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.

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Boundary corrected kernel density

Density estimation is now a trivial one-liner script in all modern software. What is not so easy is to become comfortable with the result, how well is is my density estimated? we rarely know. One reason is the lack of ground-truth. Density estimation falls under unsupervised learning, we don’t actually observe the actual underlying truth. Another reason is that the theory around density estimation is seldom useful for the particular case you have at hand, which means that trial-and-error is a requisite.

Standard kernel density estimation is by far the most popular way for density estimation. However, it is biased around the edges of the support. In this post I show what does this bias imply, and while not the only way, a simple way to correct for this bias. Practically, you could present density curves which makes sense, rather than apologizing (as I often did) for your estimate making less sense around the edges of the chart; that is, when you use a standard software implementation.

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R + Python = Rython

Enough! Enough with that pointless R versus Python debate. I find it almost as pointless as the Bayesian vs Frequentist “dispute”. I advocate here what I advocated there (“..don’t be a Bayesian, nor be a Frequenist, be opportunist“).

Nowadays even marginally tedious computation is being sent to faster, minimum-overhead languages like C++. So it’s mainly syntax administration we insist to insist on. What does it matter if we have this:

Or that

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R tips and tricks, on-screen colors

I like using Rlogo for many reasons. Two of those are (1) easy integration with almost whichever software you can think of, and (2) for its graphical powers. Color-wise, I dare to assume you probably plotted, re-specified your colors, plotted again, and iterated until you found what works for your specific chart. Here you can find modern visualization so you are able to quickly find the colors you look for, and to quickly see how it looks on screen. See below for quick demo.

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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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