This post complements the previous post by demonstrating a real-world solution of a problem using the neural network defined in that post. This is a solution of Udacity’s Deep Learning Nanodegree’s first project. Most of the text in this notebook is also borrowed from the same project. We’ll be using the sample neural network we created in the previous post to predict daily bike rental ridership. We didn’t use any DL framework (Tensorflow, Pytorch, etc) just to demonstrate the ideas behind the basic concepts in neural networks. The neural network implementation in this kernel elaborates the concepts like backward propagation, stochastic gradient descent, and hyperparameters.

You can find this post as a Jupyter Notebook on Github and Kaggle. Feel free to submit a pull request if you want to change/improve anything.

The dataset

We’re going to take on a forecasting problem to illustrate the concepts of deep learning. In this Kaggle dataset, we have the hourly and daily count of rental bikes between the years 2011 and 2012 in the Capital bike-share system in Washington, DC with the corresponding weather and seasonal information. We have a lot of parameters in this data that can influence Bike-sharing. We’ll model and train a neural network to harness this information to forecast the hourly count of rental bikes for the future.

Notebook Config

%matplotlib inline
%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = 'retina'

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

Load and prepare the data

A critical step in working with neural networks is preparing the data correctly. Variables on different scales make it difficult for the network to efficiently learn the correct weights. Below, we’ve written the code to load and prepare the data.

data_path = '../input/bike-sharing-dataset/hour.csv'

rides = pd.read_csv(data_path)
rides.head()
No. instant dteday season yr mnth hr holiday weekday workingday weathersit temp atemp hum windspeed casual registered cnt
0 1 2011-01-01 1 0 1 0 0 6 0 1 0.24 0.2879 0.81 0.0 3 13 16
1 2 2011-01-01 1 0 1 1 0 6 0 1 0.22 0.2727 0.80 0.0 8 32 40
2 3 2011-01-01 1 0 1 2 0 6 0 1 0.22 0.2727 0.80 0.0 5 27 32
3 4 2011-01-01 1 0 1 3 0 6 0 1 0.24 0.2879 0.75 0.0 3 10 13
4 5 2011-01-01 1 0 1 4 0 6 0 1 0.24 0.2879 0.75 0.0 0 1 1

Checking out the data

This dataset has the number of riders for each hour of each day from January 1 2011 to December 31 2012. The number of riders is split between casual and registered, summed up in the cnt column. You can see the first few rows of the data above.

Below is a plot showing the number of bike riders over the first 10 days or so in the data set. (Some days don’t have exactly 24 entries in the data set, so it’s not exactly 10 days.) You can see the hourly rentals here. This data is pretty complicated! The weekends have lower overall ridership and there are spikes when people are biking to and from work during the week. Looking at the data above, we also have information about temperature, humidity, and wind-speed, all of these parameters likely affecting the number of riders. We’ll be trying to capture all this with our model.

rides[:24*10].plot(x='dteday', y='cnt')
<matplotlib.axes._subplots.AxesSubplot at 0x7fccd5bca668>

png

Dummy variables

Here we have some categorical variables like the season, weather, month, etc. To include these in our model, we’ll need to make binary dummy variables. This is simple to do with Pandas; thanks to get_dummies().

dummy_fields = ['season', 'weathersit', 'mnth', 'hr', 'weekday']
for each in dummy_fields:
    dummies = pd.get_dummies(rides[each], prefix=each, drop_first=False)
    rides = pd.concat([rides, dummies], axis=1)

fields_to_drop = ['instant', 'dteday', 'season', 'weathersit', 
                  'weekday', 'atemp', 'mnth', 'workingday', 'hr']
data = rides.drop(fields_to_drop, axis=1)
data.head()
yr holiday temp hum windspeed casual registered cnt season_1 season_2 ... hr_21 hr_22 hr_23 weekday_0 weekday_1 weekday_2 weekday_3 weekday_4 weekday_5 weekday_6
0 0 0 0.24 0.81 0.0 3 13 16 1 0 ... 0 0 0 0 0 0 0 0 0 1
1 0 0 0.22 0.80 0.0 8 32 40 1 0 ... 0 0 0 0 0 0 0 0 0 1
2 0 0 0.22 0.80 0.0 5 27 32 1 0 ... 0 0 0 0 0 0 0 0 0 1
3 0 0 0.24 0.75 0.0 3 10 13 1 0 ... 0 0 0 0 0 0 0 0 0 1
4 0 0 0.24 0.75 0.0 0 1 1 1 0 ... 0 0 0 0 0 0 0 0 0 1

5 rows × 59 columns

Scaling target variables

To make training the network easier, we’ll standardize each of the continuous variables. That is, we’ll shift and scale the variables such that they have zero mean and a standard deviation of 1.

The scaling factors are saved so we can go backward when we use the network for predictions.

quant_features = ['casual', 'registered', 'cnt', 'temp', 'hum', 'windspeed']
# Store scalings in a dictionary so we can convert back later
scaled_features = {}
for each in quant_features:
    mean, std = data[each].mean(), data[each].std()
    scaled_features[each] = [mean, std]
    data.loc[:, each] = (data[each] - mean)/std

Splitting the data into training, testing, and validation sets

We’ll save the data for approximately the last 21 days to use as a test set after we’ve trained the network. We’ll use this set to make predictions and compare them with the actual number of riders.

# Save data for approximately the last 21 days 
test_data = data[-21*24:]

# Now remove the test data from the data set 
data = data[:-21*24]

# Separate the data into features and targets
target_fields = ['cnt', 'casual', 'registered']
features, targets = data.drop(target_fields, axis=1), data[target_fields]
test_features, test_targets = test_data.drop(target_fields, axis=1), test_data[target_fields]

We’ll split the data into two sets, one for training and one for validating as the network is being trained. Since this is time-series data, we’ll train on historical data, then try to predict future data (the validation set).

# Hold out the last 60 days or so of the remaining data as a validation set
train_features, train_targets = features[:-60*24], targets[:-60*24]
val_features, val_targets = features[-60*24:], targets[-60*24:]

Time to build the network

This is the same neural network we created in the previous post. You can refer to that post to understand the details of how this network has been implemented. I’m adding it here for the sake of completeness.

class NeuralNetwork(object):
    def __init__(self, input_nodes, hidden_nodes, output_nodes, learning_rate):
        # Set number of nodes in input, hidden and output layers.
        self.input_nodes = input_nodes
        self.hidden_nodes = hidden_nodes
        self.output_nodes = output_nodes

        # Initialize weights
        self.weights_input_to_hidden = np.random.normal(0.0, self.input_nodes**-0.5, 
                                       (self.input_nodes, self.hidden_nodes))

        self.weights_hidden_to_output = np.random.normal(0.0, self.hidden_nodes**-0.5, 
                                       (self.hidden_nodes, self.output_nodes))
        self.lr = learning_rate
        
        # Sigmoid activation function
        self.activation_function = lambda x : (1/(1+np.exp(-x)))  
                    
    def train(self, features, targets):
        ''' Train the network on batch of features and targets. 
        
            Arguments
            ---------
            
            features: 2D array, each row is one data record, each column is a feature
            targets: 1D array of target values
        
        '''
        n_records = features.shape[0]
        delta_weights_i_h = np.zeros(self.weights_input_to_hidden.shape)
        delta_weights_h_o = np.zeros(self.weights_hidden_to_output.shape)
        for X, y in zip(features, targets):
            
            final_outputs, hidden_outputs = self.forward_pass_train(X) 
            
            delta_weights_i_h, delta_weights_h_o = self.backpropagation(final_outputs, hidden_outputs, X, y, 
                                                                        delta_weights_i_h, delta_weights_h_o)
        self.update_weights(delta_weights_i_h, delta_weights_h_o, n_records)


    def forward_pass_train(self, X):
        ''' The forward pass while training 
         
            Arguments
            ---------
            X: features batch

        '''

        hidden_inputs = np.dot(X, self.weights_input_to_hidden) # signals into hidden layer
        hidden_outputs = self.activation_function(hidden_inputs) # signals from hidden layer

        final_inputs = np.dot(hidden_outputs, self.weights_hidden_to_output) # signals into final output layer
        final_outputs = final_inputs # signals from final output layer
        
        return final_outputs, hidden_outputs

    def backpropagation(self, final_outputs, hidden_outputs, X, y, delta_weights_i_h, delta_weights_h_o):
        ''' The backpropagation implementation
         
            Arguments
            ---------
            final_outputs: output from forward pass
            y: target (i.e. label) batch
            delta_weights_i_h: change in weights from input to hidden layers
            delta_weights_h_o: change in weights from hidden to output layers

        '''

        error = y-final_outputs # Output layer error is the difference between desired target and actual output.
        
        # The hidden layer's contribution to the error
        hidden_error = np.dot(self.weights_hidden_to_output, error)
        
        #Backpropagated error terms
        output_error_term = error * 1
        
        hidden_error_term = hidden_error * hidden_outputs * (1 - hidden_outputs)
        
        # Weight step (input to hidden)
        delta_weights_i_h += hidden_error_term * X[:,None]
        # Weight step (hidden to output)
        delta_weights_h_o += (output_error_term * hidden_outputs[:,None])
        return delta_weights_i_h, delta_weights_h_o

    def update_weights(self, delta_weights_i_h, delta_weights_h_o, n_records):
        ''' Update weights on gradient descent step
         
            Arguments
            ---------
            delta_weights_i_h: change in weights from input to hidden layers
            delta_weights_h_o: change in weights from hidden to output layers
            n_records: number of records

        '''
        # update hidden-to-output weights with gradient descent step
        self.weights_hidden_to_output += self.lr * delta_weights_h_o/n_records 

        # update input-to-hidden weights with gradient descent step
        self.weights_input_to_hidden += self.lr * delta_weights_i_h/n_records 

    def run(self, features):
        ''' Run a forward pass through the network with input features 
        
            Arguments
            ---------
            features: 1D array of feature values
        '''
        
        #Hidden layer
        hidden_inputs = np.dot(features, self.weights_input_to_hidden) # signals into hidden layer
        hidden_outputs = self.activation_function(hidden_inputs) # signals from hidden layer
        
        #Output layer 
        final_inputs = np.dot(hidden_outputs, self.weights_hidden_to_output) # signals into final output layer
        final_outputs = final_inputs # signals from final output layer 
        
        return final_outputs

def MSE(y, Y):
    return np.mean((y-Y)**2)

Unit tests

The following unit tests ensure the correctness of the neural network implementation above.

network = NeuralNetwork(3, 2, 1, 0.5)
network.activation_function(0.5)
0.6224593312018546
import unittest

inputs = np.array([[0.5, -0.2, 0.1]])
targets = np.array([[0.4]])
test_w_i_h = np.array([[0.1, -0.2],
                       [0.4, 0.5],
                       [-0.3, 0.2]])
test_w_h_o = np.array([[0.3],
                       [-0.1]])

class TestMethods(unittest.TestCase):
    
    ##########
    # Unit tests for data loading
    ##########
    
    def test_data_path(self):
        # Test that file path to dataset has been unaltered
        self.assertTrue(data_path.lower() == '../input/bike-sharing-dataset/hour.csv')
        
    def test_data_loaded(self):
        # Test that data frame loaded
        self.assertTrue(isinstance(rides, pd.DataFrame))
    
    ##########
    # Unit tests for network functionality
    ##########

    def test_activation(self):
        network = NeuralNetwork(3, 2, 1, 0.5)
        # Test that the activation function is a sigmoid
        self.assertTrue(np.all(network.activation_function(0.5) == 1/(1+np.exp(-0.5))))

    def test_train(self):
        # Test that weights are updated correctly on training
        network = NeuralNetwork(3, 2, 1, 0.5)
        network.weights_input_to_hidden = test_w_i_h.copy()
        network.weights_hidden_to_output = test_w_h_o.copy()
        
        network.train(inputs, targets)
        self.assertTrue(np.allclose(network.weights_hidden_to_output, 
                                    np.array([[ 0.37275328], 
                                              [-0.03172939]])))
        self.assertTrue(np.allclose(network.weights_input_to_hidden,
                                    np.array([[ 0.10562014, -0.20185996], 
                                              [0.39775194, 0.50074398], 
                                              [-0.29887597, 0.19962801]])))

    def test_run(self):
        # Test correctness of run method
        network = NeuralNetwork(3, 2, 1, 0.5)
        network.weights_input_to_hidden = test_w_i_h.copy()
        network.weights_hidden_to_output = test_w_h_o.copy()

        self.assertTrue(np.allclose(network.run(inputs), 0.09998924))

suite = unittest.TestLoader().loadTestsFromModule(TestMethods())
unittest.TextTestRunner().run(suite)
.....
----------------------------------------------------------------------
Ran 5 tests in 0.059s

OK

<unittest.runner.TextTestResult run=5 errors=0 failures=0>

Training the network

The strategy here is to find hyperparameters such that the error on the training set is low, while we’re not overfitting to the data. If we train the network too long or have too many hidden nodes, it can become overly specific to the training set and will fail to generalize to the validation set (overfitting). That is, the loss in the validation set will start increasing as the training set loss drops.

We’ve also been using a method known as Stochastic Gradient Descent (SGD) to train the network. The idea is that for each training pass, you grab a random sample of the data instead of using the whole data set. You use many more training passes than with normal gradient descent, but each pass is much faster. This ends up training the network more efficiently.

The number of iterations

This is the number of batches of samples from the training data we’ll use to train the network. The more iterations we use, the better the model will fit the data. However, this process can have sharply diminishing returns and can waste computational resources if we use too many iterations. We want to find a number here where the network has a low training loss, and the validation loss is at a minimum. The ideal number of iterations would be a level that stops shortly after the validation loss is no longer decreasing.

The learning rate

This scales the size of weight updates. If this is too big, the weights tend to explode and the network fails to fit the data. Normally a good choice to start at is 0.1; however, if we effectively divide the learning rate by n_records, we can have a bigger learning rate. In either case, if the network has problems fitting the data, reducing the learning rate can effect it positively. Note that the lower the learning rate, the smaller the steps are in the weight updates and the longer it takes for the neural network to converge. It’s always a tradeoff between the exploding gradient and longer computational times.

The number of hidden nodes

In a model where all the weights are optimized, the more hidden nodes we have, the more accurate the predictions of the model will be. A fully optimized model could have weights of zero, after all. However, the more hidden nodes we have, the harder it will be to optimize the weights of the model, and the more likely it will be that suboptimal weights will lead to overfitting. With overfitting, the model will memorize the training data instead of learning the true pattern, and won’t generalize well to unseen data.

We can look at the losses dictionary for a metric of the network performance. If the number of hidden units is too low, then the model won’t have enough space to learn and if it is too high there are too many options for the direction that the learning can take. The trick here is to find the right balance in the number of hidden units we choose. We’ll generally find that the best number of hidden nodes to use ends up being between the number of input and output nodes.

I have chosen the following hyperparameters. Feel free to play with these parameters by forking this notebook.

iterations = 3500
learning_rate = 0.7
hidden_nodes = 15
output_nodes = 1
import sys

N_i = train_features.shape[1]
network = NeuralNetwork(N_i, hidden_nodes, output_nodes, learning_rate)

losses = {'train':[], 'validation':[]}
for ii in range(iterations):
    # Go through a random batch of 128 records from the training data set
    batch = np.random.choice(train_features.index, size=128)
    X, y = train_features.iloc[batch].values, train_targets.iloc[batch]['cnt']
                             
    network.train(X, y)
    
    # Printing out the training progress
    train_loss = MSE(network.run(train_features).T, train_targets['cnt'].values)
    val_loss = MSE(network.run(val_features).T, val_targets['cnt'].values)
    sys.stdout.write("\rProgress: {:2.1f}".format(100 * ii/float(iterations)) \
                     + "% ... Training loss: " + str(train_loss)[:5] \
                     + " ... Validation loss: " + str(val_loss)[:5])
    sys.stdout.flush()
    
    losses['train'].append(train_loss)
    losses['validation'].append(val_loss)
Progress: 100.0% ... Training loss: 0.058 ... Validation loss: 0.129
plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.legend()
_ = plt.ylim()

png

Check out the predictions

We’re using the test data to view how well our network is modeling the data.

fig, ax = plt.subplots(figsize=(8,4))

mean, std = scaled_features['cnt']
predictions = network.run(test_features).T*std + mean
ax.plot(predictions[0], label='Prediction')
ax.plot((test_targets['cnt']*std + mean).values, label='Data')
ax.set_xlim(right=len(predictions))
ax.legend()

dates = pd.to_datetime(rides.iloc[test_data.index]['dteday'])
dates = dates.apply(lambda d: d.strftime('%b %d'))
ax.set_xticks(np.arange(len(dates))[12::24])
_ = ax.set_xticklabels(dates[12::24], rotation=45)

png

Conclusion

We can see the prediction accuracy of even this simple neural network and appreciate the heuristics of a machine. The simple mathematical model explained in the previous post solves a seemingly complex forecasting problem.

The predictions are pretty good until 22nd December. After that, you can see the predictions don’t coincide with the actual data at that time. This can be due to the fact that from 22nd December till new year eve, we have the holiday season which can impact the bike ridership. The model couldn’t learn this holiday behavior because it didn’t have enough training data to fit for the holiday season. Had the data been for more than 5 years, the model might have fitted the holiday times accurately as well.

Appendix

How to convert Jupyter Notebook to other formats

I used Jupyter’s command-line tool ipython to convert this notebook to the markdown format. You can convert it to other formats as well using the same command: ipython nbconvert --to markdown notebook.ipynb. Refer to the documentation for details.