Neural Network implementation

In this project, I've built my first neural network, in order to use it to predict daily flow of a bike rental shop.

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

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

Loading and preparing 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.

data_path = 'Bike-Sharing-Dataset/hour.csv'

rides = pd.read_csv(data_path)

rides.head()

	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 over all 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 windspeed, all of these likely affecting the number of riders. Let's capture all this with a model.

rides[:24*10].plot(x='dteday', y='cnt')

<matplotlib.axes._subplots.AxesSubplot at 0x27d5368b438>

Dummy variables

Here we have some categorical variables like season, weather, month. 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 backwards 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 the last approximately 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 on 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

Time to build the Neural Network and also set the hyperparameters: the learning rate, the number of hidden units, and the number of training passes.

The network has two layers, a hidden layer and an output layer. The hidden layer will use the sigmoid function for activations. The output layer has only one node and is used for the regression, the output of the node is the same as the input of the node. That is, the activation function is $f(x)=x$. A function that takes the input signal and generates an output signal, but takes into account the threshold, is called an activation function. We work through each layer of our network calculating the outputs for each neuron. All of the outputs from one layer become inputs to the neurons on the next layer. This process is called forward propagation.

We use the weights to propagate signals forward from the input to the output layers in a neural network. We use the weights to also propagate error backwards from the output back into the network to update our weights. This is called backpropagation.

To do so we'll need the derivative of the output activation function ($f(x) = x$) for the backpropagation implementation. This function is equivalent to the equation $y = x$. What is the slope of that equation? That is the derivative of $f(x)$.

Below:

1. Implemented the sigmoid function to use as the activation function. Set self.activation_function in __init__ to your sigmoid function.
2. Implemented the forward pass in the train method.
3. Implemented the backpropagation algorithm in the train method, including calculating the output error.
4. Implemented the forward pass in the run method.

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):
            ### Implementing Forward Pass ###
            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

            ### Implementing Backward Pass ###
            output_error = y - final_outputs # Output layer error is the difference between desired target and actual output
            
            output_error_term = output_error
            hidden_error = np.dot(self.weights_hidden_to_output, output_error_term)
            hidden_error_term = hidden_error * (hidden_outputs * (1 - hidden_outputs))

            # Weight step (input to hidden)
            delta_weights_i_h +=  hidden_error_term.T * X[:, None]
            # Weight step (hidden to output)
            delta_weights_h_o += output_error_term * hidden_outputs [:, None]

        # Updating weights
        self.weights_hidden_to_output += self.lr * delta_weights_h_o / n_records
        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
        '''
        
        #### Implement the forward pass here ####
        ## 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

Run these unit tests to check the correctness of the network implementation. This will help you be sure that the network was implemented correctly before trying to train it.

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() == '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 0.016s

OK





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

Training the network

Here we set the hyperparameters for the network. The strategy here is to find hyperparameters such that the error on the training set is low, but data is not being overfitted. 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. That is, the loss on the validation set will start increasing as the training set loss drops.

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

Choose 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, if we use too many iterations, then the model with not generalize well to other data, this is called overfitting. We want to find a number here where the network has a low training loss, and the validation loss is at a minimum. As you start overfitting, we'll see the training loss continue to decrease while the validation loss starts to increase.

Choose 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. A good choice to start at is 0.1. If the network has problems fitting the data, try reducing the learning rate. 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.

Choose the number of hidden nodes

The more hidden nodes we have, the more accurate predictions the model makes. Try a few different numbers and to see how it affects the performance. 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 number of hidden units we choose.

import sys

### Set the hyperparameters here ###
iterations = 1000
learning_rate = 0.1
hidden_nodes = 2
output_nodes = 1

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.ix[batch].values, train_targets.ix[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: 0.4% ... Training loss: 1.208 ... Validation loss: 1.283

C:\Users\DNastri\AppData\Local\Continuum\Anaconda3\envs\dlnd\lib\site-packages\ipykernel_launcher.py:16: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#deprecate_ix
  app.launch_new_instance()


Progress: 99.9% ... Training loss: 0.318 ... Validation loss: 0.534

plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.legend()
_ = plt.ylim()

Check out predictions

Here we use the test data to view how well your 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.ix[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)

C:\Users\DNastri\AppData\Local\Continuum\Anaconda3\envs\dlnd\lib\site-packages\ipykernel_launcher.py:10: DeprecationWarning: 
.ix is deprecated. Please use
.loc for label based indexing or
.iloc for positional indexing

See the documentation here:
http://pandas.pydata.org/pandas-docs/stable/indexing.html#deprecate_ix
  # Remove the CWD from sys.path while we load stuff.