Python for Next Generation Data Analytics¶
Dr. Yves J. Hilpisch
The Python Quants GmbH
BI Forum, Budapest – 06. November 2013
Python for Analytics¶
Corporations, decision makers and analysts nowadays generally face a number of problems with data:
- sources: data typically comes from different sources, like from the Web, from in-house databases or it is generated in-memory, e.g. for simulation purposes
- formats: data is generally available in different formats, like SQL databases/tables, Excel files, CSV files, arrays, proprietary formats
- structure: data typically comes differently structured, be it unstructured, simply indexed, hierarchically indexed, in table form, in matrix form, in multidimensional arrays
- completeness: real-world data is generally incomplete, i.e. there is missing data (e.g. along an index) or multiple series of data cannot be aligned correctly (e.g. two time series with different time indexes)
- conventions: for some types of data there a many “competing” conventions with regard to formatting, like for dates and time
- interpretation: some data sets contain information that can be easily and intelligently interpreted, like a time index, others not, like texts
- performance: reading, streamlining, aligning, analyzing – i.e. processing – (big) data sets might be slow
In addition to these data-oriented problems, there typically are organizational issues that have to be considered:
- departments: the majority of companies is organized in departments with different technologies, databases, etc., leading to “data silos”
- analytics skills: analytical and business skills in general are possessed by people working in line functions (e.g. production) or administrative functions (e.g. finance)
- technical skills: technical skills, like retrieving data from databases and visualizing them, are generally possessed by people in technology functions (e.g. development, systems operations)
This presentation focuses on
- Python as a general purpose analytics environment
- interactive analytics examples
- prototyping-like Python usage
It does not address such important issues like
- architectural issues regarding hardware and software
- development processes, testing, documentation and production
- real world problem modeling
Some Python Fundamentals¶
Fundamental Python Libraries¶
A fundamental Python stack for interactive data analytics and visualization should at least contain the following libraries tools:
- Python – the Python interpreter itself
- NumPy – high performance, flexible array structures and operations
- SciPy – collection of scientific modules and functions (e.g. for regression, optimization, integration)
- pandas – time series and panel data analysis and I/O
- PyTables – hierarchical, high performance database (e.g. for out-of-memory analytics)
- matplotlib – 2d and 3d visualization
- IPython – interactive data analytics, visualization, publishing
It is best to use either the Python distribution Anaconda or the Web-based analytics environment Wakari. Both provide almost "complete" Python environments.
For example, pandas can, among others, help with the following data-related problems:
- sources: pandas reads data directly from different data sources such as SQL databases, Excel files or JSON based APIs
- formats: pandas can process input data in different formats like CSV files or Excel files; it can also generate output in different formats like CSV, XLS, HTML or JSON
- structure: pandas' strength lies in structured data formats, like time series and panel data
- completeness: pandas automatically deals with missing data in most circumstances, e.g. computing sums even if there are a few or many “not a number”, i.e. missing, values
- conventions/interpretation: for example, pandas can interpret and convert different date-time formats to Python datetime objects and/or timestamps
- performance: the majority of pandas classes, methods and functions is implemented in a performance-oriented fashion making heavy use of the Python/C compiler Cython
First Interactive Steps with Python¶
As a simple example let's generate a NumPy array with five sets of 1000 (pseudo-)random numbers each.
import numpy as np # this imports the NumPy library
data = np.random.standard_normal((5, 1000)) # generate 5 sets with 1000 rn each
data[:, :5].round(3) # print first five values of each set rounded to 3 digits
Let's plot a histogram of the 1st, 2nd and 3rd data set.
import matplotlib.pyplot as plt # this imports matplotlib.pyplot
plt.hist([data[0], data[1], data[2]], label=['Set 0', 'Set 1', 'Set 2'])
plt.grid(True) # grid for better readability
plt.legend()
We then want to plot the 'running' cumulative sum of each set.
plt.figure() # initialize figure object
plt.grid(True)
for data_set in enumerate(data): # iterate over all rows
plt.plot(data_set[1].cumsum(), label='Set %s' % data_set[0])
# plot the running cumulative sums for each row
plt.legend(loc=0) # write legend with labels
Some fundamental statistics from our data sets.
data.mean(axis=1) # average value of the 5 sets
data.std(axis=1) # standard deviation of the 5 sets
np.corrcoef(data) # correltion matrix of the 5 data sets
General I/O Capabilities of Python¶
SQL Databases¶
Currently, and in the foreseeable future as well, most corporate data is stored in SQL databases. We take this as a starting point and generate a SQL table with dummy data.
filename = 'numbers'
import sqlite3 as sq3 # imports the SQLite library
# Remove files if they exist
import os
try:
os.remove(filename + '.db')
os.remove(filename + '.h5')
os.remove(filename + '.csv')
os.remove(filename + '.xls')
except:
pass
We create a table and populate the table with the dummy data.
query = 'CREATE TABLE numbs (no1 real, no2 real, no3 real, no4 real, no5 real)'
con = sq3.connect(filename + '.db')
con.execute(query)
con.commit()
# Writing Random Numbers to SQLite3 Database
rows = 500000 # 500,000 rows for the SQL table (ca. 25 MB in size)
nr = np.random.standard_normal((rows, 5))
%time con.executemany('INSERT INTO numbs VALUES(?, ?, ?, ?, ?)', nr)
con.commit()
We can now read the data into a NumPy array.
# Reading a couple of rows from SQLite3 database table
array(con.execute('SELECT * FROM numbs').fetchmany(3)).round(3)
# Reading the whole table into an NumPy array
%time result = array(con.execute('SELECT * FROM numbs').fetchall())
Reading and Writing Data with pandas¶
pandas is optimized both for convenience and performance when it comes to analytics and I/O operations.
import pandas as pd
import pandas.io.sql as pds
%time data = pds.read_frame('SELECT * FROM numbs', con)
con.close()
data.head()
The plotting of larger data sets often is a problem. This can easily be accomplished here by only plotting every 500th data row, for instance.
%time data[::500].cumsum().plot(grid=True, legend=True) # every 500th row is plotted
There are some reasons why you may want to write the DataFrame to disk:
- Database Performance: you do not want to query the SQL database everytime anew
- IO Speed: you want to increase data/information retrievel
- Data Structure: maybe you want to store additional data with the original data set or you have generated a DataFrame out of multiple sources
pandas is tightly integrated with PyTables, a library that implements the HDF5 standard for Python. It makes data storage and I/O operations not only convenient but quite fast as well.
h5 = pd.HDFStore(filename + '.h5', 'w')
%time h5['data'] = data # writes the DataFrame to HDF5 database
Now, we can read the data from the HDF5 file into another DataFrame object. Performance of such an operation generally is quite high.
%time data_new = h5['data']
%time data.sum() - data_new.sum() # checking if the DataFrames are the same
The data is easily exported to different standard data file formats like comma separated value files (CSV).
%time data.to_csv(filename + '.csv') # writes the whole data set as CSV file
csv_file = open(filename + '.csv')
for i in range(5):
print csv_file.readline(),
pandas also interacts easily with Microsoft Excel spreadsheet files (XLS/XLSX).
%time data.ix[:65000].to_excel(filename + '.xls') # first 65,000 rows written in a Excel file
With pandas you can in one step read data from an Excel file and e.g. plot it.
%time pd.read_excel(filename + '.xls', 'sheet1').hist(grid=True)
One can also generate HTML code, e.g. for automatic Web-based report generation.
html_code = data.ix[:2].to_html()
html_code
from IPython.core.display import HTML
HTML(html_code)
Some final checks.
ll numb*
Parallel Execution of Algorithms¶
Parallelization of Algorithms¶
First, let us define the function we want to evaluate on a large array.
from math import *
def f(x):
return abs(cos(x)) ** 0.5 + sin(2 + 3 * sqrt(abs(x)))
x = pi
f(x)
Second, as a benchmark, a pure Python implementation.
I = 10000000
x = range(I)
def f1(x):
return [f(x) for x in x]
%time r = f1(x)
print r[:5]
A better implementation uses NumPy for faster looping via vectorization.
import numpy as np
x = np.arange(I)
def f2(x):
return (np.abs(np.cos(x)) ** 0.5 +
np.sin(2 + 3 * np.sqrt(np.abs(x))))
%time r = f2(x)
print r[:5]
An even better and parallel version uses the numexpr library.
import numexpr as ne
ex = 'abs(cos(x)) ** 0.5 + sin(2 + 3 * sqrt(abs(x)))'
def f3(x):
ne.set_num_threads(4)
return ne.evaluate(ex)
%time r = f3(x)
print r[:5]
Let us compare the performance of the different functions in a bit more detail.
from perf_comp import perf_comp
func_list = [f1, f2, f3]
perf_comp(func_list, 'x', rep=3)
The parallel version with numexpr is magnitudes of order faster than pure Python.
Using IPython for Distributed Computing¶
The algorithm we (arbitrarily) choose is the Monte Carlo estimator for a European option value in the Black-Scholes-Merton set-up. In this set-up the underlying follows the stochastic differential equation:
\[dS_t = r S_t dt + \sigma S_t dZ_t\]
The Monte Carlo estimator for a call option value looks like follows:
\(C_0 = e^{-rT} \frac{1}{I} \sum_{i=1}^{I} \max[S_T(i) - K, 0]\)
where \(S_T(i)\) is the \(i\)-th simulated value of the underlying at maturity \(T\).
A dynamic implementation of the BSM Monte Carlo valuation could look like follows.
def BSM_MCS(S0):
''' Dynamic Black-Scholes-Merton MCS Estimator for European Calls. '''
import numpy as np
K = 100.; T = 1.0; r = 0.05; vola = 0.2
M = 50; I = 20000
dt = T / M
rand = np.random.standard_normal((M + 1, I))
S = np.zeros((M + 1, I)); S[0] = S0
for t in range(1, M + 1):
S[t] = S[t-1] * np.exp((r - 0.5 * vola ** 2) * dt + vola * np.sqrt(dt) * rand[t])
BSM_MCS_Value = np.sum(np.maximum(S[-1] - K, 0)) / I
return BSM_MCS_Value
The Sequential Calculation¶
The benchmark case of sequential valuation of \(n\) options.
def seqValue(n):
optionValues = []
for S in linspace(80, 100, n):
optionValues.append(BSM_MCS(S))
return optionValues
The Parallel Calculation¶
First, start a (local) cluster.
# in the shell ...
####
# ipcluster start -n 4
####
# or maybe more cores
Second, enable parallel computing capabilities.
from IPython.parallel import Client
c = Client(profile="default")
view = c.load_balanced_view()
The function for the parallel execution of a number \(n\) of different options valuations.
def parValue(n):
optionValues = []
for S in linspace(80, 100, n):
value = view.apply_async(BSM_MCS, S)
# asynchronously calculate the option values
optionValues.append(value)
c.wait(optionValues)
return optionValues
Let us again compare performance of the two different approaches.
n = 100 # number of option valuations
perf_comp((seqValue, parValue), data='n', rep=3)
Performance scales linearly with the number of CPU cores available.
Analyzing Financial Time Series Data¶
Historical Correlation between EURO STOXX 50 and VSTOXX¶
It is a stylized fact that stock indexes and related volatility indexes are highly negatively correlated. The following example analyzes this stylized fact based on the EURO STOXX 50 stock index and the VSTOXX volatility index using Ordinary Least-Squares regession (OLS).
First, we collect historical data for both the EURO STOXX 50 stock and the VSTOXX volatility index.
import pandas as pd
from urllib import urlretrieve
try:
open('es.txt')
open('vs.txt')
except:
es_url = 'http://www.stoxx.com/download/historical_values/hbrbcpe.txt'
vs_url = 'http://www.stoxx.com/download/historical_values/h_vstoxx.txt'
urlretrieve(es_url, 'es.txt')
urlretrieve(vs_url, 'vs.txt')
The EURO STOXX 50 data is not yet in the right format. Some house cleaning is necessary.
lines = open('es.txt').readlines() # reads the whole file line-by-line
new_file = open('es50.txt', 'w') # opens a new file
new_file.writelines('date' + lines[3][:-1].replace(' ', '') + ';DEL' + lines[3][-1])
# writes the corrected third line of the orginal file as first line of new file
new_file.writelines(lines[4:-1]) # writes the remaining lines of the orginial file
print list(open('es50.txt'))[:5] # opens the new file for inspection
Now, the data can be safely read into a DataFrame object.
es = pd.read_csv('es50.txt', index_col=0, parse_dates=True, sep=';', dayfirst=True)
del es['DEL'] # delete the helper column
es
The VSTOXX data can be read without touching the raw data.
vs = pd.read_csv('vs.txt', index_col=0, header=2,
parse_dates=True, sep=',', dayfirst=True)
# you can alternatively read from the Web source directly
# without saving the csv file to disk:
# vs = pd.read_csv(vs_url, index_col=0, header=2,
# parse_dates=True, sep=',', dayfirst=True)
We now merge the data for further analysis.
from datetime import datetime
data = pd.DataFrame({'EUROSTOXX' :
es['SX5E'][es.index > datetime(1999, 12, 31)]})
data = data.join(pd.DataFrame({'VSTOXX' :
vs['V2TX'][vs.index > datetime(1999, 12, 31)]}))
data
Let's inspect the two time series.
data.head()
A picture can tell the complete story.
data.plot(subplots=True, grid=True, style='b')
We now generate log returns for both time series.
rets = np.log(data / data.shift(1))
rets.head()
To this new data set, also stored in a DataFrame object, we apply ordinary least-square regression (OLS).
xdat = rets['EUROSTOXX']
ydat = rets['VSTOXX']
model = pd.ols(y=ydat, x=xdat)
model
Again, we want to see how our results look graphically.
plt.plot(xdat, ydat, 'r.')
ax = plt.axis() # grab axis values
x = np.linspace(ax[0], ax[1] + 0.01)
plt.plot(x, model.beta[1] + model.beta[0] * x, 'b', lw=2)
plt.grid(True)
plt.axis('tight')
Analyzing High Frequency Data¶
Using pandas, the code that follows reads intraday, high-frequency data from a Web source, plots it and resamples it.
try:
AAPL = pd.read_csv('aapl.csv', index_col=0, header=0, parse_dates=True)
except:
date = datetime.now()
url1 = 'http://hopey.netfonds.no/posdump.php?'
url2 = 'date=%s%s%s&paper=AAPL.O&csv_format=csv' % ('2013', '10', '31')
# you may have to adjust the date since only recent dates are available
urlretrieve(url1 + url2, 'aapl.csv')
AAPL = pd.read_csv('aapl.csv', index_col=0, header=0, parse_dates=True)
AAPL
The intraday evolution of the Apple stock price.
AAPL['bid'].plot()
A resampling of the data is easily accomplished with pandas.
import datetime as dt
# this resamples the record frequency to 5 minutes, using mean as aggregation rule
AAPL_resam = AAPL.resample(rule='5min', how='mean')
AAPL_resam.head()
Let's have a graphical look at the new data set.
AAPL_resam['bid'].plot()
Obvioulsly, there hasn't been trading data for every interval.
pandas has a number of approaches to fill in missing data.
AAPL_resam['bid'].fillna(method='ffill').plot() # this forward fills missing data
Compute-Intensive Applications¶
The simulation of the movement of N bodies in relation to each other, where \(N\) is a large number and a body can be a star/planet or a small particle, is an important problem in Physics.
The basic simulation algorithm involves a number of iterations ("loops") since one has to determine the force that every single body exerts on all the other bodies in the system to be simulated (e.g. a galaxy or a system of atomic particles).
The algorithm lends iteself pretty well for parallization. However, the example code that follows does not use any parallelization techniques. It illustrates rather what speed-ups are achievable by
- using vectorization approaches with NumPy and
- just-in-time compiling capabilities of Numba
A description and discussion of the algorithm is found in the recent article Vladimirov, Andrey and Vadim Karpusenko (2013): "Test-Driving Intel Xeon-Phi Coprocessors with a Basic N-Body Simulation." White Paper, Colfax International.
Pure Python¶
We first use pure Python to implement the algorithm. We start by generating random vectors of particle locations and a zero vector for the speed of each particle.
import random
nParticles = 500
particle = []
for i in range(nParticles):
particle.append([random.gauss(0.0, 1.0) for j in range(3)])
particlev = [[0, 0, 0] for x in particle]
Next, we define the simulation function for the movements of each particle given their gravitational interaction.
def nbody_py(particle, particlev): # lists as input
nSteps = 15; dt = 0.01
for step in range(1, nSteps + 1, 1):
for i in range(nParticles):
Fx = 0.0; Fy = 0.0; Fz = 0.0
for j in range(nParticles):
if j != i:
dx = particle[j][0] - particle[i][0]
dy = particle[j][1] - particle[i][1]
dz = particle[j][2] - particle[i][2]
drSquared = dx * dx + dy * dy + dz * dz
drPowerN32 = 1.0 / (drSquared + sqrt(drSquared))
Fx += dx * drPowerN32
Fy += dy * drPowerN32
Fz += dz * drPowerN32
particlev[i][0] += dt * Fx
particlev[i][1] += dt * Fy
particlev[i][2] += dt * Fz
for i in range(nParticles):
particle[i][0] += particlev[i][0] * dt
particle[i][1] += particlev[i][1] * dt
particle[i][2] += particlev[i][2] * dt
We then execute the function to measure execution speed.
%time nbody_py(particle, particlev)
NumPy Solution¶
The basic algorithm allows for a number of vectorizations by using NumPy.
import numpy as np
particle = np.random.standard_normal((nParticles, 3))
particlev = np.zeros_like(particle)
Using NumPy arrays and matplotlib's 3d plotting capabilities, we can easily visualize the intial positions of the particles in the space.
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(particle[:, 0], particle[:, 1], particle[:, 2],
c='r', marker='.')
The respective NumPy simulation function is considerably more compact.
def nbody_np(particle, particlev): # NumPy arrays as input
nSteps = 15; dt = 0.01
for step in range(1, nSteps + 1, 1):
Fp = np.zeros((nParticles, 3))
for i in range(nParticles):
dp = particle - particle[i]
drSquared = np.sum(dp ** 2, axis=1)
h = drSquared + np.sqrt(drSquared)
drPowerN32 = 1. / np.maximum(h, 1E-10)
Fp += -(dp.T * drPowerN32).T
particlev += dt * Fp
particle += particlev * dt
It is also considerably faster.
%time nbody_np(particle, particlev)
Let's check how the particles moved. The following is a snapshot of the final positions of all particles.
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(particle[:, 0], particle[:, 1], particle[:, 2],
c='r', marker='.')
Dynamically Compiled Version¶
Numba is an open-source, NumPy-aware optimizing compiler for Python sponsored by The Python Quants GmbH It uses the remarkable LLVM compiler infrastructure to compile Python byte-code to machine code especially for use in the NumPy run-time and SciPy modules.
For this case, we first implement a function which is somehow a hybrid function of the pure Python (loop-heavy) and the NumPy array class-based version.
def nbody_hy(particle, particlev): # NumPy arrays as input
nSteps = 15; dt = 0.01
for step in range(1, nSteps + 1, 1):
for i in range(nParticles):
Fx = 0.0; Fy = 0.0; Fz = 0.0
for j in range(nParticles):
if j != i:
dx = particle[j,0] - particle[i,0]
dy = particle[j,1] - particle[i,1]
dz = particle[j,2] - particle[i,2]
drSquared = dx * dx + dy * dy + dz * dz
drPowerN32 = 1.0 / (drSquared + sqrt(drSquared))
Fx += dx * drPowerN32
Fy += dy * drPowerN32
Fz += dz * drPowerN32
particlev[i, 0] += dt * Fx
particlev[i, 1] += dt * Fy
particlev[i, 2] += dt * Fz
for i in range(nParticles):
particle[i,0] += particlev[i,0] * dt
particle[i,1] += particlev[i,1] * dt
particle[i,2] += particlev[i,2] * dt
This function is then compiled with Numba.
import numba as nb
nbody_nb = nb.autojit(nbody_hy)
nbody_nb(particle, particlev)
This new, compiled version of the hybrid function is yet faster.
particle = np.random.standard_normal((nParticles, 3))
particlev = np.zeros_like(particle)
%time nbody_nb(particle, particlev)
Speed Comparisons¶
The dynamically compiled version is by far the fastest.
func_list = [nbody_py, nbody_np, nbody_nb]
perf_comp(func_list, data='particle, particlev', rep=3)
Python for Next Generation Data Analytics & Visualization?!¶
10 years ago, Python was considered exotic in the analytics space – at best. Languages/packages like R and Matlab dominated the scene. Today, Python has become a major force in data analytics & visualization due to a number of characteristics:
- multi-purpose: prototyping, development, production, sytems administration – Python is one for all
- libraries: there is a library for almost any task or problem you face
- efficiency: Python speeds up all IT development tasks for analytics applications and reduces maintenance costs
- performance: Python has evolved from a scripting language to a 'meta' language with bridges to all high performance environments (e.g. LLVM, multi-core CPUs, GPUs, clusters)
- interoperalbility: Python seamlessly integrates with almost any other language and technology
- interactivity: Python allows domain experts to get closer to their business and financial data pools and to do real-time analytics
- collaboration: solutions like Wakari with IPython Notebook allow the easy sharing of code, data, results, graphics, etc.
One of the easiest ways to deploy Python today across a whole organization for collaboration with a heterogenous IT infrastructure is via Wakari.io, Continuum's Web-/Browser- and Python-based Data Analytics environment. It is availble both as a cloud as well as an enterprise solution for in-house deployment.
The Python Quants GmbH – Experts for Python in Quant finance¶
The Python Quants – the company Web site
Dr. Yves J. Hilpisch – my personal Web site
Derivatives Analytics with Python – my new book
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