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.
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%matplotlib inline
%config InlineBackend.figure_format = 'retina'
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
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!
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data_path = 'Bike-Sharing-Dataset/hour.csv'
rides = pd.read_csv(data_path)
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rides.head()
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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. You'll be trying to capture all this with your model.
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rides[:24*10].plot(x='dteday', y='cnt')
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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()
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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.
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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
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# 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).
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# 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:]
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:
self.activation_function in __init__ to your sigmoid function.train method.train method, including calculating the output error.run method.
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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
#### TODO: Set self.activation_function to your implemented sigmoid function ####
# Replace 0 with your sigmoid calculation.
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):
#### Implement the forward pass here ####
### Forward pass ###
# hidden_inputs: signals into hidden layer
# hidden_outputs: signals from hidden layer
# final_inputs: signals into final output layer
# final_outputs: signals from final output layer
hidden_inputs, hidden_outputs, final_inputs, final_outputs = self.forward_pass(X)
#### Implement the backward pass here ####
### Backward pass ###
error, output_error_term, hidden_error, hidden_error_term = self.backward_pass_errors(y, final_outputs, hidden_outputs)
# Weight step (input to hidden)
# the hidden_error_term by the transpose(inputs)
delta_weights_i_h += hidden_error_term * X[:, None]
# Weight step (hidden to output)
# the output_error_term by the transpose(hidden_outputs)
delta_weights_h_o += error * hidden_outputs[:, None]
# TODO: Update the weights - Replace these values with your calculations.
# 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 backward_pass_errors(self, targets, final_outputs, hidden_outputs):
"""Calculate the errors and error terms for the network, output layer, and hidden layer
Notes about the shapes
Numbers
# i: number of input units, 3 in our tests
# j: number of hidden units, 2 in our tests
# k: number of output units, 1 in our tests
Matrices
# y: a 1 x k matrix of target outputs
# final_outputs: a 1 x k matrix of network outputs
# hidden_outputs: sigmoid of hidden_inputs, a 1 x j matrix
# error: target - final_outputs, a 1 x k matrix for each of the k output units
# output_error_term = error, a 1 x k matix
# hidden_error = error DOT transpose(w_h_o) or 1 x k DOT k x j, yields a 1 x j matrix
for each of the j hidden units
# hidden_error_term = hidden_error * activation derivative of hidden_outputs,
1 x j * 1 x j, yields a 1 x j matrix for each of the j hidden units
"""
# Output layer error is the difference between desired target and actual output.
error = targets - final_outputs
# Get the output error, which is just the error
output_error_term = error
# Take the output error term and scale by the weights from the hidden layer to that output
hidden_error = np.dot(output_error_term, self.weights_hidden_to_output.T)
# Use derivative of activation for the hidden outputs
hidden_error_term = hidden_error * hidden_outputs * (1 - hidden_outputs)
return error, output_error_term, hidden_error, hidden_error_term
def forward_pass(self, features):
"""Calculate the values for the inputs and outputs of the hidden and final layers
Notes about the shapes
Numbers
# i: number of input units, 3 in our tests
# j: number of hidden units, 2 in our tests
# k: number of output units, 1 in our tests
Matrices
# features: a 1 x i row vector
# w_i_h: i x j matrix of weights from input units to hidden units
# w_h_o: j x k matrix of weights from hidden units to output units
# hidden_inputs: features DOT w_i_h, yeilds a 1 x j matrix, for each of the j hidden units
# hidden_outputs: sigmoid of hidden_inputs, also 1 x j matrix
# final_inputs: hidden_outputs DOT w_h_o, so 1 x j DOT j x k, yields 1 x k matrix
# final_outputs: same as the final inputs, so 1 x k matrix
"""
hidden_inputs = np.dot(features, self.weights_input_to_hidden)
hidden_outputs = self.activation_function(hidden_inputs)
final_inputs = np.dot(hidden_outputs, self.weights_hidden_to_output)
final_outputs = final_inputs
return hidden_inputs, hidden_outputs, final_inputs, final_outputs
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_inputs, hidden_outputs, final_inputs, final_outputs = self.forward_pass(features)
return final_outputs
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def MSE(y, Y):
return np.mean((y-Y)**2)
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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(verbosity=1).run(suite)
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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.
This is the number of batches of samples from the training data we'll use to train the network. The more iterations you use, the better the model will fit the data. However, if you use too many iterations, then the model with not generalize well to other data, this is called overfitting. You 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, you'll see the training loss continue to decrease while the validation loss starts to increase.
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.
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.
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import sys
### Set the hyperparameters here ###
# Results from earlier hyperparameter settings
# (iterations, learning_rate, hidden_nodes) -> (Training Loss, Validation Loss)
# (5000, 0.8, 8) -> (0.059, 0.148)
iterations = 5000 #100
learning_rate = 0.9 #0.1
hidden_nodes = 8 #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)
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plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.legend()
_ = plt.ylim()
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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)
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
In my first submsission after setting the following hyperparameters:
learning_rate = 0.4 #0.1
hidden_nodes = 8 #2
After the first review I was advised to lower the learning rate:
You notice your loss plot can be a bit spiky? That indicates a high learning rate usually (but more prominent in more complex models). At such high learning rates, the network doesn't usually converge because the weight update steps are too large and the weights don't end up converging. So there's overshooting of the minimum of the cost function. You can experiment with different values here (lower it a bit and try).
In my second submission, I followed the recommendation and lowered my learning rate to:
learning_rate = 0.18 #0.1
After the second review I was instead advised to raise the learning rate
Please note that the learning rate is the learn rate divided by the n_records, here you're aiming at a very very low learn rate. I would suggest since you anyway have to change other things, to change this one as well. Set it to ~0.8 or 0.9.
It seems if I set the learning_rate to 0.9, things work out fine, but I don't know if in general I should see a spiky loss plot as a clue to lower the learning rate or use the recommended guide to set the learning rate higher. Any additional recommendations are welcome.