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Independent component analysis

Two microphones are placed in a room where two conversations are taking place simultaneously. Given these two recordings, can one “remix” them in some prescribed way to isolate the individual conversations? Yes! In this post, we review one simple approach to solving this type of problem, Independent Component Analysis (ICA). We share an ipython document implementing ICA and link to a youtube video illustrating its application to audio de-mixing.

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Maximum-likelihood asymptotics

In this post, we review two facts about maximum-likelihood estimators: 1) They are consistent, meaning that they converge to the correct values given a large number of samples, $N$, and 2) They satisfy the Cramer-Rao lower bound for unbiased parameter estimates in this same limit — that is, they have the lowest possible variance of any unbiased estimator, in the $N\gg 1$ limit.

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Principal component analysis

We review the two essentials of principal component analysis (“PCA”): 1) The principal components of a set of data points are the eigenvectors of the correlation matrix of these points in feature space. 2) Projecting the data onto the subspace spanned by the first $k$ of these — listed in descending eigenvalue order — provides the best possible $k$-dimensional approximation to the data, in the sense of captured variance.

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NBA 2015-16!!!

NBA is back this Tuesday! The dashboard and weekly predictions are now live*, once again. These will each be updated daily, with game winner predictions, hypothetical who-would-beat-whom daily matchup predictions, and more. For a discussion on how we make our predictions, see our first post on this topic. Note that our approach does not make use of any bookie predictions (unlike many other sites), and so provide an independent look on the game.

This season, we hope to crack 70% accuracy!

* Note that we have left up last season’s completed games results, for review purposes. Once every team has played one game, we’ll switch it over to the current season’s results.

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Support Vector Machines for classification

To whet your appetite for support vector machines, here’s a quote from machine learning researcher Andrew Ng:

“SVMs are among the best (and many believe are indeed the best) ‘off-the-shelf’ supervised learning algorithms.”

Andrew Ng

Professor Ng covers SVMs in his excellent Machine Learning MOOC, a gateway for many into the realm of data science, but leaves out some details, motivating us to put together some notes here to answer the question:

“What are the support vectors in support vector machines?”

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A review of parameter regularization and Bayesian regression

Here, we review parameter regularization, which is a method for improving regression models through the penalization of non-zero parameter estimates. Why is this effective? Biasing parameters towards zero will (of course!) unfavorably bias a model, but it will also reduce its variance. At times the latter effect can win out, resulting in a net reduction in generalization error. We also review Bayesian regressions — in effect, these generalize the regularization approach, biasing model parameters to any specified prior estimates, not necessarily zero.

This is the second of a series of posts expounding on topics discussed in the text, “An Introduction to Statistical Learning”. Here, we cover material from its Chapters 2 and 6. See prior post here.

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Getting started with Pandas

We have made use of Python’s Pandas package in a variety of posts on the site. These have showcased some of Pandas’ abilities including the following:

  • DataFrames for data manipulation with built in indexing
  • Handling of missing data
  • Data alignment
  • Melting/stacking and Pivoting/unstacking data sets
  • Groupby feature allowing split -> apply -> combine operations on data sets
  • Data merging and joining

Pandas is also a high performance library, with much of its code written in Cython or C. Unfortunately, Pandas can have a bit of a steep learning curve — In this post, I’ll cover some introductory tips and tricks to help one get started with this excellent package.

Notes:

  • This post was partially inspired by Tom Augspurger’s Pandas tutorial, which has a youtube video that can be viewed along side it. We also suggest some other excellent resource materials — where relevant — below.
  • The notebook we use below can be downloaded from our github page. Feel free to grab it and follow along.


Follow us on twitter for new submission alerts!

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Stochastic geometric series

Let $a_1, a_2, \ldots$ be an infinite set of non-negative samples taken from a distribution $P_0(a)$, and write
$$\tag{1} \label{problem}
S = 1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \ldots.
$$
Notice that if the $a_i$ were all the same, $S$ would be a regular geometric series, with value $S = \frac{1}{1-a}$. How will the introduction of $a_i$ randomness change this sum? Will $S$ necessarily converge? How is $S$ distributed? In this post, we discuss some simple techniques to answer these questions.

Note: This post covers work done in collaboration with my aged p, S. Landy.

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Build a web scraper for a literature search – from soup to nuts

Code, references, and examples of this project are on Github.

In this post, I’ll describe the soup to nuts process of automating a literature search in Pubmed Central using R.

It feels deeply satisfying to sit back and let the code do the dirty work.

Is it as satisfying as a bowl of red-braised beef noodle soup with melt-in-your-mouth tendons from Taipei’s Yong Kang Restaurant (featured image)?

If you have to do a lit search like this more than once, then I have to say the answer is yes — unequivocally, yes.
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Leave-one-out cross-validation

This will be the first of a series of short posts relating to subject matter discussed in the text, “An Introduction to Statistical Learning”. This is an interesting read, but it often skips over statement proofs — that’s where this series of posts comes in! Here, I consider the content of Section 5.1.2: This gives a lightning-quick “short cut” method for evaluating a regression’s leave-one-out cross-validation error. The method is applicable to any least-squares linear fit.

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