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 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.
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rides[:24*7].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.tail()
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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 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).
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# 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:]
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.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
self.activation_function = sigmoid
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
1. Hidden inputs, matrix multiple weights and inputs
2. Activation function (Sigmoid)
3. Output layer, matrix multiple weights and hidden inputs
No activation function as it's an output layer
"""
hidden_inputs = np.dot(self.weights_input_to_hidden, inputs)
hidden_outputs = self.activation_function(hidden_inputs)
final_inputs = np.dot(self.weights_hidden_to_output, hidden_outputs)
final_outputs = final_inputs
# Credit http://iamtrask.github.io/2015/07/28/dropout/
# if(do_dropout):
# final_outputs *= np.random.binomial([np.ones((len(final_outputs),hidden_dim))],1-dropout_percent)[0] * (1.0/(1-dropout_percent))
# print(final_outputs)
"""
Backward pass
1. Error function, compare target error to final output
2. Hidden gradient, as hidden_outputs (1 - hidden_outputs)
3. Hidden error, matrix multiple weights by output errors
Transpose weights to align with final_outputs that was
transposed earlier.
4. Update hidden weights, learning rate * matrix multiple
output errors and the transpose of the hidden_outputs.
5. Update input weights, learning rate * the matrix
multiple of the (hidden gradient and the hidden errors),
and the transposed inputs.
The matrix multiplier in 4 and 5 is needed to allow for
an n demensial array of hidden layers.
What is backpropagation?
Imagine 3 car trying to pass through a narrow tunnel. The first time
the cars try to go through, they all bump into each other.
The driver's realize that 3 can't fit at once. (This is the error)
They reflect on a better way to do it, and say "hah!, what
if we tried 2 at a time?" So they try 2 at a time, they still don't
fit. But this time they get a little further into the tunnel.
In effect the error has been reduced. This little change,
from 3 cars not fitting to 2 cars sorta fitting, can be imagined
as a line, sloping down. (1,3 -> 2,2) If you want to be fancy you call
it dee-cars/dee-fit. A little bit of a change in the number of cars
trying to go through with respect to how well the cars can fit.
(the derivative/or hidden_gradient in this case). Finnally,
the drivers yell at each other to go 1 at a time. (Weight update.),
and they all make it through! :)
"""
output_errors = (targets - final_outputs)
hidden_grad = hidden_outputs * (1-hidden_outputs)
hidden_errors = np.dot(self.weights_hidden_to_output.T, output_errors)
self.weights_hidden_to_output += self.lr * np.dot(output_errors, hidden_outputs.T)
self.weights_input_to_hidden += self.lr * np.dot(hidden_grad * hidden_errors, inputs.T)
# Uncomment line to see shapes.
# print(hidden_errors.shape, hidden_grad.shape,inputs.T.shape)
def run(self, inputs_list):
# Run a forward pass through the network
inputs = np.array(inputs_list, ndmin=2).T
#### forward pass ####
# print(inputs.shape)
# print(self.weights_input_to_hidden.shape)
# print(self.weights_hidden_to_output.shape)
hidden_inputs = np.dot(self.weights_input_to_hidden, inputs)
hidden_outputs = self.activation_function(hidden_inputs)
# print("Forward pass, Hidden_inputs \n", hidden_inputs)
# Output layer
# signals into final output layer
final_inputs = np.dot(self.weights_hidden_to_output, hidden_outputs)
# print("Forward pass, final inputs \n", final_inputs)
# signals from final output layer
final_outputs = final_inputs
return final_outputs
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def MSE(y, Y):
return np.mean((y-Y)**2)
def sigmoid(x):
return 1 / (1 + np.exp(-x))
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import sys
### Set the hyperparameters here ###
epochs = 10000
learning_rate = 0.1
hidden_nodes = 16
output_nodes = 1
N_i = train_features.shape[1]
network = NeuralNetwork(N_i, hidden_nodes, output_nodes, learning_rate)
# Credit http://iamtrask.github.io/2015/07/28/dropout/
# alpha,hidden_dim,dropout_percent,do_dropout = (0.5,1,0.2,True)
losses = {'train':[], 'validation':[]}
epoch_count = 1
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']):
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)
# Reduce learning rate over time,
learning_rate = min(.0001, learning_rate/epoch_count)
epoch_count = epoch_count + 1
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 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.
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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# plt.plot(losses['train'], label='Training loss')
plt.plot(losses['validation'], label='Validation loss')
plt.plot(losses['train'], label='Train loss')
plt.legend()
plt.ylim(ymax=1)
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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)
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?
Not great. It seems to capture the general trends however doesn't do well with with the "low" extremes (ie Dec 23-26).
Added an exponential decay to the learning rate and that seemed to help a little. It's kinda converging.
If the model can't accruately fit the data it would indicate an under capacity. Capacity is the model's ability to represent the data effectively. If the difference between training and validation is high it would indicate over fitting. Over fitting is when the model learns the training data well, but fails to generalize. Gerneralize means the model will work on new previously unseen data. Both problems can be improved with more sophisticated models (ie Dropout, Hinton et al), more assumptions on the staistical learning domain of the model (ie restict model to week days or weekends perhaps), and simply more time tuning hyper parameters.
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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)
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