Every so often I read about the kernel trick. Each time I read about it I need to relearn what it is. Now I am thinking “Eran, don’t you have this fancy blog of yours where you write about statistics you don’t want to forget?” and then: “why indeed I do have a fancy blog where I write about statistics I don’t want to forget”. So in this post I explain the “trick” in kernel trick and why it is useful.
Category: Statistics and Econometrics
Local Linear Forests
Random forests is one of the most powerful pure-prediction algorithms; immensely popular with modern statisticians. Despite the potent performance, improvements to the basic random forests algorithm are still possible. One such improvement is put forward in a recent paper called Local Linear Forests which I review in this post. To enjoy the read you need to be already familiar with the basic version of random forests.
Publication in Significance – code
Couple of months ago I published a paper in Significance – couple of pages describing the essence of deep learning algorithms, and why they are so popular. I got a few requests for the code which generated the figures in that paper. This weekend I reviewed my code and was content to see that I used a pseudorandom numbers, with a seed (as oppose to completely random numbers; without a seed). So now the figures are exactly reproducible. The actual code to produce the figures, and the figures themselves (e.g. for teaching purposes) are provided below.
A New Parameterization of Correlation Matrices
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?
What’s the big idea? Deep learning algorithms
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.
You can download the two-pager from Significance, specifically here (subscription required).
Bootstrap Standard Error Estimates – good news
More good news for the statistical bootstrap. A new paper in the prestigious Econometrica journal makes two interesting points.
Bayesian vs. Frequentist in Practice, part 3
This post is inspired by Leo Breiman’s opinion piece “No Bayesians in foxholes”. The saying “there are no atheists in foxholes” refers to the fact that if you are in the foxhole (being bombarded..), you pray! Leo’s paraphrase indicates that when complex, real problems are present, there are no Bayesian to be found.
Random forest importance measures are NOT important
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.
Beta in the tails
Every form of strength is also a form of weakness*. I love statistics, but I focus to much on methodology, which is not for everyone. Some people (right or wrong) question: “wonderful sir, but what can I do with it?”.
A new paper titled “Beta in the tails” is a showcase application for why we should focus on correlation structure rather than on average correlation. They discuss the question: Do hedge funds hedge? The reply: No, they don’t!
The paper “Beta in the tails” was published in the Journal of Econometrics but you can find a link to a working paper version below. We start with a figure replicated from the paper, go through the meaning and interpretation of it, and explain the methods used thereafter.
How flexible neural networks really are?
Very!
A distinctive power of neural networks (neural nets from here on) is their ability to flex themselves in order to capture complex underlying data structure. This post shows that the expressive power of neural networks can be quite swiftly taken to the extreme, in a bad way.
What does it mean? A paper from 1989 (universal approximation theorem, reference below) shows that any reasonable function can be approximated arbitrarily well by fairly a shallow neural net.
Speaking freely, if one wants to abuse the data, to overfit it like there is no tomorrow, then neural nets is the way to go; with neural nets you can perfectly map your fitted values to any data shape. Let’s code an example and explain the meaning of this.
Correlation and correlation structure (5) – a new coefficient of correlation
This is the fifth post which is concerned with quantifying the dependence between variables. When talking correlations one usually thinks about linear correlation, aka Pearson’s correlation. One serious limitation of linear correlation is that it’s, well.. linear. By construction it’s not useful for detecting non-monotonic relation between variables. Here I share some recent academic research, a new way to detect associations that are not monotonic.
Understanding Variance Explained in PCA – Matrix Approximation
Principal component analysis (PCA from here on) is performed via linear algebra functions called eigen decomposition or singular value decomposition. Since you are actually reading this, you may well have used PCA in the past, at school or where you work. There is a strong link between PCA and the usual least squares regression (previous posts here and here). More recently I explained what does variance explained by the first principal component actually means.
This post offers a matrix approximation perspective. As a by-product, we also show how to compare two matrices, to see how different they are from each other. Matrix approximation is a bit math-hairy, but we keep it simple here I promise. For this fascinating field itself I suspect a rise in importance. We are constantly stretching what we can do computationally, and by using approximations rather than the actual data, we can ease that burden. The price for using approximation is decrease in accuracy (à la “garbage in garbage out”), but with good approximation the tradeoff between the accuracy and computational time is favorable.
Why complex models are data-hungry?
If you regularly read this blog then you know I am not one to jump on the “AI Bandwagon”, being quickly weary of anyone flashing the “It’s Artificial Intelligence” joker card. Don’t get me wrong, I understand it is a sexy term I, but to me it always feels a bit like a sales pitch.
If the machine does anything (artificially) intelligent it means that the model at the back is complex, and complex models need massive (massive I say) amounts of data. This is because of the infamous Curse of dimensionality.
I know it. You know it. Complex models need a lot of data. You have read this fact, even wrote it at some point. But why is it the case? “So we get a good estimate of the parameter, and a good forecast thereafter”, you reply. I accept. But.. what is it about simple models that they could suffice themselves with much less data compared to complex models? Why do I always recommend to start simple? and why the literature around shrinkage and overfitting is as prolific as it is?
Correlation and correlation structure (4) – asymmetric correlations of equity portfolios
Here I share a refreshing idea from the paper “Asymmetric correlations of equity portfolios” which was published in the Journal of financial Economics, a top tier journal in this field. The question is how much the observed conditional correlation on the downside (say) differs from the conditional correlation you would expect from a symmetrical distribution. You can find here an explanation for the H-statistic developed in the aforementioned paper and some code for illustration.
Understanding Spectral Clustering
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.