1-26. Risk Factor Models with PCA

Let’s summarize. What I’m trying to tell you is that the goal of maximizing these links, the projections of our data onto the first dimension can be written this way, where this thing in here is the variance of this set of projections of the X vectors onto the W direction, up to a constant … Read more

Variance, okay we know the formula for that. It’s just the sum of the squared deviations from the mean, divided by n minus one. But if the mean of the original data coordinates is zero, so is the mean of their projections onto the new direction. So, we have that this term I call mu … Read more

Now for simplicity, let’s look at just one of these mean centered data points. We’re looking for a new basis for the data and we want to write down the coordinates of this point in a new basis. This is the same thing we discussed earlier, we are looking for two new basis vectors for … Read more

Okay. So, what I told you is that we want to maximize the variance of the data in the new basis direction. Now I want to show you exactly how what I showed you in pictures translates into symbols that we write down. I don’t think this is as essential for you to know, because … Read more

Okay. Let’s get back to what we really want to talk about, PCA. In a nutshell, what is PCA? PCA is a series of calculations that gives us a new and special basis for our data. Why is it special? Well, the first dimension is the dimension along which the data points are the most … Read more

Let’s take a close look at these red and blue vectors. In fact, let’s try looking at them in the original i hat j hat basis. In the original basis, the first vector can be written (1,0.5). That is to say it can be created with one i hat and a half a j hat. … Read more

PCA, amounts to finding a new and special basis for your dataset. But hang on, what does it mean to find a new basis? I’m guessing that you’ve heard about and even done this yourself before. But before we move on, let’s just briefly review exactly what this means. Let’s return to thinking about vectors … Read more

Okay, we’re doing math now, so let’s start with something super simple; a vector. Oh my gosh you’re thinking, a vector dah, I’ve seen this a million times. Well, I just want to remind you that there are a couple of different ways of thinking about a vector, that we will transition between when discussing … Read more

Well, I hope you’re excited because you have just learned a whole stack of things. You’ve learned about a really cool and commonly used algorithm, and also about one way it’s used in finance to model risk. Now, hold on to your hats because next we’re going to talk about possibly the most exciting part … Read more

So, let’s return to the main subject of this lesson, which is factor models of risk. So, how do we use PCA to create a factor model of risk? What are our data? They are a set of time series of stock returns for many, many companies. Our main motivations for using PCA are to … Read more

Hello and welcome. In this notebook, we will learn how to use principal component analysis for dimensionality reduction. Dimensionality reduction, is one of the main applications of PCA. In the previous lessons, you’ve already learned how PCA works and about eigenvectors and eigenvalues. In this notebook, we will see how to apply PCA to a … Read more

So, you have a sense of what the PCs are. Now, I have an important thing to tell you about how we use them. In fact, we frequently don’t use all the PCs. Instead, we decide to use some fraction of them starting with the first, which we think explain most of the variance, and … Read more

Great. So, we found this new and special set of basis vectors, which are the principal components. What now? Well, let’s look at what we have. We have these new basis dimensions. What are the significance of these? How do we interpret them? What do they represent? Well, we’ve been talking about vectors in geometric … Read more

By now, you’ve seen a few important types of factor models. You know that the goal of factor models is to model roughly things you think have a similar underlying effect on your variables of interest. You want to represent a variable or variables in terms of several important underlying variables or factors. In our … Read more