1. Python Crash Course Scipy 3 rd year Bachelors V1.0 dd 05-09-2013 Hour 3

2. SciPy SciPy is an Open Source library of scientific tools for Python. It depends on the NumPy library, and it gathers a variety of high level science and engineering modules together as a single package. SciPy provides modules for –file input/output –statistics –optimization –numerical integration –linear algebra –Fourier transforms –signal processing –image processing –ODE solvers –special functions –and more... See http://docs.scipy.org/doc/scipy/reference/tutorial/

3. SciPy: Statistics scipy.stats The main public methods for continuous Random Variables are: rvs: Random Variates pdf: Probability Density Function cdf: Cumulative Distribution Function sf: Survival Function (1-CDF) ppf: Percent Point Function (Inverse of CDF) isf: Inverse Survival Function (Inverse of SF) stats: Return mean, variance, (Fisher’s) skew, or (Fisher’s) kurtosis moment: non-central moments of the distribution >>> from scipy import stats >>> dir(stats)

4. SciPy: Statistics scipy.stats The >>> s = sp.randn(100) # same as numpy.random.randn >>> print len(s) 100 >>> print("Mean : {0:8.6f}".format(s.mean())) Mean : -0.105170 >>> print("Minimum : {0:8.6f}".format(s.min())) Minimum : -3.091940 >>> print("Maximum : {0:8.6f}".format(s.max())) Maximum : 2.193828 >>> print("Variance : {0:8.6f}".format(s.var())) Variance : 0.863475 >>> print("Std. deviation : {0:8.6f}".format(s.std())) Std. deviation : 0.929233 >>> from scipy import stats >>> n, min_max, mean, var, skew, kurt = stats.describe(s) >>> print("Number of elements: {0:d}".format(n)) Number of elements: 100 >>> print("Minimum: {0:8.6f} Maximum: {1:8.6f}".format(min_max[0], min_max[1])) Minimum: -3.091940 Maximum: 2.193828 >>> print("Mean: {0:8.6f}".format(mean)) Mean: -0.105170 >>> print("Variance: {0:8.6f}".format(var)) Variance: 0.872197 >>> print("Skew : {0:8.6f}".format(skew)) Skew : -0.146500 >>> print("Kurtosis: {0:8.6f}".format(kurt)) Kurtosis: 0.117884

5. Example: Numerical integration Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n+1/2). For example: –Electromagnetic waves in a cylindrical waveguide –Heat conduction in a cylindrical object –Modes of vibration of a thin circular (or annular) artificial membrane (such as a drum or other membranophone) –Diffusion problems on a lattice –Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle –Solving for patterns of acoustical radiation –Frequency-dependent friction in circular pipelines Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).

6. Example: Numerical integration Integrate a Bessel function jv(2.5,x) from 0 to 4.5: >>> import numpy as np >>> import scipy as sp >>> from scipy import integrate # scipy sub-packages have to be imported separately >>> from scipy import special >>> result = integrate.quad(lambda x: special.jv(2.5,x), 0, 4.5) >>> print result (1.1178179380783249, 7.866317182537226e-09) >>> I = sqrt(2/pi)*(18.0/27*sqrt(2)*cos(4.5)-4.0/27*sqrt(2)*sin(4.5)+ :... sqrt(2*pi)*special.fresnel(3/sqrt(pi))[0]) >>> print I 1.11781793809 >>> print abs(result[0]-I) 1.03761443881e-11

7. SciPy: Linear Algebra scipy.linalg SciPy is built using the optimized ATLAS LAPACK (Linear Algebra PACKage) and BLAS (Basic Linear Algebra Subprograms) libraries, it has very fast linear algebra capabilities. All of these linear algebra routines expect an object that can be converted into a 2-dimensional array. The matrix class is initialized with the SciPy command mat which is just convenient short-hand for matrix. >>> A = matrix('1.0 2.0; 3.0 4.0') >>> A [[ 1. 2.] [ 3. 4.]] >>> type(A) # file where class is defined >>> B = mat('[1.0 2.0; 3.0 4.0]') >>> B [[ 1. 2.] [ 3. 4.]] >>> type(B) # file where class is defined

8. SciPy – Linear Algebra >>> A = mat('[1 3 5; 2 5 1; 2 3 8]') >>> A matrix([[1, 3, 5], [2, 5, 1], [2, 3, 8]]) >>> A.I matrix([[-1.48, 0.36, 0.88], [ 0.56, 0.08, -0.36], [ 0.16, -0.12, 0.04]]) >>> from scipy import linalg >>> linalg.inv(A) array([[-1.48, 0.36, 0.88], [ 0.56, 0.08, -0.36], [ 0.16, -0.12, 0.04]]) Matrix inversion.

9. SciPy – Linear Algebra >>> A = mat('[1 3 5; 2 5 1; 2 3 8]') >>> b = mat('[10;8;3]') >>> A.I*b matrix([[-9.28], [ 5.16], [ 0.76]]) >>> linalg.solve(A,b) array([[-9.28], [ 5.16], [ 0.76]]) Solving linear system S = A -1 B where S=[ x y z] and B = [ 10 8 3]

10. SciPy – Linear Algebra >>> A = mat('[1 3 5; 2 5 1; 2 3 8]') >>> linalg.det(A) -25.000000000000004 Finding Determinant

11. SciPy – Linear Algebra Solving linear least-squares problems The model Minimize After some algebra, vector/matrix representaions of this problem As an example solve for model:

12. SciPy – Linear Algebra from numpy import * from scipy import linalg import matplotlib.pyplot as plt c1,c2= 5.0,2.0 i = r_[1:11] xi = 0.1*i yi = c1*exp(-xi)+c2*xi zi = yi + 0.05*max(yi)*random.randn(len(yi)) A = c_[exp(-xi)[:,newaxis],xi[:,newaxis]] c,resid,rank,sigma = linalg.lstsq(A,zi) xi2 = r_[0.1:1.0:100j] yi2 = c[0]*exp(-xi2) + c[1]*xi2 plt.plot(xi,zi,'x',xi2,yi2) plt.axis([0,1.1,3.0,5.5]) plt.xlabel('$x_i$') plt.title('Data fitting with linalg.lstsq') plt.show()

13. SciPy – Singular value decomposition >>> A = mat('[1 3 2; 1 2 3]') >>> M,N = A.shape >>> U,s,Vh = linalg.svd(A) >>> Sig = mat(linalg.diagsvd(s,M,N)) >>> U, Vh = mat(U), mat(Vh) >>> print U [[-0.70710678 -0.70710678] [-0.70710678 0.70710678]] >>> print Sig [[ 5.19615242 0. 0. ] [ 0. 1. 0. ]] >>> print Vh [[ -2.72165527e-01 -6.80413817e-01 -6.80413817e-01] [ -6.18652536e-16 -7.07106781e-01 7.07106781e-01] [ -9.62250449e-01 1.92450090e-01 1.92450090e-01]] Singular Value Decompostion (SVD) can be thought of as an extension of the eigenvalue problem to matrices that are not square. Let A be an M x N matrix. SVD of A can be written as With U = A H A and V= AA H and  is the matrix of eigenvectors

14. Example: Linear regression import scipy as sp from scipy import stats import pylab as plt n=50 # number of points x=sp.linspace(-5,5,n) # create x axis data a, b=0.8, -4 y=sp.polyval([a,b],x) yn=y+sp.randn(n) #add some noise (ar,br)=sp.polyfit(x,yn,1) yr=sp.polyval([ar,br],x) err=sp.sqrt(sum((yr-yn)**2)/n) #compute the mean square error print('Linear regression using polyfit') print(‘Input parameters: a=%.2f b=%.2f’ % (a,b)) print(‘Regression: a=%.2f b=%.2f, ms error= %.3f' % (ar,br,err)) plt.title('Linear Regression Example') plt.plot(x,y,'g--') plt.plot(x,yn,'k.') plt.plot(x,yr,'r-') plt.legend(['original','plus noise', 'regression']) plt.show() (a_s,b_s,r,xx,stderr)=stats.linregress(x,yn) print('Linear regression using stats.linregress') print('parameters: a=%.2f b=%.2f’ % (a,b)) print(‘regression: a=%.2f b=%.2f, std error= %.3f' % (a_s,b))

15. Example: Least squares fit from pylab import * from numpy import * from matplotlib import * from scipy.optimize import leastsq fp = lambda v, x: v[0]/(x**v[1])*sin(v[2]*x) # parametric function v_real = [1.7, 0.0, 2.0] fn = lambda x: fp(v_real, x) # fn to generate noisy data e = lambda v, x, y: (fp(v,x)-y) # error function n, xmin, xmax = 30, 0.1, 5 # Generate noisy data to fit x = linspace(xmin,xmax,n) y = fn(x) + rand(len(x))*0.2*(fn(x).max()-fn(x).min()) v0 = [3., 1, 4.] # Initial parameter values v, success = leastsq(e, v0, args=(x,y), maxfev=10000) # perform fit print ‘Fit parameters: ', v print ‘Original parameters: ', v_real X = linspace(xmin,xmax,n*5) # plot results plot(x,y,'ro', X, fp(v,X)) show()

16. SciPy: Ordinary Differential Equations Simple differential equations can be solved numerically using the Euler-Cromer method, but more complicated differential equations may require a more sophisticated method. The scipy library for Python contains numerous functions for scientific computing and data analysis. It includes the function odeint for numerically solving sets of first-order, ordinary differential equations (ODEs) using a sophisticated algorithm. rewritten as the following two first-order differential equations, The function y will be the first element y[0] (remember that the lowest index of an array is zero, not one) and the derivative y ′ will be the second element y[1]. You can think of the index as how many derivatives are taken of the function. In this notation, the differential equiations are from scipy import odeint from pylab import * # for plotting commands def deriv(y,t): # return derivatives of the array y a = -2.0 b = -0.1 return array([ y[1], a*y[0]+b*y[1] ]) time = linspace(0.0,10.0,1000) yinit = array([0.0005,0.2]) # initial values y = odeint(deriv,yinit,time) figure() plot(time,y[:,0]) # y[:,0] is the first column of y xlabel(‘t’) ylabel(‘y’) show()

17. Introduction to language End