number crunching in python

LOREMI P S U M

NUMBER CRUNCHING IN PYTHONEnrico Franchi ([email protected]) &Valerio Maggio ([email protected])

mailto:[email protected]




DOLORS I T OUTLINE

• Scientific and Engineering Computing

• Common FP pitfalls

• Numpy NDArray (Memory and Indexing)

• Case Studies

number-crunching: n. [common] Computations of a numerical nature, esp. those that make extensive use of floating-point numbers. This term is in widespread informal use outside hackerdom and even in mainstream slang, but has additional hackish connotations: namely, that the computations are mindless and involve massive use of brute force. This is not always evil, esp. if it involves ray tracing or fractals or some other use that makes pretty pictures, esp. if such pictures can be used as screen backgrounds. See also crunch.


We are not evil.


We are not evil. Just chaotic neutral.

AMETM E N T I

T U M ALTERNATIVES• Matlab (IDE, numeric computations oriented, high quality algorithms,

lots of packages, poor GP programming support, commercial)

• Octave (Matlab clone)

• R (stats oriented, poor general purpose programming support)

• Fortran/C++ (very low level, very fast, more complex to use)

• In general, these tools either are low level GP or high level DSLs

HIS EX,T E M P O

R PYTHON• Numpy (low-level numerical computations) +

Scipy (lots of additional packages)

• IPython (wonderfull command line interpreter) + IPython Notebook (“Mathematica-like” interactive documents)

• HDF5 (PyTables, H5Py), Databases

• Specific libraries for machine learning, etc.

• General Purpose Object Oriented Programming

TOOLSCUS E D

DENIQUE

G U B E RG R E N

Our Code

Numpy

Atlas/MKL

Improvements

Improvements

Algorithms are fast because of highly optimized C/Fortran code

4 30 LOAD_GLOBAL 1 (dot) 33 LOAD_FAST 0 (a) 36 LOAD_FAST 1 (b) 39 CALL_FUNCTION 2 42 STORE_FAST 2 (c)

NUMPY STACKc = a · b

ndar

ray

ndarray

Memory

behavior

shape, stride, flags

(i0, . . . , in�1) ! I

Shape: (d0, …, dn-1)

4x3

An n-dimensional array references some (usually contiguous memory area)

An n-dimensional array has property such as its shape or the

data-type of the elements containes

Is an object, so there is some behavior, e.g., the def. of __add__ and similar stuff

N-dimensional arrays are homogeneous

(i0, . . . , in�1) ! I

C-contiguousF-contiguous

Shape: (d0, …, dn)

IC =n�1X

k=0

ik

n�1Y

j=k+1

dj

IF =n�1X

k=0

ik

k�1Y

j=0

dj

Shape: (d0, …, dk ,…, dn-1)

Shape: (d0, …, dk ,…, dn-1)

IC = i0 · d0 + i14x3

IF = i0 + i1 · d1

Elem

ent L

ayou

t in

Mem

ory

Strid

e

C-contiguous F-contiguous

sF (k) =k�1Y

j=0

dj

IF =nX

k=0

ik · sF (k)

sC(k) =n�1Y

j=k+1

dj

IC =n�1X

k=0

ik · sC(k)

Stride

C-contiguousF-contiguous

C-contiguous

(s0 = d0, s1 = 1) (s0 = 1, s1 = d1)

IC =n�1X

k=0

ik

n�1Y

j=k+1

dj IF =n�1X

k=0

ik

k�1Y

j=0

dj

ndarray

Memory

behavior


ndarray

behavior


View View

View View

View

s

C-contiguous

ndarray

behavior

(1,4)

Memory

ndarray

Memory

behavior


matrix

Memory

behavior


ndarray

matrix

Basic

Inde

xing

Adva

nced

Inde

xing

Broa

dcas

ting!

Adva

nced

Inde

xing

2

Vect

orize

!

Don’t use explicit for loops unless you have to!

PART II: NUMBER CRUNCHING IN ACTION

General Disclaimer: All the Maths appearing in the next slides is only intended to better introduce the considered case studies. Speakers are not responsible for any possible disease or “brain consumption” caused by too much formulas.

So BEWARE; use this information at your own risk! It's intention is solely educational. We would strongly encourage you to use this information in cooperation with a medical or health professional.

Awfu

l Mat

hs

BEFORE STARTINGWhat do you need to get started:

• A handful Unix Command-line tool:

• Linux / Mac OSX Users: Your’re done.

• Windows Users: It should be the time to change your OS :-)

• [I]Python (You say?!)

• A DBMS:

• Relational: e.g., SQLite3, PostgreSQL

• No-SQL: e.g., MongoDB

MINIMS C R I PT O R E M

LOREMI P S U M

BENCHMARKING

LOREMI P S U M

• Vectorization (NumPy vs. “pure” Python

• Loops and Math functions (i.e., sin(x))

• Matrix-Vector Product

• Different implementations of Matrix-Vector Product

CASE STUDIES ON NUMERICAL EFFICIENCY

Hw In

fo

Vect

oriza

tion:

sin

(x)

NumPy, Winssi

n(x)

: Res

ults

NumPy, Winsfatality

sin(

x): R

esul

ts

Mat

rix-V

ecto

r Pro

duct

NumPy, Winsdo

t: R

esul

ts

NumPy, Winsfatality

dot:

Res

ults

LOREMI P S U M

NUMBER CRUNCHING APPLICATIONS

MACHINE LEARNING• Machine Learing = Learning by Machine(s)

• Algorithms and Techniques to gain insights from data or a dataset

• Supervised or Unsupervised Learning

• Machine Learning is actively being used today, perhaps in many more places than you’d expected

• Mail Spam Filtering

• Search Engine Results Ranking

• Preference Selection

• e.g., Amazon “Customers Who Bought This Item Also Bought”

NAM IN,S E A

N O

LOREMI P S U M

CLUSTERING: BRIEF INTRODUCTION

• Clustering is a type of unsupervised learning that automatically forms clusters (groups) of similar things. It’s like automatic classification. You can cluster almost anything, and the more similar the items are in the cluster, the better your clusters are.

• k-means is an algorithm that will find k clusters for a given dataset.

• The number of clusters k is user defined.

• Each cluster is described by a single point known as the centroid.

• Centroid means it’s at the center of all the points in the cluster.

from scipy.cluster.vq import kmeans, vqK-

mea

ns

K-m

eans

plo

tfrom scipy.cluster.vq import kmeans, vq

LOREMI P S U M

EXAMPLE:CLUSTERING POINTS ON A MAP

Here’s the situation: your friend <NAME> wants you to take him out in the greater Portland, Oregon, area (US) for his birthday. A number of other friends are going to come also, so you need to provide a plan that everyone can follow. Your friend has given you a list of places he wants to go. This list is long; it has 70 establishments in it.

Yaho

o AP

I: ge

oGra

b

�s�s �f�fLatitude and Longitude Coordinates of two points (s and f)

�� Corresponding differences

��̂ = arccos(sin�s sin�f + cos�s cos�f cos��)Spherical Distance Measure

Sphe

rical

Dist

ance

Mea

sure

kmea

ns w

ith dis

tLSC

• Problem: Given an input matrix A, calculate if possible, its inverse matrix.

• Definition: In linear algebra, a n-by-n (square) matrix A is invertible (a.k.a. is nonsingular or nondegenerate) if there exists a n-by-n matrix B (A-1) such that: AB = BA = In

TRIVIAL EXAMPLE:INVERSE MATRIX

✓ Eigen Decomposition: • If A is nonsingular, i.e., it can be eigendecomposed and none of its

eigenvalue is equal to zero

✓ Cholesky Decomposition:• If A is positive definite, where is the Conjugate transpose matrix

of L (i.e., L is a lower triangular matrix)

✓ LU Factorization: (with L and U Lower (Upper) Triangular Matrix)

✓ Analytic Solution: (writing the Matrix of Cofactors), a.k.a. Cramer Method

A�1 = Q⇤Q�1

A�1 = (L⇤)�1L�1

A�1 = 1det(A) (C

T )i,j =1

det(A) (Cji) =1

det(A)

0

BBB@

C1,1 C1,2 · · · C1,n

C2,1 C2,2 · · · C2,n...

.... . .

...Cm,1 Cm,2 · · · Cm,n

1

CCCA

L⇤

A = LU

Solu

tion(

s)

C =

0

@C1,1 C1,2 C1,3

C2,1 C2,2 C2,3

C3,1 C3,2 C3,3

1

A

Exam

ple

C =

0

@C1,1 C1,2 C1,3

C2,1 C2,2 C2,3

C3,1 C3,2 C3,3

1

A

Exam

ple

C�1 =1

det(C)⇤

⇤

0

@(C2,2C3,3 � C2,3C3,2) (C1,3C3,2 � C1,2C3,3) (C1,2C2,3 � C1,3C2,2)(C2,3C3,1 � C2,1C3,3) (C1,1C3,3 � C1,3C3,1) (C1,3C2,1 � C1,1C2,3)(C2,1C3,2 � C2,2C3,1) (C3,1C1,2 � C1,1C3,2) (C1,1C2,2 � C1,2C2,1)

1

A

C =

0

@C1,1 C1,2 C1,3

C2,1 C2,2 C2,3

C3,1 C3,2 C3,3

1

A

Exam

pledet(C) = C1,1(C2,2C3,3 � C2,3C3,2)

+C1,2(C1,3C3,2 � C1,2C3,3)

+C1,3(C1,2C2,3 � C1,3C2,2)

C�1 =1

det(C)⇤

⇤

0

@(C2,2C3,3 � C2,3C3,2) (C1,3C3,2 � C1,2C3,3) (C1,2C2,3 � C1,3C2,2)(C2,3C3,1 � C2,1C3,3) (C1,1C3,3 � C1,3C3,1) (C1,3C2,1 � C1,1C2,3)(C2,1C3,2 � C2,2C3,1) (C3,1C1,2 � C1,1C3,2) (C1,1C2,2 � C1,2C2,1)

1

A

Hom

e M

ade

Duplicated Code

Hom

e M

ade

Duplicated CodeTemplate Method Pattern

Hom

e M

ade

Duplicated CodeTemplate Method Pattern

However, we still have to implementfrom scratch computational functions!!

Reinventing the wheel!

Hom

e M

ade

Num

pyfrom numpy import linalg

Type: functionString Form:<function inv at 0x105f72b90>File: /Library/Python/2.7/site-packages/numpy/linalg/linalg.pyDefinition: linalg.inv(a)Source:def inv(a): """ Compute the (multiplicative) inverse of a matrix. [...]

Parameters ---------- a : array_like, shape (M, M) Matrix to be inverted.

Returns ------- ainv : ndarray or matrix, shape (M, M) (Multiplicative) inverse of the matrix `a`.

Raises ------ LinAlgError If `a` is singular or not square.

[...] """ a, wrap = _makearray(a) return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))Unde

r the

hoo

d

• Alternative built-in solutions to the same problem:

Num

py A

ltern

ative

s

Thanks for your kind attention.

Vect

oriza

tion:

i+=

2

NumPy, Winsi+

=2: R

esul

ts

fatalityNumPy, Wins

i+=2

: Res

ults

Create k points for starting centroids (often randomly)

While any point has changed cluster assignment for every point in dataset: for every centroid:

d = distance(centroid,point) assign(point, nearest(cluster))

for each cluster: mean = average(cluster) centroid[cluster] = mean

K-m

eans

number crunching in python

Technology

contiguous shape

ccontiguous ndarray

ags ndarray behavior

dj shape

widespread informal

e n t i t u

common computations

contiguous ccontiguous