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Scipy Lecture Notes.pdf
I Getting started with Python for science
Scientific computing with tools and workflow
Why Python?
Scientific Python building blocks
The interactive workflow: IPython and a text editor
The Python language
First steps
Basic types
Control Flow
Defining functions
Reusing code: scripts and modules
Input and Output
Standard Library
Exception handling in Python
Object-oriented programming (OOP)
NumPy: creating and manipulating numerical data
The Numpy array object
Numerical operations on arrays
More elaborate arrays
Advanced operations
Some exercises
Matplotlib: plotting
Introduction
Simple plot
Figures, Subplots, Axes and Ticks
Other Types of Plots: examples and exercises
Beyond this tutorial
Quick references
Scipy : high-level scientific computing
File input/output: scipy.io
Special functions: scipy.special
Linear algebra operations: scipy.linalg
Fast Fourier transforms: scipy.fftpack
Optimization and fit: scipy.optimize
Statistics and random numbers: scipy.stats
Interpolation: scipy.interpolate
Numerical integration: scipy.integrate
Signal processing: scipy.signal
Image processing: scipy.ndimage
Summary exercises on scientific computing
Getting help and finding documentation
II Advanced topics
Advanced Python Constructs
Iterators, generator expressions and generators
Decorators
Context managers
Advanced Numpy
Life of ndarray
Universal functions
Interoperability features
Array siblings: chararray, maskedarray, matrix
Summary
Contributing to Numpy/Scipy
Debugging code
Avoiding bugs
Debugging workflow
Using the Python debugger
Debugging segmentation faults using gdb
Optimizing code
Optimization workflow
Profiling Python code
Making code go faster
Writing faster numerical code
Sparse Matrices in SciPy
Introduction
Storage Schemes
Linear System Solvers
Other Interesting Packages
Image manipulation and processing using Numpy and Scipy
Opening and writing to image files
Displaying images
Basic manipulations
Image filtering
Feature extraction
Measuring objects properties: ndimage.measurements
Mathematical optimization: finding minima of functions
Knowing your problem
A review of the different optimizers
Practical guide to optimization with scipy
Special case: non-linear least-squares
Optimization with constraints
Interfacing with C
Introduction
Python-C-Api
Ctypes
SWIG
Cython
Summary
Further Reading and References
Exercises
III Packages and applications
Statistics in Python
Data representation and interaction
Hypothesis testing: comparing two groups
Linear models, multiple factors, and analysis of variance
More visualization: seaborn for statistical exploration
Testing for interactions
Sympy : Symbolic Mathematics in Python
First Steps with SymPy
Algebraic manipulations
Calculus
Equation solving
Linear Algebra
Scikit-image: image processing
Introduction and concepts
Input/output, data types and colorspaces
Image preprocessing / enhancement
Image segmentation
Measuring regions' properties
Data visualization and interaction
Feature extraction for computer vision
Traits: building interactive dialogs
Introduction
Example
What are Traits
3D plotting with Mayavi
Mlab: the scripting interface
Interactive work
Slicing and dicing data: sources, modules and filters
Animating the data
Making interactive dialogs
Putting it together
scikit-learn: machine learning in Python
Loading an example dataset
Classification
Clustering: grouping observations together
Dimension Reduction with Principal Component Analysis
Putting it all together: face recognition
Linear model: from regression to sparsity
Model selection: choosing estimators and their parameters
Index
PythonMatplotlibSciKitsNumpySciPyIPythonIP[y]:Cython2015EDITIONEdited byGaël VaroquauxEmmanuelle GouillartOlaf VahtrasScipyLecture Noteswww.scipy-lectures.orgGaël Varoquaux • Emmanuelle Gouillart • Olav VahtrasValentin Haenel • Nicolas P. Rougier • Ralf GommersFabian Pedregosa • Zbigniew Jędrzejewski-Szmek • Pauli VirtanenChristophe Combelles • Didrik Pinte • Robert CimrmanAndré Espaze • Adrian Chauve • Christopher Burns
Contents
I Getting started with Python for science
1 Scientific computing with tools and workflow
1.1 Why Python?
1.2
1.3 The interactive workflow: IPython and a text editor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scientific Python building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2
4
4
5
6
2 The Python language
10
First steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1
2.2 Basic types
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Control Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Defining functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Reusing code: scripts and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6
2.7
Standard Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.8 Exception handling in Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.9 Object-oriented programming (OOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 NumPy: creating and manipulating numerical data
43
3.1 The Numpy array object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Numerical operations on arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 More elaborate arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Advanced operations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Some exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5
4 Matplotlib: plotting
82
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2
Simple plot
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3
Figures, Subplots, Axes and Ticks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Other Types of Plots: examples and exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5 Beyond this tutorial
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6 Quick references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Scipy : high-level scientific computing
101
File input/output: scipy.io . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.1
Special functions: scipy.special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2
Linear algebra operations: scipy.linalg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3
Fast Fourier transforms: scipy.fftpack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4
5.5 Optimization and fit: scipy.optimize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Statistics and random numbers: scipy.stats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.6
Interpolation: scipy.interpolate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.7
5.8 Numerical integration: scipy.integrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Signal processing: scipy.signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.9
5.10 Image processing: scipy.ndimage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
i
5.11 Summary exercises on scientific computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Getting help and finding documentation
II Advanced topics
137
140
7 Advanced Python Constructs
142
Iterators, generator expressions and generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Decorators
7.3 Context managers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Advanced Numpy
159
8.1
Life of ndarray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2 Universal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.4 Array siblings: chararray, maskedarray, matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.6 Contributing to Numpy/Scipy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Interoperability features
9 Debugging code
192
9.1 Avoiding bugs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.2 Debugging workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.3 Using the Python debugger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9.4 Debugging segmentation faults using gdb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10 Optimizing code
203
10.1 Optimization workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.2 Profiling Python code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.3 Making code go faster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10.4 Writing faster numerical code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
11 Sparse Matrices in SciPy
210
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
11.2 Storage Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.3 Linear System Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
11.4 Other Interesting Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
12 Image manipulation and processing using Numpy and Scipy
230
12.1 Opening and writing to image files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
12.2 Displaying images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
12.3 Basic manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
12.4 Image filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
12.5 Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
12.6 Measuring objects properties: ndimage.measurements . . . . . . . . . . . . . . . . . . . . . . . . 243
13 Mathematical optimization: finding minima of functions
248
13.1 Knowing your problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
13.2 A review of the different optimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
13.3 Practical guide to optimization with scipy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
13.4 Special case: non-linear least-squares
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
13.5 Optimization with constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
14 Interfacing with C
263
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
14.2 Python-C-Api . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
14.3 Ctypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
14.4 SWIG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
14.5 Cython . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
14.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
14.7 Further Reading and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
ii
14.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
III Packages and applications
282
15 Statistics in Python
284
15.1 Data representation and interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
15.2 Hypothesis testing: comparing two groups
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
15.3 Linear models, multiple factors, and analysis of variance . . . . . . . . . . . . . . . . . . . . . . . . 292
15.4 More visualization: seaborn for statistical exploration . . . . . . . . . . . . . . . . . . . . . . . . . . 297
15.5 Testing for interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
16 Sympy : Symbolic Mathematics in Python
301
16.1 First Steps with SymPy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
16.2 Algebraic manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
16.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
16.4 Equation solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
16.5 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
17 Scikit-image: image processing
308
17.1 Introduction and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
17.2 Input/output, data types and colorspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
17.3 Image preprocessing / enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
17.4 Image segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
17.5 Measuring regions’ properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
17.6 Data visualization and interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
17.7 Feature extraction for computer vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
18 Traits: building interactive dialogs
322
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
18.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
18.3 What are Traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
19 3D plotting with Mayavi
340
19.1 Mlab: the scripting interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
19.2 Interactive work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
19.3 Slicing and dicing data: sources, modules and filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
19.4 Animating the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
19.5 Making interactive dialogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
19.6 Putting it together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
20 scikit-learn: machine learning in Python
353
20.1 Loading an example dataset
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
20.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
20.3 Clustering: grouping observations together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
20.4 Dimension Reduction with Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . 359
20.5 Putting it all together: face recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
20.6 Linear model: from regression to sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
20.7 Model selection: choosing estimators and their parameters . . . . . . . . . . . . . . . . . . . . . . . 362
Index
363
iii
Scipy lecture notes, Edition 2015.2
Contents
1
Part I
Getting started with Python for science
2
This part of the Scipy lecture notes is a self-contained introduction to everything that is needed to use Python
for science, from the language itself, to numerical computing or plotting.
Scipy lecture notes, Edition 2015.2
3
CHAPTER 1
Scientific computing with tools and workflow
Authors: Fernando Perez, Emmanuelle Gouillart, Gaël Varoquaux, Valentin Haenel
1.1 Why Python?
1.1.1 The scientist’s needs
• Get data (simulation, experiment control),
• Manipulate and process data,
• Visualize results (to understand what we are doing!),
• Communicate results: produce figures for reports or publications, write presentations.
1.1.2 Specifications
• Rich collection of already existing bricks corresponding to classical numerical methods or basic actions:
we don’t want to re-program the plotting of a curve, a Fourier transform or a fitting algorithm. Don’t
reinvent the wheel!
• Easy to learn: computer science is neither our job nor our education. We want to be able to draw a curve,
smooth a signal, do a Fourier transform in a few minutes.
• Easy communication with collaborators, students, customers, to make the code live within a lab or a
company: the code should be as readable as a book. Thus, the language should contain as few syn-
tax symbols or unneeded routines as possible that would divert the reader from the mathematical or
scientific understanding of the code.
• Efficient code that executes quickly... but needless to say that a very fast code becomes useless if we
spend too much time writing it. So, we need both a quick development time and a quick execution time.
• A single environment/language for everything, if possible, to avoid learning a new software for each new
problem.
1.1.3 Existing solutions
Which solutions do scientists use to work?
Compiled languages: C, C++, Fortran, etc.
• Advantages:
– Very fast. Very optimized compilers. For heavy computations, it’s difficult to outperform these
languages.
4
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