# Learning to become a better coder with list comprehensions

One of the problems with self-teaching code as you go along is that you then often miss out on tricks (at least for me). I started out coding with MATLAB during my Undergraduate and moved over to Python once I started my PhD, and so have very much learnt on the job with coding on a need to know basis. However, as my code is needing to handle more and more data, I am frequently faced with having to go back and optimise in order to make it faster. And so, my new favourite things to improve my code are list comprehensions in place of loops.

List comprehensions are a way of transforming any list (or anything iterable) into another list, in a much faster way than a for loop can.

First, a simple example. Imagine that you want to know what the result of multiplying each number from 0 to 9, by 2. This could be done with a loop as follows:

Loop

numbers=[]
for x in range(10):
numbers.append(x*2)
print numbers
> [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]


And this works fine – it appends each x*2 term to the numbers array from the range function, which goes between 0 and 9. However, if you were to scale this up to do more than a simple multiplication, your computer may start to struggle.

This is where we can use list comprehension. This above loop can be easily translated:

List comprehension

numbers = [x * 2 for x in range(10)]
print numbers
> [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]


What this does is it takes the first statement within the square brackets (x*2 in our case), and goes through the range function to find what our value of x is, and applies the squared term, giving the same result for the numbers array.

From using the list comprehensions first very simply, it is then possible to expand them for more complex functions, as they can also handle nested loops. Thus you can greatly optimise the speed of your code by replacing any suitable loops with them. Initially I was put off by them because I struggled to understand how to use them with my own code, and so I thought I would put up a blog in the off chance that anyone else could benefit from this too.

My greatest improvement was when I was creating my synthetic seismograms. To do this, I was creating noise, and then working out interevent times to add in my events. In order to merge the two, I needed to add the noise onto the event at the corresponding times (as otherwise the event would be literally the same each time and I was trying to make this as realistic as possible) and have the event at a magnitude determined by another variable.

Whenever I need to do something sequential, I always default to loops, as it seemed simple enough to go through each interevent time,  get the relevant noise, and add the event (with a predetermined magnitude) to the noise portion to create a Stream of Traces. With 200 events, this step was taking me roughly 25 minutes! But now, with using list comprehension, I have sped up this function to only take 7 seconds. Great improvement!

I have included both my use of list comprehensions and loops to show how the code can change between the two. From taking the time to understand how they work, I have now been able to implement these into many more areas of my code. So other aspects like my cross-correlations are much faster because of this small change.

There are many great tutorials out there, and so you should look into these if you are interested in optimising your code (if it was full of loops like mine). I’m always discovering new ways in which I can improve my code, so I believe that the process of learning to become a better coder is very much ongoing for me!

— Roseanne

### Using list comprehensions : 7 seconds

# make the noise trace into a numpy array
testnoise = np.array(noise_trace)

# create a Stream of the events (with Gutenberg Richter magnitudes)
st_events_poisson = Stream([Trace(testnoise[i:i+len(st_event)]
+ (j * np.array(st_event.data)))
for i,j in zip(poisson_times.astype('int'), g)])

# loop through times to change the stats
for i in range(0, len(poisson_times)):
st_events_poisson[i].stats.starttime = st_events_poisson[i].stats.starttime + poisson_times[i]
st_events_poisson[i].stats.sampling_rate = samp_rate
st_events_poisson[i].stats.delta = delta


### Using loops : 25 minutes

# loop through for each interevent time
for i in range(0, len(poisson_times)):
# make the noise trace into a numpy array
testnoise = np.array(noise_trace)

# find the noise portion for where the event occurs
noise_portion = testnoise[(poisson_times[i]*int(samp_rate)):
(poisson_times[i]*int(samp_rate))
+ int(len(st_event))]

# add the event (multiplied by a Gutenberg-Richter magnitude)
# onto the noise portion
noise_plus_event_arr = noise_portion
+ (np.array(st_event.data) * g[i])

# make this into a Trace and assign stats
noise_plus_event = Trace(noise_plus_event_arr)
noise_plus_event.stats.sampling_rate = samp_rate
noise_plus_event.stats.delta = delta
noise_plus_event.stats.starttime = noise_trace.stats.starttime
+ poisson_times[i]

# create Stream of events
st_events_poisson.append(noise_plus_event)


# Correlation does not imply causation.. but it does give you a hint

This is just a short note on plotting a correlation matrix using the seaborn package within Python. I’ve found that this is the best way of showing the similarity between arrays to people who are unfamiliar with correlations. It also allows you to add some colour into your plots, which is always a nice thing! It can be used for a multitude of purposes, so I have left the variable names in my code (at the bottom) as general as possible, so that it can be copy and pasted for other users.

For those who have not seen these matrices before, what it shows is the similarity between different arrays. If two arrays have a correlation value of 1.0, this means that they have a perfect correlation (i.e. they are exactly the same), and a correlation value of 0.0 means that there is absolutely no similarity between the two. This can be used to compare datasets with one another if you are looking for a similar pattern.

Also, it is worth noting that one of the principal statements made in statistics is that,

“Correlation does not imply causation”

So you should also have some further information to back-up the correlation between arrays.

An example of one of these correlation matrices can be seen below, which shows the comparison of 54 arrays with each other (i.e. I have taken each array and cross-correlated it with the other 53 arrays). The squares with a darker tone have a higher correlation than those with a lighter tone.

Correlation matrix for 54 arrays

Your first step is putting your correlation values into a pandas.DataFrame format, you can then just use the code below in order to create the matrix! This table should contain the full dataset, and this code can then create it into this triangle shape (as otherwise you will end up with the mirror image of this on the identity axis). I have used absolute values as I didn’t want to deal with negative correlation at this stage (this is when it is a perfect match but reversed in the x-axis).

If you don’t have any correlation values, I’d recommend reading up on cross-correlation, which is a function where you can obtain these correlation values. I might produce a blog post on this at a later date, but it is worth reading into it yourself so that you can fully understand the output.

— Roseanne

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(font_scale=1.5)

def corr_mat_plot(correlation_mat, show = True, outfile = None):
"""
Plots the correlation matrix in an image plot to show where the
highest correlation between arrays is.
"""
# Make the mask for the upper triangle so that it doesn't mirror image the values

# Set up the figure
fig, ax = plt.subplots(figsize=(10, 10))
sns.set(font_scale=1.5)

# Draw matrix
sns.heatmap(np.abs(correlation_mat), cmap = sns.cubehelix_palette(8, as_cmap=True),
cbar_kws = {"shrink": .8, "label" : ("Correlation value")}, ax=ax)

plt.title("Correlation between the arrays")

if show:
plt.show()

if outfile:
fig.savefig(outfile)
elif show:
plt.show()
else:
return fig



# Gutenberg-Richter and fish?

This post is for explaining the basics behind two key statistical seismology terms: Gutenberg-Richter and Poisson distributions.

Gutenberg-Richter

The Gutenberg-Richter law is a relationship which every seismologist knows – for those who are not so aware (like me just over a year ago), it refers to an expression which relates the total number of earthquakes in any given region to the magnitude, by the following equation:

$log_{10} N = a - bM$

where $N$ is the total number of earthquakes, $a$ is a constant (usually 1), $b$ is another constant which depends on the seismicity in the area (close to 1 in seismically active areas), and $M$ is the magnitude. This can also be seen by the plot below.

This shows the Gutenberg-Richter distribution for a b value of 1. Code for this is at the end of the post.

What this expression does, is relate the frequency of earthquakes with their magnitude, i.e., there are lots of small earthquakes, and very few large earthquakes – makes sense.

At the moment, I am creating synthetic seismograms (see Make some noise for how to make the seismic noise), and as I am trying to make my seismograms as realistic as possible, it is only logical to want to have my seismic events follow a Gutenberg-Richter distribution as well. I have also added in a term for setting a minimum magnitude, as quite often there is a ‘fall-off’ of the magnitudes in the lower end, as it is sometimes harder to actually pick up these magnitudes in real-life.

Poisson distribution

You are probably wondering where the fish part of my title comes into play – well that’s because when I add my events, I am doing so with Poisson spaced inter-event times (also below), with magnitudes that follow this distribution (i.e., lots of small and few large earthquakes). For those still not following, Poisson = fish in French.. (ba dum tss)

Anyways, Poisson is used for the spacing of inter-event times as it is said that earthquakes follow a Poisson distribution. This is a rule which assigns probabilities to the number of occurrences, with a known average rate. This can be seen by the mathematical formula below,

$P (n >1, t, \tau) = 1 - e^{-t/ \tau}$

where the left term says the probability of at least one earthquake occurring in the time $t$, where there is an average recurrence time $\tau$ – this can also be referred to with $\tau = \frac{1}{\lambda}$, where $\lambda$ is the rate (i.e., $P = 1 - e^{- \lambda t}$), can be estimated.

So, if we were to say that there were an average recurrence time of 31 days, then after 25 days, there would be a 55% probability of an event. A Poisson distribution can be easily incorporated, as we just need to produce random numbers which scale to this $\lambda$ term, as seen in the code at the end of this post.

In summary, I utilise both Gutenberg-Richter and Poisson statistics for my events, where the magnitude is scaled to Gutenberg-Richter, and are spaced as per Poisson distribution. I have supplied both functions (including how to do the Gutenberg-Richter plot) below.

— Roseanne

def gutenberg_richter(b=1.0, size=1, mag_min = 0.0):
"""Generate sequence of earthquake magnitudes
according to G-R law. logN = a-bM
Includes both the G-R magnitudes, and the
normalised version.
"""
g = mag_min + np.log10(-np.random.rand(size) + 1.0) / (-1*b)
gn = g/g.max()

return g, gn

# code for plotting the G-R distribution
testn = gutenberg_richter(size = 10**8)
y, bine = np.histogram(testn)
binc = 0.5 * (bine[1:] + bine[:-1])
plt.plot(binc, y, '.-')
plt.yscale('log', nonposy='clip')
plt.xlabel("Magnitude")
plt.ylabel("Log Cumulative frequency")

def poisson_interevent(lamb, number_of_events,st_event_2, samp_rate):
""" Finds the interevent times using Poisson, for the events, by choosing
lamb and number_of_events. We can use the random.expovariate function in
Python, as this generates exponentially distributed random numbers with
a rate of lambda for the first x number of events
( [int(random.expovariate(lamb)) for i in range(number_of_events)] ).
By taking the cumulative sum of these values, we then have the times at
which to place the events with Poisson inter-event times.
Here we create an array with a list of times which are spaced at a Poisson
rate of lambda. This will then be used as the times of the noise in which
we place the event at.
lamb = lambda value for Poisson
number_of_events = how many events you want
samp_rate = sampling rate
"""
poisson_values = 0
while (poisson_values == 0):
poisson_values = [int(random.expovariate(lamb)) for i in range(number_of_events)]
poisson_times = np.cumsum(poisson_values)
for i in range(len(poisson_times)-1):
if poisson_times[i+1] - poisson_times[i] <= len(st_event_2)/samp_rate:
poisson_values = 0

return poisson_values, poisson_times


# Make some noise

I spent a long time looking at how to ‘create noise’ in order to make some synthetic seismograms, so I thought that I would put up my code in case anyone ends up in the same spot as me! I take several steps in order to model this:

• Load in some typical seismic noise (I have taken mine from a quiet day near the Tunguruhua volcano in Equador), which has been detrended and demeaned.
• Taking the Fast Fourier Transform (FFT) of this (this puts the data into the frequency domain).
• Smooth the FFT data.
• Multiply this by the FFT of white noise.
• Take the Inverse Fast Fourier Transform (IFFT) of this (takes it back into the time domain).

The results of this are shown below, where the green is our white noise, the blue is our real seismic noise, and the pink is our synthetic seismic noise.

Creating seismic noise

There are a few other intermediate steps to this code (such as looping through so that it is in segments), however it is quite a simple process! A few other libraries are loaded into this beforehand, such as Obspy and Numpy, however you will probably have loaded these in already if you are doing this.

Now go and make some noise!

— Roseanne


def noise_segmenting(poisson_times, st_event_2, st_t, noise_level, samp_rate, delta):
""" Creates the noise array so that it is big enough to host all of the events.
Creating the noise by multiplying white noise by the seismic noise, in the frequency domain.
We then inverse FFT it and scale it to whatever SNR level is defined to output the full
noise array.
poisson_times = array of times where we then put in the seismic events (boundary for the noise)
st_event_2 = size of events that we are putting in later (again, this is a boundary)
st_t = seismic noise array that you are basing your synthetic on
samp_rate, delta = trace properties of st_t
"""
# end time for noise to cover all events
noise_lim = (poisson_times[-1] + len(st_event_2)) *2 #gives some time after last event
# load in seismic noise to base the synthetic type on
st_noise_start_t = UTCDateTime("2015-01-22T01:00:00")
st_noise_end_t = UTCDateTime("2015-01-22T01:02:00")
test_trace = st_t[0].slice(st_noise_start_t, st_noise_end_t)
test_trace_length = int(len(test_trace) / test_trace.stats.sampling_rate)
# setting the boundary for how many loops etc
minutes_long = (noise_lim)/st_event_2.stats.sampling_rate
noise_loops = int(np.ceil(minutes_long/2.0)) #working out how many 2 minute loops we need
# zero array
noise_array = np.zeros([noise_loops, len(test_trace)])
# loop for the amount of noise_loops needed (in segments)
for j in range(noise_loops):
# we average the seismic noise over twenty 2 minute demeaned samples
tung_n_fft = np.zeros([20, int(np.ceil((len(test_trace)/2.0)))])
for i in range(20):
st_noise = st_t[0].slice(st_noise_start_t+(i*test_trace_length),st_noise_end_t+(i*test_trace_length))
noise_detrended = st_noise.detrend()
noise_demeaned = mlab.demean(noise_detrended)
noise_averaging = Trace(noise_demeaned).normalize()
tung_n_fft[i] = np.fft.rfft(noise_averaging.data)

# work out the average fft
ave = np.average(tung_n_fft, axis=0)
# smooth the data
aves = movingaverage(ave,20)
# create white noise
whitenoise = np.random.normal(0, 1, len(noise_averaging))
whitenoise_n = Trace(whitenoise).normalize()
# FFT the white noise
wn_n_fft = np.fft.rfft(whitenoise_n.data)
# multiply the FFT of white noise and the FFT smoothed seismic noise
newnoise_fft = wn_n_fft * aves
# IFFT the product
newnoise = ifft(newnoise_fft, n = len(st_noise))

noise_array[j] = np.real(newnoise)
# transform the noise into an Obspy trace
full_noise_array = np.ravel(noise_array)
full_noise_array_n = Trace(np.float32(full_noise_array)).normalize()
full_noise_array_n_scaled = Trace(np.multiply(full_noise_array_n, noise_level))
full_noise_array_n_scaled.stats.sampling_rate = samp_rate
full_noise_array_n_scaled.stats.delta = delta

return full_noise_array_n_scaled