Curse of Dimensionality part 4: Distance Metrics

Many machine learning algorithms rely on distances between data points as their input, sometimes the only input, especially so for clustering and ranking algorithms. The celebrated k-nearest neighbors (KNN) algorithm is our example chief, but distances are also frequently used as an input in the natural language processing domain; “You shall know a word by the company it keeps” (Firth, J. R. 1957:11); e.g. the word “jaguar” refers to the animal if words like “zoo” or “safari” are also in the neighborhood. But would refer to a mark of a car if words such as “parking” or “highway” are nearby. But (and a big one), ‘in the neighborhood’ means one thing in a low-dimension settings, and another thing in high-dimensional settings. This post emphasizes this important difference- another example of the curse of dimensionality; measuring distance in high dimension.

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Understanding Pointwise Mutual Information in Statistics

Intro

The term mutual information is drawn from the field of information theory. Information theory is busy with the quantification of information. For example, a central concept in this field is entropy, which we have discussed before.

If you google the term “mutual information” you will land at some page which if you understand it, there would probably be no need for you to google it in the first place. For example:

Not limited to real-valued random variables and linear dependence like the correlation coefficient, mutual information (MI) is more general and determines how different the joint distribution of the pair (X,Y) is to the product of the marginal distributions of X and Y. MI is the expected value of the pointwise mutual information (PMI).

which makes sense at first read only for those who don’t need to read it. It’s the main motivation for this post: to provide a clear intuition behind the pointwise mutual information term and equations, for everyone. At the end of this page, you would understand what mutual information metric actually measures, and how you should interpret it. We start with the easier concept of conditional probability and work our way through to the concept of pointwise mutual information.

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Most popular posts – 2019

As every year, I checked my analytics so that I can let you know what was popular. This year I have also experimented with a survey where I asked one question at the end of each relevant post. About 120 replies recieved, but the free Survey Monkey account (the survey provider I went with) only lets out the first 100 replies, and no exports*. Here are the results:

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CUR matrix decomposition for improved data analysis

I have recently been reading about more modern ways to decompose a matrix. Singular value decomposition is a popular way, but there are more. I went down the rabbit whole. After a couple of “see references therein” I found something which looks to justify spending time on this. An excellent paper titled “CUR matrix decomposition for improved data analysis”. This post describes how to single-out the most important variables from the data in an unsupervised manner. Unsupervised here means without a target variable in mind.

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Forecast Combination talk

Courtesy of R Consortium, you can view my forecast combination talk (16 mins) given in France few months ago, below.

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Understanding Variance Explained in PCA

Principal component analysis (PCA) is one of the earliest multivariate techniques. Yet not only it survived but it is arguably the most common way of reducing the dimension of multivariate data, with countless applications in almost all sciences.

Mathematically, PCA is performed via linear algebra functions called eigen decomposition or singular value decomposition. By now almost nobody cares how it is computed. Implementing PCA is as easy as pie nowadays- like many other numerical procedures really, from a drag-and-drop interfaces to prcomp in R or from sklearn.decomposition import PCA in Python. So implementing PCA is not the trouble, but some vigilance is nonetheless required to understand the output.

This post is about understanding the concept of variance explained. With the risk of sounding condescending, I suspect many new-generation statisticians/data-scientists simply echo what is often cited online: “the first principal component explains the bulk of the movement in the overall data” without any deep understanding. What does “explains the bulk of the movement in the overall data” mean exactly, actually?

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The Distribution of the Sample Maximum

Where I work we are now hiring. We took few time-consuming actions to make sure we have a large pool of candidates to choose from. But what is the value in having a large pool of candidates? Intuitively, the more candidates you have the better the chance that you will end up with a strong prospective candidate in terms of experience, talent and skill set (call this one candidate “the maximum”). But what are we talking about? is this meaningful? If there is a big difference between 10 candidates versus 1500 candidates, but very little difference between 10 candidates versus 80 candidates it means that our publicity and screening efforts are not very fruitful\efficient. Perhaps it would be better running quickly over a small pool, few dozens candidates, and choose the best fit. Below I try to cast this question in terms of the distribution of the sample maximum (think: how much better is the best candidate as the number of candidates grow).

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Portfolio Construction Tilting towards Higher Moments

When you build your portfolio you must decide what is your risk profile. A pension fund’s risk profile is different than that of a hedge fund, which is different than that of a family office. Everyone’s goal is to maximize returns given the risk. Sinfully but commonly risk is defined as the variability in the portfolio, and so we feed our expected returns and expected risk to some optimization procedure in order to find the optimal portfolio weights. Risk serves as a decision variable. You choose the risk, and (hope to) get the returns.

A new paper from Kris Boudt, Dries Cornilly, Frederiek Van Hollee and Joeri Willems titled Algorithmic Portfolio Tilting to Harvest Higher Moment Gains makes good progress in terms of our definition of risk, and risk-return trade-off. They propose a quantified way in which you can adjust your portfolio to account not only for the variance, but also for higher moments, namely skewness and kurtosis. They do that in two steps. The first is to simply set your portfolio based on whichever approach you follow (e.g. minvol, equal risk contribution or other). In the second step you tilt the portfolio such that the higher moments are brought into focus and get the attention they deserve. This is done by deviating from the original optimization target so that higher moments are utility-improved: less variance, better skew and lower kurtosis.

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Adaptive Huber Regression

Many years ago, when I was still trying to beat the market, I used to pair-trade. In principle it is quite straightforward to estimate the correlation between two stocks. The estimator for beta is very important since it determines how much you should long the one and how much you should short the other, in order to remain market-neutral. In practice it is indeed very easy to estimate, but I remember I never felt genuinely comfortable with the results. Not only because of instability over time, but also because the Ordinary Least Squares (OLS from here on) estimator is theoretically justified based on few text-book assumptions, most of which are improper in practice. In addition, the OLS estimator it is very sensitive to outliers. There are other good alternatives. I have described couple of alternatives here and here. Here below is another alternative, provoked by a recent paper titled Adaptive Huber Regression.

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Day of the week and the cross-section of returns

I just finished reading an interesting paper by Justin Birru titled: “Day of the week and the cross-section of returns” (reference below). The story is much too simple to be true, but it looks to be so. In fact, I would probably altogether skip it without the highly ranked Journal of Financial Economics stamp of approval. However, by the end of the paper I was as convinced as one can be without actually running the analysis.

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