PDF, as you recall, we are assuming that our targets, which form the true

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time series, are samples from the normal, where the mean is the model prediction and the variance

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is some constant sigma squared.

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Also recall that this is equivalent to saying that the target Y of T is equal to the mean Y had of T

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plus Epsilon T, where in this case Epsilon T is an ID sample from the normal with mean zero and variance

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Sigma Squared also for completion sake.

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Well also suppose that Y had of T is some function of the previous plague's and some model parameters

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which we will just call theta.

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This part isn't necessary, but it helps to frame this in the same way as our auto regressive models

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from this course.

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OK, so from here we can write our likelihood of function as big of theta.

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You should recognize the formula inside the product, which is just the normal PDF.

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As you know, the next step is to take the log, which simplifies the optimization.

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So as an exercise, you may want to go through these steps slowly by yourself in case you forgot exactly

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how this works.

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Now, from this log, likelihood, we know that this expression can be simplified.

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Firstly, let's recognize that our objective is to find the theta which maximizes this objective.

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In other words, the Theta we want to call it theta star is the Amax of the log likelihood.

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Now you'll recognize that Theta doesn't actually appear on the right side.

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Just remember that Y hat is a function of theta.

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I'm just not showing it here anyway.

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Once we have this, it's easy to see how this can be simplified.

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Notice that the long term does not depend on theta at all.

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To Pi is just a number, as is the variance sigma squared.

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So if we simply ignored this term, the optimal theta star would still be the same.

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The next thing to notice is that the one 1/2 and the Sigma are just the multiplicative constants.

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For example, the extreme point of X squared is the same as the extreme point of two X squared.

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Multiplying by a constant makes no difference.

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So we can remove those as well.

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The final thing to recognize is that the minus sign can also be removed.

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Basically, we know that maximizing minus X squared is the same as minimizing X squared.

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We get the same X either way.

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So to simplify this, we can remove the minus sign and change the max to an argument.

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Of course, this is just the sum of squared errors as promised.

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OK, so at this point, we recognize that in the previous derivation of the squared error, from the

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likelihood we made one critical assumption, we assume that the variance of the error is constant.

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Of course, this whole section is about exploring what happens when the variance of the error is not

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constant.

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So in order to understand how Ghazi's trained, we simply have to go back a few steps.

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This is our log likelihood again, but with one small difference.

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Now I've written Sigma with an argument t this is to signify that Sigma is no longer a constant.

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On the left side, we can think of Theta as a collection of model parameters, whether they are for

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Y hat or sigma or both.

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So this time, because Sigma is not constant, it can no longer be dropped from the Amax.

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One small simplification we can make is to drop the two pi since that part is still constant.

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We can also drop the one half and the negative signs and turn the objective into something we want to

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minimize.

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The final step we can take to make this look more like Darch is to recall that our main model is normally

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just zero and hence on the numerator we simply have Ypsilanti squared, which is the error time series.

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So this is the objective function you would use if you wanted to train a garbage.

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So essentially, this is all that you need to know if you want to train your own garbage at this point,

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you can plug this objective into any stock optimizer such as LBJ's.

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I encourage you to check the CPI documentation if you want to learn more.

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Now, as you recall, the objective is not the only thing you need to do the optimization.

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You also need to include the constraints to make sure that the variance is always positive.

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And of course, many of these stock optimizers allow you to include these constraints.

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So this is not a concern.

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Also, recall that we derived this objective, assuming that the error is normally distributed.

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You can if you want to replace that with any other distribution you like, such as the student T or

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the Scooty or anything else.

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Now, if you've studied deep learning and neural networks, then your immediate question should be why

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do we need to stick with Gurche?

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Gurche, as you know, is a linear model and we've seen that linear models can be limited.

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So a natural question to ask is why not simply parameterize Sigma T using a neural network?

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In fact, this is entirely possible.

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So this is an easy but also powerful way to extend Gargash without much extra work that is provided.

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You know how to build custom functions using Tenzer floor piter.

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One thing that's not immediately clear is how you can constrain the output of your neural network to

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be positive.

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As you recall, the final layer of a neural network is typically just linear regression, which can

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output any real number.

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So the way to handle this is to use an activation function that enforces positivity.

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For example, the soft plus by using the soft plus, you can ensure that the output of your neural network

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is a positive value.

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One final topic to discuss in this lecture, which follows directly from what we've just discussed,

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is this idea of combining Arima with garbage.

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So earlier we wrote our model down as the mean A plus the error.

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So it may seem as if we can simply train and arima to learn the mean subtracted from the time series

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to get the error and then train Aghajan the error.

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In fact, this approach appears in a few Python blogs.

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Unfortunately, this approach is suboptimal.

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As you saw earlier, the lost function actually incorporates both the mean model and the variance model

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at the same time.

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What this means is that you find the optimal answer.

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Both have to be optimized jointly.

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As mentioned, there is no Arima in the Python Arch Library, but there is one in our Yuga, which is

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a library for the language.

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So if you really need to use a Rhema garage, that would be an option.

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One fact to recognise, however, is that it is often the case that an eye one is the most parsimonious

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arima for stock prices, as you recall.

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This corresponds to a random walk.

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When this is the case, there are no parameters to optimize for the Arima power since anyone does not

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have any parameters.

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So if it is the case that any one is the best model for the price, then there is no difference whether

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you choose to fit the model jointly or separately.