1. Python Crash Course Scipy

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

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 :
>>> print("Minimum : {0:8.6f}".format(s.min()))
Minimum :
>>> print("Maximum : {0:8.6f}".format(s.max()))
Maximum :
>>> print("Variance : {0:8.6f}".format(s.var()))
Variance :
>>> print("Std. deviation : {0:8.6f}".format(s.std()))
Std. deviation :
>>> 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: Maximum:
>>> print("Mean: {0:8.6f}".format(mean))
Mean:
>>> print("Variance: {0:8.6f}".format(var))
Variance:
>>> print("Skew : {0:8.6f}".format(skew))
Skew :
>>> print("Kurtosis: {0:8.6f}".format(kurt))
Kurtosis:

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
( , e-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
>>> print abs(result[0]-I)
e-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(' ; ')
>>> A
[[ 1. 2.]
[ 3. 4.]]
>>> type(A) # file where class is defined
<class 'numpy.matrixlib.defmatrix.matrix'>
>>> B = mat('[ ; ]')
>>> B
>>> type(B) # file where class is defined

8. SciPy – Linear Algebra Matrix inversion.
[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], 8

9. SciPy – Linear Algebra Solving linear system
S = A-1 B where S=[ x y z] and B = [ ]
>>> 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], 9

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

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: 11

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 *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() 12

13. SciPy – Singular value decomposition
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 = AHA and V= AAH and  is the matrix of eigenvectors
>>> 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
[[ ]
[ ]]
>>> print Sig
[[ ]
[ ]]
>>> print Vh
[[ e e e-01]
[ e e e-01]
[ e e e-01]] 13

14. Example: Linear regression
import scipy as sp
from scipy import stats
import pylab as plt
n= # 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, # 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.
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()
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

17. Introduction to language
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