Numpy (✗)
Makine Öğrenmesi ve Derin Öğrenme için gerekli Numpy konuları.
import numpy as np
# Set seed for reproducibility
np.random.seed(seed=1234)
Let's take a took at how to create tensors with NumPy.
- Tensor: collection of values
# Scalar
x = np.array(6) # scalar
print ("x: ", x)
# Number of dimensions
print ("x ndim: ", x.ndim)
# Dimensions
print ("x shape:", x.shape)
# Size of elements
print ("x size: ", x.size)
# Data type
print ("x dtype: ", x.dtype)
# 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
# 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)
# 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)
NumPy also comes with several functions that allow us to create tensors quickly.
# 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)))
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).
# Indexing
x = np.array([1, 2, 3])
print ("x: ", x)
print ("x[0]: ", x[0])
x[0] = 0
print ("x: ", x)
# 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])
# 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)
# 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])
# 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
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.
# 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} · {b.shape} = {c.shape}")
print (c)
We can also do operations across a specific axis.
# 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
# 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))
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.
# Broadcasting
x = np.array([1,2]) # vector
y = np.array(3) # scalar
z = x + y
print ("z:\n", z)
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.
# 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)
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.
# 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)
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].
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)
When we naively do a reshape, we get the right shape but the values are not what we're looking for.
# Unintended reshaping
z_incorrect = np.reshape(x, (x.shape[1], -1))
print ("z_incorrect:\n", z_incorrect)
print ("z_incorrect.shape: ", z_incorrect.shape)
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. 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.
# 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)
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.
# 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
# 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
- NumPy reference manual: We don't have to memorize anything here and we will be taking a closer look at NumPy in the later lessons. If you want to learn more checkout the NumPy reference manual.
