dlnd-your-first-neural-network


Your first neural network

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.


In [1]:
%matplotlib inline
%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. You'll learn more about this soon!


In [2]:
data_path = 'Bike-Sharing-Dataset/hour.csv'

rides = pd.read_csv(data_path)

In [3]:
rides.head()


Out[3]:
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 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.


In [4]:
rides[:24*10].plot(x='dteday', y='cnt')


Out[4]:
<matplotlib.axes._subplots.AxesSubplot at 0x15c7c11e908>

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().


In [5]:
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()


Out[5]:
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.


In [87]:
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 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.


In [7]:
# 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).


In [8]:
# 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:

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

In [430]:
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def sigmoid_derivative(x):
    return sigmoid(x) * (1 - sigmoid(x))

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))

        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 = sigmoid
        
        # Activation function derivative
        # We could use the activation_function directly in the code, but that would make it very
        # dependent on the actual function being used. For class flexibility and clarity of steps,
        # let's leave it defined separately.
        self.activation_function_derivative = sigmoid_derivative
    
    def train(self, inputs_list, targets_list):
        # Convert inputs list to 2d array
        inputs = np.array(inputs_list, ndmin=2).T
        targets = np.array(targets_list, ndmin=2).T
        
        ### Forward pass ###
        
        ## Hidden layer
        
        # inputs to hidden layer: weights_input_to_hidden . inputs
        # this should already give us a matrix of the right dimensionality for the hidden layer
        hidden_inputs = np.dot(self.weights_input_to_hidden, inputs)        
        # outputs of hidden_layer f(hidden_inputs) => sigmoid(hidden_inputs)
        # activation functions do not change the dimensionality of the data
        hidden_outputs = self.activation_function(hidden_inputs)
        
        ## Output layer
        
        # input to output layer: weights_hidden_to_output . hidden_layer_outputs
        output_layer_in = np.dot(self.weights_hidden_to_output, hidden_outputs)
        # output of output layer: f(output_layer_in) = output_layer_in (f(x) = x)
        output = output_layer_in
        
        ### Backward pass ###
        
        # absolute error: desired - predicted
        output_errors = targets - output
        # Pending question: Should we take the squared of the error? Doing so produces divergence. Why?
        
        # hidden_errors: errors distributed to the hidden layer, according to the weights interconnecting the hidden and
        # output layer
        hidden_errors = np.dot(self.weights_hidden_to_output.T, output_errors)
        
        # hidden_grad: gradient of the error for each of the weights interconnecting the hidden and output layer.
        # As a gradient, we need to multiply it by the derivative of the activation function for this very same layer,
        # based on the input that was originally given to it.
        # Since this derivative is the sigmoid function, another way to write this would be:
        # hidden_grad = hidden_errors * hidden_inputs * (1 - hidden_inputs)
        # Notice that these are all element-wise multiplications, so they do not change the dimensionality of the
        # result. This means that hidden_errors contains an error for each weight, and hidden_grad will contain a gradient
        # for each weight too.
        # hidden_grad = hidden_errors * self.activation_function_derivative(hidden_inputs)        
        
        # Project review note:
        # > Note: In this case, we're calling the hidden gradient just the derivative of the sigmoid function, so please change hidden_grad to be just the derivative of the sigmoid function.
        # > Remove hidden errors during calculation.
        # > Also update delta_w_i_h accordingly. Your Implementation should just be as is. Just some code refactoring.
        hidden_grad = self.activation_function_derivative(hidden_inputs)
        
        # delta for weights for output layer: learningRate * error * input to output layer (hidden_outputs)
        # transposing hidden_outputs so that it takes the shape required for our weights marix
        delta_w_h_o = self.lr * np.dot(output_errors, hidden_outputs.T)
        # delta for weights for hidden layer: learningRate * errorGradient * input layer (inputs)
        # transposing inputs so that it takes the shape required for our weights matrix
        delta_w_i_h = self.lr * hidden_errors * np.dot(hidden_grad, inputs.T)
        
        # update weights
        self.weights_hidden_to_output += delta_w_h_o        
        self.weights_input_to_hidden += delta_w_i_h
                        
    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 ####
        # Hidden layer
        hidden_inputs = np.dot(self.weights_input_to_hidden, inputs) # signals into hidden layer
        hidden_outputs = self.activation_function(hidden_inputs) # signals from hidden layer
        
        # Output layer
        final_inputs = np.dot(self.weights_hidden_to_output, hidden_outputs) # signals into final output layer
        # activation function for final layer is f(x) = x, no changes required
        final_outputs = final_inputs # signals from final output layer 
        
        return final_outputs

In [431]:
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.


In [432]:
import sys

### Set the hyperparameters here ###
epochs = 1000 # bigger epochs don't make a bigger difference in the loss
learning_rate = 0.15 # bigger than 0.2 seems to oscilate too much

#hidden_nodes = 7 # 7 seems to be a "good" number between loss and performance. Bigger numbers don't increase accuracy.
# Project review note:
# > I think you can still do better here
# > Refer to this question
# > https://www.quora.com/How-do-I-decide-the-number-of-nodes-in-a-hidden-layer-of-a-neural-network
hidden_nodes = 28 # thread encourages to use geometric mean between input and output, which is also 7. Arithmetic mean is 28, also recommended.
output_nodes = 1

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

losses = {'train':[], 'validation':[]}
for e in range(epochs):
    # Go through a random batch of 128 records from the training data set
    batch = np.random.choice(train_features.index, size=128)
    for record, target in zip(train_features.ix[batch].values, 
                              train_targets.ix[batch]['cnt']):
        debug = (e == 1) or (e == 0)
        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: 0.057 ... Validation loss: 0.153

In [433]:
plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.legend()
plt.ylim(ymax=0.5)


Out[433]:
(-0.020686886070368879, 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.


In [434]:
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 model very accurately predicts the data when it roams around the 1.5 mark. This very likely means that there is a pattern in the data of the parameters and inputs leading up to that range. However, the predictions do still keep that learnt pattern and those number of sales don't apply to the time range, which was a lot lower.

The lower numbers in the actual data correspond to Christmas and New Year's Eve, which could explain in the real world why the pattern changes. (Maybe people don't buy / repair bikes during the festivities?)

Regardless of the reason, these are aspects that our data did not include and as such, the neural network could nto learn.

Unit tests

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


In [416]:
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
Out[416]:
<unittest.runner.TextTestResult run=5 errors=0 failures=0>

In [ ]: