Add session 2

Practical_sessions/Session_2/nn_classification-completed.py

+import matplotlib.pyplot as plt
+import numpy as np
+
+def load_data(file_name, delimiter=','):
+    """ Reads the file containing the data and returns the matrices corresponding to it
+
+    Parameters
+    ----------
+    file_name : name of the file containing the data
+    delimiter : character that separates the columns in the file (default is ",")
+
+    Returns
+    -------
+    x : data matrix of dimension [N, nb_var]
+    d : matrix containing the target variable values of dimension [N, nb_target]
+    N : number of elements
+    nb_var : number of predictor variables
+    nb_target : number of target variables
+
+    """
+    
+    data = np.loadtxt(file_name, delimiter=delimiter)
+    
+    nb_target = 1
+    nb_var = data.shape[1] - nb_target
+    N = data.shape[0]
+
+    x = data[:, :nb_var]
+    d = data[:, nb_var:].reshape(N,1)
+    
+    return x, d, N, nb_var, nb_target
+
+
+def normalization(x):
+    """ Normalizes the data by centering and scaling the predictor variables
+    
+    Parameters
+    ----------
+    X : data matrix of dimension [N, nb_var]
+    
+    with N : number of elements and nb_var : number of predictor variables
+
+    Returns
+    -------
+    X_norm : normalized data matrix of dimension [N, nb_var]
+    mu : mean of the variables of dimension [1, nb_var]
+    sigma : standard deviation of the variables of dimension [1, nb_var]
+    
+    """
+    
+    mu = np.mean(x, 0)
+    sigma = np.std(x, 0)
+    x_norm = (x - mu) / sigma
+
+    return x_norm, mu, sigma
+
+def split_data(x,d,prop_val=0.2, prop_test=0.2):
+    """ Splits the original data into three distinct subsets: training, validation, and test
+    
+    Parameters
+    ----------
+    x : data matrix of dimension [N, nb_var]
+    d : target values matrix [N, nb_target]
+    prop_val : proportion of validation data in the total data (between 0 and 1)
+    prop_test : proportion of test data in the total data (between 0 and 1)
+    
+    with N : number of elements, nb_var : number of predictor variables, nb_target : number of target variables
+
+    Returns
+    -------
+    x_train : training data matrix
+    d_train : target values matrix for training
+    x_val : validation data matrix
+    d_val : target values matrix for validation
+    x_test : test data matrix
+    d_test : target values matrix for test
+
+    """
+    assert prop_val + prop_test < 1.0
+
+    N = x.shape[0]
+    indices = np.arange(N)
+    np.random.shuffle(indices)
+    nb_val = int(N*prop_val)
+    nb_test = int(N*prop_test)
+    nb_train = N - nb_val - nb_test
+
+    x = x[indices,:]
+    d = d[indices,:]
+
+    x_train = x[:nb_train,:]
+    d_train = d[:nb_train,:]
+
+    x_val = x[nb_train:nb_train+nb_val,:]
+    d_val = d[nb_train:nb_train+nb_val,:]
+
+    x_test = x[N-nb_test:,:]
+    d_test = d[N-nb_test:,:]
+
+    return x_train, d_train, x_val, d_val, x_test, d_test
+
+def compute_cross_entropy_cost(y, d):
+    """ Computes the value of the cross-entropy cost function
+    
+    Parameters
+    ----------
+    y : predicted data matrix (softmax)
+    d : actual data matrix encoded by 1
+    
+    Returns
+    -------
+    cost : value corresponding to the cost function
+
+    """
+
+    N = y.shape[1]
+    
+    cost = - np.sum(d*np.log(y)) / N
+
+    return cost
+
+def forward_pass(x, W, b, activation):
+    """ Performs a forward pass in the neural network
+    
+    Parameters
+    ----------
+    x : input matrix, dimension nb_var x N
+    W : list containing the weight matrices of the network
+    b : list containing the bias matrices of the network
+    activation : list containing the activation functions of the network layers
+
+    with N : number of elements, nb_var : number of predictor variables 
+
+    Returns
+    -------
+    a : list containing the input potentials of the network layers
+    h : list containing the outputs of the network layers
+
+    """
+    h = [x]
+    a = []
+    for i in range(len(b)):
+        a.append( W[i].dot(h[i]) + b[i] )
+        h.append( activation[i](a[i]) ) 
+
+    return a, h
+
+def backward_pass(delta_h, a, h, W, activation):
+    """ Performs a backward pass in the neural network (backpropagation)
+    
+    Parameters
+    ----------
+    delta_h : matrix containing the gradient of the cost with respect to the network output
+    a : list containing the input potentials of the network layers
+    h : list containing the outputs of the network layers
+    W : list containing the weight matrices of the network
+    activation : list containing the activation functions of the network layers
+
+    Returns
+    -------
+    delta_W : list containing the gradient matrices of the network's weight layers
+    delta_b : list containing the gradient matrices of the network's bias layers
+
+    """
+
+    delta_b = []
+    delta_W = []
+
+    for i in range(len(W)-1,-1,-1):
+
+        delta_a = delta_h * activation[i](a[i], True)
+     
+        delta_b.append( delta_a.mean(1).reshape(-1,1) ) 
+        delta_W.append( delta_a.dot(h[i].T) ) 
+
+        delta_h = (W[i].T).dot(delta_a) 
+
+    delta_b = delta_b[::-1]
+    delta_W = delta_W[::-1]
+
+    return delta_W, delta_b
+
+def sigmoid(z, deriv=False):
+    """ Computes the value of the sigmoid function or its derivative applied to z
+    
+    Parameters
+    ----------
+    z : can be a scalar or a matrix
+    deriv : boolean. If False returns the value of the sigmoid function, if True returns its derivative
+
+    Returns
+    -------
+    s : value of the sigmoid function applied to z or its derivative. Same dimension as z
+
+    """
+
+    s = 1 / (1 + np.exp(-z))
+    if deriv:
+        return s * (1 - s)
+    else :
+        return s
+
+def linear(z, deriv=False):
+    """ Computes the value of the linear function or its derivative applied to z
+    
+    Parameters
+    ----------
+    z : can be a scalar or a matrix
+    deriv : boolean. If False returns the value of the linear function, if True returns its derivative
+
+
+    Returns
+    -------
+    s : value of the linear function applied to z or its derivative. Same dimension as z
+
+    """
+    if deriv:       
+        return 1     
+    else :
+        return z
+
+
+def relu(z, deriv=False):
+    """ Computes the value of the ReLU function or its derivative applied to z
+    
+    Parameters
+    ----------
+    z : can be a scalar or a matrix
+    deriv : boolean. If False returns the value of the ReLU function, if True returns its derivative
+
+    Returns
+    -------
+    s : value of the ReLU function applied to z or its derivative. Same dimension as z
+
+    """
+
+    r = np.zeros(z.shape)
+    if deriv:
+        pos = np.where(z>=0)
+        r[pos] = 1.0
+        return r
+    else :    
+        return np.maximum(r,z)
+
+def softmax(z, deriv=False):
+    """ Computes the value of the softmax function or its derivative applied to z
+    
+    Parameters
+    ----------
+    z : data matrix
+    deriv : boolean. If False returns the value of the softmax function, if True returns its derivative
+
+    Returns
+    -------
+    s : value of the softmax function applied to z or its derivative. Same dimension as z
+
+    """
+    
+    if deriv:
+        return 1
+    else :        
+        return np.exp(z) / np.sum(np.exp(z),axis=0)
+
+def one_hot_encoding(d):
+    """ Performs a one-hot encoding: for the output neurons of the network, only 1 will have the value 1, all others will be 0
+    
+    Parameters
+    ----------
+    d : matrix containing the values of the target variable (class of the elements) of dimension [N, 1]
+    with N : number of elements
+
+    Returns
+    -------
+    e : encoded data matrix of dimension [N, nb_classes]
+    with N : number of elements and nb_classes the number of classes (maximum+1) of the values in d
+
+    """    
+
+    d = d.astype(int).flatten()
+    N = d.shape[0]
+    nb_classes = d.max() + 1
+
+    e = np.zeros((N,nb_classes))
+    e[range(N),d] = 1
+
+    return e
+
+def classification_accuracy(y,d):
+    """ Computes the classification accuracy (proportion of correctly classified elements)
+    
+    Parameters
+    ----------
+    y : network outputs matrix of dimension [nb_output_neurons x N]
+    d : true values matrix [nb_output_neurons x N]
+    
+    with N : number of elements and nb_output_neurons : number of neurons in the output layer
+
+    Returns
+    -------
+    t : classification accuracy
+
+    """
+
+    ind_y = np.argmax(y,axis=0)
+    ind_d = np.argmax(d,axis=0)
+    t = np.mean(ind_y == ind_d)
+    
+    return t
+
+
+# ===================== Part 1: Reading and Normalizing the Data =====================
+print("Reading the data ...")
+
+x, d, N, nb_var, nb_target = load_data("iris.txt")
+# x, d, N, nb_var, nb_target = load_data("scores.txt")
+
+# Display the first 10 examples of the dataset
+print("Displaying the first 10 examples of the dataset: ")
+for i in range(0, 10):
+    print(f"x = {x[i,:]}, d = {d[i]}")
+    
+# Normalization of the variables (centering and scaling)
+print("Normalizing the variables ...")
+x, mu, sigma = normalization(x)
+d = one_hot_encoding(d)
+
+# Split the data into training, validation, and test subsets
+x_train, d_train, x_val, d_val, x_test, d_test = split_data(x,d)
+
+# ===================== Part 2: Training =====================
+
+# Learning rate and number of iterations
+alpha = 0.0001
+nb_iters = 10000
+training_costs = np.zeros(nb_iters)
+validation_costs = np.zeros(nb_iters)
+
+# Network dimensions
+D_c = [nb_var, 15, 15,  d_train.shape[1]] # list containing the number of neurons for each layer 
+activation = [relu, relu, softmax] # list containing the activation functions for hidden layers and output layer
+
+# Random initialization of network weights
+W = []
+b = []
+for i in range(len(D_c)-1):    
+    W.append(2 * np.random.random((D_c[i+1], D_c[i])) - 1)
+    b.append(np.zeros((D_c[i+1],1)))
+
+x_train = x_train.T # Data is presented as column vectors at the network input
+d_train = d_train.T 
+
+x_val = x_val.T # Data is presented as column vectors at the network input
+d_val = d_val.T 
+
+x_test = x_test.T # Data is presented as column vectors at the network input
+d_test = d_test.T 
+
+for t in range(nb_iters):
+
+    #############################################################################
+    # Forward pass: calculate predicted output y on validation data #
+    #############################################################################
+    a, h = forward_pass(x_val, W, b, activation)
+    y_val = h[-1] # Predicted output
+
+    ###############################################################################
+    # Forward pass: calculate predicted output y on training data #
+    ###############################################################################
+    a, h = forward_pass(x_train, W, b, activation)
+    y_train = h[-1] # Predicted output
+
+    ###########################################
+    # Compute Mean Squared Error loss function #
+    ###########################################
+    training_costs[t] = compute_cross_entropy_cost(y_train,d_train)
+    validation_costs[t] = compute_cross_entropy_cost(y_val,d_val)
+
+    ####################################
+    # Backward pass: backpropagation #
+    ####################################
+    delta_h = (y_train-d_train) # For the last layer 
+    delta_W, delta_b = backward_pass(delta_h, a, h, W, activation)
+  
+    #############################################
+    # Update weights and biases  ##### #
+    ############################################# 
+    for i in range(len(b)-1,-1,-1):
+        b[i] -= alpha * delta_b[i]
+        W[i] -= alpha * delta_W[i]
+
+print("Final cost on the training set: ", training_costs[-1])
+print("Classification accuracy on the training set: ", classification_accuracy(y_train, d_train))
+
+print("Final cost on the validation set: ", validation_costs[-1])
+print("Classification accuracy on the validation set: ", classification_accuracy(y_val, d_val))
+
+# Display cost function evolution during backpropagation
+plt.figure(0)
+plt.title("Cost function evolution during backpropagation")
+plt.plot(np.arange(training_costs.size), training_costs, label="Training")
+plt.plot(np.arange(validation_costs.size), validation_costs, label="Validation")
+plt.legend(loc="upper left")
+plt.xlabel("Number of iterations")
+plt.ylabel("Cost")
+plt.show()
+
+# ===================== Part 3: Evaluation on the test set =====================
+
+#######################################################################
+# Forward pass: calculate predicted output y on test data #
+#######################################################################
+a, h = forward_pass(x_test, W, b, activation)
+y_test = h[-1] # Predicted output
+
+cost = compute_cross_entropy_cost(y_test,d_test)
+print("Cost on the test set: ", cost)
+print("Classification accuracy on the test set: ", classification_accuracy(y_test, d_test))