LASSO, LASSO, LASSO

LASSO stands for Least Absolute Shrinkage and Selection Operator. It was first introduced 21 years ago by Robert Tibshirani (Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B). In 2004 the four statistical masters: Efron, Hastie, Johnstone and Tibshirani joined together to write the paper Least angle regression published in the Annals of statistics. It is that paper that sent the LASSO to the podium. The reason? they removed a computational barrier. Armed with a new ingenious geometric interpretation, they presented an algorithm for solving the LASSO problem. The algorithm is as simple as solving an OLS problem, and with computer code to accompany their paper, the LASSO was set for its liftoff*.

The LASSO overall reduces model complexity. It does this by completely excluding some variables, using only a subset of the original potential explanatory variables. Since this can add to the story of the model, the reduction in complexity is a desired property. Clarity of authors’ exposition and well rehashed computer code are further reasons for the fully justified, full fledged LASSO flareup.

This is not a LASSO tutorial. Google-search results, undoubtedly refined over years of increased popularity, are clear enough by now. Also, if you are still reading this I imagine you already know what is the LASSO and how it works. To continue from this point, what follows is a selective list of milestones from the academic literature- some theoretical and practical extensions.

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Density Estimation Using Regression

Density estimation using regression? Yes we can!

I like regression. It is one of those simple yet powerful statistical methods. You always know exactly what you are doing. This post is about density estimation, and how to get an estimate of the density using (Poisson) regression.

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Computer Age Statistical Inference – now free

If you consider yourself Econometrician\Statistician or one of those numerous buzz word synonyms that are floating around these days, Computer Age Statistical Inference: Algorithms, Evidence and Data Science by Bradley Efron and Trevor Hastie is a book you can’t miss, and now nor should you. You can download the book for free.

My first inclination is to deliver an unequivocal recommendation. But in truth, my praises would probably fall short of what was already written.

So what can I give you? I can say that there are currently 6 amazon reviews, with a 4.5 average. One of the reviewers writes that there is some overlap with previous work. I agree. But it doesn’t matter. It reads so well, call it a refresher. Let’s face it, it is not as if you always have it so clear in your head such that you can afford to skip sections because you read something similar before.

I can also tell you why I think it is a special book.

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Statistical Shrinkage

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Shrinkage in statistics has increased in popularity over the decades. Now statistical shrinkage is commonplace, explicitly or implicitly.

But when is it that we need to make use of shrinkage? At least partly it depends on signal-to-noise ratio.

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Understanding False Discovery Rate

False Discovery Rate is an unintuitive name for a very intuitive statistical concept. The math involved is as elegant as possible. Still, it is not an easy concept to actually understand. Hence i thought it would be a good idea to write this short tutorial.

We reviewed this important topic in the past, here as one of three Present-day great statistical discoveries, here in the context of backtesting trading strategies, and here in the context of scientific publishing. This post target the casual reader, explaining the concept of False Discovery Rate in plain words.

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Understanding K-Means Clustering

Introduction

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.

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Outliers and Loss Functions

A few words about outliers

In statistics, outliers are as thorny topic as it gets. Is it legitimate to treat the observations seen during global financial crisis as outliers? or are those simply a feature of the system, and as such are integral part of a very fat tail distribution?

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Density Confidence Interval

Density estimation belongs with the literature of non-parametric statistics. Using simple bootstrapping techniques we can obtain confidence intervals (CI) for the whole density curve. Here is a quick and easy way to obtain CI’s for different risk measures (VaR, expected shortfall) and using what follows, you can answer all kind of relevant questions.

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Trim your mean

The mean is arguably the most commonly used measure for central tendency, no no, don’t fall asleep! important point ahead.

We routinely compute the average as an estimate for the mean. All else constant, how much return should we expect the S&P 500 to deliver over some period? the average of past returns is a good answer. The average is the Maximum Likelihood (ML) estimate under Gaussianity. The average is a private case of least square minimization (a regression with no explanatory variables). It is a good answer. BUT:

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Optimism of the Training Error Rate

We all use models. We all continuously working to improve and validate our models. Constant effort is made trying to estimate: how good our model actually is?

A general term for this estimate is error rate. Low error rate is better than high error rate, it means our model is more accurate.

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Modeling Tail Behavior with EVT

Extreme Value Theory (EVT) and Heavy tails

Extreme Value Theory (EVT) is busy with understanding the behavior of the distribution, in the extremes. The extreme determine the average, not the reverse. If you understand the extreme, the average follows. But, getting the extreme right is extremely difficult. By construction, you have very few data points. By way of contradiction, if you have many data points then it is not the extreme you are dealing with.

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Multivariate Volatility Forecast Evaluation

The evaluation of volatility models is gracefully complicated by the fact that, unlike other time series, even the realization is not observable. Two researchers would never disagree about what was yesterday’s stock price, but they can easily disagree about what was yesterday’s stock volatility. Because we don’t observe volatility directly, each of us uses own proxy of choice. There are many ways to skin this cat (more on volatility proxy here).

In a previous post Univariate volatility forecast evaluation we considered common ways in which we can evaluate how good is our volatility model, dealing with one time-series at a time. But how do we evaluate, or compare two models in a multivariate settings, with two covariance matrices?

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Why bad trading strategies may perform well? Mathematical explanation

You probably know that even a trading strategy which is actually no different from a random walk (RW henceforth) can perform very well. Perhaps you chalk it up to short-run volatility. But in fact there is a deeper reason for this to happen, in force. If you insist on using and continuously testing a RW strategy, you will find, at some point with certainty, that it has significant outperformance.

This post explains why is that.

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