Category Archives: Randomness

Random Number Generation Part II – Some Data Analysis

Following on from my last post, I’ve been trying to read the voltage values that are generated from the random number generator I put together. I was initially using an arduino unit so I could control the voltage output down to below 1 volt. Ultimately I want to read the values with an ESP8266 unit running micropython so I can upload the values to an internet based data logger and the ESP8266 is only 1 volt tolerant on its analog-to-digital pin. So, the voltage has to be controlled and I didn’t mind blowing up one of my arduinos (which are 5 volt tolerant). The voltage divider is now composed of a 10k resistor from positive rail to analog out and a 1k resistor from analog out to ground.

The ardunio was giving me readings up to about 180 unit out of 1024 which gives about 180/1024 * 5V = 0.8789 volts… nice and safe.

Reading these voltages with the ESP8266 gave some interesting results. The code to do this was very simple:

from machine import ADC

import time
adc = ADC(0)            # create ADC object on ADC pin

for i in range(1000000):
    print(adc.read(), end=',')
    time.sleep_ms(10)

This reads a voltage every 10 milliseconds from the analog pin and reports one million of them. It took about 2.8 hours and the output was redirected to a file of type csv with this command on my mac:

ampy --port /dev/tty.SLAB_USBtoUART run readanalog.py > avalanche_noise.csv

‘ampy’ is a nice program from Adafruit, which you can learn more about here.

I then analyzed this data in python using the jupyter web interface. A histogram plot of the values looks like this:

The values range from 135 to 674 with a mean of ~292.6. This distribution is obviously skewed too so it’s not gaussian distributed – maybe Poisson?  But the most striking thing is the lines of empty space (no blue) – is this a plotting problem or are certain values skipped during the reads? Well lets zoom in around the top of the distribution.

So its true, some values are just never recorded. Its hard to believe the voltages from the circuit would do this. Also, its clear from the numbers that every 12th value is skipped. I think this must be an error in the way micropython reads the register so I’ll probably post this somewhere in the micropython forums and see if this is true or if I’m just doing something wrong.

Next I tried to fit this distribution to a gaussian curve. It failed. Here is the python code:

import math
import numpy as np
import matplotlib.pyplot as plt
import scipy as scipy
import scipy.stats as stats
from scipy.optimize import curve_fit
from scipy.misc import factorial
from scipy.stats import norm
%matplotlib inline          # needed for plotting in jupyter

data_p = data # not really needed

fig = plt.figure(figsize=(20, 8))
entries, bin_edges, patches = plt.hist(data_p, bins=(int(np.amax(data_p)-np.amin(data_p))), range=[np.amin(data_p),np.amax(data_p)], normed=True)
bin_middles = 0.5*(bin_edges[1:] + bin_edges[:-1])

# poisson function, parameter lamb is the fit parameter
def gauss(x, *p):
 A, mu, sigma = p
 return A*np.exp(-(x-mu)**2/(2.*sigma**2))

# fit with curve_fit
p0 = [1., 260., 30.] # initial guesses for height, mean and SD

parameters, cov_matrix = curve_fit(gauss, bin_middles, entries, p0 = p0) 
print parameters # print results
print np.sqrt(np.diag(cov_matrix)) # print errors of fit

# plot poisson-deviation with fitted parameter
x_plot = np.linspace(0, np.amax(data_p), 1000)
plt.plot(x_plot, gauss(x_plot, *parameters), 'r-', lw=4)
plt.show()

Output:

This doesn’t fit very well. But it does reveal the skew present in the data. It does in some ways look poisson distributed (you can read more about this distribution here) and electrical noise is typically poisson distributed because of the discrete nature of electrical charge (read about this and shot noise here).

Mathematical aside:

One way I like to think of the Poisson distribution and Poisson processes is as follows. They arise from discrete events that must always have a positive value – this is not true for Gaussian distributions. So, in our circuit, current (electrons) flow or do not flow. When they flow, a discrete (integer) number of them flow. There is a lower limit on the number that can jump the gap. That number is zero. The gaussian distribution is the limit of a binomial distribution as the number of events goes to infinity. The binomial distribution is based on the idea that an event either happens or does not. (N.B. this is different from the Poisson case because in that case, events either happen 1 or more times or do not happen). If we accumulate binomial events, there is no limit to how many ‘yes’ or ‘no’ events can happen so some of the distribution must extend as far as the number of events recorded – for gaussian this limit goes to infinity and negative infinity. The Poisson distribution doesn’t act this way. If we try and model the Poisson distribution as an infinite binomial distribution we quickly realize that while we can get an infinite number of ‘zero value’ events as well (with low probability) there is more than one other alternative. So the distribution must take into account these many possible values which stretches the distribution in the positive direction while there is a still a hard limit at zero. We can shift the Poisson distribution so ‘N’ zero-value trials will be plotted at ‘-N’ (like we would for a binomial distribution) but on the positive size, the curve would extend past ‘+N’ because some of those trials can have a value of more than 1.

Enough qualitative description of distributions…

Using a poisson function instead like this:

def poisson(k, lamb):
 return (lamb**k/factorial(k)) * np.exp(-lamb)

and we get this:

It doesn’t converge at all.

Yikes, whats going on?

Well my thought was that the numbers I’m reading don’t match the number of electrons actually flowing. This signal is amplified by the transistors and then quantized, not in nature, but by the ADC converter in the ESP8266. My hope is that this signal is proportional to the number of electrons that flowed during signal acquisition. But this signal is not the same as the number of electrons which will be behaving as a Poisson process. But the recorded numbers should be proportional to the number of electrons. So if we divide these values by some constant, can we get the fit to work, and at what optimal division factor.

Long story short, if I divide the 1000000 million points by 23.9 I get an optimal fit in terms of the error reported for the fitted parameter. That parameter, which is the mean value, is ~6.558336. Does this mean, that on average 6.5 electrons pass through the transistor while the ESP8266 is taking an ADC measurement? I think it might be! Here is the fit:

If I take these same numbers and try and fit a Gaussian curve, it doesn’t do as well.

Conclusions? The electrons that flow across the junction in reverse bias are behaving as a Poisson process as expected. The distribution is not flat. I’ve seen some discussion on the net where people seem to assume this would be the case. It does seem to be random! One of the next steps is to convert these numbers to a flat distribution or at least make it generate a binary sequence. It seems to be that XORing  or Von Neumann filtering will not do a good job of removing the biasing that the Poisson distribution will introduce.

Random Number Generation Part I – The Hardware

Intro:

The notion of randomness has consistently intrigued me, so I have always wanted to build a random number generator and play with it. Just how easy is it to generate truly random numbers as opposed to pseudo-random numbers? First of all, pseudo-random numbers are based on an algorithm so computationally they are easy to generate but also easy to copy or determine the nature of the sequence. They also ‘repeat’ their pattern eventually, even if the repeat cycle might be very large. No, no, no. I want to generate random numbers from an unpredictable natural source such as radioactive decay, cosmic rays, radio noise or ‘avalanche noise’ (hint: not the noise of snow falling down a mountain).

The Source:

In short, avalanche noise is the noisy current flow when a diode is reverse biased (voltage applied the wrong way), once that voltage is high enough to make electrons jump over the semiconductor gap the wrong way.  The nifty thing is transistors have these diode junctions and so current can flow, for example, from the base to the emitter in an NPN transistor once it is reverse biased with high enough voltage. So what many circuits do to generate random noise is set up a transistor in this way, and then amplifying the current that flows, which for ‘quantum energy gap’ reasons is noisy.

An Example Circuit:

Browsing the net I came across Rob Seward’s attempts at doing this and set up his circuit since I had all the components on hand. The circuit is below:

This is my understanding of this circuit: Here, Q1 is reverse biased with 12 V of EMF via the 4.7k resistor. Q2 is forward biased and so it should conduct with a small voltage drop (~0.9 V) across its base to emitter junction so the reverse bias in Q1 is actually ~11.1 volts through the 4.7k resistor. This should be enough to jump the gap as outlined in this great summary of this phenomenon by Giorgio Vazzana. If current randomly jumps the gap in Q1, this current will flow into the base of Q2 where it will be amplified by a common emitter setup and passed into the 0.1 uF capacitor. These spikes in current will pass through the capacitor and into the base of Q3, which is highly biased by the 1.5M resistor. I think whats going on here is this transistor is set up to be a switch – this high bias turns the collector to emitter current essentially off, so any current that flows into its base will turn it on. Thus a voltage appears at its collector (its also in common emitter mode). This voltage is divided by the two resistors of 10k and 4.7k and presented as an analog out signal.

I put this together and applied the analog out and a ground to some earbuds I had lying around. Faint noise!

I quickly wired up an arduino to take sample measurements of the voltage via an analogRead(). The values were peaking out at around 260 out of 1023 where 1023 would be a voltage of 5V. So Im seeing peak voltage here of about 1.25V. Ultimately I want to read these voltages with an ESP8266 unit or a raspberry pi which can only safely sample an analog voltage of 1V so I needed to play with the voltage divider. I’m not entirely sure why this works but I replaced the 10k resistor with a 22k resistor and the 4.7k resistor with a 10k resistor, which in turn dropped the analog reads to maximum reads of about 110 – or a little over half a volt. This is nice and safe.

The code I used for the arduino is here:

/*
 Read analog signal on pin A0

*/

const int analogInPin = A0; // Analog pin that noise is fed at

int sensorValue = 0; // Inital sensor value
 int low = 30; // Set a low value to be surpassed
 int high = 40; // Set a high value to be surpassed
 void setup() {
 // initialize serial communications at 9600 bps:
 Serial.begin(9600);
 }

void loop() {
 // read the analog in value:
 sensorValue = analogRead(analogInPin);

// if value is higher than ever recorded, lets note it
 if (sensorValue > high) {
 Serial.print("high: ");
 Serial.println(sensorValue);
 high = sensorValue;
 }
 // if value is lower than ever recorded, lets note it
 if (sensorValue < low) {
 Serial.print("low: ");
 Serial.println(sensorValue);
 low = sensorValue;
 }

// wait 2 milliseconds before the next read
 // thats 500 samples per second
 delay(2);
 }

Whats next:

The next step before making an extensive set of analog reads is to add an OP amp so I can drive a speaker and listen to the noise out loud and make some recordings for here. Until then…

A quick heads up (I have a cold and sound awful):

NMR, Compressed Sensing and sampling randomness Part II

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.

Histogram

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 Megamillions lottery 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.

Histogram

 

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.

Details will be in the next post.

NMR, Compressed Sensing and sampling randomness Part I

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 can collect 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.