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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 building a simple neural network
A net takes in inputs, them multiplies each input by a corresponding weight and sums them together and adds this to a constant. This is then passed through a function to spit out an output.
Here, we're mostly using the sigmoid function.
the sigmoid function
<img src="images/sigmoid.png" width=500 align="left">
The sigmoid function
Step1:  Gradient descent
A simple way to think of this
Step3:  simple Gradient descent exercise
Below, you'll calculate one gradient descent step for the weights of a simple network with two inputs and one output unit with a sigmoid activation function.
Your goal here is to calculate the correct weight step using gradient descent. Remember that the weight step is the learning rate times the error times the input values
Step4:  Implementing gradient descent
use gradient descent to train a network on graduate school admissions data (found at http
Step6:  the actual implmentation
the dataset
Step8:  Multilayer perceptrons
for a single neuron, you just have one set of weights, but for multiple layers, you need a set of weights per layer.
so you go from a an array to a matrix.
Step10:  Back Propogate
Similar to how inputs are propageted through the network, back propogate is sending the errors backwards through the network.
A short overview
Step12:  implement backprop.py
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Python Code:
x = [i for i in range(-10,10)]
#print(x)
def sigmoid(num):
    return 1.0 / (1.0 + np.exp(-num))
plt.plot(range(-10,10), [sigmoid(i) for i in x])
plt.show()
Explanation: building a simple neural network
A net takes in inputs, them multiplies each input by a corresponding weight and sums them together and adds this to a constant. This is then passed through a function to spit out an output.
Here, we're mostly using the sigmoid function.
the sigmoid function
<img src="images/sigmoid.png" width=500 align="left">
The sigmoid function:
End of explanation
from IPython.display import YouTubeVideo
YouTubeVideo('29PmNG7fuuM', width="560")
# Defining the sigmoid function for activations
def sigmoid(x):
    return 1/(1+np.exp(-x))
# Derivative of the sigmoid function
def sigmoid_prime(x):
    return sigmoid(x) * (1 - sigmoid(x))
x = np.array([0.1, 0.3])
y = 0.2
weights = np.array([-0.8, 0.5])
# The learning rate, eta in the weight step equation
learnrate = 0.5
# The neural network output
nn_output = sigmoid(x[0]*weights[0] + x[1]*weights[1])
# or nn_output = sigmoid(np.dot(x, w))
# output error
error = y - nn_output
# error gradient
error_grad = error * sigmoid_prime(np.dot(x,weights))
# Gradient descent step
del_w = [ learnrate * error_grad * x[0],
          learnrate * error_grad * x[1]]
# or del_w = learnrate * error_grad * x
Explanation: Gradient descent
A simple way to think of this: If you are standing on top of a mountain of errors, then one way to get to the solution is to take a tiny step towards the correct direction.
End of explanation
def sigmoid(x):
    
    Calculate sigmoid
    
    return 1/(1+np.exp(-x))
learnrate = 0.5
x = np.array([1, 2])
y = np.array(0.5)
# Initial weights
w = np.array([0.5, -0.5])
# Calculate one gradient descent step for each weight
# TODO: Calculate output of neural network
nn_output = sigmoid(np.dot(x,w)) 
# TODO: Calculate error of neural network
error = y - nn_output
# TODO: Calculate change in weights
del_w = learnrate * error * nn_output * (1 - nn_output) * x
print('Neural Network output:')
print(nn_output)
print('Amount of Error:')
print(error)
print('Change in Weights:')
print(del_w)
Explanation: simple Gradient descent exercise
Below, you'll calculate one gradient descent step for the weights of a simple network with two inputs and one output unit with a sigmoid activation function.
Your goal here is to calculate the correct weight step using gradient descent. Remember that the weight step is the learning rate times the error times the input values:
End of explanation
admissions = pd.read_csv('data/binary.csv')
admissions.head()
data = pd.concat([admissions, pd.get_dummies(admissions['rank'], prefix='rank')], axis=1)
data = data.drop('rank', axis=1)
data.head()
# Standarize features
for field in ['gre', 'gpa']:
    mean, std = data[field].mean(), data[field].std()
    data.loc[:,field] = (data[field]-mean)/std
data.head()
# Split off random 10% of the data for testing
np.random.seed(42)
sample = np.random.choice(data.index, size=int(len(data)*0.9), replace=False)
data, test_data = data.ix[sample], data.drop(sample)
data.shape, test_data.shape
# Split into features and targets
features, targets = data.drop('admit', axis=1), data['admit']
features_test, targets_test = test_data.drop('admit', axis=1), test_data['admit']
Explanation: Implementing gradient descent
use gradient descent to train a network on graduate school admissions data (found at http://www.ats.ucla.edu/stat/data/binary.csv. This dataset has three input features: GRE score, GPA, and the rank of the undergraduate school (numbered 1 through 4). Institutions with rank 1 have the highest prestige, those with rank 4 have the lowest.
End of explanation
def sigmoid(x):
    
    Calculate sigmoid
    
    return 1 / (1 + np.exp(-x))
# Use to same seed to make debugging easier
np.random.seed(42)
n_records, n_features = features.shape
last_loss = None
# Initialize weights
weights = np.random.normal(scale=1 / n_features**.5, size=n_features)
# Neural Network hyperparameters
epochs = 10000
learnrate = -0.015
for e in range(epochs):
    del_w = np.zeros(weights.shape)
    for x, y in zip(features.values, targets):
        # Loop through all records, x is the input, y is the target
        # TODO: Calculate the output
        output = sigmoid(np.dot(x,weights))
        # TODO: Calculate the error
        error = output - y
        # TODO: Calculate change in weights
        del_w += error * output * (1 - output) * x
        # TODO: Update weights
        weights += (del_w * learnrate) / n_records
    # Printing out the mean square error on the training set
    if e % (epochs / 10) == 0:
        out = sigmoid(np.dot(features, weights))
        loss = np.mean((out - targets) ** 2)
        if last_loss and last_loss < loss:
            print("Train loss: ", loss, "  WARNING - Loss Increasing")
        else:
            print("Train loss: ", loss)
        last_loss = loss
# Calculate accuracy on test data
tes_out = sigmoid(np.dot(features_test, weights))
predictions = tes_out > 0.5
accuracy = np.mean(predictions == targets_test)
print("Prediction accuracy: {:.3f}".format(accuracy))
Explanation: the actual implmentation
the dataset:
features, targets, features_test, targets_test
End of explanation
# Number of records and input units
n_records, n_inputs = features.shape
# Number of hidden units
n_hidden = 2
weights = np.random.normal(0, 1/n_inputs**-2, size=(n_inputs, n_hidden))
weights
import numpy as np
def sigmoid(x):
    
    Calculate sigmoid
    
    return 1/(1+np.exp(-x))
# Network size
N_input = 4
N_hidden = 3
N_output = 2
np.random.seed(42)
# Make some fake data
X = np.random.randn(4)
weights_in_hidden = np.random.normal(0, scale=0.1, size=(N_input, N_hidden))
weights_hidden_out = np.random.normal(0, scale=0.1, size=(N_hidden, N_output))
# TODO: Make a forward pass through the network
hidden_layer_in = np.dot(X, weights_in_hidden)
hidden_layer_out = sigmoid(hidden_layer_in)
print('Hidden-layer Output:')
print(hidden_layer_out)
output_layer_in = np.dot(hidden_layer_out, weights_hidden_out)
output_layer_out = sigmoid(output_layer_in)
print('Output-layer Output:')
print(output_layer_out)
Explanation: Multilayer perceptrons
for a single neuron, you just have one set of weights, but for multiple layers, you need a set of weights per layer.
so you go from a an array to a matrix.
End of explanation
def sigmoid(x):
    
    Calculate sigmoid
    
    return 1 / (1 + np.exp(-x))
x = np.array([0.5, 0.1, -0.2])
target = 0.6
learnrate = 0.5
weights_input_hidden = np.array([[0.5, -0.6],
                                 [0.1, -0.2],
                                 [0.1, 0.7]])
weights_hidden_output = np.array([0.1, -0.3])
## Forward pass
hidden_layer_input = np.dot(x, weights_input_hidden)
hidden_layer_output = sigmoid(hidden_layer_input)
output_layer_in = np.dot(hidden_layer_output, weights_hidden_output)
output = sigmoid(output_layer_in)
## Backwards pass
## TODO: Calculate error
error = target - output
# TODO: Calculate error gradient for output layer
del_err_output = error * output * (1 - output)
# TODO: Calculate error gradient for hidden layer
del_err_hidden = np.dot(del_err_output, weights_hidden_output) * \
                 hidden_layer_output * (1 - hidden_layer_output)
# TODO: Calculate change in weights for hidden layer to output layer
delta_w_h_o = learnrate * del_err_output * hidden_layer_output
# TODO: Calculate change in weights for input layer to hidden layer
delta_w_i_o = learnrate * del_err_hidden * x[:, None]
print('Change in weights for hidden layer to output layer:')
print(delta_w_h_o)
print('Change in weights for input layer to hidden layer:')
print(delta_w_i_o)
Explanation: Back Propogate
Similar to how inputs are propageted through the network, back propogate is sending the errors backwards through the network.
A short overview:
Udacity's 1min overview
series of short videos on neural networks
long lecture, useful
End of explanation
import numpy as np
#from data_prep import features, targets, features_test, targets_test
np.random.seed(42)
def sigmoid(x):
    
    Calculate sigmoid
    
    return 1 / (1 + np.exp(-x))
# Hyperparameters
n_hidden = 3  # number of hidden units
epochs = 500
learnrate = -0.02
n_records, n_features = features.shape
last_loss = None
# Initialize weights
weights_input_hidden = np.random.normal(scale=1 / n_features ** .5,
                                        size=(n_features, n_hidden))
weights_hidden_output = np.random.normal(scale=1 / n_features ** .5,
                                         size=n_hidden)
for e in range(epochs):
    del_w_input_hidden = np.zeros(weights_input_hidden.shape)
    del_w_hidden_output = np.zeros(weights_hidden_output.shape)
    for x, y in zip(features.values, targets):
        ## Forward pass ##
        # TODO: Calculate the output
        hidden_input = np.dot(x, weights_input_hidden)
        hidden_activations = sigmoid(hidden_input)
        output = np.dot(hidden_activations, weights_hidden_output)
        ## Backward pass ##
        # TODO: Calculate the error
        error = y - output
        # TODO: Calculate error gradient in output unit
        output_error = error * output * (1 - output)
        # TODO: propagate errors to hidden layer
        hidden_error = np.dot(output_error, weights_hidden_output) \
        * hidden_activations * (1 - hidden_activations)
        # TODO: Update the change in weights
        del_w_hidden_output += hidden_error * hidden_activations
        del_w_input_hidden += hidden_error * x[:, None]
    # TODO: Update weights
    weights_input_hidden += learnrate * del_w_input_hidden / n_records
    weights_hidden_output += learnrate * del_w_hidden_output / n_records
    # Printing out the mean square error on the training set
    if e % (epochs / 10) == 0:
        hidden_activations = sigmoid(np.dot(x, weights_input_hidden))
        out = sigmoid(np.dot(hidden_activations,
                             weights_hidden_output))
        loss = np.mean((out - targets) ** 2)
        if last_loss and last_loss < loss:
            print("Train loss: ", loss, "  WARNING - Loss Increasing")
        else:
            print("Train loss: ", loss)
        last_loss = loss
# Calculate accuracy on test data
hidden = sigmoid(np.dot(features_test, weights_input_hidden))
out = sigmoid(np.dot(hidden, weights_hidden_output))
predictions = out > 0.5
accuracy = np.mean(predictions == targets_test)
print("Prediction accuracy: {:.3f}".format(accuracy))
Explanation: implement backprop.py
End of explanation
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