NumPy Tutorial for Beginners

NumPy Tutorial for Beginners

If you want to make a career in big data, you need to learn NumPy. Read on to get started with one of Python's most popular libraries.

I will walk you through the basics of NumPy. If you want to do machine learning then knowledge of NumPy is necessary. It one of the most widely used Python libraries. It is the most useful library if you are dealing with numbers in Python. NumPy guarantees great execution speed compared to standard Python libraries. It comes with a great number of built-in functions.

Advantages of using NumPy with Python:

  • Array-oriented computing.
  • Efficiently implemented multi-dimensional arrays.
  • Designed for scientific computation.

First, let’s talk about its installation. NumPy is not part of the basic Python installation. We need to install it after the installation of Python in our system. We can do it by the pip using command, pip install NumPy, or by installing Conda.

We are done with the installation and now we can jump right into NumPy. First, let’s start with the most important object in NumPy, the ndarray or multi-dimensional array. A multi-dimensional array is an array of arrays. In multi-dimensional arrays, this array, [1,2,3], is a one-dimensional array because it contains only one row. The below is array is a two-dimensional array, as it contains multiple rows as well as multiple columns.

[[1 2 3]

[4 5 6]

[7 8 9]]

Let’s do some coding now. Here I am using Jupyter Notebook to run my code; you can use any IDE available and best suited to you.

We start with import NumPy.

In the following code, I am renaming the package to np for convenience sake.

import numpy as np

Now, in order to create an array in NumPy, we use its array function as shown below:

array = np.array([1,2,3])

print(array)

Output: [1 2 3]

This an example of a one-dimensional array.

Another way to create an array in NumPy is by using the zeros function.

zeros = np.zeros(3)

print(zeros)

Output: [0. 0. 0.]

If you look closely at the output, the generated array contains three zeros, but the type of the value is a float and, by default, NumPy creates the array of float values.

type(zeros[0])

Output: numpy.float64

Going back to the first example inside NumPy’s array function, we pass a list so we can also pass the list variable inside the array function and the output will be the same.

my_list = [1,2,3]

array = np.array(my_list)

print(array)

Output: [1 2 3]

Now, let’s look into how to create a two-dimensional array using NumPy. Instead of passing the list now we have to pass a list of tuples or list of lists as mentioned below.

two_dim_array = np.array([(1,2,3), (4,5,6), (7,8,9)])

print(two_dim_array)

Output:

[[1 2 3]

[4 5 6]

[7 8 9]]

Note that the number of columns should be equal, otherwise NumPy will create an array of a list.

arr = np.array([[1,2,3], [4,6], [7,8,9]])

print(arr)

Output: [list([1, 2, 3]) list([4, 6]) list([7, 8, 9])]

Now, to create an array of a range, which is very good for making plots, we use the linspace function.

range_array = np.linspace(0, 10, 4)

print(range_array)

Output: [ 0. 3.33333333 6.66666667 10. ]

Here, the first argument is the starting point and next is the endpoint and the last argument defines how many elements you want in your array.

Now, to create random arrays we can use the random function. Here, I’ve created an array of random integers, and, therefore, used randint where first I specified the maximum value and then the size of my array.

random_array = np.random.randint(15, size=10)

print(random_array)

Output: [ 7 11 8 2 6 4 9 6 10 9]

Now we know the basics of how to create arrays in NumPy. Now let’s look into some of its basic operations. First, we will start by finding the size and shape of an array. Size will give the number of elements in an array whereas shape will give us the shape of an array.

For a one dimensional array, the shape would be (n, ), where n is the number of elements in your array.

For a two dimensional array, the shape would be (n,m), where n is the number of rows and m is the number of columns in your array

print(array.size)

Output: 3

print(array.shape)

Output: (3,)

print(multi_dim_array.size)

Output: 9

print(multi_dim_array.shape)

Output: (3, 3)

If we want to change the shape of an array we can easily do it with the reshape function. It will look like something like this:

two_dim_array = np.array([(1,2,3,4), (5,6,7,8)])

two_dim_array = two_dim_array.reshape(4,2)

print(two_dim_array)

Output:

[[1 2]

[3 4]

[5 6]

[7 8]]

We need to make sure that the rows and columns can be interchangeable. For example, here, we can change rows and columns from (2,4) to (4,2) but can not change them to (4,3) because, for that, we’d need 12 elements and we have only 8. Doing so will give an error as shown below.

ValueError: cannot reshape array of size 8 into shape (4,3)

To check the dimensions of our array. we can use the ndim function.

print(two_dim_array.ndim)

Output: 2

Now, to get values from an array, a process known as slicing can be done in various ways. For example, array[1] will fetch the second element of my array, but if we want a range we can use array[0:1], which will give us the first two elements. For the last value of the array, we can use array[-1], which is similar to the standard method of getting elements from a list in Python.

Now to find the sum all we have to use is the sum(), function but if we want to find the sum of the axis we can pass an argument for the axis.

print(two_dim_array.sum(axis=0))

Output: [ 6 8 10 12]

print(two_dim_array.sum(axis=1))

Output: [10 26]

Now to add two arrays all we have to use if + operator. For example:

print(two_dim_array + two_dim_array)

Output:

[[ 2 4 6 8]

[10 12 14 16]]

Similarly, we can use other operands as well, like multiple, subtract, and divide.

We have many other operations present in NumPy like sqrt, which will give us the square root of every element, and std, which is used to find the standard deviation. To explore more about these operations visit the NumPy’s documentation.

And that’s it for the introduction of NumPy.


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Neural Network Using Python and Numpy

Neural Network Using Python and Numpy

Understanding neural networks using Python and Numpy by coding

Motivation

If you are a junior data scientist who sort of understands how neural nets work, or a machine learning enthusiast who only knows a little about deep learning, this is the article that you cannot miss. Here is how you can build a neural net from scratch using NumPy in 9 steps — from data pre-processing to back-propagation — a must-do practice.

Basic understanding of machine learning, artificial neural network, Python syntax, and programming logic is preferred (but not necessary as you can learn on the go).

Codes are available on Github.

1. Initialization Numpy

Step one. Import NumPy. Seriously.

import numpy as np 
np.random.seed(42) # for reproducibility
2. Data Generation

Deep learning is data-hungry. Although there are many clean datasets available online, we will generate our own for simplicity — for inputs a and b, we have outputs a+b, a-b, and |a-b|. 10,000 datum points are generated.

X_num_row, X_num_col = [2, 10000] # Row is no. of feature, col is no. of datum points
X_raw = np.random.rand(X_num_row,X_num_col) * 100
y_raw = np.concatenate(([(X_raw[0,:] + X_raw[1,:])], [(X_raw[0,:] - X_raw[1,:])], np.abs([(X_raw[0,:] - X_raw[1,:])])))
# for input a and b, output is a+b; a-b and |a-b|
y_num_row, y_num_col = y_raw.shape
3. Train-test Splitting

Our dataset is split into training (70%) and testing (30%) set. Only training set is leveraged for tuning neural networks. Testing set is used only for performance evaluation when the training is complete.

train_ratio = 0.7
num_train_datum = int(train_ratio*X_num_col)
X_raw_train = X_raw[:,0:num_train_datum]
X_raw_test = X_raw[:,num_train_datum:]
y_raw_train = y_raw[:,0:num_train_datum]
y_raw_test = y_raw[:,num_train_datum:]
4. Data Standardization

Data in the training set is standardized so that the distribution for each standardized feature is zero-mean and unit-variance. The scalers generated from the abovementioned procedure can then be applied to the testing set.

class scaler:
    def __init__(self, mean, std):
        self.mean = mean
        self.std = std

def get_scaler(row):
mean = np.mean(row)
std = np.std(row)
return scaler(mean, std)

def standardize(data, scaler):
return (data - scaler.mean) / scaler.std

def unstandardize(data, scaler):
return (data * scaler.std) + scaler.mean

Construct scalers from training set

X_scalers = [get_scaler(X_raw_train[row,:]) for row in range(X_num_row)]
X_train = np.array([standardize(X_raw_train[row,:], X_scalers[row]) for row in range(X_num_row)])

y_scalers = [get_scaler(y_raw_train[row,:]) for row in range(y_num_row)]
y_train = np.array([standardize(y_raw_train[row,:], y_scalers[row]) for row in range(y_num_row)])

Apply those scalers to testing set

X_test = np.array([standardize(X_raw_test[row,:], X_scalers[row]) for row in range(X_num_row)])
y_test = np.array([standardize(y_raw_test[row,:], y_scalers[row]) for row in range(y_num_row)])

Check if data has been standardized

print([X_train[row,:].mean() for row in range(X_num_row)]) # should be close to zero
print([X_train[row,:].std() for row in range(X_num_row)]) # should be close to one

print([y_train[row,:].mean() for row in range(y_num_row)]) # should be close to zero
print([y_train[row,:].std() for row in range(y_num_row)]) # should be close to one

The scaler therefore does not contain any information from our testing set. We do not want our neural net to gain any information regarding testing set before network tuning.

We have now completed the data pre-processing procedures in 4 steps.

5. Neural Net Construction


Photo by freestocks.org on Unsplash

We objectify a ‘layer’ using class in Python. Every layer (except the input layer) has a weight matrix W, a bias vector b, and an activation function. Each layer is appended to a list called neural_net. That list would then be a representation of your fully connected neural network.

class layer:
def init(self, layer_index, is_output, input_dim, output_dim, activation):
self.layer_index = layer_index # zero indicates input layer
self.is_output = is_output # true indicates output layer, false otherwise
self.input_dim = input_dim
self.output_dim = output_dim
self.activation = activation

    # the multiplication constant is sorta arbitrary
    if layer_index != 0:
        self.W = np.random.randn(output_dim, input_dim) * np.sqrt(2/input_dim) 
        self.b = np.random.randn(output_dim, 1) * np.sqrt(2/input_dim)
Change layers_dim to configure your own neural net!

layers_dim = [X_num_row, 4, 4, y_num_row] # input layer --- hidden layers --- output layers
neural_net = []

Construct the net layer by layer

for layer_index in range(len(layers_dim)):
if layer_index == 0: # if input layer
neural_net.append(layer(layer_index, False, 0, layers_dim[layer_index], 'irrelevant'))
elif layer_index+1 == len(layers_dim): # if output layer
neural_net.append(layer(layer_index, True, layers_dim[layer_index-1], layers_dim[layer_index], activation='linear'))
else:
neural_net.append(layer(layer_index, False, layers_dim[layer_index-1], layers_dim[layer_index], activation='relu'))

Simple check on overfitting

pred_n_param = sum([(layers_dim[layer_index]+1)*layers_dim[layer_index+1] for layer_index in range(len(layers_dim)-1)])
act_n_param = sum([neural_net[layer_index].W.size + neural_net[layer_index].b.size for layer_index in range(1,len(layers_dim))])
print(f'Predicted number of hyperparameters: {pred_n_param}')
print(f'Actual number of hyperparameters: {act_n_param}')
print(f'Number of data: {X_num_col}')

if act_n_param >= X_num_col:
raise Exception('It will overfit.')

Finally, we do a sanity check on the number of hyperparameters using the following formula, and by counting. The number of datums available should exceed the number of hyperparameters, otherwise it will definitely overfit.


N^l is number of hyperparameters at l-th layer, L is number of layers (excluding input layer)

6. Forward Propagation

We define a function for forward propagation given a certain set of weights and biases. The connection between layers is defined in matrix form as:


σ is element-wise activation function, superscript T means transpose of a matrix

Activation functions are defined one by one. ReLU is implemented as a → max(a,0), whereas sigmoid function should return a → 1/(1+e^(-a)), and its implementation is left as an exercise to the reader.

def activation(input_, act_func):
if act_func == 'relu':
return np.maximum(input_, np.zeros(input_.shape))
elif act_func == 'linear':
return input_
else:
raise Exception('Activation function is not defined.')

def forward_prop(input_vec, layers_dim=layers_dim, neural_net=neural_net):
neural_net[0].A = input_vec # Define A in input layer for for-loop convenience
for layer_index in range(1,len(layers_dim)): # W,b,Z,A are undefined in input layer
neural_net[layer_index].Z = np.add(np.dot(neural_net[layer_index].W, neural_net[layer_index-1].A), neural_net[layer_index].b)
neural_net[layer_index].A = activation(neural_net[layer_index].Z, neural_net[layer_index].activation)
return neural_net[layer_index].A


Photo by Holger Link on Unsplash

7. Back-propagation

This is the most tricky part where many of us simply do not understand. Once we have defined a loss metric e for evaluating performance, we would like to know how the loss metric change when we perturb each weight or bias.

We want to know how sensitive each weight and bias is with respect to the loss metric.

This is represented by partial derivatives ∂e/∂W (denoted dW in code) and ∂e/∂b (denoted db in code) respectively, and can be calculated analytically.


⊙ represents element-wise multiplication

These back-propagation equations assume only one datum y is compared. The gradient update process would be very noisy as the performance of each iteration is subject to one datum point only. Multiple datums can be used to reduce the noise where ∂W(y1, y2, …) would be the mean of ∂W(y_1), ∂W(y_2), …, and likewise for ∂b. This is not shown above in those equations, but is implemented in the code below.

def get_loss(y, y_hat, metric='mse'):
if metric == 'mse':
individual_loss = 0.5 * (y_hat - y) ** 2
return np.mean([np.linalg.norm(individual_loss[:,col], 2) for col in range(individual_loss.shape[1])])
else:
raise Exception('Loss metric is not defined.')

def get_dZ_from_loss(y, y_hat, metric):
if metric == 'mse':
return y_hat - y
else:
raise Exception('Loss metric is not defined.')

def get_dactivation(A, act_func):
if act_func == 'relu':
return np.maximum(np.sign(A), np.zeros(A.shape)) # 1 if backward input >0, 0 otherwise; then diaganolize
elif act_func == 'linear':
return np.ones(A.shape)
else:
raise Exception('Activation function is not defined.')

def backward_prop(y, y_hat, metric='mse', layers_dim=layers_dim, neural_net=neural_net, num_train_datum=num_train_datum):
for layer_index in range(len(layers_dim)-1,0,-1):
if layer_index+1 == len(layers_dim): # if output layer
dZ = get_dZ_from_loss(y, y_hat, metric)
else:
dZ = np.multiply(np.dot(neural_net[layer_index+1].W.T, dZ),
get_dactivation(neural_net[layer_index].A, neural_net[layer_index].activation))
dW = np.dot(dZ, neural_net[layer_index-1].A.T) / num_train_datum
db = np.sum(dZ, axis=1, keepdims=True) / num_train_datum

    neural_net[layer_index].dW = dW
    neural_net[layer_index].db = db

8. Iterative Optimization

We now have every building block for training a neural network.

Once we know the sensitivities of weights and biases, we try to minimize (hence the minus sign) the loss metric iteratively by gradient descent using the following update rule:

W = W - learning_rate * ∂W
b = b - learning_rate * ∂b


Photo by Rostyslav Savchyn on Unsplash

learning_rate = 0.01
max_epoch = 1000000

for epoch in range(1,max_epoch+1):
y_hat_train = forward_prop(X_train) # update y_hat
backward_prop(y_train, y_hat_train) # update (dW,db)

for layer_index in range(1,len(layers_dim)):        # update (W,b)
    neural_net[layer_index].W = neural_net[layer_index].W - learning_rate * neural_net[layer_index].dW
    neural_net[layer_index].b = neural_net[layer_index].b - learning_rate * neural_net[layer_index].db

if epoch % 100000 == 0:
    print(f'{get_loss(y_train, y_hat_train):.4f}')

Training loss should be going down as it iterates

9. Testing

The model generalizes well if the testing loss is not much higher than the training loss. We also make some test cases to see how the model performs.

# test loss

print(get_loss(y_test, forward_prop(X_test)))

def predict(X_raw_any):
X_any = np.array([standardize(X_raw_any[row,:], X_scalers[row]) for row in range(X_num_row)])
y_hat = forward_prop(X_any)
y_hat_any = np.array([unstandardize(y_hat[row,:], y_scalers[row]) for row in range(y_num_row)])
return y_hat_any

predict(np.array([[30,70],[70,30],[3,5],[888,122]]).T)


The Takeaway

This is how you can build a neural net from scratch using NumPy in 9 steps. Some of you might have already built neural nets using some high-level frameworks such as TensorFlow, PyTorch, or Keras. However, building a neural net using only low-level libraries enable us to truly understand the mathematics behind the mystery.

My implementation by no means is the most efficient way to build and train a neural net. There is so much room for improvement but that is a story for another day. Codes are available on Github. Happy coding!

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