220 KiB
220 KiB
In [1]:
import numpy as npIn [2]:
n = 10
x = np.random.normal(size=n)
print(x)[-0.65354422 0.31335664 0.77994763 -0.26292935 -0.01705695 -2.16552497 -1.06272254 -0.20520318 -1.70367841 -0.29687889]
In [3]:
import numpy as np
x = np.array([1, 2, 3])
print(x)[1 2 3]
In [4]:
import numpy as np
x = np.log(np.array([4, 7, 8]))
print(x)[1.38629436 1.94591015 2.07944154]
In [5]:
import numpy as np
from math import log
x = np.array([4, 7, 8])
for i in range(0, len(x)):
x[i] = log(x[i])
print(x)[1 1 2]
In [6]:
import numpy as np
x = np.log(np.array([4, 7, 8], dtype = np.float64))
print(x)[1.38629436 1.94591015 2.07944154]
In [7]:
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x)[1.38629436 1.94591015 2.07944154]
In [8]:
import numpy as np
x = np.log(np.array([4.0, 7.0, 8.0]))
print(x.itemsize)8
In [9]:
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
print(A)[[1.38629436 1.94591015 2.07944154] [1.09861229 2.30258509 2.39789527] [1.38629436 1.60943791 1.94591015]]
In [10]:
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
# print the first column, row-major order and elements start with 0
print(A[:,0])[1.38629436 1.09861229 1.38629436]
In [11]:
import numpy as np
A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
# print the first column, row-major order and elements start with 0
print(A[1,:])[1.09861229 2.30258509 2.39789527]
In [12]:
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to zero
A = np.zeros( (n, n) )
print(A)[[0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.] [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]
In [13]:
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to one
A = np.ones( (n, n) )
print(A)[[1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.] [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]]
In [14]:
import numpy as np
n = 10
# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
A = np.random.rand(n, n)
print(A)[[0.6564482 0.70711514 0.85049285 0.39741875 0.3000498 0.81984817 0.08300526 0.16674251 0.85142265 0.15424679] [0.18013978 0.97321835 0.83785062 0.4313428 0.7889793 0.22625657 0.54214936 0.00995699 0.81329201 0.11991488] [0.10520393 0.5823186 0.92301743 0.56913182 0.64344541 0.3466389 0.20656353 0.16072855 0.08068207 0.13625288] [0.83693088 0.18251883 0.47900239 0.66340579 0.98244726 0.68291639 0.53361935 0.2938162 0.74412221 0.32232922] [0.31743443 0.47830372 0.63116786 0.86885547 0.20263701 0.96029461 0.50744967 0.68012971 0.16181877 0.4742433 ] [0.22426265 0.20640822 0.83408369 0.20353821 0.84269323 0.76183628 0.957171 0.8509859 0.57000682 0.39184027] [0.03692749 0.36845893 0.17101958 0.31888254 0.67849271 0.60837213 0.69164986 0.18146425 0.57604071 0.88445111] [0.10371767 0.87228174 0.52154966 0.04596002 0.52212442 0.73390423 0.05999828 0.61076053 0.12394921 0.20365164] [0.90210098 0.59206051 0.09303501 0.51019911 0.00733216 0.04041644 0.3615115 0.02784163 0.46893501 0.68008931] [0.80147746 0.00959357 0.80835002 0.63542565 0.20005403 0.21894241 0.96907179 0.84745525 0.43713984 0.60568686]]
In [15]:
# Importing various packages
import numpy as np
n = 100
x = np.random.normal(size=n)
print(np.mean(x))
y = 4+3*x+np.random.normal(size=n)
print(np.mean(y))
z = x**3+np.random.normal(size=n)
print(np.mean(z))
W = np.vstack((x, y, z))
Sigma = np.cov(W)
print(Sigma)
Eigvals, Eigvecs = np.linalg.eig(Sigma)
print(Eigvals)0.007557567939261264 3.9057119872872414 0.43208935911369845 [[0.8631855 2.56690562 2.08698097] [2.56690562 8.47187403 6.28078417] [2.08698097 6.28078417 9.31159615]] [15.90162133 0.07253999 2.67249437]
In [16]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import sparse
eye = np.eye(4)
print(eye)
sparse_mtx = sparse.csr_matrix(eye)
print(sparse_mtx)
x = np.linspace(-10,10,100)
y = np.sin(x)
plt.plot(x,y,marker='x')
plt.show()[[1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 1. 0.] [0. 0. 0. 1.]] (0, 0) 1.0 (1, 1) 1.0 (2, 2) 1.0 (3, 3) 1.0
Meet the Pandas
Figure 1:
Another useful Python package is pandas, which is an open source library providing high-performance, easy-to-use data structures and data analysis tools for Python. pandas stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data. pandas has two major classes, the DataFrame class with two-dimensional data objects and tabular data organized in columns and the class Series with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. pandas allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.
The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of pandas, in particular in connection with classification of data.
In [17]:
import pandas as pd
from IPython.display import display
data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
'Date of Birth T.A.': [2968, 2890, 2931, 2980]
}
data_pandas = pd.DataFrame(data)
display(data_pandas)| First Name | Last Name | Place of birth | Date of Birth T.A. | |
|---|---|---|---|---|
| 0 | Frodo | Baggins | Shire | 2968 |
| 1 | Bilbo | Baggins | Shire | 2890 |
| 2 | Aragorn II | Elessar | Eriador | 2931 |
| 3 | Samwise | Gamgee | Shire | 2980 |
In [18]:
data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
display(data_pandas)| First Name | Last Name | Place of birth | Date of Birth T.A. | |
|---|---|---|---|---|
| Frodo | Frodo | Baggins | Shire | 2968 |
| Bilbo | Bilbo | Baggins | Shire | 2890 |
| Aragorn | Aragorn II | Elessar | Eriador | 2931 |
| Sam | Samwise | Gamgee | Shire | 2980 |
In [19]:
display(data_pandas.loc['Aragorn'])First Name Aragorn II Last Name Elessar Place of birth Eriador Date of Birth T.A. 2931 Name: Aragorn, dtype: object
In [20]:
new_hobbit = {'First Name': ["Peregrin"],
'Last Name': ["Took"],
'Place of birth': ["Shire"],
'Date of Birth T.A.': [2990]
}
data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
display(data_pandas)[0;31m---------------------------------------------------------------------------[0m
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[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/pandas/core/generic.py[0m in [0;36m?[0;34m(self, name)[0m
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[0;31mAttributeError[0m: 'DataFrame' object has no attribute 'append'In [21]:
import numpy as np
import pandas as pd
from IPython.display import display
np.random.seed(100)
# setting up a 10 x 5 matrix
rows = 10
cols = 5
a = np.random.randn(rows,cols)
df = pd.DataFrame(a)
display(df)
print(df.mean())
print(df.std())
display(df**2)In [22]:
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
df.index = np.arange(10)
display(df)
print(df['Second'].mean() )
print(df.info())
print(df.describe())
from pylab import plt, mpl
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'
df.cumsum().plot(lw=2.0, figsize=(10,6))
plt.show()
df.plot.bar(figsize=(10,6), rot=15)
plt.show()In [23]:
b = np.arange(16).reshape((4,4))
print(b)
df1 = pd.DataFrame(b)
print(df1)In [24]:
# Importing various packages
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
x = np.random.rand(100,1)
y = 2*x+np.random.randn(100,1)
linreg = LinearRegression()
linreg.fit(x,y)
xnew = np.array([[0],[1]])
ypredict = linreg.predict(xnew)
plt.plot(xnew, ypredict, "r-")
plt.plot(x, y ,'ro')
plt.axis([0,1.0,0, 5.0])
plt.xlabel(r'$x$')
plt.ylabel(r'$y$')
plt.title(r'Simple Linear Regression')
plt.show()Warning:
Output truncated. This notebook contains too many cells to display efficiently.