Create nn_regression-completed.py

Practical_sessions/Session_2/nn_regression-completed.py

+
+import matplotlib.pyplot as plt
+import numpy as np
+
+def read_data(file_name, delimiter=','):
+    """ Reads the file containing the data and returns the corresponding matrices
+
+    Parameters
+    ----------
+    file_name : name of the file containing the data
+    delimiter : character separating columns in the file ("," by default)
+
+    Returns
+    -------
+    x : data matrix of size [N, num_vars]
+    d : matrix containing the target variable values of size [N, num_targets]
+    N : number of elements
+    num_vars : number of predictor variables
+    num_targets : number of target variables
+
+    """
+    
+    data = np.loadtxt(file_name, delimiter=delimiter)
+    
+    num_targets = 1
+    num_vars = data.shape[1] - num_targets
+    N = data.shape[0]
+
+    x = data[:, :num_vars]
+    d = data[:, num_vars:].reshape(N,1)
+    
+    return x, d, N, num_vars, num_targets
+
+def normalization(x):
+    """ Normalizes the data by centering and scaling the predictor variables
+    
+    Parameters
+    ----------
+    X : data matrix of size [N, num_vars]
+    
+    with N : number of elements and num_vars : number of predictor variables
+
+    Returns
+    -------
+    X_norm : centered-scaled data matrix of size [N, num_vars]
+    mu : mean of the variables of size [1, num_vars]
+    sigma : standard deviation of the variables of size [1, num_vars]
+    
+    """
+    
+    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, val_prop=0.2, test_prop=0.2):
+    """ Splits the initial data into three distinct subsets for training, validation, and testing
+    
+    Parameters
+    ----------
+    x : data matrix of size [N, num_vars]
+    d : matrix of target values [N, num_targets]
+    val_prop : proportion of validation data over the entire dataset (between 0 and 1)
+    test_prop : proportion of test data over the entire dataset (between 0 and 1)
+    
+    with N : number of elements, num_vars : number of predictor variables, num_targets : number of target variables
+
+    Returns
+    -------
+    x_train : training data matrix
+    d_train : training target values matrix
+    x_val : validation data matrix
+    d_val : validation target values matrix
+    x_test : test data matrix
+    d_test : test target values matrix
+
+    """
+    assert val_prop + test_prop < 1.0
+
+    N = x.shape[0]
+    indices = np.arange(N)
+    np.random.shuffle(indices)
+    num_val = int(N*val_prop)
+    num_test = int(N*test_prop)
+    num_train = N - num_val - num_test
+
+    x = x[indices,:]
+    d = d[indices,:]
+
+    x_train = x[:num_train,:]
+    d_train = d[:num_train,:]
+
+    x_val = x[num_train:num_train+num_val,:]
+    d_val = d[num_train:num_train+num_val,:]
+
+    x_test = x[N-num_test:,:]
+    d_test = d[N-num_test:,:]
+
+    return x_train, d_train, x_val, d_val, x_test, d_test
+
+def calculate_mse_cost(y, d):
+    """ Calculates the value of the MSE (mean squared error) cost function
+    
+    Parameters
+    ----------
+    y : matrix of predicted data 
+    d : matrix of actual data 
+    
+    Returns
+    -------
+    cost : value corresponding to the MSE cost function (mean squared error)
+
+    """
+
+    N = y.shape[1]    
+    cost = np.square(y - d).sum() / 2 / N
+
+    return cost
+
+def forward_pass(x, W, b, activation):
+    """ Performs a forward pass in the neural network
+    
+    Parameters
+    ----------
+    x : input matrix, of size num_vars 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, num_vars : 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 output of the network
+    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 layer weights
+    delta_b : list containing the gradient matrices of the network layer biases
+
+    """
+
+    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):
+    """ Calculates 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):
+    """ Calculates 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):
+    """ Calculates 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)
+
+
+# ===================== Part 1: Data Reading and Normalization =====================
+print("Reading data ...")
+
+x, d, N, num_vars, num_targets = read_data("food_truck.txt")
+# x, d, N, num_vars, num_targets = read_data("houses.txt")
+
+# Displaying the first 10 examples from the dataset
+print("Displaying the first 10 examples from the dataset: ")
+for i in range(0, 10):
+    print(f"x = {x[i,:]}, d = {d[i]}")
+    
+# Normalizing the variables (centering and scaling)
+print("Normalizing the variables ...")
+x, mu, sigma = normalization(x)
+dmax = d.max()
+d = d / dmax
+
+# Splitting 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 =====================
+
+# Choosing the learning rate and number of iterations
+alpha = 0.001
+num_iters = 500
+train_costs = np.zeros(num_iters)
+val_costs = np.zeros(num_iters)
+
+# Network dimensions
+D_c = [num_vars, 5, 10, num_targets] # list containing the number of neurons for each layer 
+activation = [relu, sigmoid, linear] # list containing the activation functions for the hidden layers and the output layer 
+
+# Random initialization of the 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 input of the network
+d_train = d_train.T 
+
+x_val = x_val.T # Data is presented as column vectors at the input of the network
+d_val = d_val.T 
+
+x_test = x_test.T # Data is presented as column vectors at the input of the network
+d_test = d_test.T 
+
+for t in range(num_iters):
+
+    #############################################################################
+    # Forward pass: calculating predicted output y on validation data #
+    #############################################################################
+    a, h = forward_pass(x_val, W, b, activation)
+    y_val = h[-1] # Predicted output
+
+    ###############################################################################
+    # Forward pass: calculating predicted output y on training data #
+    ###############################################################################
+    a, h = forward_pass(x_train, W, b, activation)
+    y_train = h[-1] # Predicted output
+
+    ###########################################
+    # Calculating the MSE loss function #
+    ###########################################
+    train_costs[t] = calculate_mse_cost(y_train, d_train)
+    val_costs[t] = calculate_mse_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)
+  
+    #############################################
+    # Updating 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: ", train_costs[-1])
+print("Final cost on the validation set: ", val_costs[-1])
+
+# Plotting the evolution of the cost function during backpropagation
+plt.figure(0)
+plt.title("Evolution of the cost function during backpropagation")
+plt.plot(np.arange(train_costs.size), train_costs, label="Training")
+plt.plot(np.arange(val_costs.size), val_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: calculating predicted output y on test data #
+#######################################################################
+a, h = forward_pass(x_test, W, b, activation)
+y_test = h[-1] # Predicted output
+
+cost = calculate_mse_cost(y_test, d_test)
+print("Test set cost: ", cost)