Learning Objectives: Learn how to create, transform and visualize multidimensional data of a single type using Numpy.
NumPy is the foundation for scientific computing and data science in Python. Its more data object is a multidimensional array with the following characteristics:
While this notebook doesn't focus on plotting, Matplotlib will be used to make a few basic plots.
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%matplotlib inline
from matplotlib import pyplot as plt
import seaborn as sns
The vizarray package will be used to visualize NumPy arrays:
import antipackage from github.ellisonbg.misc import vizarray as va
This is the canonical way you should import Numpy:
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import numpy as np
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data = [0,2,4,6]
a = np.array(data)
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type(a)
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a
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The vz.vizarray function can be used to visualize a 1d or 2d NumPy array using a colormap:
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va.vizarray(a)
The shape of the array:
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a.shape
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The number of array dimensions:
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a.ndim
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The number of array elements:
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a.size
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The number of bytes the array takes up:
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a.nbytes
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The dtype attribute describes the "data type" of the elements:
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a.dtype
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Arrays can be created with nested lists or tuples:
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data = [[0.0,2.0,4.0,6.0],[1.0,3.0,5.0,7.0]]
b = np.array(data)
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b
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va.vizarray(b)
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b.shape, b.ndim, b.size, b.nbytes
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The arange function is similar to Python's builtin range function, but creates an array:
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c = np.arange(0.0, 10.0, 1.0) # Step size of 1.0
c
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The linspace function is similar, but allows you to specify the number of points:
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e = np.linspace(0.0, 5.0, 11) # 11 points
e
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There are also empty, zeros and ones functions:
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np.empty((4,4))
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np.zeros((3,3))
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np.ones((3,3))
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See also:
empty_like, ones_like, zeros_likeeye, identity, diagArrays have a dtype attribute that encapsulates the "data type" of each element. It can be set:
dtype argument to an array creation functionHere is an integer valued array:
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a = np.array([0,1,2,3])
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a, a.dtype
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All array creation functions accept an optional dtype argument:
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b = np.zeros((2,2), dtype=np.complex64)
b
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c = np.arange(0, 10, 2, dtype=np.float)
c
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You can use the astype method to create a copy of the array with a given dtype:
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d = c.astype(dtype=np.int)
d
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IPython's tab completion is useful for exploring the various available dtypes:
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np.float*?
The NumPy documentation on dtypes describes the many other ways of specifying dtypes.
Basic mathematical operations are elementwise for:
Fill an array with a value:
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a = np.empty((3,3))
a.fill(0.1)
a
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b = np.ones((3,3))
b
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Addition is elementwise:
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a+b
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Division is elementwise:
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b/a
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As are powers:
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a**2
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Scalar multiplication is also elementwise:
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np.pi*b
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Indexing and slicing provide an efficient way of getting the values in an array and modifying them.
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a = np.random.rand(10,10)
The enable function is part of vizarray and enables a nice display of arrays:
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va.enable()
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a
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List Python lists and tuples, NumPy arrays have zero-based indexing and use the [] syntax for getting and setting values:
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a[0,0]
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An index of -1 refers to the last element along that axis:
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a[-1,-1] == a[9,9]
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Extract the 0th column using the : syntax, which denotes all elements along that axis.
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a[:,0]
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The last row:
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a[-1,:]
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You can also slice ranges:
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a[0:2,0:2]
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Assignment also works with slices:
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a[0:5,0:5] = 1.0
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a
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Note how even though we assigned the value to the slice, the original array was changed. This clarifies that slices are views of the same data, not a copy.
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va.disable()
Arrays can be indexed using other arrays that have boolean values.
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ages = np.array([23,56,67,89,23,56,27,12,8,72])
genders = np.array(['m','m','f','f','m','f','m','m','m','f'])
Boolean expressions involving arrays create new arrays with a bool dtype and the elementwise result of the expression:
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ages > 30
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genders == 'm'
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Boolean expressions provide an extremely fast and flexible way of querying arrays:
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(ages > 10) & (ages < 50)
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You can use a boolean array to index into the original or another array. This selects the ages of all females in the genders array:
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mask = (genders == 'f')
ages[mask]
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ages[ages>30]
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va.enable()
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a = np.random.rand(3,4)
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a
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The T atrribute contains the transpose of the original array:
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a.T
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The reshape method can be used to change the shape and even the number of dimensions:
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a.reshape(2,6)
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a.reshape(6,2)
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The ravel method strings the array out in one dimension:
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a.ravel()
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va.disable()
Universal function, or "ufuncs," are functions that take and return arrays or scalars. They have the following characteristics:
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va.set_block_size(5)
va.enable()
Here is a linear sequence of values"
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t = np.linspace(0.0, 4*np.pi, 100)
t
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Take the $sin$ of each element of the array:
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np.sin(t)
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As the next two examples show, multiple ufuncs can be used to create complex mathematical expressions that can be computed efficiently:
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np.exp(np.sqrt(t))
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va.disable()
va.set_block_size(30)
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plt.plot(t, np.exp(-0.1*t)*np.sin(t))
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In general, you should always try to use ufuncs rather than do computations using for loops. These types of array based computations are referred to as vectorized.
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ages = np.array([23,56,67,89,23,56,27,12,8,72])
genders = np.array(['m','m','f','f','m','f','m','m','m','f'])
Numpy has a basic set of methods and function for computing basic quantities about data.
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ages.min(), ages.max()
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Compute the mean:
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ages.mean()
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Compute the variance and standard deviation:
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ages.var(), ages.std()
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The bincount function counts how many times each value occurs in the array:
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np.bincount(ages)
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The cumsum and cumprod methods compute cumulative sums and products:
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ages.cumsum()
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ages.cumprod()
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Most of the functions and methods above take an axis argument that will apply the action along a particular axis:
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a = np.random.randint(0,10,(3,4))
a
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With axis=0 the action takes place along rows:
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a.sum(axis=0)
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With axis=1 the action takes place along columns:
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a.sum(axis=1)
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The unique function is extremely useful in working with categorical data:
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np.unique(genders)
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np.unique(genders, return_counts=True)
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The where function allows you to apply conditional logic to arrays. Here is a rough sketch of how it works:
def where(condition, if_false, if_true):
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np.where(ages>30, 0, 1)
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The if_false and if_true values can be arrays themselves:
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np.where(ages<30, 0, ages)
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NumPy has a a number of different function to reading and writing arrays to and from disk.
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a = np.random.rand(10)
a
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Save the array to a binary file named array1.npy:
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np.save('array1', a)
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ls
Using %pycat to look at the file shows that it is binary:
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%pycat array1.npy
Load the array back into memory:
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a_copy = np.load('array1.npy')
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a_copy
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b = np.random.randint(0,10,(5,3))
b
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The savetxt function saves arrays in a simple, textual format that is less effecient, but easier for other languges to read:
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np.savetxt('array2.txt', b)
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ls
Using %pycat to look at the contents shows that the files is indeed a plain text file:
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%pycat array2.txt
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np.loadtxt('array2.txt')
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The savez function provides an efficient way of saving multiple arrays to a single file:
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np.savez('arrays.npz', a=a, b=b)
The load function returns a dictionary like object that provides access to the individual arrays:
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a_and_b = np.load('arrays.npz')
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a_and_b['a']
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a_and_b['b']
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NumPy has excellent linear algebra capabilities.
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a = np.random.rand(5,5)
b = np.random.rand(5,5)
Remember that array operations are elementwise. Thus, this is not matrix multiplication:
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a*b
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To get matrix multiplication use np.dot:
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np.dot(a, b)
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Or, NumPy as a matrix subclass for which matrix operations are the default:
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m1 = np.matrix(a)
m2 = np.matrix(b)
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m1*m2
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The np.linalg package has a wide range of fast linear algebra operations.
Here is determinant:
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np.linalg.det(a)
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Matrix inverse:
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np.linalg.inv(a)
Eigenvalues:
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np.linalg.eigvals(a)
NumPy can be built against fast BLAS/LAPACK implementation for these linear algebra operations.
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c = np.random.rand(2000,2000)
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%timeit -n1 -r1 evs = np.linalg.eigvals(c)
NumPy has functions for creating arrays of random numbers from different distributions in np.random, as well as handling things like permutation, shuffling, and choosing.
Here is the numpy.random documentation.
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plt.hist(np.random.random(250))
plt.title('Uniform Random Distribution $[0,1]$')
plt.xlabel('value')
plt.ylabel('count')
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plt.hist(np.random.randn(250))
plt.title('Standard Normal Distribution')
plt.xlabel('value')
plt.ylabel('count')
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The shuffle function shuffles an array in place:
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a = np.arange(0,10)
np.random.shuffle(a)
a
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The permutation function does the same thing but first makes a copy:
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a = np.arange(0,10)
print(np.random.permutation(a))
print(a)
The choice function provides a powerful way of creating synthetic data sets of discrete data:
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np.random.choice(['m','f'], 20, p=[0.25,0.75])
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