obviously doesn’t converge. Right? This sum doesn’t actually evaluate to a number, does it? Well I wouldn’t think so – no part of my intuition suggests this series is limited. So imagine how surprised I was (and perhaps you too) when I learned this Maths conundrum has been spreading around the internet early this year due to a few videos from the Numberphile guys. Their answer?
What? Really? ? There is something odd here, clearly. How can this be true? This is even beyond breaking intuition, this is just plain wrong. So why is this even taken seriously? The ‘proof’ comes from a field of Maths called Mathematical Analysis and infinite series. I wont go over the proof, there are examples here, here and here. However, they all depend on the following equality:
This is true for a Cesàro summation. That is, its true because of a definition of how to sum a series that doesn’t actually converge to a value. This is fairly well accepted and I suppose if you define the process of Cesàro summation to be a method of summing non-converging sequences then that is just fine… by definition. For me, however, this doesn’t sit right. This is not a sum that comes from the axioms that lead to arithmetic, the so called Peano axioms (I don’t have problems with the Peano axioms – I doubt you do/would too). But I do have a problem with Cesàro summation as a definition. Since Cesàro summations have results like 1-1+1-1… = 0.5 that contradict other forms of the sum like…
the only way the sum could be evaluated to 0.5 is by a specific definition that excludes the above sums. But why should I select this definition? Do I accept it as an axiom? If I do, how do I account for the fact that this axiom has two other intuitive sums? Obviously it is a bad axiom and therefore a bad definition. I really don’t understand why anyone would accept the Cesàro summation as anything useful at all. It immediately leads to inconsistencies. Its useless.
Using Compressed Sensing: Selecting which points to collect.
NMR data is collected as a vector or matrix of points (depending how many dimensions the experiment has). Compressed sensing permits us to collect a subset of the actual points that are usually required. The question is, how do you decide which points to collect.
TL;DR Summary: Compressed sensing (CS) allows us to sample a subset of the points normally required for an NMR spectrum. CS theory suggests we should sample these points randomly. Random sampling leads to problems with the distribution of the gaps between samples. There is a better solution for NMR data.
The problem of Random Sampling.
Formal Compressed Sensing (CS) theory says we should collect a random spread of points between 1 and the final point (N). We could make a schedule of points by constructing a “dart-throwing” algorithm that randomly selects a point between 1 and N. If the algorithm selects a point twice, we just ask it to select another time. We run this until we have N/5 points. Surprisingly, it turns out this does not work well. Lets see why…
Below is an example of what happens when you select randomly 64 points out of 256 points. First of all, to make sure I selected points in an unbiased way, I took the numbers 1 through 256 into a vector and did a Fisher-Yates shuffle. Then I simply select the first 64 points in the vector after the shuffle. It has been shown that the Fisher-Yates shuffle is unbiased and I think it is a better solution to sampling purely randomly than a dart-throwing algorithm. Since I have 64 random points out of 256, you would expect that the average distance between numbers should be:
(256 – 64)/65 = 2.95385
This is because we need to eliminate 64 out of 256 (256 – 64) because they are selected and not part of gaps, then we must divide by 65 because by selecting 64 points we actually create 65 gaps. This means that the average gap size (that is the gap between selected numbers) is 2.95385.
I did 45 simulations like this, generating 64 * 45 = 2880 gaps. The average gap was 2.96319 so this basically looks good. Now, the Standard Deviation was 3.35334 which shows something funky is happening right away. This is a very large standard deviation for a mean value of 2.95385 and permits negative numbers to appear within one standard deviation. This can’t even happen for a gap as gaps can’t be negative and there are no negative gap values in my numbers. How can this standard deviation be? Clearly the gaps are not normally distributed. Lets look at a histogram of the gap sizes.
Yikes, its an exponential distribution. This has been noted before for random gap sizes, but I was amazed when I first saw this – in hind sight I’m not, but at first I was a little shocked. To confirm that this really does happen for random sampling I went looking for another example of a subset of random numbers selected from a pool of numbers. Lottery results of course are a good example.
I acquired the complete set of Megamillionslottery numbers from about June 2005 to October 2013 (the nature of the pool of numbers did not change in that time) and calculated the gaps between these random numbers. In this game you had to pick 5 numbers from 1-56 (the rules have since changed). So the average distance between numbers is
(56 -5) / 6 = 8.5
My calculated gap distance is 8.42009 and the histogram of the gap sizes is below.
So, why does this matter? Well what is happening is the gaps are distributed in a way which can cause problems when we try and fill in the data we skipped in the gaps. To begin with look at the above histograms. When we randomly select points we end up making gaps such that the most common gap size is actually no gap at all. Thats pretty remarkable when you think about it. Then the second most common gap is the smallest gap, a gap of 1. By the time we have reach the average gap (around 8 for Megamillions, or a little less than 3 for selecting 64 points from 256) one half of our gaps have been accounted for, by definition. The reason so many gaps fall here is because there is a lower limit to the size of the gap. They can’t be less than zero. Now, all these small gaps are great, because they should be easy to fill in. The problem is the upper limit on the gap size is the size of the sampling space minus the number of samples. This number is always going to be several times larger than the average gap size. Simply put, this will always give the exponential distribution seen in the histograms, resulting in some gaps that are much larger than the average gap size. These large gaps present problems when trying to fill in data with predictive methods.
So how do we do better than this when ‘randomly’ selecting which samples to take? First of all, we don’t use a straight random algorithm – it makes gaps that are too large. Instead we introduced some randomness around the average gap size by using a poisson distribution. We call this Poisson Gap sampling, a method developed by Sven Hyberts in my lab.
One of the things I spend my time thinking about is the problem of shortening the time it takes to acquire Nuclear Magnetic Resonance data on biomolecular macromolecules like proteins and nucleic acids. Below is an attempt to describe the problem non-technically and how we can speed up data acquisition by ‘randomly’ sampling a subset of the data normally required.
TL;DR Summary:Acquiring nuclear magnetic resonance data takes a long time. This is because to get good resolution traditionally a certain number of data points must be sampled. Can we skip some of these points? The answer is yes, the next question is which points?
The problem: NMR takes a long time.
NMR spectra have great power in describing macromolecules at the atomic level when collected in 2, 3 or even 4 dimensions. Each dimension represents a different kind of information (say, nucleus type, location in a repeating unit of the polymer, distance from another nucleus) – so multiple-dimension spectra are data-heavy but help isolate specific atomic groups really well. The information in each dimension is actually frequency information – it is the frequency of each atom in the molecule. The down-side is these spectra take a long time to acquire. In fact, to acquire 3 and 4 dimensional data, experiments are usually shortened by not acquiring an ideal number of samples. That is, most of the dimensions are truncated in time which leads to poor frequency discrimination in that dimension.
Now, each dimension beyond the first dimension is acquired slowly for a number of technical reasons I wont discuss here. However, lets say that in one of these dimensions we would ideally like to acquire N points, but we really only have time to acquire N/4 points. This means our frequency resolution will drop 4 times. For further technical reasons, our frequency resolution is not just a function of N but also a function of some time delays between the sampled points. These time delays (we call them evolution delays) are actually very fast, its just that the time between when we cancollect these points is slow (blah blah blah – further technical reason). This means we don’t have to wait a long time to actually collect the Nth point above, it just takes a long time to get to N because we must collect 1, 2, 3… N-2, N-1, N points along the way. This is a window of opportunity here if we can actually skip points and quickly get to the Nth point and maintain the same high frequency resolution expected when collecting all N points.
This can be done and in NMR and is called non-uniform sampling. It is also a type of compressed sensing.
Compressed Sensing: Collecting some data and discarding most.
Several techniques have been developed to allow collection of data out to distance points without having to collect all the points in between. Programmatically it is fairly easy to get an NMR spectrometer to do this. The problem lies in processing the data into a spectrum that contains few, if any, significant artifacts. The regular FFT (Fast Fourier Transform), for example, can be used by simply setting the non-collected data points to zero. This however results in significant artifacts. The problem is how to reconstruct the missing data and minimize the artifacts. Compressed sensing (CS) is a theory that describes a way to do this. Originally CS was developed for image processing but it has successfully been applied to NMR data. Assuming signals are sparse (true for NMR data) and that noise is not significant (mostly true for NMR data), compressed sensing algorithms can reconstruct the skipped data.
How do we decide which points to skip?
I will talk more about this in the next post… stay tuned.