Endogenously Detecting Structural Breaks in a Time Series: Implementation in R

The most conventional approach to determine structural breaks in longitudinal data seems to be the Chow Test.

From Wikipedia,

The Chow test, proposed by econometrician Gregory Chow in 1960, is a test of whether the coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known a priori (for instance, a major historical event such as a war). In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.

As shown in the figure below, regressions on the 2 sub-intervals seem to have greater explanatory power than a single regression over the data.


For the data above, determining the sub-intervals is an easy task. However, things may not look that simple in reality. Conducting a Chow test for structural breaks leaves the data scientist at the mercy of his subjective gaze in choosing a null hypothesis for a break point in the data.

Instead of choosing the breakpoints in an exogenous manner, what if the data itself could learn where these breakpoints lie? Such an endogenous technique is what Bai and Perron came up with in a seminal paper published in 1998 that could detect multiple structural breaks in longitudinal data. A later paper in 2003 dealt with the testing for breaks empirically, using a dynamic programming algorithm based on the Bellman principle.

I will discuss a quick implementation of this technique in R.

Brief Outline:

Assuming you have a ts object (I don’t know whether this works with zoo, but it should) in R, called ts. Then implement the following:

An illustration 

I started with data on India’s rice crop productivity between 1950 (around Independence from British Colonial rule) and 2008. Here’s how it looks:


You can download the excel and CSV files here and here respectively.

Here’s the way to go using R:

Voila, this is what you get:


The dotted vertical lines indicated the break dates; the horizontal red lines indicate their confidence intervals.

This is a quick and dirty implementation. For a more detailed take, check out the documentation on the R package called strucchange.

Abu Mostafa’s Machine Learning MOOC – Now on EdX

This was in the pipeline for quite some time now. I have been waiting for his lectures on a platform such as EdX or Coursera, and the day has arrived. You can enroll and start with week 1’s lectures as they’re live now.

This course is taught by none other than Dr. Yaser S. Abu – Mostafa, whose textbook on machine learning, Learning from Data is #1 bestseller textbook (Amazon) in all categories of Computer Science. His online course has been offered earlier over here.


Dr. Abu-Mostafa received the Clauser Prize for the most original doctoral thesis at Caltech. He received the ASCIT Teaching Awards in 1986, 1989 and 1991, the GSC Teaching Awards in 1995 and 2002, and the Richard P. Feynman prize for excellence in teaching in 1996.

Live ‘One-take’ Recordings

The lectures have been recorded from a live broadcast (including Q&A, which will let you gauge the level of CalTech students taking this course). In fact, it almost seems as though Abu Mostafa takes a direct jab at Andrew Ng’s popular Coursera MOOC by stating the obvious on his course page.

A real Caltech course, not a watered-down version


Again, while enrolling note that this is what Abu Mostafa had to say about the online course:  “A Caltech course does not cater to short attention spans, and it may not provide instant gratification…[like] many MOOCs out there that are quite simple and have a ‘video game’ feel to them.” Unsurprisingly, many online students have dropped out in the past, but some of those students who “complained early on but decided to stick with the course had very flattering words to say at the end”.


  • Basic probability
  • Basic matrices
  • Basic calculus
  • Some programming language/platform (I choose Python!)

If you’re looking for a challenging machine learning course, this is probably one you must take.


Implementing Undirected Graphs in Python

There are 2 popular ways of representing an undirected graph.

Adjacency List
Each list describes the set of neighbors of a vertex in the graph.


Adjacency Matrix
The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.


Here’s an implementation of the above in Python:


Deterministic Selection Algorithm Python Code

Through this post, I’m sharing Python code implementing the median of medians algorithm, an algorithm that resembles quickselect, differing only in the way in which the pivot is chosen, i.e, deterministically, instead of at random.

Its best case complexity is O(n) and worst case complexity O(nlog2n)

I don’t have a formal education in CS, and came across this algorithm while going through Tim Roughgarden’s Coursera MOOC on the design and analysis of algorithms. A video covering that algorithm is shown below, followed by my implementation in Python.

I get the following output:

100 loops, best of 3: 2.38 ms per loop

Note that on the same input, quickselect is faster, giving us:

1000 loops, best of 3: 254 µs per loop

scikit-learn Linear Regression Example

Here’s a quick example case for implementing one of the simplest of learning algorithms in any machine learning toolbox – Linear Regression. You can download the IPython / Jupyter notebook here so as to play around with the code and try things out yourself.

I’m doing a series of posts on scikit-learn. Its documentation is vast, so unless you’re willing to search for a needle in a haystack, you’re better off NOT jumping into the documentation right away. Instead, knowing chunks of code that do the job might help.

Sharing IPython / Jupyter Notebooks via WordPress

In order to share (a static version of) your IPython / Jupyter notebook on your WordPress site, follow three straightforward steps.

Step 1: Let’s say your Jupyter Notebook looks like this:


Open this notebook in a text editor and copy the content which may look like so:


Step 2: Ctrl + A and Ctrl + C this content. Then Ctrl + V this to a GitHub Gist that you should create, like so:


Step 3: Now simply Create public gist and embed the gist like you always embed gists on WordPress, viz., go to the HTML editor and add like so:


I followed the exact steps that I’ve mentioned above to get the following result:


Randomized Selection Algorithm (Quickselect) – Python Code

Find the kth smallest element in an array without sorting.

That’s basically what this algorithm does. It piggybacks on the partition subroutine from the Quick Sort. If you don’t know what that is, you can check out more about the Quick Sort algorithm here and here, and understand the usefulness of partitioning an unsorted array around a pivot.

Animated visualization of the randomized selection algorithm selecting the 22nd
smallest value

Python Implementation

Related Posts
Quick Sort Python Code
Computing Work Done (Total Pivot Comparisons) by Quick Sort