CME 193: Introduction to Scientific Python Lecture 6: Numpy, Scipy, Matplotlib
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1 CME 193: Introduction to Scientific Python Lecture 6: Numpy, Scipy, Matplotlib Nolan Skochdopole stanford.edu/class/cme193 6: Numpy, Scipy, Matplotlib 6-1
2 Contents Homeworks and Project Numpy Scipy Matplotlib Exercises 6: Numpy, Scipy, Matplotlib 6-2
3 Homeworks Homework 1 was due last night. Hope that went well. I will post homework 2 tomorrow - will be longer than the first. Will be due roughly two weeks later - will post exact date with it. 6: Numpy, Scipy, Matplotlib 6-3
4 Project Welcome to do your own project instead of the homework. If you do, please give a brief proposal (1-2 paragraphs) by this Friday. Project will be due same time as homework 2. 6: Numpy, Scipy, Matplotlib 6-4
5 Contents Homeworks and Project Numpy Scipy Matplotlib Exercises 6: Numpy, Scipy, Matplotlib 6-5
6 Numpy Fundamental package for scientific computing with Python N-dimensional array object Linear algebra, Fourier transform, random number capabilities Building block for other packages (e.g. Scipy) Open source 6: Numpy, Scipy, Matplotlib 6-6
7 Numpy Fundamental package for scientific computing with Python N-dimensional array object Linear algebra, Fourier transform, random number capabilities Building block for other packages (e.g. Scipy) Open source 6: Numpy, Scipy, Matplotlib 6-7
8 import numpy as np Basics: import numpy as np A = np.array([[1, 2, 3], [4, 5, 6]]) print A # [[1 2 3] # [4 5 6]] print A[0,0] # 1 print A[0,1:3] # [2 3] Af = np.array([1, 2, 3], float) Slicing as usual. 6: Numpy, Scipy, Matplotlib 6-8
9 More basics np.arange(0, 1, 0.2) # array([ 0., 0.2, 0.4, 0.6, 0.8]) np.linspace(0, 2*np.pi, 4) # array([ 0.0, 2.09, 4.18, 6.28]) A = np.zeros((2,3)) # array([[ 0., 0., 0.], # [ 0., 0., 0.]]) # np.ones, np.diag A.shape # (2, 3) 6: Numpy, Scipy, Matplotlib 6-9
10 More basics np.random.random((2,3)) # array([[ , , ], # [ , , ]]) a = np.random.normal(loc=1.0, scale=2.0, size=(2,2)) # array([[ , ], # [ , ]]) np.savetxt("a_out.txt", a) # save to file b = np.loadtxt("a_out.txt") # read from file 6: Numpy, Scipy, Matplotlib 6-10
11 Arrays are mutable A = np.zeros((2, 2)) # array([[ 0., 0.], # [ 0., 0.]]) C = A C[0, 0] = 1 print A # [[ 1. 0.] # [ 0. 0.]] 6: Numpy, Scipy, Matplotlib 6-11
12 Array attributes a = np.arange(10).reshape((2,5)) a.ndim a.shape a.size a.t a.dtype # 2 dimension # (2, 5) shape of array # 10 # of elements # transpose # data type 6: Numpy, Scipy, Matplotlib 6-12
13 Basic operations Arithmetic operators: elementwise application a = np.arange(4) # array([0, 1, 2, 3]) b = np.array([2, 3, 2, 4]) a * b # array([ 0, 3, 4, 12]) b - a # array([2, 2, 0, 1]) c = [2, 3, 4, 5] a * c # array([ 0, 3, 8, 15]) Also, we can use += and *=. 6: Numpy, Scipy, Matplotlib 6-13
14 Array broadcasting When operating on two arrays, numpy compares shapes. Two dimensions are compatible when 1. They are of equal size 2. One of them is 1 6: Numpy, Scipy, Matplotlib 6-14
15 Array broadcasting 6: Numpy, Scipy, Matplotlib 6-15
16 Array broadcasting with scalars This also allows us to add a constant to a matrix or multiply a matrix by a constant A = np.ones((3,3)) print 3 * A - 1 # [[ ] # [ ] # [ ]] 6: Numpy, Scipy, Matplotlib 6-16
17 Vector operations inner product outer product dot product (matrix multiplication) # note: numpy automatically converts lists u = [1, 2, 3] v = [1, 1, 1] np.inner(u, v) # 6 np.outer(u, v) # array([[1, 1, 1], # [2, 2, 2], # [3, 3, 3]]) np.dot(u, v) # 6 6: Numpy, Scipy, Matplotlib 6-17
18 Matrix operations First, define some matrices: A = np.ones((3, 2)) # array([[ 1., 1.], # [ 1., 1.], # [ 1., 1.]]) A.T # array([[ 1., 1., 1.], # [ 1., 1., 1.]]) B = np.ones((2, 3)) # array([[ 1., 1., 1.], # [ 1., 1., 1.]]) 6: Numpy, Scipy, Matplotlib 6-18
19 Matrix operations np.dot(a, B) # array([[ 2., 2., 2.], # [ 2., 2., 2.], # [ 2., 2., 2.]]) np.dot(b, A) # array([[ 3., 3.], # [ 3., 3.]]) np.dot(b.t, A.T) # array([[ 2., 2., 2.], # [ 2., 2., 2.], # [ 2., 2., 2.]]) np.dot(a, B.T) # Traceback (most recent call last): # File "<stdin>", line 1, in <module> # ValueError: shapes (3,2) and (3,2) not aligned:... #... 2 (dim 1)!= 3 (dim 0) 6: Numpy, Scipy, Matplotlib 6-19
20 Operations along axes a = np.random.random((2,3)) # array([[ , , ], # [ , , ]]) a.sum() # a.sum(axis=0) # column sum # array([ , , ]) a.cumsum() # array([ , , , , # , ]) a.cumsum(axis=1) # cumulative row sum # array([[ , , ], # [ , , ]]) a.min() # a.max(axis=0) # array([ , , ]) 6: Numpy, Scipy, Matplotlib 6-20
21 Slicing arrays More advanced slicing a = np.random.random((4,5)) a[2, :] # third row, all columns a[1:3] # 2nd, 3rd row, all columns a[:, 2:4] # all rows, columns 3 and 4 6: Numpy, Scipy, Matplotlib 6-21
22 Iterating over arrays Iterating over multidimensional arrays is done with respect to the first axis: for row in A Looping over all elements: for element in A.flat 6: Numpy, Scipy, Matplotlib 6-22
23 Reshaping Reshape using reshape. Total size must remain the same. Resize using resize, always works: chopping or appending zeros First dimension has priority, so beware of unexpected results Try it! 6: Numpy, Scipy, Matplotlib 6-23
24 Reshaping Reshape using reshape. Total size must remain the same. Resize using resize, always works: chopping or appending zeros First dimension has priority, so beware of unexpected results Try it! 6: Numpy, Scipy, Matplotlib 6-24
25 Reshaping Reshape using reshape. Total size must remain the same. Resize using resize, always works: chopping or appending zeros First dimension has priority, so beware of unexpected results Try it! 6: Numpy, Scipy, Matplotlib 6-25
26 Matrix operations eye(3) trace(a) column_stack((a,b)) row_stack((a,b,a)) Identity matrix Trace Stack column wise Stack row wise 6: Numpy, Scipy, Matplotlib 6-26
27 Linear algebra import numpy.linalg qr cholesky inv(a) solve(a,b) Computes the QR decomposition Computes the Cholesky decomposition Inverse Solves Ax = b for A full rank lstsq(a,b) Solves arg min x Ax b 2 eig(a) Eigenvalue decomposition eig(a) Eigenvalue decomposition for symmetric or hermitian eigvals(a) Computes eigenvalues. svd(a, full) Singular value decomposition pinv(a) Computes pseudo-inverse of A 6: Numpy, Scipy, Matplotlib 6-27
28 Fourier transform import numpy.fft fft 1-dimensional DFT fft2 2-dimensional DFT fftn N-dimensional DFT ifft 1-dimensional inverse DFT (etc.) rfft Real DFT (1-dim) ifft Imaginary DFT (1-dim) 6: Numpy, Scipy, Matplotlib 6-28
29 Random sampling import numpy.random rand(d0,d1,...,dn) randn(d0, d1,...,dn) randint(lo, hi, size) choice(a, size, repl, p) shuffle(a) permutation(a) Random values in a given shape Random standard normal Random integers [lo, hi) Sample from a Permutation (in-place) Permutation (new array) 6: Numpy, Scipy, Matplotlib 6-29
30 import numpy.random Distributions in random The list of distributions to sample from is quite long, and includes beta binomial chisquare exponential dirichlet gamma laplace lognormal pareto poisson power 6: Numpy, Scipy, Matplotlib 6-30
31 Exercise Write a script that creates a random square matrix (A) with standard normal random variables and a random column vector (b) of the same size (also standard normal random variables). Solve the system Ax = b. Compute both the 2-norm and infinity norm of x as well as the Frobenius norm of A. Hint see documentation for np.linalg.norm 6: Numpy, Scipy, Matplotlib 6-31
32 Exercise import numpy as np n = 200 A = np.random.randn(n,n) b = np.random.randn(n,1) x = np.linalg.solve(a,b) print np.linalg.norm(x) print np.linalg.norm(x, np.inf) print np.linalg.norm(a, fro ) 6: Numpy, Scipy, Matplotlib 6-32
33 Contents Homeworks and Project Numpy Scipy Matplotlib Exercises 6: Numpy, Scipy, Matplotlib 6-33
34 What is SciPy? SciPy is a library of algorithms and mathematical tools built to work with NumPy arrays. linear algebra - scipy.linalg statistics - scipy.stats optimization - scipy.optimize sparse matrices - scipy.sparse signal processing - scipy.signal etc. 6: Numpy, Scipy, Matplotlib 6-34
35 Scipy Linear Algebra Slightly different from numpy.linalg. Always uses BLAS/LAPACK support, so could be faster. Some more functions. Functions can be slightly different. 6: Numpy, Scipy, Matplotlib 6-35
36 Scipy Optimization General purpose minimization: CG, BFGS, least-squares Constrainted minimization; non-negative least-squares Minimize using simulated annealing Scalar function minimization Root finding Check gradient function Line search 6: Numpy, Scipy, Matplotlib 6-36
37 Scipy Statistics Mean, median, mode, variance, kurtosis Pearson correlation coefficient Hypothesis tests (ttest, Wilcoxon signed-rank test, Kolmogorov-Smirnov) Gaussian kernel density estimation See also SciKits (or scikit-learn). 6: Numpy, Scipy, Matplotlib 6-37
38 Scipy sparse Sparse matrix classes: CSC, CSR, etc. Functions to build sparse matrices sparse.linalg module for sparse linear algebra sparse.csgraph for sparse graph routines 6: Numpy, Scipy, Matplotlib 6-38
39 Scipy signal Convolutions B-splines Filtering Continuous-time linear system Wavelets Peak finding 6: Numpy, Scipy, Matplotlib 6-39
40 Scipy IO Methods for loading and saving data Matlab files Matrix Market files (sparse matrices) Wav files 6: Numpy, Scipy, Matplotlib 6-40
41 Example from scipy import optimize def f(x): return [x[0] * (x[0] - x[1])**3-1.0, 0.5 * (x[1] - x[0])**3 + x[1]] x0 = [0, 0] # initial guess sol = optimize.root(f, x0) print sol.x print sol.success 6: Numpy, Scipy, Matplotlib 6-41
42 Exercise Create a matrix (A) of random entries (your choice on distribution) with m > n (more rows than columns). Create a column vector b R m. Find x that minimizes Ax b 2. What is the norm of the residual? Hint: use scipy.linalg.lstsq 6: Numpy, Scipy, Matplotlib 6-42
43 Exercise import numpy as np from scipy import linalg n = 100 m = 200 A = np.random.randn(m,n) b = np.random.randn(m,1) x = linalg.lstsq(a, b) print linalg.norm(np.dot(a,x[0])-b) 6: Numpy, Scipy, Matplotlib 6-43
44 Contents Homeworks and Project Numpy Scipy Matplotlib Exercises 6: Numpy, Scipy, Matplotlib 6-44
45 What is Matplotlib? Plotting library for Python Works well with Numpy Syntax similar to Matlab 6: Numpy, Scipy, Matplotlib 6-45
46 Scatter Plot import numpy as np import matplotlib. pyplot as plt x = np. linspace (0, 10, 1000) y = np.power(x, 2) plt. plot (x, y) plt.show() 6: Numpy, Scipy, Matplotlib 6-46
47 Seaborn makes plot pretty import numpy as np import matplotlib. pyplot as plt import seaborn as sns x = np. linspace (0, 10, 1000) y = np.power(x, 2) plt. plot (x, y) plt.show() 6: Numpy, Scipy, Matplotlib 6-47
48 plt.show() Calling plt.show() displays the figure objects to screen. To create/display multiple figures, use plt.figure(), very similar to Matlab. 6: Numpy, Scipy, Matplotlib 6-48
49 plt.figure() import numpy as np import matplotlib.pyplot as plt import seaborn as sns x = np.linspace(0, 10, 1000) y1 = np.power(x, 2) y2 = np.power(x, 3) f1 = plt.figure() plt.plot(x, y1) f2 = plt.figure() plt.plot(x, y2) plt.show() If we had not used separate figure calls, it would default to plotting both curves on same figure. 6: Numpy, Scipy, Matplotlib 6-49
50 Scatter Plot Adding titles and labels import numpy as np import matplotlib.pyplot as plt import seaborn as sns f, ax = plt.subplots(1, 1, figsize=(5,4)) x = np.linspace(0, 10, 1000) y = np.power(x, 2) ax.plot(x, y) ax.set_xlim((1, 5)) ax.set_ylim((0, 30)) ax.set_xlabel( my x label ) ax.set_ylabel( my y label ) ax.set_title( plot title, including $\Omega$ ) plt.tight_layout() plt.savefig( line_plot_plus.pdf ) 6: Numpy, Scipy, Matplotlib 6-50
51 Scatter Plot 30 plot title, including 25 my y label my x label 6: Numpy, Scipy, Matplotlib 6-51
52 Scatter Plot Adding multiple lines and a legend x = np. linspace (0, 10, 50) y1 = np.power(x, 2) y2 = np.power(x, 3) plt. plot (x, y1, b, label= $x^2$ ) plt. plot (x, y2, go, label= $x^3$ ) plt. xlim ((1, 5)) plt. ylim ((0, 30)) plt. xlabel ( my x label ) plt. ylabel ( my y label ) plt. t i t l e ( plot t i t l e, including $\Omega$ ) plt. legend () plt. savefig ( line_plot_plus2. pdf ) 6: Numpy, Scipy, Matplotlib 6-52
53 Scatter Plot plot title, including x 2 x 3 20 my y label my x label 6: Numpy, Scipy, Matplotlib 6-53
54 Histogram data = np. random. randn(1000) f, (ax1, ax2) = plt. subplots (1, 2, figsize =(6,3)) # histogram (pdf) ax1. hist (data, bins=30, normed=true, color= b ) # empirical cdf ax2. hist (data, bins=30, normed=true, color= r, cumulative=true) plt. savefig ( histogram. pdf ) 6: Numpy, Scipy, Matplotlib 6-54
55 Histogram : Numpy, Scipy, Matplotlib 6-55
56 Box Plot samp1 = np.random.normal(loc=0., scale=1., size=100) samp2 = np.random.normal(loc=1., scale=2., size=100) samp3 = np.random.normal(loc=0.3, scale=1.2, size=100) f, ax = plt.subplots(1, 1, figsize=(5,4)) ax.boxplot((samp1, samp2, samp3)) ax.set_xticklabels([ sample 1, sample 2, sample 3 ]) plt.savefig( boxplot.pdf ) 6: Numpy, Scipy, Matplotlib 6-56
57 Box Plot sample 1 sample 2 sample 3 6: Numpy, Scipy, Matplotlib 6-57
58 Image Plot A = np.random.random((100, 100)) plt.imshow(a) plt.hot() plt.colorbar() plt.savefig( imageplot.pdf ) 6: Numpy, Scipy, Matplotlib 6-58
59 Image Plot : Numpy, Scipy, Matplotlib 6-59
60 Wire Plot matplotlib toolkits extend funtionality for other kinds of visualization from mpl_toolkits.mplot3d import axes3d ax = plt.subplot(111, projection= 3d ) X, Y, Z = axes3d.get_test_data(0.1) ax.plot_wireframe(x, Y, Z, linewidth=0.1) plt.savefig( wire.pdf ) 6: Numpy, Scipy, Matplotlib 6-60
61 Wire Plot : Numpy, Scipy, Matplotlib 6-61
62 Possibilities A lot is possible, but not always easy to figure out how... 6: Numpy, Scipy, Matplotlib 6-62
63 Exercise Plot the following function on the interval [ 2, 2]. f (x) = sin(x)cos(x π)e x 6: Numpy, Scipy, Matplotlib 6-63
64 Exercise import numpy as np import matplotlib.pyplot as plt import seaborn as sns def f(x): return np.sin(x)*np.cos(x-np.pi)*np.exp(-1.*x) x = np.linspace(-2,2) y = f(x) plt.plot(x, y) plt.show() 6: Numpy, Scipy, Matplotlib 6-64
65 Contents Homeworks and Project Numpy Scipy Matplotlib Exercises 6: Numpy, Scipy, Matplotlib 6-65
66 Exercises See course website for exercises for this lecture. 6: Numpy, Scipy, Matplotlib 6-66
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