The Čech Complex and the Vietoris-Rips Complex

It’s about time we got back to computational topology. Previously in this series we endured a lightning tour of the fundamental group and homology, then we saw how to compute the homology of a simplicial complex using linear algebra.

What we really want to do is talk about the inherent shape of data. Homology allows us to compute some qualitative features of a given shape, i.e., find and count the number of connected components or a given shape, or the number of “2-dimensional holes” it has. This is great, but data doesn’t come in a form suitable for computing homology. Though they may have originated from some underlying process that follows nice rules, data points are just floating around in space with no obvious connection between them.

Here is a cool example of Thom Yorke, the lead singer of the band Radiohead, whose face was scanned with a laser scanner for their music video “House of Cards.”

radiohead-still

Radiohead’s Thom Yorke in the music video for House of Cards (click the image to watch the video).

Given a point cloud such as the one above, our long term goal (we’re just getting started in this post) is to algorithmically discover what the characteristic topological features are in the data. Since homology is pretty coarse, we might detect the fact that the point cloud above looks like a hollow sphere with some holes in it corresponding to nostrils, ears, and the like. The hope is that if the data set isn’t too corrupted by noise, then it’s a good approximation to the underlying space it is sampled from. By computing the topological features of a point cloud we can understand the process that generated it, and Science can proceed.

But it’s not always as simple as Thom Yorke’s face. It turns out the producers of the music video had to actually degrade the data to get what you see above, because their lasers were too precise and didn’t look artistic enough! But you can imagine that if your laser is mounted on a car on a bumpy road, or tracking some object in the sky, or your data comes from acoustic waves traveling through earth, you’re bound to get noise. Or more realistically, if your data comes from thousands of stock market prices then the process generating the data is super mysterious. It changes over time, it may not follow any discernible pattern (though speculators may hope it does), and you can’t hope to visualize the entire dataset in any useful way.

But with persistent homology, so the claim goes, you’d get a good qualitative understanding of the dataset. Your results would be resistant to noise inherent in the data. It also wouldn’t be sensitive to the details of your data cleaning process. And with a dash of ingenuity, you can come up with a reasonable mathematical model of the underlying generative process. You could use that model to design algorithms, make big bucks, discover new drugs, recognize pictures of cats, or whatever tickles your fancy.

But our first problem is to resolve the input data type error. We want to use homology to describe data, but our data is a point cloud and homology operates on simplicial complexes. In this post we’ll see two ways one can do this, and see how they’re related.

The Čech complex

Let’s start with the Čech complex. Given a point set X in some metric space and a number \varepsilon > 0, the Čech complex C_\varepsilon is the simplicial complex whose simplices are formed as follows. For each subset S \subset X of points, form a (\varepsilon/2)-ball around each point in S, and include S as a simplex (of dimension |S|) if there is a common point contained in all of the balls in S. This structure obviously satisfies the definition of a simplicial complex: any sub-subset S' \subset S of a simplex S will be also be a simplex. Here is an example of the epsilon balls.

Image credit: Robert Ghrist

An example of a point cloud (left) and a corresponding choice of (epsilon/2)-balls. To get the Cech complex, we add a k-simplex any time we see a subset of k points with common intersection.  [Image credit: Robert Ghrist]

Let me superscript the Čech complex to illustrate the pieces. Specifically, we’ll let C_\varepsilon^{j} denote all the simplices of dimension up to j. In particular, C_\varepsilon^1 is a graph where an edge is placed between x,y if d(x,y) < \varepsilon, and C_{\varepsilon}^2 places triangles (2-simplices) on triples of points whose balls have a three-way intersection.

A topologist will have a minor protest here: the simplicial complex is supposed to resemble the structure inherent in the underlying points, but how do we know that this abstract simplicial complex (which is really hard to visualize!) resembles the topological space we used to make it? That is, X was sitting in some metric space, and the union of these epsilon-balls forms some topological space X(\varepsilon) that is close in structure to X. But is the Čech complex C_\varepsilon close to X(\varepsilon)? Do they have the same topological structure? It’s not a trivial theorem to prove, but it turns out to be true.

The Nerve Theorem: The homotopy types of X(\varepsilon) and C_\varepsilon are the same.

We won’t remind the readers about homotopy theory, but suffice it to say that when two topological spaces have the same homotopy type, then homology can’t distinguish them. In other words, if homotopy type is too coarse for a discriminator for our dataset, then persistent homology will fail us for sure.

So this theorem is a good sanity check. If we want to learn about our point cloud, we can pick a \varepsilon and study the topology of the corresponding Čech complex C_\varepsilon. The reason this is called the “Nerve Theorem” is because one can generalize it to an arbitrary family of convex sets. Given some family F of convex sets, the nerve is the complex obtained by adding simplices for mutually overlapping subfamilies in the same way. The nerve theorem is actually more general, it says that with sufficient conditions on the family F being “nice,” the resulting Čech complex has the same topological structure as F.

The problem is that Čech complexes are tough to compute. To tell whether there are any 10-simplices (without additional knowledge) you have to inspect all subsets of size 10. In general computing the entire complex requires exponential time in the size of X, which is extremely inefficient. So we need a different kind of complex, or at least a different representation to compensate.

The Vietoris-Rips complex

The Vietoris-Rips complex is essentially the same as the Čech complex, except instead of adding a d-simplex when there is a common point of intersection of all the (\varepsilon/2)-balls, we just do so when all the balls have pairwise intersections. We’ll denote the Vietoris-Rips complex with parameter \varepsilon as VR_{\varepsilon}.

Here is an example to illustrate: if you give me three points that are the vertices of an equilateral triangle of side length 1, and I draw (1/2)-balls around each point, then they will have all three pairwise intersections but no common point of intersection.

intersection

Three balls which intersect pairwise, but have no point of triple intersection. With appropriate parameters, the Cech and V-R complexes are different.

So in this example the Vietoris-Rips complex is a graph with a 2-simplex, while the Čech complex is just a graph.

One obvious question is: do we still get the benefits of the nerve theorem with Vietoris-Rips complexes? The answer is no, obviously, because the Vietoris-Rips complex and Čech complex in this triangle example have totally different topology! But everything’s not lost. What we can do instead is compare Vietoris-Rips and Čech complexes with related parameters.

Theorem: For all \varepsilon > 0, the following inclusions hold

\displaystyle C_{\varepsilon} \subset VR_{\varepsilon} \subset C_{2\varepsilon}

So if the Čech complexes for both \varepsilon and 2\varepsilon are good approximations of the underlying data, then so is the Vietoris-Rips complex. In fact, you can make this chain of inclusions slightly tighter, and if you’re interested you can see Theorem 2.5 in this recent paper of de Silva and Ghrist.

Now your first objection should be that computing a Vietoris-Rips complex still requires exponential time, because you have to scan all subsets for the possibility that they form a simplex. It’s true, but one nice thing about the Vietoris-Rips complex is that it can be represented implicitly as a graph. You just include an edge between two points if their corresponding balls overlap. Once we want to compute the actual simplices in the complex we have to scan for cliques in the graph, so that sucks. But it turns out that computing the graph is the first step in other more efficient methods for computing (or approximating) the VR complex.

Let’s go ahead and write a (trivial) program that computes the graph representation of the Vietoris-Rips complex of a given data set.

import numpy
def naiveVR(points, epsilon):
   points = [numpy.array(x) for x in points]   
   vrComplex = [(x,y) for (x,y) in combinations(points, 2) if norm(x - y) < 2*epsilon]
   return numpy.array(vrComplex)

Let’s try running it on a modestly large example: the first frame of the Radiohead music video. It’s got about 12,000 points in \mathbb{R}^4 (x,y,z,intensity), and sadly it takes about twenty minutes. There are a couple of ways to make it more efficient. One is to use specially-crafted data structures for computing threshold queries (i.e., find all points within \varepsilon of this point). But those are only useful for small thresholds, and we’re interested in sweeping over a range of thresholds. Another is to invoke approximations of the data structure which give rise to “approximate” Vietoris-Rips complexes.

Other stuff

In a future post we’ll implement a method for speeding up the computation of the Vietoris-Rips complex, since this is the primary bottleneck for topological data analysis. But for now the conceptual idea of how Čech complexes and Vietoris-Rips complexes can be used to turn point clouds into simplicial complexes in reasonable ways.

Before we close we should mention that there are other ways to do this. I’ve chosen the algebraic flavor of topological data analysis due to my familiarity with algebra and the work based on this approach. The other approaches have a more geometric flavor, and are based on the Delaunay triangulation, a hallmark of computational geometry algorithms. The two approaches I’ve heard of are called the alpha complex and the flow complex. The downside of these approaches is that, because they are based on the Delaunay triangulation, they have poor scaling in the dimension of the data. Because high dimensional data is crucial, many researchers have been spending their time figuring out how to speed up approximations of the V-R complex. See these slides of Afra Zomorodian for an example.

Until next time!

What does it mean for an algorithm to be fair?

In 2014 the White House commissioned a 90-day study that culminated in a report (pdf) on the state of “big data” and related technologies. The authors give many recommendations, including this central warning.

Warning: algorithms can facilitate illegal discrimination!

Here’s a not-so-imaginary example of the problem. A bank wants people to take loans with high interest rates, and it also serves ads for these loans. A modern idea is to use an algorithm to decide, based on the sliver of known information about a user visiting a website, which advertisement to present that gives the largest chance of the user clicking on it. There’s one problem: these algorithms are trained on historical data, and poor uneducated people (often racial minorities) have a historical trend of being more likely to succumb to predatory loan advertisements than the general population. So an algorithm that is “just” trying to maximize clickthrough may also be targeting black people, de facto denying them opportunities for fair loans. Such behavior is illegal.

Payday-Loans

On the other hand, even if algorithms are not making illegal decisions, by training algorithms on data produced by humans, we naturally reinforce prejudices of the majority. This can have negative effects, like Google’s autocomplete finishing “Are transgenders” with “going to hell?” Even if this is the most common question being asked on Google, and even if the majority think it’s morally acceptable to display this to users, this shows that algorithms do in fact encode our prejudices. People are slowly coming to realize this, to the point where it was recently covered in the New York Times.

There are many facets to the algorithm fairness problem one that has not even been widely acknowledged as a problem, despite the Times article. The message has been echoed by machine learning researchers but mostly ignored by practitioners. In particular, “experts” continually make ignorant claims such as, “equations can’t be racist,” and the following quote from the above linked article about how the Chicago Police Department has been using algorithms to do predictive policing.

Wernick denies that [the predictive policing] algorithm uses “any racial, neighborhood, or other such information” to assist in compiling the heat list [of potential repeat offenders].

Why is this ignorant? Because of the well-known fact that removing explicit racial features from data does not eliminate an algorithm’s ability to learn race. If racial features disproportionately correlate with crime (as they do in the US), then an algorithm which learns race is actually doing exactly what it is designed to do! One needs to be very thorough to say that an algorithm does not “use race” in its computations. Algorithms are not designed in a vacuum, but rather in conjunction with the designer’s analysis of their data. There are two points of failure here: the designer can unwittingly encode biases into the algorithm based on a biased exploration of the data, and the data itself can encode biases due to human decisions made to create it. Because of this, the burden of proof is (or should be!) on the practitioner to guarantee they are not violating discrimination law. Wernick should instead prove mathematically that the policing algorithm does not discriminate.

While that viewpoint is idealistic, it’s a bit naive because there is no accepted definition of what it means for an algorithm to be fair. In fact, from a precise mathematical standpoint, there isn’t even a precise legal definition of what it means for any practice to be fair. In the US the existing legal theory is called disparate impact, which states that a practice can be considered illegal discrimination if it has a “disproportionately adverse” effect on members of a protected group. Here “disproportionate” is precisely defined by the 80% rule, but this is somehow not enforced as stated. As with many legal issues, laws are broad assertions that are challenged on a case-by-case basis. In the case of fairness, the legal decision usually hinges on whether an individual was treated unfairly, because the individual is the one who files the lawsuit. Our understanding of the law is cobbled together, essentially through anecdotes slanted by political agendas. A mathematician can’t make progress with that. We want the mathematical essence of fairness, not something that can be interpreted depending on the court majority.

The problem is exacerbated for data mining because the practitioners often demonstrate a poor understanding of statistics, the management doesn’t understand algorithms, and almost everyone is lulled into a false sense of security via abstraction (remember, “equations can’t be racist”). Experts in discrimination law aren’t trained to audit algorithms, and engineers aren’t trained in social science or law. The speed with which research becomes practice far outpaces the speed at which anyone can keep up. This is especially true at places like Google and Facebook, where teams of in-house mathematicians and algorithm designers bypass the delay between academia and industry.

And perhaps the worst part is that even the world’s best mathematicians and computer scientists don’t know how to interpret the output of many popular learning algorithms. This isn’t just a problem that stupid people aren’t listening to smart people, it’s that everyone is “stupid.” A more politically correct way to say it: transparency in machine learning is a wide open problem. Take, for example, deep learning. A far-removed adaptation of neuroscience to data mining, deep learning has become the flagship technique spearheading modern advances in image tagging, speech recognition, and other classification problems.

A typical example of how a deep neural network learns to tag images. Image source: http://engineering.flipboard.com/2015/05/scaling-convnets/

A typical example of how a deep neural network learns to tag images. Image source: http://engineering.flipboard.com/2015/05/scaling-convnets/

The picture above shows how low level “features” (which essentially boil down to simple numerical combinations of pixel values) are combined in a “neural network” to more complicated image-like structures. The claim that these features represent natural concepts like “cat” and “horse” have fueled the public attention on deep learning for years. But looking at the above, is there any reasonable way to say whether these are encoding “discriminatory information”? Not only is this an open question, but we don’t even know what kinds of problems deep learning can solve! How can we understand to what extent neural networks can encode discrimination if we don’t have a deep understanding of why a neural network is good at what it does?

What makes this worse is that there are only about ten people in the world who understand the practical aspects of deep learning well enough to achieve record results for deep learning. This means they spent a ton of time tinkering the model to make it domain-specific, and nobody really knows whether the subtle differences between the top models correspond to genuine advances or slight overfitting or luck. Who is to say whether the fiasco with Google tagging images of black people as apes was caused by the data or the deep learning algorithm or by some obscure tweak made by the designer? I doubt even the designer could tell you with any certainty.

Opacity and a lack of interpretability is the rule more than the exception in machine learning. Celebrated techniques like Support Vector Machines, Boosting, and recent popular “tensor methods” are all highly opaque. This means that even if we knew what fairness meant, it is still a challenge (though one we’d be suited for) to modify existing algorithms to become fair. But with recent success stories in theoretical computer science connecting security, trust, and privacy, computer scientists have started to take up the call of nailing down what fairness means, and how to measure and enforce fairness in algorithms. There is now a yearly workshop called Fairness, Accountability, and Transparency in Machine Learning (FAT-ML, an awesome acronym), and some famous theory researchers are starting to get involved, as are social scientists and legal experts. Full disclosure, two days ago I gave a talk as part of this workshop on modifications to AdaBoost that seem to make it more fair. More on that in a future post.

From our perspective, we the computer scientists and mathematicians, the central obstacle is still that we don’t have a good definition of fairness.

In the next post I want to get a bit more technical. I’ll describe the parts of the fairness literature I like (which will be biased), I’ll hypothesize about the tension between statistical fairness and individual fairness, and I’ll entertain ideas on how someone designing a controversial algorithm (such as a predictive policing algorithm) could maintain transparency and accountability over its discriminatory impact. In subsequent posts I want to explain in more detail why it seems so difficult to come up with a useful definition of fairness, and to describe some of the ideas I and my coauthors have worked on.

Until then!

Finding the majority element of a stream

Problem: Given a massive data stream of n values in \{ 1, 2, \dots, m \} and the guarantee that one value occurs more than n/2 times in the stream, determine exactly which value does so.

Solution: (in Python)

def majority(stream):
   held = next(stream)
   counter = 1

   for item in stream:
      if item == held:
         counter += 1
      elif counter == 0:
         held = item
         counter = 1
      else:
         counter -= 1

   return held

Discussion: Let’s prove correctness. Say that s is the unknown value that occurs more than n/2 times. The idea of the algorithm is that if you could pair up elements of your stream so that distinct values are paired up, and then you “kill” these pairs, then s will always survive. The way this algorithm pairs up the values is by holding onto the most recent value that has no pair (implicitly, by keeping a count how many copies of that value you saw). Then when you come across a new element, you decrement the counter and implicitly account for one new pair.

Let’s analyze the complexity of the algorithm. Clearly the algorithm only uses a single pass through the data. Next, if the stream has size n, then this algorithm uses O(\log(n) + \log(m)) space. Indeed, if the stream entirely consists of a single value (say, a stream of all 1’s) then the counter will be n at the end, which takes \log(n) bits to store. On the other hand, if there are m possible values then storing the largest requires \log(m) bits.

Finally, the guarantee that one value occurs more than n/2 times is necessary. If it is not the case the algorithm could output anything (including the most infrequent element!). And moreover, if we don’t have this guarantee then every algorithm that solves the problem must use at least \Omega(n) space in the worst case. In particular, say that m=n, and the first n/2 items are all distinct and the last n/2 items are all the same one, the majority value s. If you do not know s in advance, then you must keep at least one bit of information to know which symbols occurred in the first half of the stream because any of them could be s. So the guarantee allows us to bypass that barrier.

This algorithm can be generalized to detect k items with frequency above some threshold n/(k+1) using space O(k \log n). The idea is to keep k counters instead of one, adding new elements when any counter is zero. When you see an element not being tracked by your k counters (which are all positive), you decrement all the counters by 1. This is like a k-to-one matching rather than a pairing.

RealityMining, a Case Study in the Woes of Data Processing

This post is intended to be a tutorial on how to access the RealityMining dataset using Python (because who likes Matlab?), and a rant on how annoying the process was to figure out.

RealityMining is a dataset of smart-phone data logs from a group of about one hundred MIT students over the course of a year. The data includes communication and cell tower data, the latter being recorded every time a signal changes from one tower to the next. They also conducted a survey of perceived friendship, personal attributes, and a few other things. See the website for more information about what data is included. Note that the website lies: there is no lat/long location data. This is but the first sign that this is going to be a rough road.

In order to access the data yourself, you must fill out a form on the RealityMining website and wait for an email, but I’ve found their turnaround time to giving a download link is quite quick. The data comes in the form of a .mat file, “realitymining.mat”, which is a proprietary Matlab data format (warning sign number two, and largely the cause of all my problems).

Before we get started let me state my goals: to extract the person-to-person voice call data, actual proximity data (whether two subjects were connected to the same cell tower at the same time), and perceived friendship/proximity data. The resulting program is free to use and modify, and is posted on this blog’s Github page.

Loading realitymining.mat into Python

Once I found out that the Python scipy library has a function to load the Matlab data into Python, I was hopeful that things would be nice and user friendly. Boy was I wrong.

To load the matlab data into Python,

import scipy.io

matlab_filename = ...
matlab_obj = scipy.io.loadmat(matlab_filename)

Then you get a variable matlab_obj which, if you try to print it stalls the python process and you have to frantically hit the kill command to wrestle back control.

Going back a second time and being more cautious, we might inspect the type of the imported object:

>>> type(matlab_obj)
<class 'dict'>

Okay so it’s a dictionary, probably why it tried to print out its entire 2GB contents when I printed it. What are it’s keys?

>>> matlab_obj.keys()
dict_keys(['network', '__header__', '__version__', '__globals__', 's'])

So far so good (the only ones that matter are ‘network’ and ‘s’). So what is ‘s’?

>>> type(matlab_obj['s'])
<class 'numpy.ndarray'>
>>> len(matlab_obj['s'])
1

Okay it’s an array of length 1… What’s its first element?

>>> type(matlab_obj['s'][0])
<class 'numpy.ndarray'>
>>> len(matlab_obj['s'][0])
106

Okay so ‘s’ is an array of one element containing the array of actual data (I remember the study has 106 participants). Now let’s look at the first participant.

>>> type(matlab_obj['s'][0][0])
<class 'numpy.void'>

Okay… what’s a numpy.void? Searching Google for “What’s a numpy.void” gives little, but combing through Stackoverflow answers gives the following:

From what I understand, void is sort of like a Python list since it can hold objects of different data types, which makes sense since the columns in a structured array can be different data types.

So basically, numpy arrays that aren’t homogeneous are ‘voids’ and you can’t tell any information about the values. Great. Well I suppose we could just try *printing* out the value, at the cost of potentially having to kill the process and restart.

>>> matlab_obj['s'][0][0]
([], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [[]], [], [[]], [[0]], [], [[0]], [], [[0]], [], [], [], [], [], [], [], [], [], [], [], [], [[413785662906.0]], [], [], [[]], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [])

Okay… so the good news is we didn’t die, but the bad news is that there is almost no data, and there’s no way to tell what the data represents!

As a last straw let’s try printing out the second user.

>>> matlab_obj['s'][0][1]
([[(array([[ 732338.86915509]]), array([[354]], dtype=uint16), array([[-1]], dtype=int16), array(['Packet Data'],
dtype='&lt;U11'), array(['Outgoing'],
dtype='&lt;U8'), array([[0]], dtype=uint8),
...
* a thousand or so lines later *
...
[2, 0], [2, 0], [2, 0], [2, 0], [2, 0], [2, 0], [2, 0], [2, 0], [2, 0], [0, 0], [2, 0], [3, 0], [3, 0], [3, 0], [3, 0], [3, 0]])

So this one has data, but still we have the problem that we have no idea what the various pieces of data represent! We could guess and check, but come on we’re in the 21st century.

Hopelessness, and then Hope

At this point I gave up on a Python approach. As far as I could tell, the data was in a completely unusable format. My only recourse was to spend two weeks obtaining a copy of Matlab from my institution, spend an hour installing it, and then try to learn enough Matlab to parse out the subset of data I wanted into a universal (plaintext) format, so I could then read it back into Python and analyze it.

Two days after obtaining Matlab, I was referred to a blog post by Conrad Lee who provided the source code he used to parse the data in Python. His code was difficult to read and didn’t work out of the box, so I had to disassemble his solution and craft my own. This and a tutorial I later found, and enough elbow grease, were enough to help me figure things out.

The first big insight was that a numpy.void also acts like a dictionary! There’s no way to tell this programmatically, no keys() attribute on a numpy.void, so apparently it’s just folklore and bad documentation. To get the fields of a subject, we have to know its key (the RealityMining documentation tells you this, with enough typos to keep you guessing).

>>> subjects = matlab_obj['s'][0]
>>> subjects[77]['my_hashedNumber']
array([[78]], dtype=uint8)
>>> subjects[76]['mac']
array([[ 6.19653599e+10]])
>>> subjects[76]['mac'][0][0]
61965359909.0

The ‘mac’ and ‘my_hashedNumber’ are supposed to be identifiers for the subjects. It’s not clear why they have two such identifiers, but what’s worse is that not all participants in the study (not all subjects in the array of subjects, at least) have mac numbers or hashedNumbers. So in order to do anything I had to extract the subset of valid subjects.

def validSubjects(allSubjects):
   return [s for s in allSubjects if hasNumeric(s,'mac') and hasNumeric(s,'my_hashedNumber')]

I define the hasNumeric function in the next section. But before that I wanted to keep track of all the stupid ids and produce my own id (contiguous, starting from zero, for valid subjects only) that will replace me having to deal with the stupid matlab subject array. So I did.

def idDicts(subjects):
   return (dict((i, s) for (i,s) in enumerate(subjects)),
      dict((getNumeric(s,'mac'), (i, s)) for (i,s) in enumerate(subjects)),
      dict((getNumeric(s, 'my_hashedNumber'), (i, s)) for (i,s) in enumerate(subjects)))

Given a list of subjects (the valid ones), we create three dictionaries whose keys are the various identifiers, and whose values are the subject objects (and my own id). So we’ll pass around these dictionaries in place of the subject array.

Extracting Communication Events

The important data here is the set of person-to-person phone calls made over the course of the study. But before we get there we need to inspect how the data is stored more closely. Before I knew about what follows, I was getting a lot of confusing index errors.

Note that the numpy object that gets returned from scipy’s loadmat function stores single-type data differently from compound data, and what’s worse is that it represents a string (which in Python should be a single-type piece of data) as a compound piece of data!

So I wrote helper functions to keep track of checking/accessing these fields. I feel like such a dirty programmer writing code like what follows, but I can still blame everyone else.

Single-type data (usually numeric):

def hasNumeric(obj, field):
   try:
      obj[field][0][0]
      return True
   except:
      return False

def getNumeric(obj, field):
   return obj[field][0][0]

Compound-type data (usually Arrays, but also dicts and strings):

def hasArray(obj, field):
   try:
      obj[field][0]
      return True
   except:
      return False

def getArray(obj, field):
   return obj[field][0]

Note the difference is for single indexing (using [0]) or multiple (using [0][0]). It occurred to me much later that there is an option to loadmat that may get rid of some or all of this doubly-nested array crap, but by then I was too lazy to change everything; it would undoubtedly be the case that some things would break and other wouldn’t (and even if that’s not the case, you can’t blame me for assuming it). I will never understand why the scipy folks chose this representation as the default.

This issue crops up when inspecting a single comm event:

>>> subjects[76]['comm'][0][0]
([[732238.1947106482]], [[0]], [[-1]], ['Voice call'], ['Outgoing'], [[16]], [[4]])

Again this record is a numpy.void with folklore keys, but notice how numeric values are doubly nested in lists while string values are not. Makes me want to puke.

Communication events can be “Incoming,” “Outgoing,” or “Missed,” although the last of these does not exist in the documentation. Not all fields exist in every record. Some are empty lists and some are double arrays containing nan, but you can check by hand that everyone that has a nonempty list of comm events has the needed fields as well:

>>> len([x for x in subjects if len(x['comm']) > 0 and len(x['comm'][0]) > 0])
81
>>> len([x for x in subjects if len(x['comm']) > 0 and len(x['comm'][0]) > 0 and
... all([hasArrayField(e, 'description') for e in x['comm'][0]])])
81

The exception is the duration field. If the duration field does not exist (if the field is an empty list), then it definitely corresponds to the “Missed” direction field.

It could otherwise be that the duration field is zero but the call was not labeled “Missed.” This might happen because they truncate the seconds, and the call lated less than one second (dubious) or because the call was also missed, or rejected without being missed. The documentation doesn’t say. Moreover, “Missed” doesn’t specify if the missed call was incoming or outgoing. So incorporating that data would ruin any chance of having a directed graph representation.

One possibility is that the outgoing calls with a duration of 0 are missed outgoing calls, and incoming calls always have positive duration, but actually there are examples of incoming calls of zero duration. Madness!

After sorting all this out (really, guessing), I was finally able to write a program that extracts the person-to-person call records in a specified date range.

First, to extract all communication events:

def allCommEvents(idDictionary):
   events = []
   for subjectId, subject in idDictionary.items():
      if hasArray(subject, 'comm'):
         events.extend([(subjectId, event) for event in getArray(subject, 'comm')])

   return events

Then to extract that subset of communication events that were actually phone calls between users within the study:

def callsWithinStudy(commEvents, hashNumDict):
   return [(subjectId, e) for (subjectId, e) in commEvents
           if (getArray(e, 'description') == "Voice call" and
              getNumeric(e, 'hashNum') in hashNumDict)]

Now we can convert the calls into a nicer format (a Python dictionary) and apply some business rules to deal with the ambiguities in the data.

def processCallEvents(callEvents, hashNumDict):
   processedCallEvents = []

   for subjectId, event in callEvents:
      direction = getArray(event, 'direction')
      duration = 0 if direction == 'Missed' else getNumeric(event, 'duration')
      date = convertDatetime(getNumeric(event, 'date'))
      hashNum = getNumeric(event, 'hashNum')
      otherPartyId = hashNumDict[hashNum][0]

      eventAsDict = {'subjectId': subjectId,
                      'direction': direction,
                      'duration': duration,
                      'otherPartyId': otherPartyId,
                      'date': date}
      processedCallEvents.append(eventAsDict)

   print("%d call event dictionaries" % len(processedCallEvents))
   return processedCallEvents

All I needed to do was filter them by time now (implement the `convertDateTime` function used above)…

Converting datetimes from Matlab to Python

Matlab timestamps are floats like

834758.1231233

The integer part of the float represents the number of days since Jan 1, 0AD. The fractional part represents a fraction of a day, down to seconds I believe. To convert from the matlab Format to the Python format, use

def convertDatetime(dt):
   return datetime.fromordinal(int(dt)) + timedelta(days=dt%1) - timedelta(days=366) - timedelta(hours=5)

This is taken without permission from Conrad’s blog (I doubt he’ll care), and he documents his reasoning in a side-post.

This then allows one to filter a list of communication events by time:

def inRange(dateRange, timevalue):
   start, end = dateRange
   unixTime = int(time.mktime(timevalue.timetuple()))
   return start <= unixTime <= end

def filterByDate(dateRange, events):
   return [e for e in events if inRange(dateRange, e['date'])]

Phew! Glad I didn’t have to go through that hassle on my own.

Extracting Survey Data

The dataset includes a survey about friendship and proximity among the participants. That means we have data on

  1. Who considers each other in the same “close group of friends.”
  2. An estimate on how much time at work each person spends with each other person.
  3. An estimate on how much time outside work each person spends with each other person.

This information comes in the form of three lists. These lists sit inside a different numpy.void sitting in the original matlab_object from the beginning of the post.

matlab_obj = scipy.io.loadmat(matlab_filename)
matlab_obj['network'][0][0]

Now the access rules (getNumeric, getArray) we used for the matlab_obj[‘s’] matrix don’t necessarily apply to the values in the ‘network’ matrix. The friendship survey is accessed by

friends = network['friends']

and it’s an array, each element of which is a participant’s response to the survey (claimed to be stored as a 0-1 array). So, for example, to get the number corresponding to whether id1 considers id2 a close friend, you would use

friends = network['friends'][id1][id2]

This is all straightforward, but there are two complications (stupid trivialities, really). The first is that the matrix is not 0-1 valued, but (nan-1.0)-valued. That’s right, if the individual didn’t consider someone a friend, the record for that is numpy’s nan type, and if they did it’s the float 1.0. So you need to either replace all nan’s with zeros ahead of time, or take account for that in any computations you do with the data.

I don’t know whether this is caused by scipy’s bad Matlab data import or the people producing the data, but it’s a clear reason to not use proprietary file formats to distribute data.

The second problem is that the survey is stored as an array, where the index of the array corresponds to the person the respondent is evaluating for friendship. The problem with this is that not all people are in the survey (for whatever reason, people dropping out or just bad study design), so they include a separate array in the network object called “sub_sort” (a totally unhelpful name). Sub_sort contains in its $i$-th position the hashNum (of the user from the ‘s’ matrix) of “survey-user” $i$ (that is the user whose answer is represented in index i of the network array).

This is incredibly confusing, so let’s explain it with an example. Say that we wanted to parse user 4’s friendship considerations. We’d first access his friends list by

friends = network['friends'][4]

Then, to see whether he considers user 7 as a friend, we’d have to look up user 7 in the sub_sort array (which requries a search through said array) and then use the index of user 7 in that array to index the friends list appropriately. So

subsortArray = network['sub_sort'][0]
user7sIndex = subsortArray.indexof(7)
network['friends'][4][user7sIndex]

The reason this is such terrible design is that an array is the wrong data structure to use. They should have used a list of dictionaries or a dictionary with ordered pairs of hashNums for keys. Yet another reason that matlab (a language designed for numerical matrix manipulation) is an abysmal tool for the job.

The other two survey items are surveys on proximity data. Specifically, they ask the respondent to report how much time per day he or she spends within 10 feet of each other individual in the study.

inLabProximity = network['lab']
outLabProximity = network['outlab']

The values in these matrices are accessed in exactly the same way, with the values being in the set {nan, 1.0, 2.0, 3.0, 4.0, 5.0}. These numbers correspond to various time ranges (the documentation is a bit awkward about this too, using “at least” in place of complete time ranges), but fine. It’s no different from the friends matrix, so at least it’s consistent.

Cell Tower Data

Having learned all of the gotchas I did, I processed the cell tower data with little fuss. The only bothersome bit was that they provide two “parallel” arrays, one called “locs” containing pairs of (date, cell tower id), and a second called “loc_ids” containing numbers corresponding to a “second” set of simplified cell tower ids. These records correspond to times when a user entered into a region covered by a specific tower, and so it gives very (coarse location) data. It’s not the lat/long values promised by the website, and even though the ids for the cell towers are governed by a national standard, the lat/long values for the towers are not publicly or freely available in database form.

Why they need two sets of cell tower ids is beyond me, but the bothersome part is that they didn’t include the dates in the second array. So you must assume the two arrays are in sync (that the events correspond elementwise by date), and if you want to use the simplified ids you still have to access the “locs” array for the dates (because what use would the tower ids be without dates?).

One can access these fields by ignoring our established rules for accessing subject fields (so inconsistent!). And so to access the parallel arrays in a reasonable way, I used the following:

towerEvents = list(zip(subject['locs'], subject['loc_ids']))

Again the datetimes are in matlab format, so the date conversion function above is used to convert them to python datetimes.

Update: As I’ve continued to work with this data set, I’ve found out that, once again contrary to the documentation, the tower id 1 corresponds to being out of signal. The documentation claims that tower id 0 implies no signal, and this is true of the “locs” array, but for whatever reason they decided to change it to a 1 in the “loc_ids” array. What’s confusing is that there’s no mention of the connection. You have to guess it and check that the lengths of the corresponding lists of events are equal (which is by no means proof that the towers ids are the same, but good evidence).

The documentation is also unclear as to whether the (datestamp, tower id) pair represents the time when the user started using that tower, or the time when the user finished using that tower. As such, I must guess that it’s the former at the risk of producing bad data (and hence bad research).

Conclusions

I love free data sets. I really do, and I appreciate all the work that researchers put into making and providing their data. But all the time I encounter the most horrible, terrible, no good, very bad data sets! RealityMining is only one small example, and it was actually nice because I was able to figure things out eventually. There are other datasets that have frustrated me much more with inconsistencies and complete absence of documentation.

Is it really that hard to make data useable? Here are the reasons why this was such an annoying and frustrating task:

  1. The data was stored in a proprietary format. Please please please release your data in plaintext files!
  2. The documentation contradicted the data it was documenting, and was altogether unclear. (One contradiction is too often. This was egregious.)
  3. There was even inconsistency within the data. Sometimes missing values were empty lists, sometimes numpy.nan, and sometimes simply missing.
  4. Scipy has awful defaults for the loadmat function.
  5. Numpy data structures are apparently designed for opacity unless you know the folklore and spend hours poring over documentation.

It seems that most of this could be avoided by fixing 1 and 2, since then it’s a problem that an automated tool like Google Refine can fix (though I admit to never having tried Refine, because it crashed on the only dataset I’ve ever given it). I know scientists have more important things to worry about, and I could just blame it on the 2004 origin of the data set, but in the age of reproducibility and big data (and seeing that this data set is widely used in social network research), it’s just as important to release the data as it is to make the data fit for consumption. RealityMining is like a math paper with complicated terms defined only by example, and theorems stated without proof that are actually false. It doesn’t uphold the quality that science must, it makes me question the choices made by other researchers who have had to deal with the dataset, and it wastes everyone else’s time.

In this respect, I’ve decided to post my Python code in a github repository, as is usual on this blog. You’ll have to get the actual RealityMining dataset yourself if you want to use it, but once you have it you can run my program (it depends only on the numpy/scipy stack and the standard library; tested in Python 3.3.3) to get the data I’m interested in or modify it for your own purposes.

Moral of the story: For the sake of science, please don’t distribute a database as a Matlab matrix! Or anything else for that matter!