In volatility modelling, a typical challenge is to keep the covariance matrix estimate valid, meaning (1) symmetric and (2) positive semi definite*. A new paper published in Econometrica (citing from the paper) “introduces a novel parametrization of the correlation matrix. The reparametrization facilitates modeling of correlation and covariance matrices by an unrestricted vector, where positive definiteness is an innate property” (emphasis mine). Econometrica is known to publish ground-breaking research, and you may wonder: what is the big deal in being able to reparametrise the correlation matrix?
Deep learning algorithms are increasingly featuring in popular news outlets, large-scale media events and academic conferences. But what makes them so popular? Why now?
I recently published what I hope is an easy read for all of you modern-statistics
geeks lovers; explaining the thrust behind this machine-learning class of models.
Random Forests (RF from here onwards) is a widely used pure-prediction algorithm. This post assumes good familiarity with RF. If you are not familiar with this algorithm, stop here and see the first reference below for an easy tutorial. If you used RF before and you are familiar with it, then you probably encountered those “importance of the variables” plots. We start with a brief explanation of those plots, and the concept of importance scores calculation. Main takeaway from the post: don’t use those importance scores plots, because they are simply misleading. Those importance plots are simply a wrong turn taken by our human tendency to look for reason, whether it’s there or it’s not there.
Modern statistical methods use simulations; generating different scenarios and repeating those thousands of times over. Therefore, even trivial operations burden computational speed.
In the words of my favorite statistician Bradley Efron:
“There is some sort of law working here, whereby statistical methodology always expands to strain the current limits of computation.”
In addition to the need for faster computation, the richness of open-source ecosystem means that you often encounter different functions doing the same thing, sometimes even under the same name. This post explains how to measure the computational efficacy of a function so you know which one to use, with a couple of actual examples for reducing computational time.
Some problems are linear, but some problems are non-linear. I presume that you started your education discussing and solving linear problems which is a natural starting point. For non-linear problems solutions often involve an initial processing step. The aim of that initial step is to transform the problem such that it has, again, linear flavor.
A textbook example is the logistic regression, a tried-and-true recipe for getting the best linear boundary between two classes. In a standard neural network model, you will find logistic regression (or multinomial regression for multi-class output) applied on transformed data. Few preceding layers are “devoted” to transform a non-separable input-space into something which linear methods could handle, allowing the logistic regression to solve the problem with relative ease.
The same rationale holds for spectral clustering. Rather than working with the original inputs, work first with a transformed data which would make it easier to solve, and then link back to your original inputs.
Spectral clustering is an important and up-and-coming variant of some fairly standard clustering algorithms. It is a powerful tool to have in your modern statistics tool cabinet. Spectral clustering includes a processing step to help solve non-linear problems, such that they could be solved with those linear algorithms we are so fond of. For example, the undeniably popular K-means.
Google “K-means clustering”, and you usually you find ugly explanations and math-heavy sensational formulas*. It is my opinion that you can only understand those explanations if you don’t need them; meaning you are already familiar with the topic. Therefore, this is a more gentle introduction to K-means clustering. Here you will find out what K-Means Clustering, an algorithm, actually does. You will get only the basics, but in this particular topic, the extensions are not wildely different.