Introduction to Numpy, Scipy, SciKit¶
In this tutorial we will look into the Numpy library: http://www.numpy.org/
Numpy is a very important library for numerical computations and matrix manipulation. It has a lot of the functionality of Matlab, and some of the functionality of Pandas
We will also use the Scipy library for scientific computation: http://docs.scipy.org/doc/numpy/reference/index.html
import numpy as np
import scipy as sp
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
%matplotlib inline
import scipy.sparse as sp_sparse
import scipy.spatial.distance as sp_dist
import sklearn as sk
import sklearn.datasets as sk_data
import sklearn.metrics as metrics
from sklearn import preprocessing
import scipy.sparse.linalg as linalg
Why Numpy?¶
import time
def trad_version():
t1 = time.time()
X = range(10000000)
Y = range(10000000)
Z = [x+y for x,y in zip(X,Y)]
return time.time() - t1
def numpy_version():
t1 = time.time()
X = np.arange(10000000)
Y = np.arange(10000000)
Z = X + Y
return time.time() - t1
traditional_time = trad_version()
numpy_time = numpy_version()
print ("Traditional time = "+ str(traditional_time))
print ("Numpy time = "+ str(numpy_time))
Arrays¶
Creating Arrays¶
In Numpy data is organized into arrays. There are many different ways to create a numpy array.
For the following we will use the random library of Numpy: http://docs.scipy.org/doc/numpy-1.10.0/reference/routines.random.html
Creating arrays from lists
#1-dimensional arrays
x = np.array([2,5,18,14,4])
print ("\n Deterministic 1-dimensional array \n")
print (x)
#2-dimensional arrays
x = np.array([[2,5,18,14,4], [12,15,1,2,8]])
print ("\n Deterministic 2-dimensional array \n")
print (x)
We can also create Numpy arrays from Pandas DataFrames
d = {'A':[1., 2., 3., 4.],
'B':[4., 3., 2., 1.]}
df = pd.DataFrame(d)
x = np.array(df)
print(x)
Creating random arrays
#1-dimensional arrays
x = np.random.rand(5)
print ("\n Random 1-dimensional array \n")
print (x)
#2-dimensional arrays
x = np.random.rand(5,5)
print ("\n Random 5x5 2-dimensional array \n")
print (x)
x = np.random.randint(10,size=(2,3))
print("\n Random 2x3 array with integers")
print(x)
Transpose and get array dimensions
print("\n Matrix Dimensions \n")
print(x.shape)
print ("\n Transpose of the matrix \n")
print (x.T)
print (x.T.shape)
Special Arrays
x = np.zeros((4,4))
print ("\n 4x4 array with zeros \n")
print(x)
x = np.ones((4,4))
print ("\n 4x4 array with ones \n")
print (x)
x = np.eye(4)
print ("\n Identity matrix of size 4\n")
print(x)
x = np.diag([1,2,3])
print ("\n Diagonal matrix\n")
print(x)
Operations on arrays.¶
These are very similar to what we did with Pandas
x = np.random.randint(10, size = (2,4))
print (x)
print('\n mean value of all elements')
print (np.mean(x))
print('\n vector of mean values for columns')
print (np.mean(x,0)) #0 signifies the dimension meaning columns
print('\n vector of mean values for rows')
print (np.mean(x,1)) #1 signifies the dimension meaning rows
print('\n standard deviation of all elements')
print (np.std(x))
print('\n vector of std values for rows')
print (np.std(x,1)) #1 signifies the dimension meaning rows
print('\n median value of all elements')
print (np.median(x))
print('\n vector of median values for rows')
print (np.median(x,1))
print('\n sum of all elements')
print (np.sum(x))
print('\n vector of column sums')
print (np.sum(x,0))
print('\n product of all elements')
print (np.prod(x))
print('\n vector of row products')
print (np.prod(x,1))
Manipulating arrays¶
Accessing and Slicing
x = np.random.rand(4,3)
print(x)
print("\n element\n")
print(x[1,2])
print("\n row zero \n")
print(x[0,:])
print("\n column 2 \n")
print(x[:,2])
print("\n submatrix \n")
print(x[1:3,0:2])
print("\n entries > 0.5 \n")
print(x[x>0.5])
Changing entries
x = np.random.rand(4,3)
print(x)
x[1,2] = -5 #change an entry
x[0:2,:] += 1 #change a set of rows
x[2:4,1:3] = 0.5 #change a block
print(x)
print('\n Set entries > 0.5 to zero')
x[x>0.5] = 0
print(x)
print('\n Diagonal \n')
x = np.random.rand(4,4)
print(x)
print('\n Read Diagonal \n')
print(x.diagonal())
print('\n Fill Diagonal with 1s \n')
np.fill_diagonal(x,1)
print(x)
print('\n Fill Diagonal with vector \n')
x[np.diag_indices_from(x)] = [1,2,3,4]
print(x)
Quiz¶
We want to create a dataset of 10 users and 5 items, where each user i has selects an item j with probability 0.3.
How can we do this with matrix operations?
D = np.random.rand(10,5)
print(D)
D[D <= 0.3] = 1
D[D != 1] = 0
D
Operations with Arrays¶
Multiplication and addition with scalar
x = np.random.rand(4,3)
print(x)
#multiplication and addition with scalar value
print("\n Matrix 2x+1 \n")
print(2*x+1)
Vector-vector dot product
There are three ways to get the dot product of two vectors:
y = np.array([2,-1,3])
z = np.array([-1,2,2])
print('\n y:',y)
print(' z:',z)
print('\n vector-vector dot product')
print(y.dot(z))
print(np.dot(y,z))
print(y@z)
External product
The external product between two vectors x,y of size (n,1) and (m,1) results in a matrix M of size (n,m) with entries M(i,j) = x(i)*y(j)
print('\n y:',y)
print(' z:',z)
print('\n vector-vector external product')
print(np.outer(y,z))
Element-wise operations
print('\n y:',y)
print(' z:',z)
print('\n element-wise addition')
print(y+z)
print('\n element-wise product')
print(y*z)
print('\n element-wise division')
print(y/z)
Matrix-Vector multiplication
Again we can do the multiplication either using the dot method or the '@' operator
X = np.random.randint(10, size = (4,3))
print('Matrix X:\n',X)
y = np.array([1,0,0])
print("\n Matrix-vector right multiplication with",y,"\n")
print(X.dot(y))
print(np.dot(X,y))
print(X@y)
y = np.array([1,0,1,0])
print("\n Matrix-vector left multiplication with",y,"\n")
print(y.dot(X))
print(np.dot(y,X))
print(y@X)
Matrix-Matrix multiplication
Same for the matrix-matrix operation
Y = np.random.randint(10, size=(3,2))
print("\n Matrix-matrix multiplication\n")
print('Matrix X:\n',X)
print('Matrix Y:\n',Y)
print('Product:\n',X.dot(Y))
print('Product:\n',X@Y)
Matrix-Matrix element-wise operations
Z = np.random.randint(10, size=(3,2))+1
print('Matrix Y:\n',Y)
print('Matrix Z:\n',Z)
print("\n Matrix-matrix element-wise addition\n")
print(Y+Z)
print("\n Matrix-matrix element-wise multiplication\n")
print(Y*Z)
print("\n Matrix-matrix element-wise division\n")
print(Y/Z)
Creating Sparse Arrays¶
For sparse arrays we need to use the sp_sparse library from SciPy: http://docs.scipy.org/doc/scipy/reference/sparse.html
There are three types of sparse matrices:
- csr: compressed row format
- csc: compressed column format
- lil: list of lists format
- coo: coordinates format
</ul>
These different types have to do with the matrix implementation and the data structures used.
The csr and csc formats are fast for arithmetic operations, but slow for slicing and incremental changes.
The lil format is fast for slicing and incremental construction, but slow for arithmetic operations.
The coo format does not support arithmetic operations and slicing, but it is very fast for constructing a matrix incrementally. You should then transform it to some other format for operations.
Creation of matrix from triplets
Triplets are of the form (row, column, value)
import scipy.sparse as sp_sparse
d = np.array([[0, 0, 12],
[0, 1, 1],
[0, 5, 34],
[1, 3, 12],
[1, 2, 6],
[2, 0, 23],
[3, 4, 14],
])
row = d[:,0]
col = d[:,1]
data = d[:,2]
# a matrix M with M[row[i],col[i]] = data[i] will be created
M = sp_sparse.csr_matrix((data,(row,col)), shape=(5,6))
print(M)
print(M.toarray()) #transforms back to full matrix
Making a full matrix sparse
x = np.random.randint(2,size = (3,4))
print(x)
print('\n make x sparce')
A = sp_sparse.csr_matrix(x)
print(A)
Creating a sparse matrix incrementally
# Use lil (list of lists) representation
A = sp_sparse.lil_matrix((10, 10))
A[0, :5] = np.random.randint(10,size = 5)
A[1, 5:10] = A[0, :5]
A.setdiag(np.random.randint(10,size = 10))
A[9,9] = 99
A[9,0]=1
print(A)
print(A.toarray())
print(A.diagonal())
A = A.tocsr() # makes it a compressed column format. better for dot product.
B = A.dot(np.ones(10))
print(B)
All operations work like before
print(A.dot(A.T))
Singluar Value Decomposition¶
For the singular value decomposition we will use the libraries from Numpy and SciPy and SciKit Learn
We use sklearn to create a low-rank matrix (https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_low_rank_matrix.html). We will create a matrix with effective rank 2.
import sklearn.datasets as sk_data
data = sk_data.make_low_rank_matrix(n_samples=100, n_features=50, effective_rank=2, tail_strength=0.0, random_state=None)
#sns.heatmap(data, xticklabels=False, yticklabels=False, linewidths=0)
We will use the numpy.linalg.svd function to compute the Singular Value Decomposition of the matrix we created (http://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.svd.html).
U, s, V = np.linalg.svd(data,full_matrices = False)
print (U.shape, s.shape, V.shape)
print(s)
plt.plot(s[0:10])
plt.ylabel('singular value')
plt.xlabel('number of singular values')
We can also use the scipy.sparse.linalg libary to compute the SVD for sparse matrices (http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.sparse.linalg.svds.html)
We need to specify the number of components, otherwise it is by default k = 6. The singular values are in increasing order.
import scipy.sparse.linalg as sp_linalg
data2 = sp_sparse.csc_matrix(data)
print(data2.shape)
U,s,V = sp_linalg.svds(data2) #by default returns k=6 singular values
print (U.shape, s.shape, V.shape)
print(s)
plt.plot(s[::-1])
plt.ylabel('eigenvalue value')
plt.xlabel('number of eigenvalues')
We can also compute SVD using the library of SciKit Learn (https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.TruncatedSVD.html)
from sklearn.decomposition import TruncatedSVD
K = 6
svd = TruncatedSVD(n_components=K)
svd.fit(data2)
print(svd.components_.shape) # the V vectors
print(svd.transform(data2).shape) # the U vectors
print(svd.singular_values_)
Onbtaining a low rank approximation of the data¶
To obtain a rank-k approximation of the matrix we multiplty the k first columns of U, with the diagonal matrix with the k first (largest) singular values, with the matrix with the first k rows of V transpose
K = 6
U_k,s_k,V_k = sp_linalg.svds(data2, K, which = 'LM')
print (U_k.shape, s_k.shape, V_k.shape)
print(s_k)
plt.plot(s_k[::-1])
plt.ylabel('eigenvalue value')
plt.xlabel('number of eigenvalues')
S_k = np.diag(s_k)
reconstruction_error = []
for k in range(K,0,-1):
data_k = U_k[:,k:].dot(S_k[k:,k:]).dot(V_k[k:,:])
error = np.linalg.norm(data_k-data2,ord='fro')
reconstruction_error.append(error)
print(error)
data_k = U_k.dot(S_k).dot(V_k)
print(np.linalg.norm(data_k-data2,ord='fro'))
plt.plot(reconstruction_error)
plt.ylabel('rank-k reconstruction error')
plt.xlabel('rank')
An example¶
We will create a block diagonal matrix, with blocks of different "intensity" of values
import numpy as np
M1 = np.random.randint(1,50,(50,20))
M2 = np.random.randint(1,10,(50,20))
M3 = np.random.randint(1,10,(50,20))
M4 = np.random.randint(1,50,(50,20))
T = np.concatenate((M1,M2),axis=1)
B = np.concatenate((M3,M4),axis=1)
M = np.concatenate([T,B],axis = 0)
plt.imshow(M, cmap='hot')
plt.show()
We observe that there is a correlation between the column and row sums and the left and right singular vectors
Note: The values of the vectors are negative. We would get the same result if we make them positive.
import scipy.stats as stats
import matplotlib.pyplot as plt
(U,S,V) = np.linalg.svd(M,full_matrices = False)
#print(S)
c = M.sum(0)
r = M.sum(1)
print(stats.pearsonr(r,U[:,0]))
print(stats.pearsonr(c,V[0]))
plt.scatter(r,U[:,0])
plt.figure()
plt.scatter(c,V[0])
Using the first two signular vectors we can clearly differentiate the two blocks of rows
plt.scatter(U[:,0],U[:,1])
plt.scatter(x = U[:50,0],y = U[:50,1],color='r')
plt.scatter(x = U[50:,0],y = U[50:,1], color = 'b')
PCA using SciKit Learn¶
We will now use the PCA package from the SciKit Learn (sklearn) library. PCA is the same as SVD but now the matrix is centered: the mean is removed from the columns of the matrix.
You can read more here: https://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html
from sklearn.decomposition import PCA
pca = PCA(n_components=2)
pca.fit(M)
pca.components_
plt.scatter(pca.components_[0],pca.components_[1])
Using the operation transform we can transform the data directly to the lower-dimensional space
MPCA = pca.transform(M)
print(MPCA.shape)
plt.scatter(MPCA[:,0],MPCA[:,1])
from sklearn import datasets
iris = datasets.load_iris()
X = iris.data
y = iris.target
pca = PCA(n_components=3)
pca.fit(X)
X = pca.transform(X)
pca.explained_variance_
plt.scatter(X[:,0],X[:,1])
plt.scatter(X[y==0,0],X[y==0,1], color='b')
plt.scatter(X[y==1,0],X[y==1,1], color='r')
plt.scatter(X[y==2,0],X[y==2,1], color='g')
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.scatter(X[y==0,0],X[y==0,1], X[y==0,2], color='b')
ax.scatter(X[y==1,0],X[y==1,1], X[y==1,2], color='r')
ax.scatter(X[y==2,0],X[y==2,1], X[y==2,2], color='g')
