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NumPy

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In this lesson, we'll learn about numerical analysis with the NumPy computing library.

Set up

First we'll import the NumPy package and set seeds for reproducability so that we can receive the exact same results every time.

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import numpy as np
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# Set seed for reproducibility
np.random.seed(seed=1234)


Basics

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# Scalar
x = np.array(6)
print ("x: ", x)
print ("x ndim: ", x.ndim) # number of dimensions
print ("x shape:", x.shape) # dimensions
print ("x size: ", x.size) # size of elements
print ("x dtype: ", x.dtype) # data type
x:  6
x ndim:  0
x shape: ()
x size:  1
x dtype:  int64
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# Vector
x = np.array([1.3 , 2.2 , 1.7])
print ("x: ", x)
print ("x ndim: ", x.ndim)
print ("x shape:", x.shape)
print ("x size: ", x.size)
print ("x dtype: ", x.dtype) # notice the float datatype
x:  [1.3 2.2 1.7]
x ndim:  1
x shape: (3,)
x size:  3
x dtype:  float64
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# Matrix
x = np.array([[1,2], [3,4]])
print ("x:\n", x)
print ("x ndim: ", x.ndim)
print ("x shape:", x.shape)
print ("x size: ", x.size)
print ("x dtype: ", x.dtype)
x:
 [[1 2]
 [3 4]]
x ndim:  2
x shape: (2, 2)
x size:  4
x dtype:  int64
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# 3-D Tensor
x = np.array([[[1,2],[3,4]],[[5,6],[7,8]]])
print ("x:\n", x)
print ("x ndim: ", x.ndim)
print ("x shape:", x.shape)
print ("x size: ", x.size)
print ("x dtype: ", x.dtype)
x:
 [[[1 2]
  [3 4]]

 [[5 6]
  [7 8]]]
x ndim:  3
x shape: (2, 2, 2)
x size:  8
x dtype:  int64

NumPy also comes with several functions that allow us to create tensors quickly.

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# Functions
print ("np.zeros((2,2)):\n", np.zeros((2,2)))
print ("np.ones((2,2)):\n", np.ones((2,2)))
print ("np.eye((2)):\n", np.eye((2))) # identity matrix
print ("np.random.random((2,2)):\n", np.random.random((2,2)))

np.zeros((2,2)):
 [[0. 0.]
 [0. 0.]]
np.ones((2,2)):
 [[1. 1.]
 [1. 1.]]
np.eye((2)):
 [[1. 0.]
 [0. 1.]]
np.random.random((2,2)):
 [[0.19151945 0.62210877]
 [0.43772774 0.78535858]]

Note

Keep in mind that when indexing the row and column, indices start at 0. And like indexing with lists, we can use negative indices as well (where -1 is the last item).

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# Indexing
x = np.array([1, 2, 3])
print ("x: ", x)
print ("x[0]: ", x[0])
x[0] = 0
print ("x: ", x)
x:  [1 2 3]
x[0]:  1
x:  [0 2 3]
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# Slicing
x = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
print (x)
print ("x column 1: ", x[:, 1])
print ("x row 0: ", x[0, :])
print ("x rows 0,1 & cols 1,2: \n", x[0:2, 1:3])
[[ 1  2  3  4]
 [ 5  6  7  8]
 [ 9 10 11 12]]
x column 1:  [ 2  6 10]
x row 0:  [1 2 3 4]
x rows 0,1 & cols 1,2:
 [[2 3]
 [6 7]]
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# Integer array indexing
print (x)
rows_to_get = np.array([0, 1, 2])
print ("rows_to_get: ", rows_to_get)
cols_to_get = np.array([0, 2, 1])
print ("cols_to_get: ", cols_to_get)
# Combine sequences above to get values to get
print ("indexed values: ", x[rows_to_get, cols_to_get]) # (0, 0), (1, 2), (2, 1)
[[ 1  2  3  4]
 [ 5  6  7  8]
 [ 9 10 11 12]]
rows_to_get:  [0 1 2]
cols_to_get:  [0 2 1]
indexed values:  [ 1  7 10]
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# Boolean array indexing
x = np.array([[1, 2], [3, 4], [5, 6]])
print ("x:\n", x)
print ("x > 2:\n", x > 2)
print ("x[x > 2]:\n", x[x > 2])
x:
 [[1 2]
 [3 4]
 [5 6]]
x > 2:
 [[False False]
 [ True  True]
 [ True  True]]
x[x > 2]:
 [3 4 5 6]

Arithmetic

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# Basic math
x = np.array([[1,2], [3,4]], dtype=np.float64)
y = np.array([[1,2], [3,4]], dtype=np.float64)
print ("x + y:\n", np.add(x, y)) # or x + y
print ("x - y:\n", np.subtract(x, y)) # or x - y
print ("x * y:\n", np.multiply(x, y)) # or x * y
x + y:
 [[2. 4.]
 [6. 8.]]
x - y:
 [[0. 0.]
 [0. 0.]]
x * y:
 [[ 1.  4.]
 [ 9. 16.]]

Dot product

One of the most common NumPy operations we’ll use in machine learning is matrix multiplication using the dot product. We take the rows of our first matrix (2) and the columns of our second matrix (2) to determine the dot product, giving us an output of [2 X 2]. The only requirement is that the inside dimensions match, in this case the first matrix has 3 columns and the second matrix has 3 rows.

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# Dot product
a = np.array([[1,2,3], [4,5,6]], dtype=np.float64) # we can specify dtype
b = np.array([[7,8], [9,10], [11, 12]], dtype=np.float64)
c = a.dot(b)
print (f"{a.shape} Β· {b.shape} = {c.shape}")
print (c)
(2, 3) Β· (3, 2) = (2, 2)
[[ 58.  64.]
 [139. 154.]]

Axis operations

We can also do operations across a specific axis.

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# Sum across a dimension
x = np.array([[1,2],[3,4]])
print (x)
print ("sum all: ", np.sum(x)) # adds all elements
print ("sum axis=0: ", np.sum(x, axis=0)) # sum across rows
print ("sum axis=1: ", np.sum(x, axis=1)) # sum across columns
[[1 2]
 [3 4]]
sum all:  10
sum axis=0:  [4 6]
sum axis=1:  [3 7]
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# Min/max
x = np.array([[1,2,3], [4,5,6]])
print ("min: ", x.min())
print ("max: ", x.max())
print ("min axis=0: ", x.min(axis=0))
print ("min axis=1: ", x.min(axis=1))
min:  1
max:  6
min axis=0:  [1 2 3]
min axis=1:  [1 4]

Broadcast

Here, we’re adding a vector with a scalar. Their dimensions aren’t compatible as is but how does NumPy still gives us the right result? This is where broadcasting comes in. The scalar is broadcast across the vector so that they have compatible shapes.

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# Broadcasting
x = np.array([1,2]) # vector
y = np.array(3) # scalar
z = x + y
print ("z:\n", z)
z:
 [4 5]

Transpose

We often need to change the dimensions of our tensors for operations like the dot product. If we need to switch two dimensions, we can transpose the tensor.

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# Transposing
x = np.array([[1,2,3], [4,5,6]])
print ("x:\n", x)
print ("x.shape: ", x.shape)
y = np.transpose(x, (1,0)) # flip dimensions at index 0 and 1
print ("y:\n", y)
print ("y.shape: ", y.shape)
x:
 [[1 2 3]
 [4 5 6]]
x.shape:  (2, 3)
y:
 [[1 4]
 [2 5]
 [3 6]]
y.shape:  (3, 2)

Reshape

Sometimes, we'll need to alter the dimensions of the matrix. Reshaping allows us to transform a tensor into different permissible shapes -- our reshaped tensor has the same amount of values in the tensor. (1X6 = 2X3). We can also use -1 on a dimension and NumPy will infer the dimension based on our input tensor.

The way reshape works is by looking at each dimension of the new tensor and separating our original tensor into that many units. So here the dimension at index 0 of the new tensor is 2 so we divide our original tensor into 2 units, and each of those has 3 values.

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# Reshaping
x = np.array([[1,2,3,4,5,6]])
print (x)
print ("x.shape: ", x.shape)
y = np.reshape(x, (2, 3))
print ("y: \n", y)
print ("y.shape: ", y.shape)
z = np.reshape(x, (2, -1))
print ("z: \n", z)
print ("z.shape: ", z.shape)
[[1 2 3 4 5 6]]
x.shape:  (1, 6)
y:
 [[1 2 3]
 [4 5 6]]
y.shape:  (2, 3)
z:
 [[1 2 3]
 [4 5 6]]
z.shape:  (2, 3)

Unintended reshaping

Though reshaping is very convenient to manipulate tensors, we must be careful of their pitfalls as well. Let's look at the example below. Suppose we have x, which has the shape [2 X 3 X 4].

[[[ 1  1  1  1]
  [ 2  2  2  2]
  [ 3  3  3  3]]
 [[10 10 10 10]
  [20 20 20 20]
  [30 30 30 30]]]

We want to reshape x so that it has shape [3 X 8] which we'll get by moving the dimension at index 0 to become the dimension at index 1 and then combining the last two dimensions. But when we do this, we want our output

to look like:

[[ 1  1  1  1 10 10 10 10]
 [ 2  2  2  2 20 20 20 20]
 [ 3  3  3  3 30 30 30 30]]

and not like:

[[ 1  1  1  1  2  2  2  2]
 [ 3  3  3  3 10 10 10 10]
 [20 20 20 20 30 30 30 30]]

even though they both have the same shape [3X8].

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x = np.array([[[1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3]],
              [[10, 10, 10, 10], [20, 20, 20, 20], [30, 30, 30, 30]]])
print ("x:\n", x)
print ("x.shape: ", x.shape)
x:
 [[[ 1  1  1  1]
   [ 2  2  2  2]
   [ 3  3  3  3]]

 [[10 10 10 10]
  [20 20 20 20]
  [30 30 30 30]]]
x.shape:  (2, 3, 4)

When we naively do a reshape, we get the right shape but the values are not what we're looking for.

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# Unintended reshaping
z_incorrect = np.reshape(x, (x.shape[1], -1))
print ("z_incorrect:\n", z_incorrect)
print ("z_incorrect.shape: ", z_incorrect.shape)
z_incorrect:
 [[ 1  1  1  1  2  2  2  2]
  [ 3  3  3  3 10 10 10 10]
  [20 20 20 20 30 30 30 30]]
z_incorrect.shape:  (3, 8)

Instead, if we transpose the tensor and then do a reshape, we get our desired tensor. Transpose allows us to put our two vectors that we want to combine together and then we use reshape to join them together.

Note

Always create a dummy example like this when you’re unsure about reshaping. Blindly going by the tensor shape can lead to lots of issues downstream.

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# Intended reshaping
y = np.transpose(x, (1,0,2))
print ("y:\n", y)
print ("y.shape: ", y.shape)
z_correct = np.reshape(y, (y.shape[0], -1))
print ("z_correct:\n", z_correct)
print ("z_correct.shape: ", z_correct.shape)
y:
 [[[ 1  1  1  1]
  [10 10 10 10]]

 [[ 2  2  2  2]
  [20 20 20 20]]

 [[ 3  3  3  3]
  [30 30 30 30]]]
y.shape:  (3, 2, 4)
z_correct:
 [[ 1  1  1  1 10 10 10 10]
  [ 2  2  2  2 20 20 20 20]
  [ 3  3  3  3 30 30 30 30]]
z_correct.shape:  (3, 8)

Expanding / reducing

We can also easily add and remove dimensions to our tensors and we'll want to do this to make tensors compatible for certain operations.

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# Adding dimensions
x = np.array([[1,2,3],[4,5,6]])
print ("x:\n", x)
print ("x.shape: ", x.shape)
y = np.expand_dims(x, 1) # expand dim 1
print ("y: \n", y)
print ("y.shape: ", y.shape)   # notice extra set of brackets are added
x:
 [[1 2 3]
  [4 5 6]]
x.shape:  (2, 3)
y:
 [[[1 2 3]]
  [[4 5 6]]]
y.shape:  (2, 1, 3)
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# Removing dimensions
x = np.array([[[1,2,3]],[[4,5,6]]])
print ("x:\n", x)
print ("x.shape: ", x.shape)
y = np.squeeze(x, 1) # squeeze dim 1
print ("y: \n", y)
print ("y.shape: ", y.shape)  # notice extra set of brackets are gone
x:
 [[[1 2 3]]
  [[4 5 6]]]
x.shape:  (2, 1, 3)
y:
 [[1 2 3]
  [4 5 6]]
y.shape:  (2, 3)