### Ajout explications

parent b674df38
 ... ... @@ -30,6 +30,7 @@ "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import random\n", "\n", "from sklearn.linear_model import Lasso, ElasticNet\n", "from sklearn.pipeline import make_pipeline\n", ... ... @@ -51,13 +52,13 @@ "- m, s the parameters of the normal law used for the generation of the first feature\n", "and the outputs X and y\n", "\n", "For this purpose, we will proceed in X steps :\n", "For this purpose, we will proceed in 3 steps :\n", "\n", "- First, we will generate the first dimension of X randomly from a normal law (m, s)\n", "- For the other dimensions of X, noted i, the value will be calculated as follow :\n", " - We generate a number from a normal law N(i, 1)\n", " - We generate a number from a normal law N(i / 2, 1)\n", " - We add it to the value of the first column\n", "- For Y, the value is calculated as the mean of the values we generated for X" "- For Y, we select 2 over 3 values of X and we sum them" ] }, { ... ... @@ -66,7 +67,6 @@ "metadata": {}, "outputs": [], "source": [ "# /!\\ THIS IS A THIRD TEST VERSION, COULD (AND WILL CERTAINLY) CHANGE\n", "def generate_data(n_samples, n_features, m, s):\n", " X = np.ndarray((n_samples, n_features))\n", " y = np.ndarray((n_samples,))\n", ... ... @@ -75,14 +75,11 @@ " X[i, 0] = np.random.normal(m, s)\n", " for j in range(1, n_features):\n", " X[i, j] = X[i, 0] + np.random.normal(i / 2, 1)\n", " selected_features = [X[i, j] for j in range(n_features) if j % 3 != 0]\n", " y[i] = np.sum(selected_features)\n", " \n", " y = np.mean(X, axis=1)\n", "\n", " return X, y\n", "# /!\\ THIS IS A THIRD TEST VERSION, COULD (AND WILL CERTAINLY) CHANGE\n", "\n", "\n", "import random\n", "def generate_data_2(n_samples, n_features):\n", " X = []\n", " y = np.ndarray((n_samples,))\n", ... ... @@ -167,6 +164,15 @@ "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "These results show the instability of the Lasso regression :\n", "- We see that the number of selected feature changes from one training to another\n", "- We see that the regression model selects different features : if the first features are selected (almost) each time, the others may be selected or not (some of them are selected 1 time over two for example)." ] }, { "cell_type": "code", "execution_count": null, ... ... @@ -203,7 +209,7 @@ "source": [ "### Demonstrate stability of elastic net\n", "\n", "In a second time, we can do the same test with an elastic net regression model to highlight the difference between the two methods" "In a second time, we can do the same test with an elastic net regression model to highlight the difference between the two methods." ] }, { ... ... @@ -214,7 +220,7 @@ "source": [ "# We use the same alpha as the lasso regression\n", "# Assume we really want to select features, we give the priority to l1\n", "elastic_net = ElasticNet(alpha=3.0, l1_ratio=0.9, fit_intercept=True, max_iter=10000)\n", "elastic_net = ElasticNet(alpha=3.0, l1_ratio=0.99, fit_intercept=True, max_iter=10000)\n", "\n", "model = make_pipeline(standard_scaler, elastic_net)\n", "\n", ... ... @@ -242,6 +248,15 @@ "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "With this data, wo observe that elastic net is stable in term of selected features, but the result on this sample is not satisfying : we always use 49 of the 50 features, even with a very high (0.99) importance given to the L1 factor. The bypassed feature is always the first one.\n", "\n", "It is **like** the elastic net « found » that each \$X[i], i > 0\$ were generated from \$X\$ but did not « found » a link between the elements.\n" ] }, { "cell_type": "code", "execution_count": null, ... ... @@ -278,15 +293,13 @@ "cell_type": "markdown", "metadata": {}, "source": [ "Notes : **migh change if we found a better dataset**\n", "### Conclusion\n", "\n", "- Instability of Lasso is proved\n", "- Stability of elastic_net is OK for this sample.\n", "With our correlated data, we were able to demonstrate the instability of the Lasso regression : the number of features and the features selected from one trainig to another can vary a lot.\n", "\n", "BUT :\n", "- Feature selection w/ elastic net for this sample is not satisfying (we only remove the first one)\n", "However, if we managed to show a form of stability for the elastic net regression, the restults are not satisfying : the regression dit not select much features among the 50 ones of the sample (one for the first method, 0 a lot of time for the second one).\n", "\n", "It is **like** the elastic net « found » that each \$X[i], i > 0\$ were generated from \$X\$ but did not « found » a link between the elements.\n" "We tried various changes in the method for the generation of the data and various params for the regularization without success." ] } ], ... ... @@ -295,8 +308,7 @@ "hash": "3abb0a1ef4892304d86bb3a3dfd052bcca35057beadba016173999c775e8d3ba" }, "kernelspec": { "display_name": "Python 3", "language": "python", "display_name": "Python 3.9.7 64-bit ('AOS1-QteoCFsS': pipenv)", "name": "python3" }, "language_info": { ... ...
 %% Cell type:markdown id: tags: # AOS1 assignment ## Make elastic net outshine the Lasso authors : Mathilde Rineau, Rémy Huet ### Introduction The aim of this work is to demonstrate experimentally that the elastic net regularization outshine the Lasso regulation in some cases. We know that the Lasso regularization may be unstable when used on high-correlated data. Indeed, the Lasso regularization may lead to ignore some features (by setting their weight in the regression to 0). When the data is highly correlated, small changes on the sample could lead to changes in the selection of the features (what whe call instability). At the opposite, elastic net regression should be able to ignore some features but with more stability than Lasso. In this work, we will construct a dataset with highly correlated data to demonstrate that. %% Cell type:code id: tags: ``` python ``` import numpy as np import matplotlib.pyplot as plt import random from sklearn.linear_model import Lasso, ElasticNet from sklearn.pipeline import make_pipeline from sklearn.preprocessing import StandardScaler ``` %% Cell type:markdown id: tags: ### Data generation First, we will generate highly correlated data, containing a sample X (multidim) and a target y (one dim). We write a function for this. Its parameters are : - n_samples the number of samples - n_features the number of features in X - m, s the parameters of the normal law used for the generation of the first feature and the outputs X and y For this purpose, we will proceed in X steps : For this purpose, we will proceed in 3 steps : - First, we will generate the first dimension of X randomly from a normal law (m, s) - For the other dimensions of X, noted i, the value will be calculated as follow : - We generate a number from a normal law N(i, 1) - We generate a number from a normal law N(i / 2, 1) - We add it to the value of the first column - For Y, the value is calculated as the mean of the values we generated for X - For Y, we select 2 over 3 values of X and we sum them %% Cell type:code id: tags: ``` python # /!\ THIS IS A THIRD TEST VERSION, COULD (AND WILL CERTAINLY) CHANGE ``` def generate_data(n_samples, n_features, m, s): X = np.ndarray((n_samples, n_features)) y = np.ndarray((n_samples,)) for i in range(n_samples): X[i, 0] = np.random.normal(m, s) for j in range(1, n_features): X[i, j] = X[i, 0] + np.random.normal(i / 2, 1) y = np.mean(X, axis=1) selected_features = [X[i, j] for j in range(n_features) if j % 3 != 0] y[i] = np.sum(selected_features) return X, y # /!\ THIS IS A THIRD TEST VERSION, COULD (AND WILL CERTAINLY) CHANGE import random def generate_data_2(n_samples, n_features): X = [] y = np.ndarray((n_samples,)) X.append(np.random.geometric(p = 0.5, size = n_features)) sum_X = np.ndarray((n_features,)) for i in range(n_samples): p = random.random() temp = np.random.geometric(p = p, size = n_features) #print(temp) sum_X = sum_X + temp #print(sum_X) X.append(sum_X) X = np.array(X) y = np.mean(X, axis=1) return X, y ``` %% Cell type:markdown id: tags: ### Demonstrate instability of Lasso Now that we have a way to generate highly correlated data, we will use a lasso regression on it. The aim of this part is to demonstrate the instability of the Lasso regression on this data. For this purpose, we will use a loop to generate several datasets using **the same params** on our `generate_data` function. We will demonstrate instability by counting the number of selected features each time, and registe which features are selected. **Note :** the model auto-correct intercept by default `fit_intercept=True`. The parameter `normalize` is deprecated, so we use a pipeline to normalize the data before the regression, as suggested in the deprecation message. We set the `with_mean` parameter to `False` because we just scale the data and do not center it, we leave this work to the Lasso. %% Cell type:code id: tags: ``` python ``` # Params for data generation: n_samples = 300 n_features = 50 m = 30 s = 3 # Number of tests n_tests = 100 standard_scaler = StandardScaler(with_mean=False) lasso = Lasso(alpha=3.0, fit_intercept=True, max_iter=5000) model = make_pipeline(standard_scaler, lasso) # Arrays to store results n_selected = [] selection_count = np.zeros((n_features)) # We test our regression n_test time for i in range(n_tests): # Generate the data X, y = generate_data(n_samples, n_features, m, s) # Fit the model (pipeline with the data) model.fit(X, y) # We can now retrieve selected features : selected_features = (lasso.coef_ != 0) * 1 # (lasso.coef_ != 0) gives a matrix of True / False. * 1 transforms True in 1 and False in 0 n_selected.append(np.count_nonzero(selected_features)) selection_count += selected_features # Using n_selected, we can display the number of selected features per training uniq, count = np.unique(n_selected, return_counts=True) plt.bar(uniq, count, label='Number of selected features per training') plt.legend() plt.show() # Using selection_count, we can display the number of times each feature was selected plt.bar(range(n_features), selection_count, label='Number of time each feature was selected') plt.legend() plt.show() ``` %% Cell type:markdown id: tags: These results show the instability of the Lasso regression : - We see that the number of selected feature changes from one training to another - We see that the regression model selects different features : if the first features are selected (almost) each time, the others may be selected or not (some of them are selected 1 time over two for example). %% Cell type:code id: tags: ``` python ``` # Arrays to store results n_selected = [] selection_count = np.zeros((n_features)) # We test our regression n_test time for i in range(n_tests): # Generate the data X, y = generate_data_2(n_samples, n_features,) # Fit the model (pipeline with the data) model.fit(X, y) # We can now retrieve selected features : selected_features = (lasso.coef_ != 0) * 1 # (lasso.coef_ != 0) gives a matrix of True / False. * 1 transforms True in 1 and False in 0 n_selected.append(np.count_nonzero(selected_features)) selection_count += selected_features uniq, count = np.unique(n_selected, return_counts=True) plt.bar(uniq, count, label='Number of selected features per training') plt.show() # Using selection_count, we can display the number of times each feature was selected plt.bar(range(n_features), selection_count, label='Number of time each feature was selected') plt.legend() plt.show() ``` %% Cell type:markdown id: tags: ### Demonstrate stability of elastic net In a second time, we can do the same test with an elastic net regression model to highlight the difference between the two methods In a second time, we can do the same test with an elastic net regression model to highlight the difference between the two methods. %% Cell type:code id: tags: ``` python ``` # We use the same alpha as the lasso regression # Assume we really want to select features, we give the priority to l1 elastic_net = ElasticNet(alpha=3.0, l1_ratio=0.9, fit_intercept=True, max_iter=10000) elastic_net = ElasticNet(alpha=3.0, l1_ratio=0.99, fit_intercept=True, max_iter=10000) model = make_pipeline(standard_scaler, elastic_net) # Arrays to store results n_selected = [] selection_count = np.zeros((n_features)) # We test our regression n_test time for i in range(n_tests): # Generate the data X, y = generate_data(n_samples, n_features, m, s) # Fit the model (pipeline with the data) model.fit(X, y) # We can now retrieve selected features : selected_features = (elastic_net.coef_ != 0) * 1 n_selected.append(np.count_nonzero(selected_features)) selection_count += selected_features uniq, count = np.unique(n_selected, return_counts=True) print(f'Features selected : {uniq}, count : {count}') plt.bar(range(n_features), selection_count, label='Number of time each feature was selected') plt.legend() plt.show() ``` %% Cell type:markdown id: tags: With this data, wo observe that elastic net is stable in term of selected features, but the result on this sample is not satisfying : we always use 49 of the 50 features, even with a very high (0.99) importance given to the L1 factor. The bypassed feature is always the first one. It is **like** the elastic net « found » that each \$X[i], i > 0\$ were generated from \$X\$ but did not « found » a link between the elements. %% Cell type:code id: tags: ``` python ``` # Arrays to store results n_selected = [] selection_count = np.zeros((n_features)) # We test our regression n_test time for i in range(n_tests): # Generate the data X, y = generate_data_2(n_samples, n_features) # Fit the model (pipeline with the data) model.fit(X, y) # We can now retrieve selected features : selected_features = (elastic_net.coef_ != 0) * 1 n_selected.append(np.count_nonzero(selected_features)) selection_count += selected_features uniq, count = np.unique(n_selected, return_counts=True) plt.bar(uniq, count, label='Number of selected features per training') plt.legend() plt.show() plt.bar(range(n_features), selection_count, label='Number of time each feature was selected') #plt.legend() plt.show() ``` %% Cell type:markdown id: tags: Notes : **migh change if we found a better dataset** ### Conclusion - Instability of Lasso is proved - Stability of elastic_net is OK for this sample. With our correlated data, we were able to demonstrate the instability of the Lasso regression : the number of features and the features selected from one trainig to another can vary a lot. BUT : - Feature selection w/ elastic net for this sample is not satisfying (we only remove the first one) However, if we managed to show a form of stability for the elastic net regression, the restults are not satisfying : the regression dit not select much features among the 50 ones of the sample (one for the first method, 0 a lot of time for the second one). It is **like** the elastic net « found » that each \$X[i], i > 0\$ were generated from \$X\$ but did not « found » a link between the elements. We tried various changes in the method for the generation of the data and various params for the regularization without success. ... ...
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