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
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]:
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.
In [4]:
rides[:24*10].plot(x='dteday', y='cnt')
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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]:
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 [6]:
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
In [7]:
# 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).
In [8]:
# 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.
In [24]:
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 ####
#
# Note: in Python, you can define a function with a lambda expression,
# as shown below.
#self.activation_function = lambda x : 0 # Replace 0 with your sigmoid calculation.
### If the lambda code above is not something you're familiar with,
# You can uncomment out the following three lines and put your
# implementation there instead.
#
def sigmoid(x):
return 1 / (1 + np.exp(-x))
self.activation_function = sigmoid
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 ###
# TODO: Hidden layer - Replace these values with your calculations.
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
# TODO: Output layer - Replace these values with your calculations.
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
#### Implement the backward pass here ####
### Backward pass ###
# TODO: Output error - Replace this value with your calculations.
error = y - final_outputs # Output layer error is the difference between desired target and actual output.
# TODO: Calculate the hidden layer's contribution to the error
hidden_error = np.dot(error, self.weights_hidden_to_output.T)
# TODO: Backpropagated error terms - Replace these values with your calculations.
output_error_term = error
hidden_error_term = hidden_error * hidden_outputs * (1 - hidden_outputs)
# Weight step (input to hidden)
#delta_weights_i_h += hidden_error_term * X[:,None]
delta_weights_i_h += hidden_error_term * X[:, np.newaxis]
# Weight step (hidden to output)
delta_weights_h_o += output_error_term * hidden_outputs[:, np.newaxis]
# TODO: Update the weights - Replace these values with your calculations.
self.weights_hidden_to_output += self.lr * delta_weights_h_o / n_records # update hidden-to-output weights with gradient descent step
self.weights_input_to_hidden += self.lr * delta_weights_i_h / n_records # update input-to-hidden weights with gradient descent step
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 ####
# TODO: Hidden layer - replace these values with the appropriate calculations.
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
# TODO: Output layer - Replace these values with the appropriate calculations.
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
In [10]:
def MSE(y, Y):
return np.mean((y-Y)**2)
In [25]:
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)
Out[25]:
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.
In [33]:
import sys
### Set the hyperparameters here ###
iterations = 5000
learning_rate = 0.4
hidden_nodes = 20
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)
In [34]:
plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.legend()
_ = plt.ylim()
In [35]:
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 submission, the following hyperparameters were set:
With the high number of iterations, the network was being overfitted to this particular data. My reviewer also noted that the number of hidden nodes should ideally be between the number of input nodes and number of output nodes. They recommended 20 hidden nodes.
With this feedback, the following hyperparameters were set:
The model does a good job of predicting the data right up until the Christmas time period where it overpredicts. This is due to a dip in the number of bike rentals during this holiday period as people are off on holidays. Being limited to the data from the historic non-holiday period, the model is unable to take into account the holiday period without any additional information, hence the greater discrepancy
My second reviewer noted that 5,000 was sufficient for number of iterations as the network fully converged at around 5,000 iterations. The learning rate was deemed too high as well.
TESTING:
A training loss of 0.4 and 5000 iterations provides a good balance between training and validation losses, given the lower number of iterations compared to the 2nd submission.
The set of hyperparameters set in the third submission were found to be sufficient and passed all criteria for the project.
It looks like you are clearly overfitting since you end the training process while the validation loss plot slope is still negative. I adjusted the plot graph scale to logarithmic limits, in order to be able to see the slope using the following code: $$plt.yscale('log')$$ That makes a plot like this, for your network parameters. IMO, any training beyond 8000 epochs will overfit your network.
As a general rule, a good rule of thumb is the half way in between the number of input and output units. You could consider to use 20 in this case and try.
Also, you might want to read this link
Your observation about the Christmas Holidays is spot on. The actual bike ridership demand decreases during this period. However, it is not that we are limited to the non-holiday data during training. The model has seen one occurrence of the Christmas holidays during training. Since the model has seen this type of data only once, it is probably not being able to discern if the drop in demand is due to seasonality or an aberration. Hence, the discrepancy in the predictions. Adding more data during training for holidays should help improve the prediction.
If you want to learn about other kinds of activation functions, please refer to this.
Please note that we are using an identity function as an activation function for the final output layer as the task is a regression one.
The backward pass has been implemented correctly. However, I have a few suggestions to make:
Although the network has converged successfully, the network has been trained for far longer than required. The network was fully converged around 5000 epochs and has made no improvement after that. There is no point in training the network for longer when there is no improvement in the loss.
20 is a good choice for the number of hidden units. Anything between 8 and 20 nodes works quite well for this problem. There is no mathematical formula that will determine the number of hidden units to be used for a given problem. However, there are heuristics that provide guidelines on selecting the number of hidden units. One of the most common heuristics is to use the average of input and output nodes. However, this is not a rule and must only be used as a guideline. e.g. for this problem, the average of input and output nodes comes around 25. However, in practice, nodes in the range between 8 and 20 work better. You can read the discussion here.
The learning rate is quite high for the current configuration of the neural net. This is evident from the learning curve wherein the training and validation loss decreases rapidly and then tapers off with a neck. You will have to lower your learning rate. With 20 hidden nodes, a learning rate in the range 0.2 - 0.3 should work fine. Please note that as you lower your learning rate, you might have to adjust your epochs too. Ideally, when learning rate decreases, the epochs increase and vice-versa. However, since the number of epochs is currently already high, you might not have to increase it much. Please experiment extensively before the next submission. Below is a cartoon representation of different learning rates from cs231n class notes. It will help you make an informed decision.
Here is a bit of intuition that might help: To explain as to why the december holiday season predictions performing poor is because prior information about such trends was not captured in the data, i.e the dataset that we trained on does not have information on holiday season even for the previous year. So when a new pattern is experienced the model performance is bad. This can be solved by training the model with more data possibly randomised data so that the model captures such patterns and predicts well. Think of it like you are the manager for the bike sharing service, this is your first year there and you know about the trends from January to October, knowing that data you are most likely to anticipate that the trends from your previous experience will hold right? So you would make preparations accordingly, but once december hits the holiday starts and then people stay at their houses maybe and there is a slump, as this is your first time here you only expected as per your experience. This is the same case with the model. You will have this in mind the next year December right? If you notice properly the holiday season extends over a week, but you will find that the days near christmas which are considered holiday season are marked as non-holidays, so a new feature marking the holiday season will be really helpful along with training over additional holiday season data.
As for your request the difference between using explicit np.multiply vs operator is that there might be a problem when the dtype i.e. float64 and int64 and so on and this might result in the output being different than expected. You can find an example of an issue with here
I read through your other reviews, there is just one thing common that the reviewers expect here, i.e, the model to converge to a minima and to have a really good performance, training loss ~0.7 and validation loss ~0.17. Any combination of the hyper parameters hidden units and learning rate that achieves such results in around 5000 epochs and making good predictions overall except for the holiday season is what is expected.
