From c556eab14c1dff6efacbc4fd112ac180252deb64 Mon Sep 17 00:00:00 2001
From: Emmanuel Dellandrea <emmanuel.dellandrea@ec-lyon.fr>
Date: Mon, 13 Jan 2025 16:27:44 +0100
Subject: [PATCH] Add session 2
---
.../Session_1/linear_regression-completed.py | 185 ++++++++++
.../logistic_regression-completed.py | 261 ++++++++++++++
.../Session_2/Subject_2_Neural_Networks.pdf | Bin 0 -> 74821 bytes
Practical_sessions/Session_2/food_truck.txt | 97 ++++++
Practical_sessions/Session_2/houses.txt | 47 +++
Practical_sessions/Session_2/iris.txt | 150 ++++++++
Practical_sessions/Session_2/nn_regression.py | 323 ++++++++++++++++++
Practical_sessions/Session_2/scores.txt | 100 ++++++
8 files changed, 1163 insertions(+)
create mode 100644 Practical_sessions/Session_1/linear_regression-completed.py
create mode 100644 Practical_sessions/Session_1/logistic_regression-completed.py
create mode 100644 Practical_sessions/Session_2/Subject_2_Neural_Networks.pdf
create mode 100644 Practical_sessions/Session_2/food_truck.txt
create mode 100644 Practical_sessions/Session_2/houses.txt
create mode 100644 Practical_sessions/Session_2/iris.txt
create mode 100644 Practical_sessions/Session_2/nn_regression.py
create mode 100644 Practical_sessions/Session_2/scores.txt
diff --git a/Practical_sessions/Session_1/linear_regression-completed.py b/Practical_sessions/Session_1/linear_regression-completed.py
new file mode 100644
index 0000000..67d2999
--- /dev/null
+++ b/Practical_sessions/Session_1/linear_regression-completed.py
@@ -0,0 +1,185 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+def read_data(file_name, delimiter=','):
+ """ Read the data file and returns the corresponding matrices
+
+ Parameters
+ ----------
+ file_name : file name containg data
+ delimiter : character separating columns in the file ("," by default)
+
+ Returns
+ -------
+ X : data matrix of size [N, nb_var]
+ Y : matrix containg values of the target variable of size [N, 1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ """
+
+ data = np.loadtxt(file_name, delimiter=delimiter)
+ nb_var = data.shape[1] - 1
+ N = data.shape[0]
+
+ X = data[:, :-1]
+ Y = data[:, -1].reshape(N,1)
+
+ return X, Y, N, nb_var
+
+def normalization(X):
+ """ Normalize the provided matrix (substracts mean and divides by standard deviation)
+
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ Returns
+ -------
+ X_norm : normalized data matrix of size [N, nb_var]
+ mu : means of the variables of size [1,nb_var]
+ sigma : standard deviations of the variables of size [1,nb_var]
+
+ """
+
+ mu = np.mean(X, 0)
+ sigma = np.std(X, 0)
+ X_norm = (X - mu) / sigma
+
+ return X_norm, mu, sigma
+
+def compute_loss(X, Y, theta):
+ """ Compute the loss function value (mean square error)
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+ theta : matrix containing the theta parameters of the linear model of size [1, nb_var+1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ Returns
+ -------
+ loss : loss function value (mean square error)
+
+ """
+
+ N = X.shape[0]
+
+ loss = np.sum((X.dot(theta.T) - Y) ** 2) / (2 * N)
+
+ return loss
+
+def gradient_descent(X, Y, theta, alpha, nb_iters):
+ """ Training to compute the linear regression parameters by gradient descent
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+ theta : matrix containing the theta parameters of the linear model of size [1, nb_var+1]
+ alpha : learning rate
+ nb_iters : number of iterations
+
+ with N : number of elements and nb_var : number of predictor variables
+
+
+ Returns
+ -------
+ theta : matrix containing the theta parameters learnt by gradient descent of size [1, nb_var+1]
+ J_history : list containg the loss function values for each iteration of length nb_iters
+
+
+ """
+
+ # Initialisation de variables utiles
+ N = X.shape[0]
+ J_history = np.zeros(nb_iters)
+
+ for i in range(0, nb_iters):
+
+ error = X.dot(theta.T) - Y
+ theta -= (alpha/N)*np.sum(X*error, 0)
+
+ J_history[i] = compute_loss(X, Y, theta)
+
+
+ return theta, J_history
+
+def display(X, Y, theta):
+ """ Display in 2 dimensions of data points and of the linear regression curve defined by theta parameters
+
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+ theta : matrix containing the theta parameters of the linear model of size [1, nb_var+1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ Returns
+ -------
+ None
+
+ """
+ plt.figure(0)
+ plt.scatter(X[:, 1], Y, c='r', marker="x")
+ line1, = plt.plot(X[:, 1], X.dot(theta.T))
+ plt.title("Linear Regression")
+
+ plt.show()
+
+
+if __name__ == "__main__":
+
+ # ===================== Part 1: Data loading and normalization =====================
+ print("Data loading ...")
+
+ X, Y, N, nb_var = read_data("food_truck.txt")
+ # X, Y, N, nb_var = read_data("houses.txt")
+
+ # Print of the ten first examples of the dataset
+ print("Print of the ten first examples of the dataset : ")
+ for i in range(0, 10):
+ print(f"x = {X[i,:]}, y = {Y[i]}")
+
+ # Normalization of variables
+ print("Normalization of variables ...")
+
+ X, mu, sigma = normalization(X)
+
+ # Add one column of 1 values to X (for theta 0)
+ X = np.hstack((np.ones((N,1)), X))
+
+ # ===================== Part 2: Gradient descent =====================
+ print("Training by gradient descent ...")
+
+ # Choice of the learning rate and number of iterations
+ alpha = 0.01
+ nb_iters = 1500
+
+ # Initialization of theta and call to the gradient descent function
+ theta = np.zeros((1,nb_var+1))
+ theta, J_history = gradient_descent(X, Y, theta, alpha, nb_iters)
+
+ # Display of the loss function values obtained during gradient descent training
+ plt.figure()
+ plt.title("Loss function values obtained during gradient descent training")
+ plt.plot(np.arange(J_history.size), J_history)
+ plt.xlabel("Nomber of iterations")
+ plt.ylabel("Loss function J")
+
+ # Print of theta values
+ print(f"Theta computed by gradient descent : {theta}")
+
+ # In case of only one predictor variable, display the linear regression curve
+ if nb_var == 1 :
+ display(X,Y,theta)
+ plt.show()
+
+ print("Linear Regression completed.")
diff --git a/Practical_sessions/Session_1/logistic_regression-completed.py b/Practical_sessions/Session_1/logistic_regression-completed.py
new file mode 100644
index 0000000..826b54b
--- /dev/null
+++ b/Practical_sessions/Session_1/logistic_regression-completed.py
@@ -0,0 +1,261 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+def read_data(file_name, delimiter=','):
+ """ Read the data file and returns the corresponding matrices
+
+ Parameters
+ ----------
+ file_name : file name containg data
+ delimiter : character separating columns in the file ("," by default)
+
+ Returns
+ -------
+ X : data matrix of size [N, nb_var]
+ Y : matrix containg values of the target variable of size [N, 1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ """
+
+ data = np.loadtxt(file_name, delimiter=delimiter)
+ nb_var = data.shape[1] - 1
+ N = data.shape[0]
+
+ X = data[:, :-1]
+ Y = data[:, -1].reshape(N,1)
+
+ return X, Y, N, nb_var
+
+def normalization(X):
+ """ Normalize the provided matrix (substracts mean and divides by standard deviation)
+
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ Returns
+ -------
+ X_norm : normalized data matrix of size [N, nb_var]
+ mu : means of the variables of sizede dimension [1,nb_var]
+ sigma : standar deviations of the variables of size [1,nb_var]
+
+ """
+
+ mu = np.mean(X, 0)
+ sigma = np.std(X, 0)
+ X_norm = (X - mu) / sigma
+
+ return X_norm, mu, sigma
+
+def sigmoid(z):
+ """ Compute the value of the sigmoid function applied to z
+
+ Parameters
+ ----------
+ z : can be a scalar value or a matrix
+
+ Returns
+ -------
+ s : sigmoid value of z. Same size as z
+
+ """
+
+ s = 1 / (1 + np.exp(-z))
+
+ return s
+
+def compute_loss(X, Y, theta):
+ """ Compute the loss function value (log likelihood)
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+ theta : matrix containing the theta parameters of the linear model of size [1, nb_var+1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ Returns
+ -------
+ loss : loss function value (log likelihood)
+ """
+
+ N = X.shape[0]
+
+ loss = - (Y*np.log(sigmoid(X.dot(theta.T))) + (1-Y)*np.log(1-sigmoid(X.dot(theta.T)))).sum() / N
+
+ return loss
+
+def gradient_descent(X, Y, theta, alpha, nb_iters):
+ """ Training to compute the logistic regression parameters by gradient descent
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+ theta : matrix containing the theta parameters of the logistic model of size [1, nb_var+1]
+ alpha : learning rate
+ nb_iters : number of iterations
+
+ with N : number of elements and nb_var : number of predictor variables
+
+
+ Returns
+ -------
+ theta : matrix containing the theta parameters learnt by gradient descent of size [1, nb_var+1]
+ J_history : list containg the loss function values for each iteration of length nb_iters
+
+
+ """
+
+ # Init of useful variables
+ N = X.shape[0]
+ J_history = np.zeros(nb_iters)
+
+ for i in range(0, nb_iters):
+
+ error = sigmoid(X.dot(theta.T)) - Y
+ theta -= (alpha/N)*np.sum(X*error, 0)
+
+ J_history[i] = compute_loss(X, Y, theta)
+
+
+ return theta, J_history
+
+def prediction(X,theta):
+ """ Predict the class of each element in X
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+ theta : matrix containing the theta parameters of the logistic model of size [1, nb_var+1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+
+ Returns
+ -------
+ p : matrix of size [N,1] providing the class of each element in X (either 0 or 1)
+
+ """
+
+ p = sigmoid(X.dot(theta.T))
+ pos = np.where(p >= 0.5)
+ neg = np.where(p < 0.5)
+
+ p[pos] = 1
+ p[neg] = 0
+
+ return p
+
+def classification_rate(Ypred,Y):
+ """ Compute the classification rate (proportion of correctly classified elements)
+
+ Parameters
+ ----------
+ Ypred : matrix containing the predicted values of the class of size [N, 1]
+ Y : matrix containing the values of the target variable of size [N, 1]
+
+ with N : number of elements
+
+
+ Returns
+ -------
+ r : classification rate
+
+ """
+
+ N = Ypred.size
+ nb_errors = np.sum(np.abs(Ypred-Y))
+
+ t = (N-nb_errors) / N
+
+ return t
+
+def display(X, Y):
+ """ Display of data in 2 dimensions (2 dimensions of X) and class representation (provided by Y) by a color
+
+ Parameters
+ ----------
+ X : data matrix of size [N, nb_var+1]
+ Y : matrix containg values of the target variable of size [N, 1]
+
+ with N : number of elements and nb_var : number of predictor variables
+
+ Returns
+ -------
+ None
+
+ """
+
+ pos = np.where(Y == 1)[0]
+ neg = np.where(Y == 0)[0]
+ plt.scatter(X[pos, 1], X[pos, 2], marker="+", c='b')
+ plt.scatter(X[neg, 1], X[neg, 2], marker="o", c='r')
+
+
+if __name__ == "__main__":
+ # ===================== Part 1: Data loading and normalization =====================
+ print("Data loading ...")
+
+ X, Y, N, nb_var = read_data("scores.txt")
+
+ # Print of the ten first examples of the dataset
+ print("Print of the ten first examples of the dataset : ")
+ for i in range(0, 10):
+ print(f"x = {X[i,:]}, y = {Y[i]}")
+
+ # Normalization of variables
+ print("Normalization of variables ...")
+
+ X, mu, sigma = normalization(X)
+
+ # Add one column of 1 values to X (for theta 0)
+ X = np.hstack((np.ones((N,1)), X))
+
+ # Display in 2D of data points and actual class representation by a color
+ if nb_var == 2 :
+ plt.figure(0)
+ plt.title("Coordinates of data points in 2D - Reality")
+ display(X,Y)
+
+ # ===================== Part 2: Gradient descent =====================
+ print("Training by gradient descent ...")
+
+ # Choice of the learning rate and number of iterations
+ alpha = 0.01
+ nb_iters = 10000
+
+ # Initialization of theta and call to the gradient descent function
+ theta = np.zeros((1,nb_var+1))
+ theta, J_history = gradient_descent(X, Y, theta, alpha, nb_iters)
+
+ # Display of the loss function values obtained during gradient descent training
+ plt.figure(1)
+ plt.title("Loss function values obtained during gradient descent training")
+ plt.plot(np.arange(J_history.size), J_history)
+ plt.xlabel("Nomber of iterations")
+ plt.ylabel("Loss function J")
+
+ # Print of theta values
+ print(f"Theta computed by gradient descent : {theta}")
+
+ # Evaluation of the model
+ Ypred = prediction(X,theta)
+
+ print("Classification rate : ", classification_rate(Ypred,Y))
+
+ # Display in 2D of data points and predicted class representation by a color
+ if nb_var == 2 :
+ plt.figure(2)
+ plt.title("Coordinates of data points in 2D - Prediction")
+ display(X,Ypred)
+
+ plt.show()
+
+ print("Logistic Regression completed.")
diff --git a/Practical_sessions/Session_2/Subject_2_Neural_Networks.pdf b/Practical_sessions/Session_2/Subject_2_Neural_Networks.pdf
new file mode 100644
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diff --git a/Practical_sessions/Session_2/food_truck.txt b/Practical_sessions/Session_2/food_truck.txt
new file mode 100644
index 0000000..0f88ccb
--- /dev/null
+++ b/Practical_sessions/Session_2/food_truck.txt
@@ -0,0 +1,97 @@
+6.1101,17.592
+5.5277,9.1302
+8.5186,13.662
+7.0032,11.854
+5.8598,6.8233
+8.3829,11.886
+7.4764,4.3483
+8.5781,12
+6.4862,6.5987
+5.0546,3.8166
+5.7107,3.2522
+14.164,15.505
+5.734,3.1551
+8.4084,7.2258
+5.6407,0.71618
+5.3794,3.5129
+6.3654,5.3048
+5.1301,0.56077
+6.4296,3.6518
+7.0708,5.3893
+6.1891,3.1386
+20.27,21.767
+5.4901,4.263
+6.3261,5.1875
+5.5649,3.0825
+18.945,22.638
+12.828,13.501
+10.957,7.0467
+13.176,14.692
+22.203,24.147
+5.2524,-1.22
+6.5894,5.9966
+9.2482,12.134
+5.8918,1.8495
+8.2111,6.5426
+7.9334,4.5623
+8.0959,4.1164
+5.6063,3.3928
+12.836,10.117
+6.3534,5.4974
+5.4069,0.55657
+6.8825,3.9115
+11.708,5.3854
+5.7737,2.4406
+7.8247,6.7318
+7.0931,1.0463
+5.0702,5.1337
+5.8014,1.844
+11.7,8.0043
+5.5416,1.0179
+7.5402,6.7504
+5.3077,1.8396
+7.4239,4.2885
+7.6031,4.9981
+6.3328,1.4233
+6.3589,-1.4211
+6.2742,2.4756
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+14.908,12.054
+18.959,17.054
+7.2182,4.8852
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+20.341,20.992
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+7.3345,4.0259
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+6.5479,0.29678
+7.5386,3.8845
+5.0365,5.7014
+10.274,6.7526
+5.1077,2.0576
+5.7292,0.47953
+5.1884,0.20421
+6.3557,0.67861
+9.7687,7.5435
+6.5159,5.3436
+8.5172,4.2415
+9.1802,6.7981
+6.002,0.92695
+5.5204,0.152
+5.0594,2.8214
+5.7077,1.8451
+7.6366,4.2959
+5.8707,7.2029
+5.3054,1.9869
+8.2934,0.14454
+13.394,9.0551
+5.4369,0.61705
diff --git a/Practical_sessions/Session_2/houses.txt b/Practical_sessions/Session_2/houses.txt
new file mode 100644
index 0000000..79e9a80
--- /dev/null
+++ b/Practical_sessions/Session_2/houses.txt
@@ -0,0 +1,47 @@
+2104,3,399900
+1600,3,329900
+2400,3,369000
+1416,2,232000
+3000,4,539900
+1985,4,299900
+1534,3,314900
+1427,3,198999
+1380,3,212000
+1494,3,242500
+1940,4,239999
+2000,3,347000
+1890,3,329999
+4478,5,699900
+1268,3,259900
+2300,4,449900
+1320,2,299900
+1236,3,199900
+2609,4,499998
+3031,4,599000
+1767,3,252900
+1888,2,255000
+1604,3,242900
+1962,4,259900
+3890,3,573900
+1100,3,249900
+1458,3,464500
+2526,3,469000
+2200,3,475000
+2637,3,299900
+1839,2,349900
+1000,1,169900
+2040,4,314900
+3137,3,579900
+1811,4,285900
+1437,3,249900
+1239,3,229900
+2132,4,345000
+4215,4,549000
+2162,4,287000
+1664,2,368500
+2238,3,329900
+2567,4,314000
+1200,3,299000
+852,2,179900
+1852,4,299900
+1203,3,239500
diff --git a/Practical_sessions/Session_2/iris.txt b/Practical_sessions/Session_2/iris.txt
new file mode 100644
index 0000000..1de4bba
--- /dev/null
+++ b/Practical_sessions/Session_2/iris.txt
@@ -0,0 +1,150 @@
+5.1,3.5,1.4,0.2,0
+4.9,3.0,1.4,0.2,0
+4.7,3.2,1.3,0.2,0
+4.6,3.1,1.5,0.2,0
+5.0,3.6,1.4,0.2,0
+5.4,3.9,1.7,0.4,0
+4.6,3.4,1.4,0.3,0
+5.0,3.4,1.5,0.2,0
+4.4,2.9,1.4,0.2,0
+4.9,3.1,1.5,0.1,0
+5.4,3.7,1.5,0.2,0
+4.8,3.4,1.6,0.2,0
+4.8,3.0,1.4,0.1,0
+4.3,3.0,1.1,0.1,0
+5.8,4.0,1.2,0.2,0
+5.7,4.4,1.5,0.4,0
+5.4,3.9,1.3,0.4,0
+5.1,3.5,1.4,0.3,0
+5.7,3.8,1.7,0.3,0
+5.1,3.8,1.5,0.3,0
+5.4,3.4,1.7,0.2,0
+5.1,3.7,1.5,0.4,0
+4.6,3.6,1.0,0.2,0
+5.1,3.3,1.7,0.5,0
+4.8,3.4,1.9,0.2,0
+5.0,3.0,1.6,0.2,0
+5.0,3.4,1.6,0.4,0
+5.2,3.5,1.5,0.2,0
+5.2,3.4,1.4,0.2,0
+4.7,3.2,1.6,0.2,0
+4.8,3.1,1.6,0.2,0
+5.4,3.4,1.5,0.4,0
+5.2,4.1,1.5,0.1,0
+5.5,4.2,1.4,0.2,0
+4.9,3.1,1.5,0.1,0
+5.0,3.2,1.2,0.2,0
+5.5,3.5,1.3,0.2,0
+4.9,3.1,1.5,0.1,0
+4.4,3.0,1.3,0.2,0
+5.1,3.4,1.5,0.2,0
+5.0,3.5,1.3,0.3,0
+4.5,2.3,1.3,0.3,0
+4.4,3.2,1.3,0.2,0
+5.0,3.5,1.6,0.6,0
+5.1,3.8,1.9,0.4,0
+4.8,3.0,1.4,0.3,0
+5.1,3.8,1.6,0.2,0
+4.6,3.2,1.4,0.2,0
+5.3,3.7,1.5,0.2,0
+5.0,3.3,1.4,0.2,0
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+6.4,3.2,4.5,1.5,1
+6.9,3.1,4.9,1.5,1
+5.5,2.3,4.0,1.3,1
+6.5,2.8,4.6,1.5,1
+5.7,2.8,4.5,1.3,1
+6.3,3.3,4.7,1.6,1
+4.9,2.4,3.3,1.0,1
+6.6,2.9,4.6,1.3,1
+5.2,2.7,3.9,1.4,1
+5.0,2.0,3.5,1.0,1
+5.9,3.0,4.2,1.5,1
+6.0,2.2,4.0,1.0,1
+6.1,2.9,4.7,1.4,1
+5.6,2.9,3.6,1.3,1
+6.7,3.1,4.4,1.4,1
+5.6,3.0,4.5,1.5,1
+5.8,2.7,4.1,1.0,1
+6.2,2.2,4.5,1.5,1
+5.6,2.5,3.9,1.1,1
+5.9,3.2,4.8,1.8,1
+6.1,2.8,4.0,1.3,1
+6.3,2.5,4.9,1.5,1
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+6.4,2.9,4.3,1.3,1
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+6.8,2.8,4.8,1.4,1
+6.7,3.0,5.0,1.7,1
+6.0,2.9,4.5,1.5,1
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+5.5,2.4,3.8,1.1,1
+5.5,2.4,3.7,1.0,1
+5.8,2.7,3.9,1.2,1
+6.0,2.7,5.1,1.6,1
+5.4,3.0,4.5,1.5,1
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\ No newline at end of file
diff --git a/Practical_sessions/Session_2/nn_regression.py b/Practical_sessions/Session_2/nn_regression.py
new file mode 100644
index 0000000..e80e6c6
--- /dev/null
+++ b/Practical_sessions/Session_2/nn_regression.py
@@ -0,0 +1,323 @@
+
+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)
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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]
+
+ """
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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
+
+ """
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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)
+
+ """
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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
+
+ """
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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
+
+ """
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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
+
+ """
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # 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
+
+ """
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # return s
+
+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
+
+ """
+
+ #######################################################
+ ##### To complete (and remove the pass statement) #####
+ #######################################################
+ pass
+
+ # return s
+
+
+# ===================== 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)
diff --git a/Practical_sessions/Session_2/scores.txt b/Practical_sessions/Session_2/scores.txt
new file mode 100644
index 0000000..3a5f952
--- /dev/null
+++ b/Practical_sessions/Session_2/scores.txt
@@ -0,0 +1,100 @@
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--
GitLab