Practical 9¶
In this practical we will will practice Numpy and Matplotlib.
Numpy¶
Numpy is a fundamental library for high performance scientific
computations. It provides fast and memory efficient data structures like
ndarray
with broadcasting capabilities, standard mathematical
functions that can be applied on the arrays avoiding loops, linear
algebra functions, I/O methods and it is well integrated with
programming languages like C.
To use the functionalities of the library we always need to import it with (standard name is np):
import numpy as np
The official documentation of the numpy library can be found at http://www.numpy.org/.
Numpy ndarray¶
Numpy ndarray is an N-dimensional array object designed to contain homogeneous data (i.e. all data must have the same type). Every array (also called tensor) comes with two information: the shape (a tuple describing the dimensions of the array) and a dtype (the data type of the array).
Given a list of homogeneous values the ndarray can be obtained by using
the np.array(list)
method. Given the numpy array its shape,
dimensionality and dtype can be obtained with array.shape
,
array.ndim
and array.dtype
.
An example follows:
In [1]:
import numpy as np
Aint = np.array([[1,2,3], [4,5,6]])
Afloat = np.array([[1.1,2,3], [4.2,5,6], [1,2,3]])
print(Aint)
print(type(Aint))
print(Aint.shape)
print(Aint.dtype)
print("")
print(Afloat)
print(type(Afloat))
print(Afloat.shape)
print(Afloat.dtype)
[[1 2 3]
[4 5 6]]
<class 'numpy.ndarray'>
(2, 3)
int64
[[1.1 2. 3. ]
[4.2 5. 6. ]
[1. 2. 3. ]]
<class 'numpy.ndarray'>
(3, 3)
float64
Example:
Let’s create a ndarray holding the square root and one with the cubic root of the first five integers. Let’s put them together in a 2-d array and check its dimensions.
In [2]:
import numpy as np
import math
mysqrt = [math.sqrt(x) for x in range(0,5)]
mycrt = [x**(1/3) for x in range(0,5)]
myOtherRt = [x**(1/5) for x in range(0,5)]
print("Original lists:")
print(mysqrt)
print(mycrt)
print("")
npData = np.array(mysqrt)
print("Numpy ndarray:")
print(npData)
print("")
print("The shape:", npData.shape)
print("The dimensionality:", npData.ndim)
print("The type:", npData.dtype)
print("")
twoDarray = np.array([mysqrt, mycrt, myOtherRt])
print("The 2D array:")
print(twoDarray)
print("")
print("The shape:", twoDarray.shape)
print("The dimensionality:", twoDarray.ndim)
print("The type:", twoDarray.dtype)
Original lists:
[0.0, 1.0, 1.4142135623730951, 1.7320508075688772, 2.0]
[0.0, 1.0, 1.2599210498948732, 1.4422495703074083, 1.5874010519681994]
Numpy ndarray:
[0. 1. 1.41421356 1.73205081 2. ]
The shape: (5,)
The dimensionality: 1
The type: float64
The 2D array:
[[0. 1. 1.41421356 1.73205081 2. ]
[0. 1. 1.25992105 1.44224957 1.58740105]
[0. 1. 1.14869835 1.24573094 1.31950791]]
The shape: (3, 5)
The dimensionality: 2
The type: float64
We can also create arrays (matrices) full of zeros or ones or even diagonal matrices with the following numpy methods:
- Array:
np.zeros(N)
or matrix:np.zeros((N,M))
- Array:
np.ones(N)
or matrix:np.ones((N,M))
- Matrix:
np.eye(N)
Numpy has its own range method that is called np.arange(N)
.
Evenly spaced values in a range can be obtained also with
np.linspace(S,E, num=N, endpoint=True/False)
to obtain N linearly
spaced values from S to E (included, unless endpoint = False is
specified).
In [3]:
import numpy as np
zeros = np.zeros(3)
zMat = np.zeros((4,3))
ones = np.ones(3)
oMat = np.ones((3,2))
diag = np.eye(4)
rng = np.arange(5) #5 excluded!
print("Zero array (1x3)")
print(zeros)
print("")
D = zMat.shape
print("Zero matrix ({}x{})".format(D[0],D[1]))
print(zMat)
print("")
print("Ones array (1x3)")
print(ones)
print("")
print("Ones matrix (3x2)")
print(oMat)
print("")
print("Diagonal matrix")
print(diag)
print("")
print("Range 0-4")
print(rng)
print("A diagonal matrix:")
dm = np.diag(rng)
print(dm)
print("Its shape:")
print(dm.shape)
rng = np.arange(7)
print("Range 0-6")
print(rng)
myRange = np.linspace(-5,2.5,num =6)
print("6 linearly spaced elements in [-5 - 2.5]:")
print(myRange)
myRange = np.linspace(0,21,num =7, endpoint=False)
print("7 linearly spaced elements in [0 - 21):")
print(myRange)
Zero array (1x3)
[0. 0. 0.]
Zero matrix (4x3)
[[0. 0. 0.]
[0. 0. 0.]
[0. 0. 0.]
[0. 0. 0.]]
Ones array (1x3)
[1. 1. 1.]
Ones matrix (3x2)
[[1. 1.]
[1. 1.]
[1. 1.]]
Diagonal matrix
[[1. 0. 0. 0.]
[0. 1. 0. 0.]
[0. 0. 1. 0.]
[0. 0. 0. 1.]]
Range 0-4
[0 1 2 3 4]
A diagonal matrix:
[[0 0 0 0 0]
[0 1 0 0 0]
[0 0 2 0 0]
[0 0 0 3 0]
[0 0 0 0 4]]
Its shape:
(5, 5)
Range 0-6
[0 1 2 3 4 5 6]
6 linearly spaced elements in [-5 - 2.5]:
[-5. -3.5 -2. -0.5 1. 2.5]
7 linearly spaced elements in [0 - 21):
[ 0. 3. 6. 9. 12. 15. 18.]
Random arrays¶
Random arrays can be created from several different distributions (check here for all options). Some examples include uniform distribution, normal, binomial,…
Example:
Create a random array of 1000 values drawn from: 1. a gaussian distribution with \(\sigma = 20\) and \(\mu = 2\) 2. a uniform distribution from 0 to 5 3. a binomial distribution with \(p = 0.5\) and \(n = 12\)
and plot their histogram.
In [4]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
#get the gaussian random array
g = np.random.normal(20,2, 1000)
#a uniform random array with vals in [0,5]
u = np.random.uniform(0,5, 1000)
#get the binomial random array
b = np.random.binomial(12,0.5, 1000)
plt.hist(g, bins = 30)
plt.title("Gaussian(20,2)")
plt.xlabel("Random #")
plt.ylabel("Count")
plt.show()
plt.hist(u, bins = 10)
plt.title("Uniform(0,5)")
plt.xlabel("Random #")
plt.ylabel("Count")
plt.show()
plt.hist(b, bins = 10)
plt.title("Binomial(12,0.5)")
plt.xlabel("Random #")
plt.ylabel("Count")
plt.show()
It is also possible to specify the the size of the output (with the
tuple size
)to get a matrix instead.
In [5]:
import numpy as np
u = np.random.uniform(0,1, size = (4,3))
print("Uniform distr. in (0,1). 4x3 matrix:")
print("")
print(u)
Uniform distr. in (0,1). 4x3 matrix:
[[0.01748314 0.68196938 0.7337711 ]
[0.9061673 0.27011933 0.83324199]
[0.13474732 0.21965773 0.24174557]
[0.81930429 0.25657836 0.10198619]]
To always get the same values we need to initialize the random seed (a parameter of the pseudo-random sequence generator) always to the same value.
In [6]:
import numpy as np
u = np.random.uniform(0,1,size=(3,2))
u1 = np.random.uniform(0,1,size=(3,2))
u2 = np.random.uniform(0,1,size=(3,2))
print(" u: {}\n\n u1:{}\n\n u2:{}".format(u,u1,u2))
print("")
print("With random seed reinit.")
np.random.seed(0)
u = np.random.uniform(0,1,3)
np.random.seed(0)
u1 = np.random.uniform(0,1,3)
np.random.seed(0)
u2 = np.random.uniform(0,1,3)
print(" u: {}\n u1:{}\n u2:{}".format(u,u1,u2))
u: [[0.1182575 0.88378038]
[0.64969197 0.45585741]
[0.0245672 0.28397606]]
u1:[[0.43501974 0.6943427 ]
[0.527252 0.96579963]
[0.20484529 0.05242482]]
u2:[[0.1545874 0.89794199]
[0.75233458 0.56678759]
[0.86651043 0.90583683]]
With random seed reinit.
u: [0.5488135 0.71518937 0.60276338]
u1:[0.5488135 0.71518937 0.60276338]
u2:[0.5488135 0.71518937 0.60276338]
Numpy to and from pandas¶
Pandas Series and DataFrames are actually built on top of numpy ndarrays. Moreover, given a numpy matrix (or array) it is quite easy to convert it into a DataFrame (or Series).
In [7]:
import pandas as pd
import numpy as np
myDict = {"one" : 1, "two" : 2, "three" : 3, "four" : 4}
mySeries = pd.Series(myDict)
print(myDict)
print("")
print(type(mySeries))
print("")
print(type(mySeries.values))
print("")
myMat = np.diag(np.arange(6))
myDF = pd.DataFrame(myMat)
print("Numpy matrix")
print(myMat)
print("")
print("Pandas DataFrame")
print(myDF)
print("")
print("Reindexed DataFrame")
myDF = pd.DataFrame(myMat, index = list("ABCDEF"),
columns = ['a1','b2','c3','d4','e5','f6'])
print(myDF)
{'one': 1, 'two': 2, 'three': 3, 'four': 4}
<class 'pandas.core.series.Series'>
<class 'numpy.ndarray'>
Numpy matrix
[[0 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 2 0 0 0]
[0 0 0 3 0 0]
[0 0 0 0 4 0]
[0 0 0 0 0 5]]
Pandas DataFrame
0 1 2 3 4 5
0 0 0 0 0 0 0
1 0 1 0 0 0 0
2 0 0 2 0 0 0
3 0 0 0 3 0 0
4 0 0 0 0 4 0
5 0 0 0 0 0 5
Reindexed DataFrame
a1 b2 c3 d4 e5 f6
A 0 0 0 0 0 0
B 0 1 0 0 0 0
C 0 0 2 0 0 0
D 0 0 0 3 0 0
E 0 0 0 0 4 0
F 0 0 0 0 0 5
Reshaping¶
The shape of a numpy ndarray can be changed at any time with the methods
reshape((R,C))
where R and C are the rows and columns used to shape
an array into a matrix, and ravel
that is used to go the other way
around from matrix to array.
In [8]:
import numpy as np
myA = np.arange(12)
print("The array:")
print(myA)
print("")
myB = myA.reshape((2,6))
print("Reshaped (2x6):")
print(myB)
print("")
myC = myA.reshape((3,4))
print("Reshaped: (3x4)")
print(myC)
myD = myB.ravel()
print("")
print("Back to array:")
print(myD)
The array:
[ 0 1 2 3 4 5 6 7 8 9 10 11]
Reshaped (2x6):
[[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]]
Reshaped: (3x4)
[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]]
Back to array:
[ 0 1 2 3 4 5 6 7 8 9 10 11]
Be careful though, if shapes don’t match an error cannot reshape array is returned:
Iterating over arrays and Indexing¶
It is possible to loop over the elements of an array by element or by
row of the matrix. In the first case we use a the method
np.ndarray.flat
to make a matrix accessible as a one dimensional
array. Check out the following examples:
In [9]:
import numpy as np
myA = np.arange(12)
print(myA)
print("")
print("Matrix:")
myA = myA.reshape((4,3))
print(myA)
print("Looping through elements:")# equivalent to:
for el in myA.flat: # for el in myA.ravel():
print("Element:", el) # print("Element:",el)
print("Looping row by row:")
for el in myA:
print("Row: ", el, "is a", type(el))
for j in el:
print("\tel:", j)
[ 0 1 2 3 4 5 6 7 8 9 10 11]
Matrix:
[[ 0 1 2]
[ 3 4 5]
[ 6 7 8]
[ 9 10 11]]
Looping through elements:
Element: 0
Element: 1
Element: 2
Element: 3
Element: 4
Element: 5
Element: 6
Element: 7
Element: 8
Element: 9
Element: 10
Element: 11
Looping row by row:
Row: [0 1 2] is a <class 'numpy.ndarray'>
el: 0
el: 1
el: 2
Row: [3 4 5] is a <class 'numpy.ndarray'>
el: 3
el: 4
el: 5
Row: [6 7 8] is a <class 'numpy.ndarray'>
el: 6
el: 7
el: 8
Row: [ 9 10 11] is a <class 'numpy.ndarray'>
el: 9
el: 10
el: 11
Indexing and slicing can be performed as usual with the [] operator.
np.ndarray[i,j]
returns the element at row i
and column j
.
A multi-dimensional matrix has more than 2 dimensions. They can be thought as several 2D matrices stacked one on top of the other:
Note that np.ndarray[0,:,:]
is the whole first matrix.
np.ndarray[:,0,:]
is all the first rows, while np.ndarray[:,:,0]
is all the first columns. Regarding slicing and indexing, the same
reasoning applies to n-dimensional matrices. For example, myB
below
is a 3x3x3 matrix.
In [10]:
import numpy as np
myA = np.random.randint(0,10, size = (3,4))
print(myA)
print("")
print("myA[2,2] = ", myA[2,2])
print("myA[1,3] = ", myA[1,3])
print("myA[0,3] = ", myA[0,3])
print("second row:", myA[1,:])
print("")
print("3D matrix ",)
myB = np.random.randint(0,10, size = (3,3,3))
print(" - shape:", myB.shape)
print(myB)
print("")
print("myB[0,2,2] = ", myB[0,2,2])
print("Second matrix:")
print(myB[1,:,:])
print("Third row of second matrix:")
print(myB[1,2,:])
print("Second column of second matrix:")
print(myB[1,:,1])
[[3 7 9 3]
[5 2 4 7]
[6 8 8 1]]
myA[2,2] = 8
myA[1,3] = 7
myA[0,3] = 3
second row: [5 2 4 7]
3D matrix
- shape: (3, 3, 3)
[[[6 7 7]
[8 1 5]
[9 8 9]]
[[4 3 0]
[3 5 0]
[2 3 8]]
[[1 3 3]
[3 7 0]
[1 9 9]]]
myB[0,2,2] = 9
Second matrix:
[[4 3 0]
[3 5 0]
[2 3 8]]
Third row of second matrix:
[2 3 8]
Second column of second matrix:
[3 5 3]
Broadcasting and arithmetic functions¶
Universal functions are very efficient element-wise functions that can be applied on ndarrays. As seen in Pandas, operations involving scalar values and arrays are broadcast to all the elements of the ndarray, while operations performed on arrays having the same size are done element by element.
Note also that it is possible to perform some operations only on a sub-part of an ndarray. Moreover, if operators are applied to differently sized, but compatible, ndarrays (es. matrix 3x3x3 and vector 3x1), numpy will perform broadcast of the operator. Let’s see some examples:
In [11]:
import numpy as np
A = np.random.randint(0,10, size = (3,2))
B = np.random.randint(0,10, size = (3,3,3))
C = np.random.randint(0,10, size = (3,1))
print("Matrix A 3x2")
print(A)
print("")
print("Matrix B 3x3x3")
print(B)
print("")
print("Matrix C 3x1")
print(C)
print("")
print("A squared")
print(A**2)
print("")
print("A square-rooted")
print(np.sqrt(A))
print("")
print("B square-rooted")
print(np.sqrt(B))
print("A + C ")
print(A + C)
print("")
print("B + C ")
print(B + C)
print("")
print("B")
print(B)
print("Sub array B - 20")
B[:, 0:2 , 0:2 ] -= 20
print(B)
Matrix A 3x2
[[0 4]
[7 3]
[2 7]]
Matrix B 3x3x3
[[[2 0 0]
[4 5 5]
[6 8 4]]
[[1 4 9]
[8 1 1]
[7 9 9]]
[[3 6 7]
[2 0 3]
[5 9 4]]]
Matrix C 3x1
[[4]
[6]
[4]]
A squared
[[ 0 16]
[49 9]
[ 4 49]]
A square-rooted
[[0. 2. ]
[2.64575131 1.73205081]
[1.41421356 2.64575131]]
B square-rooted
[[[1.41421356 0. 0. ]
[2. 2.23606798 2.23606798]
[2.44948974 2.82842712 2. ]]
[[1. 2. 3. ]
[2.82842712 1. 1. ]
[2.64575131 3. 3. ]]
[[1.73205081 2.44948974 2.64575131]
[1.41421356 0. 1.73205081]
[2.23606798 3. 2. ]]]
A + C
[[ 4 8]
[13 9]
[ 6 11]]
B + C
[[[ 6 4 4]
[10 11 11]
[10 12 8]]
[[ 5 8 13]
[14 7 7]
[11 13 13]]
[[ 7 10 11]
[ 8 6 9]
[ 9 13 8]]]
B
[[[2 0 0]
[4 5 5]
[6 8 4]]
[[1 4 9]
[8 1 1]
[7 9 9]]
[[3 6 7]
[2 0 3]
[5 9 4]]]
Sub array B - 20
[[[-18 -20 0]
[-16 -15 5]
[ 6 8 4]]
[[-19 -16 9]
[-12 -19 1]
[ 7 9 9]]
[[-17 -14 7]
[-18 -20 3]
[ 5 9 4]]]
A lot of mathematical and statistical functions can be applied. Examples include trigonometric functions like sin, cos, tan,…, statistical functions like mean, std, max, min, cumsum,… see the documentation for a complete list.
Example: Let’s create and plot a sin, cos and \(\log_2 (2 + sin+cos)\) from -2Pi to 2Pi
In [12]:
import numpy as np
import matplotlib.pyplot as plt
A = np.arange(-2* np.pi, 2*np.pi, 0.01)
sA = np.sin(A)
cA = np.cos(A)
plt.plot(sA)
plt.plot(cA)
plt.plot(np.log2(2 + sA + cA))
plt.legend(["Sin(x)", "Cos(x)","$\log_2{2+sin(x)+cos(x)}$"])
plt.show()
Filtering¶
It is possible to filter np.ndarrays
to retrieve the indexes (or the
values) meeting specific conditions. The method where
provides the
index of those values. Note that if the np.ndarray
is actually a
matrix, where
returns a tuple of indexes that are respectively the i
and j coordinates of the elements fulfilling the condition.
Example: write a method that creates the np.ndarray with all the integers multiple of a certain number N that do not exceed M. The method should return also the indexes of those values in the original np.ndarray.
In [13]:
import numpy as np
def getMultiples(N,M):
all = np.arange(1,M+1)
ind = np.where(all % 27 == 0)
filt = (all % 27 == 0)
return ind, all[filt]
res = getMultiples(11, 1000)
print("Indexes: {}\nValues:{}".format(res[0],res[1]))
Indexes: (array([ 26, 53, 80, 107, 134, 161, 188, 215, 242, 269, 296, 323, 350,
377, 404, 431, 458, 485, 512, 539, 566, 593, 620, 647, 674, 701,
728, 755, 782, 809, 836, 863, 890, 917, 944, 971, 998]),)
Values:[ 27 54 81 108 135 162 189 216 243 270 297 324 351 378 405 432 459 486
513 540 567 594 621 648 675 702 729 756 783 810 837 864 891 918 945 972
999]
In [14]:
import numpy as np
myA = np.arange(12).reshape((3,4))
print(myA)
print("Elements divisible by 3")
indexes = np.where(myA % 3 == 0)
print(indexes)
for ind in range(len(indexes[0])):
i = indexes[0][ind]
j = indexes[1][ind]
print("({},{}) -> {}".format(i,j, myA[i][j]))
[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]]
Elements divisible by 3
(array([0, 0, 1, 2]), array([0, 3, 2, 1]))
(0,0) -> 0
(0,3) -> 3
(1,2) -> 6
(2,1) -> 9
Example: Let’s do some filtering on the sine and cosine series plotted above.
In [15]:
import numpy as np
import matplotlib.pyplot as plt
A = np.arange(-2* np.pi, 2*np.pi, 0.01)
sA = np.sin(A)
cA = np.cos(A)
plt.plot(sA)
plt.plot(cA)
s0A_Y = sA[sA < 0]
s0A_X = np.where(sA < 0)
c0A_Y = cA[np.all( [cA > 0.5 , cA < 0.75], axis = 0)]
c0A_X = np.where(np.all( [cA > 0.5 , cA < 0.75], axis = 0))
plt.scatter(s0A_X,s0A_Y, marker='o', c = 'red')
plt.scatter(c0A_X,c0A_Y, marker='X', c = 'green')
plt.legend(["Sin(x)", "Cos(x)","Sin(x) < 0", "Cos(x) in[0.5,0.75]"])
plt.show()
Note that in the code above, np.all
tests if the two conditions are
True at the same time (i.e. AND). If we want to test if at least one is
True we use np.any
. See
np.all
and
np.any
for more info:
In [16]:
import numpy as np
v1 = [True, False,False, True]
v2 = [False, False,True, True]
v3 = [False, False, False, True]
print("vals:")
vals = np.array([v1,v2,v3])
print(vals)
print("\nANY(vals):")
print(np.any(vals, axis=0))
print("\nANY(vals) on rows:")
print(np.any(vals, axis=1))
print("\nALL(vals):")
print(np.all(vals, axis=0))
print("\nALL(vals) on rows:")
print(np.all(vals, axis=1))
vals:
[[ True False False True]
[False False True True]
[False False False True]]
ANY(vals):
[ True False True True]
ANY(vals) on rows:
[ True True True]
ALL(vals):
[False False False True]
ALL(vals) on rows:
[False False False]
Linear algebra¶
In numpy the product operator *
is used to perform element-wise
product, and therefore the arrays in input must have the same size.
Matrix-matrix multiplication can be done through the np.dot(x,y)
method, which is also a method of the object np.ndarray
and
therefore if x
is a np.ndarray
of compatible size with y
, we
can perform their product with x.dot(y)
.
A ndarray can be transposed by using the method T
.
In [17]:
import numpy as np
a = np.array([1, 2, 3, 1, 2, 3,1, 1 ,1])
A = a.reshape((3,3))
B = np.random.randint(0,10, size = (3,2))
print("A (3x3)")
print(A)
print("")
print("B (3x2)")
print(B)
print("")
print("AxB (3x2)")
print(A.dot(B))
print("")
print("A transposed:")
print(A.T)
A (3x3)
[[1 2 3]
[1 2 3]
[1 1 1]]
B (3x2)
[[4 3]
[4 4]
[8 4]]
AxB (3x2)
[[36 23]
[36 23]
[16 11]]
A transposed:
[[1 1 1]
[2 2 1]
[3 3 1]]
Numpy has a module called linalg
providing several functions that
implement matrix decompositions, inversions, determinant calculation and
so on. Some of these methods are linalg.inv
for the matrix
inversion, linalg.qr
for the QR decomposition, linalg.cholesky
for the Cholesky decomposition, linalg.eig
to find the eigenvalues
of a matrix, linalg.solve
to solve a system of linear equations and
so on.
The complete list of all available methods can be found in the documentation.
Example: Let’s create a random 4x4 matrix and invert it. Let’s finally check that the product of the matrix by its inversion yields the 4x4 identity matrix.
In [18]:
import numpy as np
from numpy import linalg
A = np.random.randint(0,10, size = (4,4))
print("Matrix A:")
print(A)
print("")
print("inv(A)")
A_1 = linalg.inv(A)
print(A_1)
print("")
print(np.dot(A,A_1))
Matrix A:
[[3 7 5 5]
[0 1 5 9]
[3 0 5 0]
[1 2 4 2]]
inv(A)
[[ 0.23121387 0.05780347 0.38150289 -0.83815029]
[ 0.09537572 -0.10115607 -0.16763006 0.21676301]
[-0.13872832 -0.03468208 -0.02890173 0.50289017]
[ 0.06647399 0.1416185 0.03468208 -0.30346821]]
[[ 1.00000000e+00 2.77555756e-17 2.08166817e-17 -1.66533454e-16]
[-4.16333634e-17 1.00000000e+00 -6.93889390e-18 5.55111512e-17]
[ 2.77555756e-17 6.93889390e-18 1.00000000e+00 0.00000000e+00]
[-2.77555756e-17 0.00000000e+00 6.93889390e-17 1.00000000e+00]]
Example: Let’s solve the following system of linear equations:\(\begin{cases} 2x + y - 2z = 3\\ x - y - z = 0\\ x + y + 3z = 12\end{cases}\)
In [19]:
import numpy as np
from numpy import linalg
A = np.array([[2, 1, -2],[1,-1,-1], [1, 1 ,3]])
b = np.array([3, 0, 12])
print("A")
print(A)
print("")
print("b")
print(b)
print("")
#Let's find x such that: Ax = b
x = linalg.solve(A,b)
print("X")
print(x)
#Let's double check Ax = b:
print("")
print("Ax:")
print(np.dot(A,x))
A
[[ 2 1 -2]
[ 1 -1 -1]
[ 1 1 3]]
b
[ 3 0 12]
X
[3.5 1. 2.5]
Ax:
[ 3. 0. 12.]
Matplotlib¶
Matplotlib is a library created for producing publication quality plots. To use matplotlib one has to import it in the python program. Typically this is done with
import matplotlib.pyplot as plt
Documentation and several examples can be found at the matplotlib website.
Figures can also be saved to files with plt.savefig('filename')
.
Example: [adapted from documentation] Let’s plot the function \(cos(2 * \Pi * x)*sin(2 * \Pi * x)\) for x ranging from 0 to 10, annotating the point (5,0).
In [20]:
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(0, 10, 0.005)
y = np.cos(np.pi*x) * np.sin(2*np.pi*x)
fig = plt.figure()
ax = fig.add_subplot(111)
xdata = 5
ydata = 0
ax.plot(x, y)
ax.set_xlim(0, 10)
ax.set_ylim(-1, 1)
xdata = 5
ydata = 0
bbox = dict(boxstyle="round", fc="0.8")
arrowprops = dict(
arrowstyle = "->")
offset = 92
ax.annotate('(%.1f, %.1f)'%(xdata, ydata),
(xdata, ydata), xytext=(-1.5*offset, offset), textcoords='offset points',
bbox=bbox, arrowprops=arrowprops)
plt.show()
Example: Let’s plot simultaneusly \(sin(2 * \Pi * x)\), \(cos(2 \Pi x)\) and \(sin(2 \Pi x)*cos(2 \Pi x)\) for x ranging from 0 to 5.
In [21]:
import numpy as np
import matplotlib.pyplot as plt
x = np.arange(0, 5, 0.005)
y = np.sin(2*np.pi*x)
y1 = np.cos(2*np.pi*x)
y2 = np.sin(2*np.pi*x)*np.cos(2*np.pi*x)
f, (ax1, ax2, ax3) = plt.subplots(3, sharex=True, sharey=True)
ax1.plot(x, y,'r')
ax2.plot(x, y1,'g')
ax3.plot(x,y2,'b')
ax1.legend(["sin"])
ax2.legend(["cos"])
ax3.legend(["sin * cos"])
plt.show()
Exercises¶
- Create the following functions:
- createRadomList : with parameters, N, min, max. Creates a list of N random integers ranging from min to max;
- getIdentical : with parameters two lists of integers L1 and L2 having the same size. It returns the list of indexes I where L1[I] == L2[I]
- check : gets lists L1, L2, identities (as computed by getIdentical) and a number N and prints if the first N and last N values in identities correspond to indexes of identical values in L1 and L2;
- implement getIdentical using numpy.ndarrays. Call it getIdenticalNpy (hint: subtract the two arrays and find zeros).
Test the software creating two lists of 100,000 random numbers from 0 to 10. You should get something similar to:
100000
100000
9032
Index: 4 L: 1 L1: 1
Index: 21 L: 10 L1: 10
Index: 32 L: 6 L1: 6
Index: 37 L: 5 L1: 5
Index: 47 L: 1 L1: 1
Index: 99974 L: 8 L1: 8
Index: 99982 L: 6 L1: 6
Index: 99983 L: 3 L1: 3
Index: 99984 L: 4 L1: 4
Index: 99987 L: 0 L1: 0
Identities are OK!
9032
Index: 4 L: 1 L1: 1
Index: 21 L: 10 L1: 10
Index: 32 L: 6 L1: 6
Index: 37 L: 5 L1: 5
Index: 47 L: 1 L1: 1
Index: 99974 L: 8 L1: 8
Index: 99982 L: 6 L1: 6
Index: 99983 L: 3 L1: 3
Index: 99984 L: 4 L1: 4
Index: 99987 L: 0 L1: 0
Identities are OK!
Show/Hide Solution
In [22]:
%reset -s -f
import random
import numpy as np
import time
def createRandomList(N=0, min=0, max=10):
"""creates a list of N random numbers in [min,max]"""
L = []
for i in range(0,N):
L.append(random.randint(min,max))
#L = [random.randint(min,max) for i in range(0,N)]
return L
def getIdentical(list1,list2):
"""return indexes of identical values
(lists assumed of same size)"""
return [i for i in range(0,len(list1)) if list1[i] == list2[i]]
def getIdenticalNpy(list1,list2):
"""return indexes of identical values
(lists assumed of same size)"""
nda1 = np.array(list1)
nda2 = np.array(list2)
diff = nda1 - nda2
ret = np.where(diff == 0)
return ret
def check(list1, list2, identical,N=10):
ret = True
for index in identical[0:5]:
if(list1[index] !=list2[index]):
ret = False
print("Index: {} L: {} L1: {}".format(index, list1[index], list2[index]))
#assert(L[index] == L1[index])
for index in identical[-N:]:
if(list1[index] !=list2[index]):
ret = False
print("Index: {} L: {} L1: {}".format(index, list1[index], list2[index]))
#assert(L[index] == L1[index])
return ret
def checkRND(list1, list2, identical,N=10):
ret = True
for index in createRandomList(N,0,len(identical)):
I = identical[index]
if(list1[I] !=list2[I]):
ret = False
print("Index: {} L: {} L1: {}".format(index, list1[index], list2[index]))
#assert(L[index] == L1[index])
return ret
L = createRandomList(100000)
L1 = createRandomList(100000)
print(len(L))
print(len(L1))
identical = getIdentical(L,L1)
print(len(identical))
if check(L,L1,identical,5):
print("\nIdentities are OK!")
else:
print("\nSomething went wrong!")
id1 = getIdenticalNpy(L,L1)
print(len(id1[0]))
if check(L,L1,identical,5):
print("\nIdentities are OK!")
else:
print("\nSomething went wrong!")
100000
100000
9143
Index: 16 L: 2 L1: 2
Index: 23 L: 6 L1: 6
Index: 51 L: 2 L1: 2
Index: 55 L: 1 L1: 1
Index: 84 L: 8 L1: 8
Index: 99935 L: 10 L1: 10
Index: 99957 L: 2 L1: 2
Index: 99966 L: 3 L1: 3
Index: 99992 L: 4 L1: 4
Index: 99995 L: 5 L1: 5
Identities are OK!
9143
Index: 16 L: 2 L1: 2
Index: 23 L: 6 L1: 6
Index: 51 L: 2 L1: 2
Index: 55 L: 1 L1: 1
Index: 84 L: 8 L1: 8
Index: 99935 L: 10 L1: 10
Index: 99957 L: 2 L1: 2
Index: 99966 L: 3 L1: 3
Index: 99992 L: 4 L1: 4
Index: 99995 L: 5 L1: 5
Identities are OK!
Write a function that converts a numpy ndarray of temperatures expressed in Degrees Celsius into Degrees Farenheit. The formula to convert a temperature C in Celsius into F in Farenheit is the following:
\[F = C * 9/5 + 32\]Write then a function that converts a numpy ndarray of temperatures in Farenheit into Celsius.
Finally:
- apply the Celsius to Farenheit conversion on an ndarray containing the following October’s minimum and maximum temperatures in Trento: tmin = [12, 11, 11, 8, 9, 10, 3, 8, 4, 5, 10, 9, 8, 9, 8, 7, 6, 4, 5, 6, 9, 9, 3, 3, 5] and tmax =[15, 22, 18, 20, 22, 22, 20, 21, 21, 21, 21, 23, 24, 24, 24, 25, 22, 22, 20, 20, 19, 15, 20, 23, 19];
- check that both functions work correctly by converting the values from Celsius to Farenheit and back to Celsius;
- plot the minimum and maximum temperatures in celsius on the same graph. Since the temperatures refer to the first 25 days of the month of October 2017, the x coordinate can be a range(1,26);
Show/Hide Solution
In [23]:
%reset -s -f
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
def Celsius2Farenheit(CA):
FA = CA * 9/5 + 32
return FA
def Farenheit2Celsius(FA):
CA = (FA - 32) * 5/9
return CA
tmin = [12, 11, 11, 8, 9, 10, 3, 8, 4, 5, 10, 9, 8, 9, 8, 7, 6, 4, 5, 6, 9, 9, 3, 3, 5]
tmax =[15, 22, 18, 20, 22, 22, 20, 21, 21, 21, 21, 23, 24, 24, 24, 25, 22, 22, 20, 20, 19, 15, 20, 23, 19]
minC = np.array(tmin)
maxC = np.array(tmax)
minF = Celsius2Farenheit(minC)
maxF = Celsius2Farenheit(maxC)
print("T min in Celsius:")
print(minC)
print("")
print("T min in Farenheit")
print(minF)
print("T max in Celsius:")
print(maxC)
print("")
print("T max in Farenheit")
print(maxF)
print(Farenheit2Celsius(minF))
#plotting the data:
x = np.arange(1,26,1)
plt.plot(x, minC, 'r', x,maxC, 'b')
plt.title("Temperature in Trento: October 2017")
plt.ylabel('°C')
plt.xlabel('Day')
plt.show()
#plt.close()
plt.plot(x, minF, 'r', x,maxF, 'b')
plt.title("Temperature in Trento: October 2017")
plt.ylabel('°F')
plt.xlabel('Day')
plt.show()
T min in Celsius:
[12 11 11 8 9 10 3 8 4 5 10 9 8 9 8 7 6 4 5 6 9 9 3 3
5]
T min in Farenheit
[53.6 51.8 51.8 46.4 48.2 50. 37.4 46.4 39.2 41. 50. 48.2 46.4 48.2
46.4 44.6 42.8 39.2 41. 42.8 48.2 48.2 37.4 37.4 41. ]
T max in Celsius:
[15 22 18 20 22 22 20 21 21 21 21 23 24 24 24 25 22 22 20 20 19 15 20 23
19]
T max in Farenheit
[59. 71.6 64.4 68. 71.6 71.6 68. 69.8 69.8 69.8 69.8 73.4 75.2 75.2
75.2 77. 71.6 71.6 68. 68. 66.2 59. 68. 73.4 66.2]
[12. 11. 11. 8. 9. 10. 3. 8. 4. 5. 10. 9. 8. 9. 8. 7. 6. 4.
5. 6. 9. 9. 3. 3. 5.]
Write some python code to solve the following system of linear equations:
\[\begin{split}\begin{cases} 3x - 4y + 2z -5w = -9\\ -4x + 4y + 10z + w = 32\\ -x + 2y + -7z -w = -7\\ x + y + z + w = 1\end{cases}\end{split}\]Define then a function to compute the Euclidean norm of a ndarray in input. Hint: \(||X||_2 = \sqrt{{x_1}^2 + {x_2}^2 + ... + {x_n}^2}\)
Compute then the Euclidean norm of the solution.
Implement a function that takes two N-dimensional vectors and computes the euclidean distance between them. Compute the distance between the solution of the system of equations above and the vector V = [7,120,11,33]. Check also that the distance between a vector and itself is zero. The Euclidean distance between two vectors P and Q can be computed as: \(\sqrt{\sum\limits_{i=1}^n (q_i -p_i)^2}\).
Show/Hide Solution
In [24]:
import numpy as np
from numpy import linalg
def eucNorm(arr): ### equivalent to np.linalg.norm
"""computes the Euclidean norm of the array arr"""
return np.sqrt(np.power(arr,2).sum())
def eucDist(v1, v2):
"""computes the Euclidean distance of two N-dimensional vectors"""
dV = np.power(v2-v1,2)
return np.sqrt(dV.sum())
#AX = b
A = [[3, 4, 2, -5],[4, 4, 10, 1],[-1, 2, -7, -1],[1, 1, 1, 1]]
Amat = np.array(A)
b = np.array([-9, 32, -7, 1])
print("A matrix")
print(Amat)
print("")
print("b vector")
print(b)
print("")
Xsol = linalg.solve(A,b)
print("Solution X:")
print(Xsol)
print(Amat.dot(Xsol))
print("")
#Euclidean norm:
print("Euclidean norm of {}:".format(Xsol))
print(eucNorm(Xsol))
#Euclidean distance
V = np.array([7,120,11,33])
print("\nThe Euclidean distance of")
print(Xsol)
print("\nand\n")
print(V)
print("\nis:")
print(eucDist(Xsol,V))
print("\neucDist(V,V) = {}".format(eucDist(V,V)))
A matrix
[[ 3 4 2 -5]
[ 4 4 10 1]
[-1 2 -7 -1]
[ 1 1 1 1]]
b vector
[-9 32 -7 1]
Solution X:
[-15.88888889 9.28888889 5.64444444 1.95555556]
[-9. 32. -7. 1.]
Euclidean norm of [-15.88888889 9.28888889 5.64444444 1.95555556]:
19.35004386379691
The Euclidean distance of
[-15.88888889 9.28888889 5.64444444 1.95555556]
and
[ 7 120 11 33]
is:
117.35966455387272
eucDist(V,V) = 0.0
Implement the following functions working on numpy ndarrays:
- The component form of a vector
comp(A,B)
starting in point A and ending in point B is \(\overline{AB}\). Hint: if \(A=(a_1,...,a_n)\) and \(B=(b_1,...,b_n)\), \(\overline{AB} = (b_1 - a_1, ..., b_n - a_n)\). - The direction cosines of a 3D vector
dirCos(A)
. If \(P=(x,y,z)\) the direction cosines are the cosines of the angles made by the vector with the three axes. They can be computed as:
\[\frac{x}{\sqrt(x^2+y^2+z^2}, \frac{y}{\sqrt(x^2+y^2+z^2}, \frac{z}{\sqrt(x^2+y^2+z^2}\]- The angle between two vectors
angle(A,B)
. Given two vectors \(A = (a_x,a_y,a_z)\) and \(B = (b_x,b_y,b_z)\), \(cos(\alpha) = \frac{A \cdot B}{||A||_2||B||_2}\) where the Euclidean norms are computed as in the previous example.
- The volume of the pyramid described by three 3D vectors \(\overline{a},\overline{b},\overline{c}\):
the volume
pyrVol(a,b,c)
can be computed as: \(\frac{1}{6} |det(M)|\), where \(M = \begin{bmatrix}\overline{a} \\ \overline{b} \\ \overline{c}\end{bmatrix}\), det is the determinant of the matrix anx || represents the absolute value.Define some numpy arrays starting from the following lists:
- A = [10, 3, 4, 7, 12], B = [6, 7, 8, 1, 1] and test
comp(A,B)
- Va = [90,12,72] and test
dirCos(Va)
- VA = [70,9,12] and VB = [1,-10,30] and test
angle(VA,VB)
- a = [10, 20, 30], b = [ 4, 7, 21], c = [-5, 4, 12] and test
pyrVol(a,b,c)
- The component form of a vector
Check your solutions here.
Show/Hide Solution
In [25]:
import numpy as np
from numpy import linalg
def comp(p1,p2):
"""component of vector starting in p1 and ending in p2"""
return p2 - p1
def dirCos(v1):
"""direction cosines of v1"""
norm = np.sqrt(np.power(v1,2).sum())
return v1[0]/norm, v1[1]/norm, v1[2]/norm
def eucNorm(arr):
"""computes the Euclidean norm of the array arr"""
return np.sqrt(np.power(arr,2).sum())
def angle(v1,v2):
"""computes the angle between two vectors v1 and v2"""
dp = np.dot(v1,v2)
cosVal = dp / (eucNorm(v1) * eucNorm(v2))
return cosVal
def pyrVol(v1,v2,v3):
"""returns the volume of the pyramid described by v1,v2 and v3"""
matM = np.concatenate((v1,v2,v3))
matM = matM.reshape((v1.shape[0], 3))
return abs(np.linalg.det(matM))/6
#Testing the functions
#1. comp(A,B)
A = np.array([10, 3, 4, 7, 12])
B = np.array([6, 7, 8, 1, 1])
print("A")
print(A)
print("")
print("B")
print(B)
print("")
print("comp(A,B)")
print(comp(A,B))
#2. dirCos(Va)
print("")
Va = np.array([90,12,72])
print("Va")
print(Va)
print("")
print("dirCos(Va)")
print(dirCos(Va))
#3. angle(VA,VB)
VA = np.array([70,9,12])
VB = np.array([1,-10,30])
print("")
print("VA")
print(VA)
print("")
print("VB")
print(VB)
print("")
print("angle(VA,VB)")
print(angle(VA,VB))
print("")
#4. pyrVol(a,b,c)
a = np.array([10, 20, 30])
b = np.array([ 4, 7, 21])
c = np.array([-5, 4, 12])
print("a")
print(a)
print("")
print("b")
print(b)
print("")
print("c")
print(c)
print("")
print("pyrVol(a,b,c)")
print(pyrVol(a,b,c))
A
[10 3 4 7 12]
B
[6 7 8 1 1]
comp(A,B)
[ -4 4 4 -6 -11]
Va
[90 12 72]
dirCos(Va)
(0.7766705596177466, 0.10355607461569954, 0.6213364476941973)
VA
[70 9 12]
VB
[ 1 -10 30]
angle(VA,VB)
0.15011183829724634
a
[10 20 30]
b
[ 4 7 21]
c
[-5 4 12]
pyrVol(a,b,c)
254.99999999999977
- Implement the power iteration algorithm which is a method to obtain an approximation of the biggest (in absolute value) eigenvalue of a matrix A.
The algorithm starts from a random vector \(b_0\) and keeps computing \(b_{k+1} = \frac{A\cdot b_{k}}{min(A\cdot b_{k})}\). When the algorithm ends, the eigenvalue is actually the value: \(min(A\cdot b_{k})\)
The powerIteration function will have the maximum number of
iterations and dThr as parameters
powerIteration(A,maxIt, dThr)
. The computation will stop either when
the maximum number of iterations maxIt
is reached or when the
Euclidean distance between b_{k-1}
and b_k
is lower than
dThr
.
Check the result against numpy eigenvalues obtained with
np.linalg.eig()
using various matrices.
Show/Hide Solution
In [26]:
import numpy as np
from numpy import linalg
def eucDist(v1, v2):
"""computes the Euclidean distance of two N-dimensional vectors"""
dV = np.power(v2-v1,2)
return np.sqrt(dV.sum())
def powerIteration(A,maxIt,dThr):
b = np.random.randint(1,10,(A.shape[0]))
it = 0
dist = 1000000
while( it < maxIt and dist > dThr ):
tmp = np.dot(A,b)
dist = eucDist(b,tmp)
b = tmp/np.min(tmp)
it += 1
return np.min(tmp)
M = np.array([[1, 3, 7, 9], [ 4, 11, 7, 27], [22, 11, 57, 1], [4, 7, 27, 27]])
r = powerIteration(M,200, 0.0001)
print("Eigenvalue from power method: ", r)
print("FROM linalg:" , max(np.linalg.eig(M)[0]))
N = np.array([[0, 11, -5],[-2, 17, -7],[-4, 26, -10]])
print(N)
r = powerIteration(N,200, 0.0001)
print("Eigenvalue from power method: ", r)
print("FROM linalg:" , max(np.linalg.eig(N)[0]))
N1 = np.array([[2, -12],[1, -5]])
r = powerIteration(N1,200, 0.0001)
print("Eigenvalue from power method: ", r)
print("FROM linalg:" , max(np.linalg.eig(N1)[0]))
N2 = np.array([[1, 2, 0],[-2, 1, 2], [1,3,1]])
r = powerIteration(N2,200, 0.0001)
print("Eigenvalue from power method: ", r)
print("FROM linalg:" , max(np.linalg.eig(N2)[0]))
R = np.random.randint(0,20, size=(5,5))
r = powerIteration(R,200, 0.0001)
print("Eigenvalue from power method: ", r)
print("FROM linalg:" , max(np.linalg.eig(R)[0]))
Eigenvalue from power method: 68.24845123079758
FROM linalg: 68.24845123079754
[[ 0 11 -5]
[ -2 17 -7]
[ -4 26 -10]]
Eigenvalue from power method: 3.9999999999999982
FROM linalg: 4.000000000000021
Eigenvalue from power method: -2.0
FROM linalg: -0.9999999999999982
Eigenvalue from power method: 3.0
FROM linalg: (3.0000000000000004+0j)
Eigenvalue from power method: 50.69133949667758
FROM linalg: (50.69133949667753+0j)
- Implement a function
movingAvg(A,n)
where A is a numpy one dimensional array and n is the window size on which computing the average, that outputs the moving average over a numpy one dimensional array. Es. A = [1,2,3,4,5]movingAvg(A,2) = [1.5, 2.5, 3.5, 4.5]
, whilemovingAvg(A,3) = [2,3,4]
without using for loops. Hint: use cumsum and clever slicing. Assess the smoothing effect of the moving average by creating a numpy array containing a sinusoidal wave with some additional noise (i.e. use np.sin and np.random.rand) and testing several values of the window size n.
Show/Hide Solution
In [27]:
import numpy as np
import matplotlib.pyplot as plt
def movingAvg(v, n):
"""computes the moving average of n values in the array v"""
cs = np.cumsum(v)
print(cs)
print("\t",cs[n:])
print("\t", cs[:-n])
cs[n:] = cs[n:] - cs[:-n]
print("CS:", cs)
return cs[n - 1:] / n
A = np.arange(1,8,1)
print("A:")
print(A)
print(movingAvg(A,2))
print(movingAvg(A,3))
X = np.arange(0, 5, 0.005)
#B = np.random(100)
B = np.sin(2*np.pi*X)
B += np.random.rand(1000)
C = movingAvg(B,5)
D = movingAvg(B,10)
E = movingAvg(B,20)
#X = np.arange(0,B.shape[0])
plt.plot(X,B)
plt.title("No moving average")
plt.show()
plt.close()
plt.plot(np.arange(0,C.shape[0]),C)
plt.title("Window size: 5")
plt.show()
plt.close()
plt.title("Window size: 10")
plt.plot(np.arange(0,D.shape[0]),D)
plt.show()
plt.close()
plt.title("Window size: 20")
plt.plot(np.arange(0,E.shape[0]),E)
plt.show()
A:
[1 2 3 4 5 6 7]
[ 1 3 6 10 15 21 28]
[ 6 10 15 21 28]
[ 1 3 6 10 15]
CS: [ 1 3 5 7 9 11 13]
[1.5 2.5 3.5 4.5 5.5 6.5]
[ 1 3 6 10 15 21 28]
[10 15 21 28]
[ 1 3 6 10]
CS: [ 1 3 6 9 12 15 18]
[2. 3. 4. 5. 6.]
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