Udacity Deep Learning Foundations Nanodegree¶

Bike Rental Ridership Prediction with a Deep Neural Network in Python¶

In this project, you'll build your first neural network and use it to predict daily bike rental ridership. We've provided some of the code, but left the implementation of the neural network up to you (for the most part). After you've submitted this project, feel free to explore the data and the model more.

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

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. You'll learn more about this soon!

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

rides = pd.read_csv(data_path)

rides.head()

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 in the data set. 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. You'll be trying to capture all this with your model.

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

<matplotlib.axes._subplots.AxesSubplot at 0x8ea8a20>

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()

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

data.head()

Splitting the data into training, testing, and validation sets¶

We'll save the last 21 days of the data 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 the last 21 days 
test_data = data[-21*24:]
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 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¶

Below you'll build your network. We've built out the structure and the backwards pass. You'll implement the forward pass through the network. You'll 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.

Hint: You'll need the derivative of the output activation function ($f(x) = x$) for the backpropagation implementation. If you aren't familiar with calculus, this function is equivalent to the equation $y = x$. What is the slope of that equation? That is the derivative of $f(x)$.

Below, you have these tasks:

Implement the sigmoid function to use as the activation function. Set self.activation_function in __init__ to your sigmoid function.
Implement the forward pass in the train method.
Implement the backpropagation algorithm in the train method, including calculating the output error.
Implement 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.hidden_nodes**-0.5, 
                                       (self.hidden_nodes, self.input_nodes))
        #returns a numpy array of hidden nodes elements each containing input nodes elements. These are the weights to
        #multiply with each feature=input node
        # It returns a shape of (2,30)

        self.weights_hidden_to_output = np.random.normal(0.0, self.output_nodes**-0.5, 
                                       (self.output_nodes, self.hidden_nodes))
        self.lr = learning_rate
        
        #### Set this to your implemented sigmoid function ####
        # Activation function is the sigmoid function

        
        self.activation_function = lambda x: self.sigmoid(x)
        
    def sigmoid(self, x):
            return 1/(1 + np.exp(-x))
    
    def train(self, inputs_list, targets_list):
        # Convert inputs list to 2d array
        # We are actually making a column out of each element in the list,resulting in an array of lists(or arrays)
        #the resulting shape is (30,1)
        inputs = np.array(inputs_list, ndmin=2).T
        targets = np.array(targets_list, ndmin=2).T
        
        #### Implement the forward pass here ####
        ### Forward pass ###
        # TODO: Hidden layer
        
        # signals into hidden layer
        # multiply each of the weight lists with the input features
        # it results in a shape of (2,1) which is two arrays of one value each to be fed into the two hidden nodes
        hidden_inputs = np.dot(self.weights_input_to_hidden,inputs)
        # signals from hidden layer
        #Apply the activation function you get out a (2,1) array
        hidden_outputs =self.activation_function(hidden_inputs) 
        
        # TODO: Output layer
        
        # signals into final output layer
        #apply the weights_hidden_to_output now which is a (1,2) array to the output of the hidden nodes (2,1)
        #you get a (1,1) array 
        final_inputs = np.dot(self.weights_hidden_to_output, hidden_outputs) 
        # signals from final output layer
        # no function so what comes in goes out
        final_outputs = final_inputs 
        
        #### Implement the backward pass here ####
        ### Backward pass ###
        
        # TODO: Output error (Wouldn't it be better if we had three output nodes?)
        
        # Output layer error is the difference between desired target and actual output.
        # deducts from the desired feature value and return a (1,1) array 
        output_errors = targets - final_outputs
        
        # TODO: Backpropagated error
        
        # errors propagated to the hidden layer
        # going back now:  weights_hidden_to_output is a (1,2) array and output_errors is a (1,1) array , so we have to 
        # transpose the weights so that we can multiply them and we get a (2,1) array
        hidden_errors = np.dot(self.weights_hidden_to_output.T,output_errors)
                                               
        # hidden layer gradients
        # hidden_outputs is a (2,1) array that came out of the sigmoid function of the hidden layer (2 hidden nodes)
        # the hidden grad is the derivative of this output which we will use as our guide to minimize the error
        # it is also a (2,1) array
        hidden_grad = hidden_outputs * (1 - hidden_outputs)
        
        # TODO: Update the weights
        # update hidden-to-output weights with gradient descent step
        self.weights_hidden_to_output += self.lr * np.dot(output_errors, hidden_outputs.T)
        
        # update input-to-hidden weights with gradient descent step
        # this is where we apply the gradient to the hidden errors and then dot it with the inputs to get the desired shape
        self.weights_input_to_hidden += self.lr * np.dot(hidden_grad*hidden_errors, inputs.T)
           
    def run(self, inputs_list):
        # Run a forward pass through the network
        inputs = np.array(inputs_list, ndmin=2).T
        
        #### Implement the forward pass here ####
        # TODO: Hidden layer
        # signals into hidden layer
        hidden_inputs = np.dot(self.weights_input_to_hidden,inputs) 
        # signals from hidden layer
        hidden_outputs = self.activation_function(hidden_inputs)
        
        # TODO: Output layer
        # signals into final output layer
        final_inputs = np.dot(self.weights_hidden_to_output,hidden_outputs)
        final_outputs = final_inputs# signals from final output layer 
        
        return final_outputs

# Calculate the difference between the loss of the training and the validation set
def MSE(y, Y):
    return np.mean((y-Y)**2)

Training the network¶

Here you'll set the hyperparameters for the network. The strategy here is to find hyperparameters such that the error on the training set is low, but you're not overfitting to the data. If you 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.

You'll also be using a method know 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. You'll learn more about SGD later.

Choose the number of epochs¶

This is the number of times the dataset will pass through the network, each time updating the weights. As the number of epochs increases, the network becomes better and better at predicting the targets in the training set. You'll need to choose enough epochs to train the network well but not too many or you'll be overfitting.

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 you have, the more accurate predictions the model will make. Try a few different numbers and see how it affects the performance. You 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 you choose.

import sys

### Set the hyperparameters here ###
epochs = 2000
learning_rate = 0.05
hidden_nodes = 28
output_nodes = 1

#get the number of input nodes from the shape of the first row of the train_features
N_i = train_features.shape[1]
#Initiate the network
network = NeuralNetwork(N_i, hidden_nodes, output_nodes, learning_rate)

#record the losses in a dictionary of lists
losses = {'train':[], 'validation':[]}

#train the network
for e in range(epochs):
    # Go through a random batch of 128 records from the training data set
    # train_features.index is a generator of the index number of the pd dataframe
    batch = np.random.choice(train_features.index, size=128)
    for record, target in zip(train_features.ix[batch].values, 
                              train_targets.ix[batch]['cnt']):
        network.train(record, target)
    
    # Printing out the training progress
    train_loss = MSE(network.run(train_features), train_targets['cnt'].values)
    val_loss = MSE(network.run(val_features), val_targets['cnt'].values)
    sys.stdout.write("\rProgress: " + str(100 * e/float(epochs))[:4] \
                     + "% ... Training loss: " + str(train_loss)[:5] \
                     + " ... Validation loss: " + str(val_loss)[:5])
    
    losses['train'].append(train_loss)
    losses['validation'].append(val_loss)

Progress: 99.9% ... Training loss: 0.049 ... Validation loss: 0.155

plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.legend()
plt.ylim(ymax=0.5)

(0.0, 0.5)

Check out your predictions¶

Here, use the test data to view how well your network is modeling the data. If something is completely wrong here, make sure each step in your network is implemented correctly.

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

mean, std = scaled_features['cnt']
predictions = network.run(test_features)*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)

Thinking about your results¶

Answer these questions about your results. How well does the model predict the data? Where does it fail? Why does it fail where it does?

Note: You can edit the text in this cell by double clicking on it. When you want to render the text, press control + enter

Your answer below¶

The validation loss for our model and these hyper parameters ranges from o.130 to 0.143. The model does much better on the normal days versus the weekends and holiday season. The large variance of the bike sharing users over the weekend and especially on the holiday season at the end of December does not allow the model to acurately predict these values without overfitting. Probably more training data , over a larger period of time would improve the prediction for these outlyiing values.

Having tried various hyperarameter combinations, I have noticed that after around 2000 epochs the validation and training stop converging and start diverging slightly , indicating that overfitting may be taking place after this number. I did not observe any improvement in the validation loss or the model for learning rates below 0.09 or above a number of hidden nodes around 30.

Unit tests¶

Run these unit tests to check the correctness of your network implementation. These tests must all be successful to pass the project.

import unittest

inputs = [0.5, -0.2, 0.1]
targets = [0.4]
test_w_i_h = np.array([[0.1, 0.4, -0.3], 
                       [-0.2, 0.5, 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,  0.39775194, -0.29887597],
                                              [-0.20185996,  0.50074398,  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>

	instant	dteday	season	mnth	hr	weekday	weathersit	temp	atemp	hum	casual	registered	cnt
0	1	2011-01-01	1	1	0	6	1	0.24	0.2879	0.81	3	13	16
1	2	2011-01-01	1	1	1	6	1	0.22	0.2727	0.80	8	32	40
2	3	2011-01-01	1	1	2	6	1	0.22	0.2727	0.80	5	27	32
3	4	2011-01-01	1	1	3	6	1	0.24	0.2879	0.75	3	10	13
4	5	2011-01-01	1	1	4	6	1	0.24	0.2879	0.75	0	1	1

	temp	hum	windspeed	casual	registered	cnt	season_1	...	weekday_6
0	-1.334609	0.947345	-1.553844	-0.662736	-0.930162	-0.956312	1.0	...	1.0
1	-1.438475	0.895513	-1.553844	-0.561326	-0.804632	-0.823998	1.0	...	1.0
2	-1.438475	0.895513	-1.553844	-0.622172	-0.837666	-0.868103	1.0	...	1.0
3	-1.334609	0.636351	-1.553844	-0.662736	-0.949983	-0.972851	1.0	...	1.0
4	-1.334609	0.636351	-1.553844	-0.723582	-1.009445	-1.039008	1.0	...	1.0