Numpy

Numpy stands for Numerical Python, is the foundational package for scientific computing in Python. It provides

A fast and efficient multidimensional arrary object ndarray
Functions for performing element-wise computation with arrays and mathematical operations between arrays
Linear algebra operations, Fourier transform, and random number generation



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import numpy as np



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a = np.array([1, 2, 3, 4])
a



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type(a)



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2*a  # multiple ndarray by number



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b = np.array([2, 3, 4, 5])
print(a)
print(b)

a+b  # two array summation



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a*b



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np.log(a)  # apply functions to array

index and slicing
var[lower:upper:step]



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a



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a[1]



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a[1:3]



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# omitted boundaries are assumed to be the beginning (or end) of the array

a[:3]  # grap first three elements



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a[::2] # element with step size 2

Multi-Dimensional Arrays



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c = np.array([[0, 1, 2, 3],
          [10, 11, 12, 13]])
c



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c[0, 3]  # get the first row and fourth number



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c[1]  # get the second row



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c[0, 2:4]  # get the first rown and slicing the first row

Basic info of array



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print(a)
a.shape



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b = a.reshape((2, 2))
b



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b.T



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c[:, 2:4] # slice the row



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print(c)
c.shape



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a.size



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c.size

Joining arrays



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a = np.array([1, 2, 3])
b = np.array([2, 3, 4])

# Stack arrays in sequence vertically (row wise).
np.vstack([a, b])



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# Stack arrays in sequence horizontally (column wise).
np.hstack([a, b])

Array Calculation Methods



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a = np.array([[1, 2, 3],
            [4, 5, 6]])
a



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np.sum(a)



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np.sum(a, axis=0)



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np.sum(a, axis=1)



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np.max(a)



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np.min(a, axis=0)



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np.prod(a)

Array creation Functions



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np.arange(4)  # arange(start, stop=None, step=1)



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np.arange(1, 5, 2)



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np.ones(4)  # ones(shape)



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np.ones((2, 3))



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np.zeros((2, 3)) # zeros(shape)



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np.linspace(0, 1, 5)  # Generate N evenly spaced elements between start and stop values

For more detail of numpy please refer to Numpy.org



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from IPython.core.display import HTML
HTML("<iframe src=http://www.numpy.org/  width=800 height=350></iframe>")

matplotlib

matplotlib for producing plots and other 2D data visualizations. It is well-suited for creating plots suitable for publication.



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%matplotlib inline
import matplotlib.pyplot as plt



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x = np.linspace(0, 2*np.pi, 20)
y = np.sin(x)
plt.xlabel('x label')  # set label for x coordinate
plt.ylabel('y label')  # set label for y coordinates
plt.title('Sin Function')
plt.plot(x, y)

Draw two plots on one coordinate



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x = np.linspace(0, 2*np.pi, 20)
y1 = np.sin(x)
y2 = np.cos(x)
plt.plot(x, y1)
plt.plot(x, y2)

Working with multiple figures and axes

figure represents the whole picture
axes represents each plot with independ coordinate
The subplot() command specifies (numrows, numcols, fignum) where fignum ranges from 1 to numrows * numcols.



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x = np.linspace(0, 2*np.pi, 20)
y1 = np.sin(x)
y2 = np.cos(x)

fig = plt.figure()
ax1 = plt.subplot(211)
ax1.plot(x, y1)

ax2 = plt.subplot(212)
ax2.plot(x, y2)

Show the labels of each plot
Specify color of each plot



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x = np.linspace(0, 2*np.pi, 20)
y1 = np.sin(x)
y2 = np.cos(x)

fig = plt.figure()
ax1 = plt.subplot(211)
ax1.plot(x, y1, label='Sin', color='b')
ax1.legend()  # show the label of plots on ax1

ax2 = plt.subplot(212)
ax2.plot(x, y2, label='Cos', color='r')
ax2.legend()  # show the label of plotson ax2

Set boundary of each coordinate



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x = np.linspace(0, 2*np.pi, 20)
y1 = np.sin(x)
y2 = np.cos(x)

fig = plt.figure()
ax1 = plt.subplot(211)
ax1.plot(x, y1, label='Sin', color='b')
ax1.legend()  # show the label on ax1
ax1.set_xlim(-10, 10)  # set the boundary of x coordinate

ax2 = plt.subplot(212)
ax2.plot(x, y2, label='Cos', color='r')
ax2.legend()  # show the label on ax2
ax2.set_ylim(-2, 2)  # set the boundary of y coordinate

Histogram



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r1 = np.random.normal(0, 10, 1000)
_ = plt.hist(r1, bins=50, normed=True)

Scatterplot



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x = np.random.normal(0, 10, 100)
y = np.random.normal(5, 3, 100)
plt.scatter(x, y)

Boxplots



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r1 = np.random.normal(0, 10, 100)
r2 = np.random.gamma(1, 2, 100)
r3 = r1 + r2
_ = plt.boxplot(r3)

For more detail of matplotlib please refer to matplotlib.org



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HTML("<iframe src=https://matplotlib.org/index.html width=800 height=350></iframe>")

SciPy

SciPy is a collection of packages addressing a number of different standard problem domain in scientific computing. Here is some packages you may use:

scipy.integrate: numerical integration routines and differential equation solvers
scipy.linalg: linear algebra routines and matrix decompositions
scipy.optimize: function optimizers and root finding algorithms
scipy.stats: standard probability distributions (cdf, pdf, sampler and etc)

Example of linear algebra operations



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from scipy import linalg  # import linear algebra operations

arr = np.array([[1, 2],
               [3, 4]])
linalg.det(arr)  # compute the determinant of a square matrix



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linalg.inv(arr)  # compute the inverse of a squar matrix

Example of optimization algorithm



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from scipy import optimize
def f(x):
    return x**2 + 10*np.sin(x)

x = np.arange(-10, 10, 0.1)
plt.plot(x, f(x))



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result = optimize.minimize(f, x0=0)
result.x



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plt.plot(x, f(x))
plt.axvline(result.x)  # plot a vertical lines at result x

Example of probability distributions method



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from scipy import stats

samples = np.random.normal(size=100)
plt.hist(samples, bins = 20, normed=True, label='Histogram of samples')
x_space = np.linspace(*plt.xlim())

pdf = stats.norm.pdf(x_space)
plt.plot(x_space, pdf, label='PDF')
plt.legend()

For more detail of matplotlib please refer to scipy.org



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HTML("<iframe src=https://www.scipy.org/scipylib/index.html width=800 height=350></iframe>")

Exercises 1

Write a Python program to reverse an array (first element becomes last)



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a = np.array([1, 3, 5, 7, 8, 9])

Exercises 2

Write a Python program to create a (5*5) 2d array with 1 on the border and 0 inside.



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Exercises 3

Write a Python program to sample a distribution $\epsilon_1 + 2\epsilon_2$ with 100 samples, where $\epsilon_1 \sim N(0, 3)$ and $\epsilon_2 \sim N(2, 1)$. Then plot the histgram of the sample.



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Exercises 4

Write a Python program to get the values and indices of the elements that are bigger than 10 in a given array.

Original array: [[ 0 10 20] [20 30 40]] Values bigger than 10 = [20 20 30 40] Their indices are (array([0, 1, 1, 1]), array([2, 0, 1, 2]))



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