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Aurélien Geron 2021-10-10 10:56:30 +13:00
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2
01_the_machine_learning_landscape.ipynb
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@ -93,7 +93,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The code in the book expects the data files to be located in the current directory. I just tweaked it here to fetch the files in datasets/lifesat."
"The code in the book expects the data files to be located in the current directory. I just tweaked it here to fetch the files in `datasets/lifesat`."
]
},
{

138
02_end_to_end_machine_learning_project.ipynb
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@ -83,7 +83,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Get the data"
"# Get the Data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Download the Data"
]
},
{
@ -132,6 +139,13 @@
" return pd.read_csv(csv_path)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Take a Quick Look at the Data Structure"
]
},
{
"cell_type": "code",
"execution_count": 5,
@ -182,6 +196,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Create a Test Set"
]
},
{
"cell_type": "code",
"execution_count": 10,
@ -443,7 +464,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Discover and visualize the data to gain insights"
"# Discover and Visualize the Data to Gain Insights"
]
},
{
@ -455,6 +476,13 @@
"housing = strat_train_set.copy()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Visualizing Geographical Data"
]
},
{
"cell_type": "code",
"execution_count": 33,
@ -540,6 +568,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Looking for Correlations"
]
},
{
"cell_type": "code",
"execution_count": 38,
@ -585,6 +620,13 @@
"save_fig(\"income_vs_house_value_scatterplot\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Experimenting with Attribute Combinations"
]
},
{
"cell_type": "code",
"execution_count": 42,
@ -631,7 +673,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Prepare the data for Machine Learning algorithms"
"# Prepare the Data for Machine Learning Algorithms"
]
},
{
@ -644,6 +686,29 @@
"housing_labels = strat_train_set[\"median_house_value\"].copy()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Data Cleaning"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the book 3 options are listed:\n",
"\n",
"```python\n",
"housing.dropna(subset=[\"total_bedrooms\"]) # option 1\n",
"housing.drop(\"total_bedrooms\", axis=1) # option 2\n",
"median = housing[\"total_bedrooms\"].median() # option 3\n",
"housing[\"total_bedrooms\"].fillna(median, inplace=True)\n",
"```\n",
"\n",
"To demonstrate each of them, let's create a copy of the housing dataset, but keeping only the rows that contain at least one null. Then it will be easier to visualize exactly what each option does:"
]
},
{
"cell_type": "code",
"execution_count": 47,
@ -815,6 +880,13 @@
"housing_tr.head()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Handling Text and Categorical Attributes"
]
},
{
"cell_type": "markdown",
"metadata": {},
@ -910,6 +982,13 @@
"cat_encoder.categories_"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Custom Transformers"
]
},
{
"cell_type": "markdown",
"metadata": {},
@ -985,6 +1064,13 @@
"housing_extra_attribs.head()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Transformation Pipelines"
]
},
{
"cell_type": "markdown",
"metadata": {},
@ -1154,7 +1240,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Select and train a model "
"# Select and Train a Model"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Training and Evaluating on the Training Set"
]
},
{
@ -1269,7 +1362,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Fine-tune your model"
"## Better Evaluation Using Cross-Validation"
]
},
{
@ -1382,6 +1475,20 @@
"svm_rmse"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Fine-Tune Your Model"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Grid Search"
]
},
{
"cell_type": "code",
"execution_count": 99,
@ -1457,6 +1564,13 @@
"pd.DataFrame(grid_search.cv_results_)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Randomized Search"
]
},
{
"cell_type": "code",
"execution_count": 104,
@ -1488,6 +1602,13 @@
" print(np.sqrt(-mean_score), params)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Analyze the Best Models and Their Errors"
]
},
{
"cell_type": "code",
"execution_count": 106,
@ -1512,6 +1633,13 @@
"sorted(zip(feature_importances, attributes), reverse=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Evaluate Your System on the Test Set"
]
},
{
"cell_type": "code",
"execution_count": 108,

56
03_classification.ipynb
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@ -245,7 +245,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Binary classifier"
"# Training a Binary Classifier"
]
},
{
@ -296,6 +296,20 @@
"cross_val_score(sgd_clf, X_train, y_train_5, cv=3, scoring=\"accuracy\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Performance Measures"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Measuring Accuracy Using Cross-Validation"
]
},
{
"cell_type": "code",
"execution_count": 18,
@ -362,6 +376,13 @@
"* lastly, other things may prevent perfect reproducibility, such as Python dicts and sets whose order is not guaranteed to be stable across sessions, or the order of files in a directory which is also not guaranteed."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Confusion Matrix"
]
},
{
"cell_type": "code",
"execution_count": 21,
@ -394,6 +415,13 @@
"confusion_matrix(y_train_5, y_train_perfect_predictions)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Precision and Recall"
]
},
{
"cell_type": "code",
"execution_count": 24,
@ -453,6 +481,13 @@
"cm[1, 1] / (cm[1, 1] + (cm[1, 0] + cm[0, 1]) / 2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Precision/Recall Trade-off"
]
},
{
"cell_type": "code",
"execution_count": 30,
@ -625,7 +660,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# ROC curves"
"## The ROC Curve"
]
},
{
@ -757,7 +792,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Multiclass classification"
"# Multiclass Classification"
]
},
{
@ -882,7 +917,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Error analysis"
"# Error Analysis"
]
},
{
@ -969,7 +1004,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Multilabel classification"
"# Multilabel Classification"
]
},
{
@ -1018,7 +1053,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Multioutput classification"
"# Multioutput Classification"
]
},
{
@ -2339,7 +2374,14 @@
"source": [
"# if running this notebook on Colab or Kaggle, we just pip install urlextract\n",
"if IS_COLAB or IS_KAGGLE:\n",
" !pip install -q -U urlextract"
" %pip install -q -U urlextract"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Note:** inside a Jupyter notebook, always use `%pip` instead of `!pip`, as `!pip` may install the library inside the wrong environment, while `%pip` makes sure it's installed inside the currently running environment."
]
},
{

80
04_training_linear_models.ipynb
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@ -4,7 +4,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"**Chapter 4 – Training Linear Models**"
"**Chapter 4 – Training Models**"
]
},
{
@ -89,7 +89,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Linear regression using the Normal Equation"
"# Linear Regression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The Normal Equation"
]
},
{
@ -243,7 +250,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Linear regression using batch gradient descent"
"# Gradient Descent\n",
"## Batch Gradient Descent"
]
},
{
@ -330,7 +338,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Stochastic Gradient Descent"
"## Stochastic Gradient Descent"
]
},
{
@ -416,7 +424,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mini-batch gradient descent"
"## Mini-batch gradient descent"
]
},
{
@ -494,7 +502,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Polynomial regression"
"# Polynomial Regression"
]
},
{
@ -616,6 +624,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Learning Curves"
]
},
{
"cell_type": "code",
"execution_count": 35,
@ -678,7 +693,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Regularized models"
"# Regularized Linear Models"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Ridge Regression"
]
},
{
@ -772,6 +794,13 @@
"sgd_reg.predict([[1.5]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Lasso Regression"
]
},
{
"cell_type": "code",
"execution_count": 43,
@ -803,6 +832,13 @@
"lasso_reg.predict([[1.5]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Elastic Net"
]
},
{
"cell_type": "code",
"execution_count": 45,
@ -815,6 +851,13 @@
"elastic_net.predict([[1.5]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Early Stopping"
]
},
{
"cell_type": "code",
"execution_count": 46,
@ -829,13 +872,6 @@
"X_train, X_val, y_train, y_val = train_test_split(X[:50], y[:50].ravel(), test_size=0.5, random_state=10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Early stopping example:"
]
},
{
"cell_type": "code",
"execution_count": 47,
@ -1029,7 +1065,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Logistic regression"
"# Logistic Regression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Decision Boundaries"
]
},
{
@ -1166,6 +1209,13 @@
"log_reg.predict([[1.7], [1.5]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Softmax Regression"
]
},
{
"cell_type": "code",
"execution_count": 62,

145
05_support_vector_machines.ipynb
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@ -84,14 +84,16 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Large margin classification"
"# Linear SVM Classification"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The next few code cells generate the first figures in chapter 5. The first actual code sample comes after:"
"The next few code cells generate the first figures in chapter 5. The first actual code sample comes after.\n",
"\n",
"**Code to generate Figure 5–1. Large margin classification**"
]
},
{
@ -175,7 +177,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Sensitivity to feature scales"
"**Code to generate Figure 5–2. Sensitivity to feature scales**"
]
},
{
@ -220,7 +222,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Sensitivity to outliers"
"## Soft Margin Classification\n",
"**Code to generate Figure 5–3. Hard margin sensitivity to outliers**"
]
},
{
@ -278,14 +281,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Large margin *vs* margin violations"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is the first code example in chapter 5:"
"**This is the first code example in chapter 5:**"
]
},
{
@ -325,7 +321,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's generate the graph comparing different regularization settings:"
"**Code to generate Figure 5–4. Large margin versus fewer margin violations**"
]
},
{
@ -408,7 +404,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Non-linear classification"
"# Nonlinear SVM Classification"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–5. Adding features to make a dataset linearly separable**"
]
},
{
@ -471,6 +474,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Here is second code example in the chapter:**"
]
},
{
"cell_type": "code",
"execution_count": 13,
@ -490,6 +500,13 @@
"polynomial_svm_clf.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–6. Linear SVM classifier using polynomial features**"
]
},
{
"cell_type": "code",
"execution_count": 14,
@ -513,6 +530,20 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Polynomial Kernel"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Next code example:**"
]
},
{
"cell_type": "code",
"execution_count": 15,
@ -528,6 +559,13 @@
"poly_kernel_svm_clf.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–7. SVM classifiers with a polynomial kernel**"
]
},
{
"cell_type": "code",
"execution_count": 16,
@ -564,6 +602,20 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Similarity Features"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–8. Similarity features using the Gaussian RBF**"
]
},
{
"cell_type": "code",
"execution_count": 18,
@ -644,6 +696,20 @@
" print(\"Phi({}, {}) = {}\".format(x1_example, landmark, k))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Gaussian RBF Kernel"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Next code example:**"
]
},
{
"cell_type": "code",
"execution_count": 20,
@ -657,6 +723,13 @@
"rbf_kernel_svm_clf.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–9. SVM classifiers using an RBF kernel**"
]
},
{
"cell_type": "code",
"execution_count": 21,
@ -701,7 +774,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Regression\n"
"# SVM Regression"
]
},
{
@ -716,6 +789,13 @@
"y = (4 + 3 * X + np.random.randn(m, 1)).ravel()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Next code example:**"
]
},
{
"cell_type": "code",
"execution_count": 23,
@ -728,6 +808,13 @@
"svm_reg.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–10. SVM Regression**"
]
},
{
"cell_type": "code",
"execution_count": 24,
@ -807,6 +894,13 @@
"**Note**: to be future-proof, we set `gamma=\"scale\"`, as this will be the default value in Scikit-Learn 0.22."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Next code example:**"
]
},
{
"cell_type": "code",
"execution_count": 27,
@ -819,6 +913,13 @@
"svm_poly_reg.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–11. SVM Regression using a second-degree polynomial kernel**"
]
},
{
"cell_type": "code",
"execution_count": 28,
@ -855,7 +956,15 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Under the hood"
"# Under the Hood\n",
"## Decision Function and Predictions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 5–12. Decision function for the iris dataset**"
]
},
{
@ -917,7 +1026,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Small weight vector results in a large margin"
"**Code to generate Figure 5–13. A smaller weight vector results in a larger margin**"
]
},
{
@ -976,7 +1085,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Hinge loss"
"**Code to generate the Hinge Loss figure:**"
]
},
{

130
06_decision_trees.ipynb
View File

@ -89,7 +89,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Training and visualizing"
"# Training and Visualizing a Decision Tree"
]
},
{
@ -109,6 +109,13 @@
"tree_clf.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**This code example generates Figure 6–1. Iris Decision Tree:**"
]
},
{
"cell_type": "code",
"execution_count": 3,
@ -130,6 +137,20 @@
"Source.from_file(os.path.join(IMAGES_PATH, \"iris_tree.dot\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Making Predictions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6–2. Decision Tree decision boundaries**"
]
},
{
"cell_type": "code",
"execution_count": 4,
@ -181,7 +202,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Predicting classes and class probabilities"
"# Estimating Class Probabilities"
]
},
{
@ -206,7 +227,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# High Variance"
"## Regularization Hyperparameters"
]
},
{
@ -227,6 +248,13 @@
"tree_clf_tweaked.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6–8. Sensitivity to training set details:**"
]
},
{
"cell_type": "code",
"execution_count": 8,
@ -244,9 +272,16 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6–3. Regularization using min_samples_leaf:**"
]
},
{
"cell_type": "code",
"execution_count": 10,
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
@ -271,9 +306,16 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Rotating the dataset also leads to completely different decision boundaries:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
@ -290,9 +332,16 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6–7. Sensitivity to training set rotation**"
]
},
{
"cell_type": "code",
"execution_count": 12,
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": [
@ -324,12 +373,19 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Regression trees"
"# Regression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's prepare a simple linear dataset:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
@ -341,9 +397,16 @@
"y = y + np.random.randn(m, 1) / 10"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code example:**"
]
},
{
"cell_type": "code",
"execution_count": 14,
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": [
@ -353,9 +416,16 @@
"tree_reg.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6–5. Predictions of two Decision Tree regression models:**"
]
},
{
"cell_type": "code",
"execution_count": 15,
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
@ -400,9 +470,16 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6-4. A Decision Tree for regression:**"
]
},
{
"cell_type": "code",
"execution_count": 16,
"execution_count": 15,
"metadata": {},
"outputs": [],
"source": [
@ -417,16 +494,23 @@
},
{
"cell_type": "code",
"execution_count": 17,
"execution_count": 16,
"metadata": {},
"outputs": [],
"source": [
"Source.from_file(os.path.join(IMAGES_PATH, \"regression_tree.dot\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 6–6. Regularizing a Decision Tree regressor:**"
]
},
{
"cell_type": "code",
"execution_count": 18,
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
@ -512,7 +596,7 @@
},
{
"cell_type": "code",
"execution_count": 19,
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
@ -530,7 +614,7 @@
},
{
"cell_type": "code",
"execution_count": 20,
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
@ -548,7 +632,7 @@
},
{
"cell_type": "code",
"execution_count": 21,
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": [
@ -562,7 +646,7 @@
},
{
"cell_type": "code",
"execution_count": 22,
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
@ -585,7 +669,7 @@
},
{
"cell_type": "code",
"execution_count": 23,
"execution_count": 22,
"metadata": {},
"outputs": [],
"source": [
@ -618,7 +702,7 @@
},
{
"cell_type": "code",
"execution_count": 24,
"execution_count": 23,
"metadata": {},
"outputs": [],
"source": [
@ -645,7 +729,7 @@
},
{
"cell_type": "code",
"execution_count": 25,
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
@ -673,7 +757,7 @@
},
{
"cell_type": "code",
"execution_count": 26,
"execution_count": 25,
"metadata": {},
"outputs": [],
"source": [
@ -685,7 +769,7 @@
},
{
"cell_type": "code",
"execution_count": 27,
"execution_count": 26,
"metadata": {},
"outputs": [],
"source": [
@ -703,7 +787,7 @@
},
{
"cell_type": "code",
"execution_count": 28,
"execution_count": 27,
"metadata": {},
"outputs": [],
"source": [

227
07_ensemble_learning_and_random_forests.ipynb
View File

@ -89,7 +89,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Voting classifiers"
"# Voting Classifiers"
]
},
{
@ -103,6 +103,13 @@
"cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 7–3. The law of large numbers:**"
]
},
{
"cell_type": "code",
"execution_count": 3,
@ -121,6 +128,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's use the moons dataset:"
]
},
{
"cell_type": "code",
"execution_count": 4,
@ -141,6 +155,13 @@
"**Note**: to be future-proof, we set `solver=\"lbfgs\"`, `n_estimators=100`, and `gamma=\"scale\"` since these will be the default values in upcoming Scikit-Learn versions."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code examples:**"
]
},
{
"cell_type": "code",
"execution_count": 5,
@ -232,7 +253,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Bagging ensembles"
"# Bagging and Pasting\n",
"## Bagging and Pasting in Scikit-Learn"
]
},
{
@ -273,6 +295,13 @@
"print(accuracy_score(y_test, y_pred_tree))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 7–5. A single Decision Tree (left) versus a bagging ensemble of 500 trees (right):**"
]
},
{
"cell_type": "code",
"execution_count": 13,
@ -302,7 +331,9 @@
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"metadata": {
"scrolled": true
},
"outputs": [],
"source": [
"fix, axes = plt.subplots(ncols=2, figsize=(10,4), sharey=True)\n",
@ -321,7 +352,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Random Forests"
"## Out-of-Bag evaluation"
]
},
{
@ -331,8 +362,10 @@
"outputs": [],
"source": [
"bag_clf = BaggingClassifier(\n",
" DecisionTreeClassifier(max_features=\"sqrt\", max_leaf_nodes=16),\n",
" n_estimators=500, random_state=42)"
" DecisionTreeClassifier(), n_estimators=500,\n",
" bootstrap=True, oob_score=True, random_state=40)\n",
"bag_clf.fit(X_train, y_train)\n",
"bag_clf.oob_score_"
]
},
{
@ -341,13 +374,32 @@
"metadata": {},
"outputs": [],
"source": [
"bag_clf.fit(X_train, y_train)\n",
"y_pred = bag_clf.predict(X_test)"
"bag_clf.oob_decision_function_"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"scrolled": true
},
"outputs": [],
"source": [
"from sklearn.metrics import accuracy_score\n",
"y_pred = bag_clf.predict(X_test)\n",
"accuracy_score(y_test, y_pred)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Random Forests"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
@ -359,18 +411,53 @@
"y_pred_rf = rnd_clf.predict(X_test)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A Random Forest is equivalent to a bag of decision trees:"
]
},
{
"cell_type": "code",
"execution_count": 18,
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"bag_clf = BaggingClassifier(\n",
" DecisionTreeClassifier(max_features=\"sqrt\", max_leaf_nodes=16),\n",
" n_estimators=500, random_state=42)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": [
"bag_clf.fit(X_train, y_train)\n",
"y_pred = bag_clf.predict(X_test)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"np.sum(y_pred == y_pred_rf) / len(y_pred) # very similar predictions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Feature Importance"
]
},
{
"cell_type": "code",
"execution_count": 19,
"execution_count": 22,
"metadata": {},
"outputs": [],
"source": [
@ -384,16 +471,23 @@
},
{
"cell_type": "code",
"execution_count": 20,
"execution_count": 23,
"metadata": {},
"outputs": [],
"source": [
"rnd_clf.feature_importances_"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The following figure overlays the decision boundaries of 15 decision trees. As you can see, even though each decision tree is imperfect, the ensemble defines a pretty good decision boundary:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
@ -402,7 +496,7 @@
"for i in range(15):\n",
" tree_clf = DecisionTreeClassifier(max_leaf_nodes=16, random_state=42 + i)\n",
" indices_with_replacement = np.random.randint(0, len(X_train), len(X_train))\n",
" tree_clf.fit(X[indices_with_replacement], y[indices_with_replacement])\n",
" tree_clf.fit(X_train[indices_with_replacement], y_train[indices_with_replacement])\n",
" plot_decision_boundary(tree_clf, X, y, axes=[-1.5, 2.45, -1, 1.5], alpha=0.02, contour=False)\n",
"\n",
"plt.show()"
@ -412,47 +506,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Out-of-Bag evaluation"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [],
"source": [
"bag_clf = BaggingClassifier(\n",
" DecisionTreeClassifier(), n_estimators=500,\n",
" bootstrap=True, oob_score=True, random_state=40)\n",
"bag_clf.fit(X_train, y_train)\n",
"bag_clf.oob_score_"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [],
"source": [
"bag_clf.oob_decision_function_"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
"from sklearn.metrics import accuracy_score\n",
"y_pred = bag_clf.predict(X_test)\n",
"accuracy_score(y_test, y_pred)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Feature importance"
"**Code to generate Figure 7–6. MNIST pixel importance (according to a Random Forest classifier):**"
]
},
{
@ -516,7 +570,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# AdaBoost"
"# Boosting\n",
"## AdaBoost"
]
},
{
@ -542,6 +597,13 @@
"plot_decision_boundary(ada_clf, X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 7–8. Decision boundaries of consecutive predictors:**"
]
},
{
"cell_type": "code",
"execution_count": 31,
@ -583,7 +645,14 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Gradient Boosting"
"## Gradient Boosting"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let create a simple quadratic dataset:"
]
},
{
@ -597,6 +666,13 @@
"y = 3*X[:, 0]**2 + 0.05 * np.random.randn(100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's train a decision tree regressor on this dataset:"
]
},
{
"cell_type": "code",
"execution_count": 33,
@ -658,6 +734,13 @@
"y_pred"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 7–9. In this depiction of Gradient Boosting, the first predictor (top left) is trained normally, then each consecutive predictor (middle left and lower left) is trained on the previous predictor’s residuals; the right column shows the resulting ensemble’s predictions:**"
]
},
{
"cell_type": "code",
"execution_count": 39,
@ -714,6 +797,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's try a gradient boosting regressor:"
]
},
{
"cell_type": "code",
"execution_count": 41,
@ -726,6 +816,13 @@
"gbrt.fit(X, y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 7–10. GBRT ensembles with not enough predictors (left) and too many (right):**"
]
},
{
"cell_type": "code",
"execution_count": 42,
@ -763,7 +860,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Gradient Boosting with Early stopping"
"**Gradient Boosting with Early stopping:**"
]
},
{
@ -789,6 +886,13 @@
"gbrt_best.fit(X_train, y_train)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 7–11. Tuning the number of trees using early stopping:**"
]
},
{
"cell_type": "code",
"execution_count": 45,
@ -827,6 +931,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Early stopping with some patience (interrupts training only after there's no improvement for 5 epochs):"
]
},
{
"cell_type": "code",
"execution_count": 47,
@ -873,7 +984,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Using XGBoost"
"**Using XGBoost:**"
]
},
{

172
08_dimensionality_reduction.ipynb
View File

@ -84,8 +84,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Projection methods\n",
"Build 3D dataset:"
"# PCA\n",
"Let's build a simple 3D dataset:"
]
},
{
@ -110,7 +110,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## PCA using SVD decomposition"
"## Principal Components"
]
},
{
@ -146,6 +146,13 @@
"np.allclose(X_centered, U.dot(S).dot(Vt))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Projecting Down to d Dimensions"
]
},
{
"cell_type": "code",
"execution_count": 6,
@ -169,7 +176,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## PCA using Scikit-Learn"
"## Using Scikit-Learn"
]
},
{
@ -344,6 +351,13 @@
"Notice how the axes are flipped."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Explained Variance Ratio"
]
},
{
"cell_type": "markdown",
"metadata": {},
@ -406,6 +420,13 @@
"Next, let's generate some nice figures! :)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–2. A 3D dataset lying close to a 2D subspace:**"
]
},
{
"cell_type": "markdown",
"metadata": {},
@ -515,6 +536,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–3. The new 2D dataset after projection:**"
]
},
{
"cell_type": "code",
"execution_count": 25,
@ -540,8 +568,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# Manifold learning\n",
"Swiss roll:"
"**Code to generate Figure 8–4. Swiss roll dataset:**"
]
},
{
@ -551,6 +578,7 @@
"outputs": [],
"source": [
"from sklearn.datasets import make_swiss_roll\n",
"\n",
"X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)"
]
},
@ -578,6 +606,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–5. Squashing by projecting onto a plane (left) versus unrolling the Swiss roll (right):**"
]
},
{
"cell_type": "code",
"execution_count": 28,
@ -603,6 +638,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–6. The decision boundary may not always be simpler with lower dimensions:**"
]
},
{
"cell_type": "code",
"execution_count": 29,
@ -688,7 +730,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# PCA"
"**Code to generate Figure 8–7. Selecting the subspace to project on:**"
]
},
{
@ -761,7 +803,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# MNIST compression"
"## Choosing the Right Number of Dimensions"
]
},
{
@ -818,6 +860,13 @@
"d"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–8. Explained variance as a function of the number of dimensions:**"
]
},
{
"cell_type": "code",
"execution_count": 35,
@ -867,6 +916,13 @@
"np.sum(pca.explained_variance_ratio_)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## PCA for Compression"
]
},
{
"cell_type": "code",
"execution_count": 39,
@ -878,6 +934,13 @@
"X_recovered = pca.inverse_transform(X_reduced)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–9. MNIST compression that preserves 95% of the variance:**"
]
},
{
"cell_type": "code",
"execution_count": 40,
@ -930,7 +993,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Incremental PCA"
"## Randomized PCA"
]
},
{
@ -938,6 +1001,23 @@
"execution_count": 43,
"metadata": {},
"outputs": [],
"source": [
"rnd_pca = PCA(n_components=154, svd_solver=\"randomized\", random_state=42)\n",
"X_reduced = rnd_pca.fit_transform(X_train)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Incremental PCA"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [],
"source": [
"from sklearn.decomposition import IncrementalPCA\n",
"\n",
@ -952,16 +1032,23 @@
},
{
"cell_type": "code",
"execution_count": 44,
"execution_count": 45,
"metadata": {},
"outputs": [],
"source": [
"X_recovered_inc_pca = inc_pca.inverse_transform(X_reduced)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's check that compression still works well:"
]
},
{
"cell_type": "code",
"execution_count": 45,
"execution_count": 46,
"metadata": {},
"outputs": [],
"source": [
@ -975,7 +1062,7 @@
},
{
"cell_type": "code",
"execution_count": 46,
"execution_count": 47,
"metadata": {},
"outputs": [],
"source": [
@ -991,7 +1078,7 @@
},
{
"cell_type": "code",
"execution_count": 47,
"execution_count": 48,
"metadata": {},
"outputs": [],
"source": [
@ -1007,7 +1094,7 @@
},
{
"cell_type": "code",
"execution_count": 48,
"execution_count": 49,
"metadata": {},
"outputs": [],
"source": [
@ -1018,7 +1105,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Using `memmap()`"
"**Using `memmap()`:**"
]
},
{
@ -1030,7 +1117,7 @@
},
{
"cell_type": "code",
"execution_count": 49,
"execution_count": 50,
"metadata": {},
"outputs": [],
"source": [
@ -1050,7 +1137,7 @@
},
{
"cell_type": "code",
"execution_count": 50,
"execution_count": 51,
"metadata": {},
"outputs": [],
"source": [
@ -1066,7 +1153,7 @@
},
{
"cell_type": "code",
"execution_count": 51,
"execution_count": 52,
"metadata": {},
"outputs": [],
"source": [
@ -1077,21 +1164,11 @@
"inc_pca.fit(X_mm)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [],
"source": [
"rnd_pca = PCA(n_components=154, svd_solver=\"randomized\", random_state=42)\n",
"X_reduced = rnd_pca.fit_transform(X_train)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Time complexity"
"**Time complexity:**"
]
},
{
@ -1226,6 +1303,13 @@
"X_reduced = rbf_pca.fit_transform(X)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–10. Swiss roll reduced to 2D using kPCA with various kernels:**"
]
},
{
"cell_type": "code",
"execution_count": 58,
@ -1260,6 +1344,13 @@
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–11. Kernel PCA and the reconstruction pre-image error:**"
]
},
{
"cell_type": "code",
"execution_count": 59,
@ -1300,6 +1391,13 @@
"plt.grid(True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Selecting a Kernel and Tuning Hyperparameters"
]
},
{
"cell_type": "code",
"execution_count": 61,
@ -1384,6 +1482,13 @@
"X_reduced = lle.fit_transform(X)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–12. Unrolled Swiss roll using LLE:**"
]
},
{
"cell_type": "code",
"execution_count": 67,
@ -1405,7 +1510,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"# MDS, Isomap and t-SNE"
"## Other Dimensionality Reduction Techniques"
]
},
{
@ -1459,6 +1564,13 @@
"X_reduced_lda = lda.transform(X_mnist)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Code to generate Figure 8–13. Using various techniques to reduce the Swill roll to 2D:**"
]
},
{
"cell_type": "code",
"execution_count": 72,

37
09_unsupervised_learning.ipynb
View File

@ -91,7 +91,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Introduction – Classification _vs_ Clustering"
"**Introduction – Classification _vs_ Clustering**"
]
},
{
@ -320,7 +320,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Fit and Predict"
"**Fit and predict**"
]
},
{
@ -428,7 +428,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Decision Boundaries"
"**Decision Boundaries**"
]
},
{
@ -507,7 +507,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Hard Clustering _vs_ Soft Clustering"
"**Hard Clustering _vs_ Soft Clustering**"
]
},
{
@ -546,7 +546,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### K-Means Algorithm"
"### The K-Means Algorithm"
]
},
{
@ -639,7 +639,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### K-Means Variability"
"**K-Means Variability**"
]
},
{
@ -827,7 +827,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### K-Means++"
"### Centroid initialization methods"
]
},
{
@ -1432,7 +1432,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Limits of K-Means"
"## Limits of K-Means"
]
},
{
@ -1494,7 +1494,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Using clustering for image segmentation"
"## Using Clustering for Image Segmentation"
]
},
{
@ -1578,7 +1578,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Using Clustering for Preprocessing"
"## Using Clustering for Preprocessing"
]
},
{
@ -1785,7 +1785,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### Clustering for Semi-supervised Learning"
"## Using Clustering for Semi-Supervised Learning"
]
},
{
@ -2756,7 +2756,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Anomaly Detection using Gaussian Mixtures"
"## Anomaly Detection Using Gaussian Mixtures"
]
},
{
@ -2797,7 +2797,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Model selection"
"## Selecting the Number of Clusters"
]
},
{
@ -2983,7 +2983,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Variational Bayesian Gaussian Mixtures"
"## Bayesian Gaussian Mixture Models"
]
},
{
@ -3151,7 +3151,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Likelihood Function"
"**Likelihood Function**"
]
},
{
@ -3242,13 +3242,6 @@
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},

17
11_training_deep_neural_networks.ipynb
View File

@ -970,6 +970,13 @@
"model_B_on_A.add(keras.layers.Dense(1, activation=\"sigmoid\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that `model_B_on_A` and `model_A` actually share layers now, so when we train one, it will update both models. If we want to avoid that, we need to build `model_B_on_A` on top of a *clone* of `model_A`:"
]
},
{
"cell_type": "code",
"execution_count": 61,
@ -977,7 +984,9 @@
"outputs": [],
"source": [
"model_A_clone = keras.models.clone_model(model_A)\n",
"model_A_clone.set_weights(model_A.get_weights())"
"model_A_clone.set_weights(model_A.get_weights())\n",
"model_B_on_A = keras.models.Sequential(model_A_clone.layers[:-1])\n",
"model_B_on_A.add(keras.layers.Dense(1, activation=\"sigmoid\"))"
]
},
{
@ -1042,7 +1051,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Great! We got quite a bit of transfer: the error rate dropped by a factor of 4.5!"
"Great! We got quite a bit of transfer: the error rate dropped by a factor of 4.9!"
]
},
{
@ -1051,7 +1060,7 @@
"metadata": {},
"outputs": [],
"source": [
"(100 - 97.05) / (100 - 99.35)"
"(100 - 97.05) / (100 - 99.40)"
]
},
{
@ -1777,7 +1786,7 @@
"layer = keras.layers.Dense(100, activation=\"elu\",\n",
" kernel_initializer=\"he_normal\",\n",
" kernel_regularizer=keras.regularizers.l2(0.01))\n",
"# or l1(0.1) for ℓ1 regularization with a factor or 0.1\n",
"# or l1(0.1) for ℓ1 regularization with a factor of 0.1\n",
"# or l1_l2(0.1, 0.01) for both ℓ1 and ℓ2 regularization, with factors 0.1 and 0.01 respectively"
]
},

14
12_custom_models_and_training_with_tensorflow.ipynb
View File

@ -2209,7 +2209,9 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"**Note**: due to an issue introduced in TF 2.2 ([#46858](https://github.com/tensorflow/tensorflow/issues/46858)), it is currently not possible to use `add_loss()` along with the `build()` method. So the following code differs from the book: I create the `reconstruct` layer in the constructor instead of the `build()` method. Unfortunately, this means that the number of units in this layer must be hard-coded (alternatively, it could be passed as an argument to the constructor)."
"**Note**: the following code has two differences with the code in the book:\n",
"1. It creates a `keras.metrics.Mean()` metric in the constructor and uses it in the `call()` method to track the mean reconstruction loss. Since we only want to do this during training, we add a `training` argument to the `call()` method, and if `training` is `True`, then we update `reconstruction_mean` and we call `self.add_metric()` to ensure it's displayed properly.\n",
"2. Due to an issue introduced in TF 2.2 ([#46858](https://github.com/tensorflow/tensorflow/issues/46858)), we must not call `super().build()` inside the `build()` method."
]
},
{
@ -2218,20 +2220,18 @@
"metadata": {},
"outputs": [],
"source": [
"class ReconstructingRegressor(keras.models.Model):\n",
"class ReconstructingRegressor(keras.Model):\n",
" def __init__(self, output_dim, **kwargs):\n",
" super().__init__(**kwargs)\n",
" self.hidden = [keras.layers.Dense(30, activation=\"selu\",\n",
" kernel_initializer=\"lecun_normal\")\n",
" for _ in range(5)]\n",
" self.out = keras.layers.Dense(output_dim)\n",
" self.reconstruct = keras.layers.Dense(8) # workaround for TF issue #46858\n",
" self.reconstruction_mean = keras.metrics.Mean(name=\"reconstruction_error\")\n",
"\n",
" #Commented out due to TF issue #46858, see the note above\n",
" #def build(self, batch_input_shape):\n",
" # n_inputs = batch_input_shape[-1]\n",
" # self.reconstruct = keras.layers.Dense(n_inputs)\n",
" def build(self, batch_input_shape):\n",
" n_inputs = batch_input_shape[-1]\n",
" self.reconstruct = keras.layers.Dense(n_inputs)\n",
" #super().build(batch_input_shape)\n",
"\n",
" def call(self, inputs, training=None):\n",

2
13_loading_and_preprocessing_data.ipynb
View File

@ -52,7 +52,7 @@
"IS_KAGGLE = \"kaggle_secrets\" in sys.modules\n",
"\n",
"if IS_COLAB or IS_KAGGLE:\n",
" !pip install -q -U tfx==0.21.2\n",
" %pip install -q -U tfx==0.21.2\n",
" print(\"You can safely ignore the package incompatibility errors.\")\n",
"\n",
"# Scikit-Learn ≥0.20 is required\n",

22
15_processing_sequences_using_rnns_and_cnns.ipynb
View File

@ -162,12 +162,12 @@
"metadata": {},
"outputs": [],
"source": [
"def plot_series(series, y=None, y_pred=None, x_label=\"$t$\", y_label=\"$x(t)$\"):\n",
"def plot_series(series, y=None, y_pred=None, x_label=\"$t$\", y_label=\"$x(t)$\", legend=True):\n",
" plt.plot(series, \".-\")\n",
" if y is not None:\n",
" plt.plot(n_steps, y, \"bx\", markersize=10)\n",
" plt.plot(n_steps, y, \"bo\", label=\"Target\")\n",
" if y_pred is not None:\n",
" plt.plot(n_steps, y_pred, \"ro\")\n",
" plt.plot(n_steps, y_pred, \"rx\", markersize=10, label=\"Prediction\")\n",
" plt.grid(True)\n",
" if x_label:\n",
" plt.xlabel(x_label, fontsize=16)\n",
@ -175,16 +175,26 @@
" plt.ylabel(y_label, fontsize=16, rotation=0)\n",
" plt.hlines(0, 0, 100, linewidth=1)\n",
" plt.axis([0, n_steps + 1, -1, 1])\n",
" if legend and (y or y_pred):\n",
" plt.legend(fontsize=14, loc=\"upper left\")\n",
"\n",
"fig, axes = plt.subplots(nrows=1, ncols=3, sharey=True, figsize=(12, 4))\n",
"for col in range(3):\n",
" plt.sca(axes[col])\n",
" plot_series(X_valid[col, :, 0], y_valid[col, 0],\n",
" y_label=(\"$x(t)$\" if col==0 else None))\n",
" y_label=(\"$x(t)$\" if col==0 else None),\n",
" legend=(col == 0))\n",
"save_fig(\"time_series_plot\")\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Note**: in this notebook, the blue dots represent targets, and red crosses represent predictions. In the book, I first used blue crosses for targets and red dots for predictions, then I reversed this later in the chapter. Sorry if this caused some confusion."
]
},
{
"cell_type": "markdown",
"metadata": {},
@ -499,8 +509,8 @@
" n_steps = X.shape[1]\n",
" ahead = Y.shape[1]\n",
" plot_series(X[0, :, 0])\n",
" plt.plot(np.arange(n_steps, n_steps + ahead), Y[0, :, 0], \"ro-\", label=\"Actual\")\n",
" plt.plot(np.arange(n_steps, n_steps + ahead), Y_pred[0, :, 0], \"bx-\", label=\"Forecast\", markersize=10)\n",
" plt.plot(np.arange(n_steps, n_steps + ahead), Y[0, :, 0], \"bo-\", label=\"Actual\")\n",
" plt.plot(np.arange(n_steps, n_steps + ahead), Y_pred[0, :, 0], \"rx-\", label=\"Forecast\", markersize=10)\n",
" plt.axis([0, n_steps + ahead, -1, 1])\n",
" plt.legend(fontsize=14)\n",
"\n",

4
16_nlp_with_rnns_and_attention.ipynb
View File

@ -57,8 +57,8 @@
"IS_KAGGLE = \"kaggle_secrets\" in sys.modules\n",
"\n",
"if IS_COLAB:\n",
" !pip install -q -U tensorflow-addons\n",
" !pip install -q -U transformers\n",
" %pip install -q -U tensorflow-addons\n",
" %pip install -q -U transformers\n",
"\n",
"# Scikit-Learn ≥0.20 is required\n",
"import sklearn\n",

7
18_reinforcement_learning.ipynb
View File

@ -52,7 +52,8 @@
"\n",
"if IS_COLAB or IS_KAGGLE:\n",
" !apt update && apt install -y libpq-dev libsdl2-dev swig xorg-dev xvfb\n",
" !pip install -q -U tf-agents pyvirtualdisplay gym[atari,box2d]\n",
" %pip install -q -U tf-agents pyvirtualdisplay gym[box2d]\n",
" %pip install -q -U atari_py==0.2.5\n",
"\n",
"# Scikit-Learn ≥0.20 is required\n",
"import sklearn\n",
@ -219,7 +220,7 @@
"Alternatively, you can install the [pyvirtualdisplay](https://github.com/ponty/pyvirtualdisplay) Python library which wraps Xvfb:\n",
"\n",
"```bash\n",
"python3 -m pip install -U pyvirtualdisplay\n",
"%pip install -U pyvirtualdisplay\n",
"```\n",
"\n",
"And run the following code:"
@ -2811,7 +2812,7 @@
"metadata": {},
"source": [
"## 8.\n",
"_Exercise: Use policy gradients to solve OpenAI Gym's LunarLander-v2 environment. You will need to install the Box2D dependencies (`python3 -m pip install -U gym[box2d]`)._"
"_Exercise: Use policy gradients to solve OpenAI Gym's LunarLander-v2 environment. You will need to install the Box2D dependencies (`%pip install -U gym[box2d]`)._"
]
},
{

2
19_training_and_deploying_at_scale.ipynb
View File

@ -54,7 +54,7 @@
" !echo \"deb http://storage.googleapis.com/tensorflow-serving-apt stable tensorflow-model-server tensorflow-model-server-universal\" > /etc/apt/sources.list.d/tensorflow-serving.list\n",
" !curl https://storage.googleapis.com/tensorflow-serving-apt/tensorflow-serving.release.pub.gpg | apt-key add -\n",
" !apt update && apt-get install -y tensorflow-model-server\n",
" !pip install -q -U tensorflow-serving-api\n",
" %pip install -q -U tensorflow-serving-api\n",
"\n",
"# Scikit-Learn ≥0.20 is required\n",
"import sklearn\n",

2
environment.yml
View File

@ -3,7 +3,7 @@ channels:
- conda-forge
- defaults
dependencies:
- atari_py=0.2 # used only in chapter 18
- atari_py=0.2.6 # used only in chapter 18
- box2d-py=2.3 # used only in chapter 18
- ftfy=5.8 # used only in chapter 16 by the transformers library
- graphviz # used only in chapter 6 for dot files

212
math_differential_calculus.ipynb
View File

@ -439,16 +439,16 @@
"Let's look at a concrete example. Let's see if we can determine what the slope of the $y=x^2$ curve is, at any point $\\mathrm{A}$ (try to understand each line, I promise it's not that hard):\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x_\\mathrm{A}) \\, && = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim\\dfrac{f(x_\\mathrm{B}) - f(x_\\mathrm{A})}{x_\\mathrm{B} - x_\\mathrm{A}} \\\\\n",
"&& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim\\dfrac{{x_\\mathrm{B}}^2 - {x_\\mathrm{A}}^2}{x_\\mathrm{B} - x_\\mathrm{A}} \\quad && \\text{since } f(x) = x^2\\\\\n",
"&& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim\\dfrac{(x_\\mathrm{B} - x_\\mathrm{A})(x_\\mathrm{B} + x_\\mathrm{A})}{x_\\mathrm{B} - x_\\mathrm{A}}\\quad && \\text{since } {x_\\mathrm{A}}^2 - {x_\\mathrm{B}}^2 = (x_\\mathrm{A}-x_\\mathrm{B})(x_\\mathrm{A}+x_\\mathrm{B})\\\\\n",
"&& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim(x_\\mathrm{B} + x_\\mathrm{A})\\quad && \\text{since the two } (x_\\mathrm{B} - x_\\mathrm{A}) \\text{ cancel out}\\\\\n",
"&& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim x_\\mathrm{B} \\, + \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim x_\\mathrm{A}\\quad && \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"&& = x_\\mathrm{A} \\, + \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim x_\\mathrm{A} \\quad && \\text{since } x_\\mathrm{B}\\text{ approaches } x_\\mathrm{A} \\\\\n",
"&& = x_\\mathrm{A} + x_\\mathrm{A} \\quad && \\text{since } x_\\mathrm{A} \\text{ remains constant when } x_\\mathrm{B}\\text{ approaches } x_\\mathrm{A} \\\\\n",
"&& = 2 x_\\mathrm{A}\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x_\\mathrm{A}) \\, & = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim\\dfrac{f(x_\\mathrm{B}) - f(x_\\mathrm{A})}{x_\\mathrm{B} - x_\\mathrm{A}} \\\\\n",
"& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim\\dfrac{{x_\\mathrm{B}}^2 - {x_\\mathrm{A}}^2}{x_\\mathrm{B} - x_\\mathrm{A}} \\quad && \\text{since } f(x) = x^2\\\\\n",
"& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim\\dfrac{(x_\\mathrm{B} - x_\\mathrm{A})(x_\\mathrm{B} + x_\\mathrm{A})}{x_\\mathrm{B} - x_\\mathrm{A}}\\quad && \\text{since } {x_\\mathrm{A}}^2 - {x_\\mathrm{B}}^2 = (x_\\mathrm{A}-x_\\mathrm{B})(x_\\mathrm{A}+x_\\mathrm{B})\\\\\n",
"& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim(x_\\mathrm{B} + x_\\mathrm{A})\\quad && \\text{since the two } (x_\\mathrm{B} - x_\\mathrm{A}) \\text{ cancel out}\\\\\n",
"& = \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim x_\\mathrm{B} \\, + \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim x_\\mathrm{A}\\quad && \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"& = x_\\mathrm{A} \\, + \\underset{x_\\mathrm{B} \\to x_\\mathrm{A}}\\lim x_\\mathrm{A} \\quad && \\text{since } x_\\mathrm{B}\\text{ approaches } x_\\mathrm{A} \\\\\n",
"& = x_\\mathrm{A} + x_\\mathrm{A} \\quad && \\text{since } x_\\mathrm{A} \\text{ remains constant when } x_\\mathrm{B}\\text{ approaches } x_\\mathrm{A} \\\\\n",
"& = 2 x_\\mathrm{A}\n",
"\\end{align*}\n",
"$\n",
"\n",
"That's it! We just proved that the slope of $y = x^2$ at any point $\\mathrm{A}$ is $f'(x_\\mathrm{A}) = 2x_\\mathrm{A}$. What we have done is called **differentiation**: finding the derivative of a function."
@ -517,14 +517,14 @@
"Okay! Now let's use this new definition to find the derivative of $f(x) = x^2$ at any point $x$, and (hopefully) we should find the same result as above (except using $x$ instead of $x_\\mathrm{A}$):\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) \\, && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x + \\epsilon) - f(x)}{\\epsilon} \\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{(x + \\epsilon)^2 - {x}^2}{\\epsilon} \\quad && \\text{since } f(x) = x^2\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{{x}^2 + 2x\\epsilon + \\epsilon^2 - {x}^2}{\\epsilon}\\quad && \\text{since } (x + \\epsilon)^2 = {x}^2 + 2x\\epsilon + \\epsilon^2\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{2x\\epsilon + \\epsilon^2}{\\epsilon}\\quad && \\text{since the two } {x}^2 \\text{ cancel out}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim \\, (2x + \\epsilon)\\quad && \\text{since } 2x\\epsilon \\text{ and } \\epsilon^2 \\text{ can both be divided by } \\epsilon\\\\\n",
"&& = 2 x\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) \\, & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x + \\epsilon) - f(x)}{\\epsilon} \\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{(x + \\epsilon)^2 - {x}^2}{\\epsilon} \\quad && \\text{since } f(x) = x^2\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{{x}^2 + 2x\\epsilon + \\epsilon^2 - {x}^2}{\\epsilon}\\quad && \\text{since } (x + \\epsilon)^2 = {x}^2 + 2x\\epsilon + \\epsilon^2\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{2x\\epsilon + \\epsilon^2}{\\epsilon}\\quad && \\text{since the two } {x}^2 \\text{ cancel out}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim \\, (2x + \\epsilon)\\quad && \\text{since } 2x\\epsilon \\text{ and } \\epsilon^2 \\text{ can both be divided by } \\epsilon\\\\\n",
"& = 2 x\n",
"\\end{align*}\n",
"$\n",
"\n",
"Yep! It works out."
@ -705,13 +705,13 @@
"One very important rule is that **the derivative of a sum is the sum of the derivatives**. More precisely, if we define $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$. This is quite easy to prove:\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon) + h(x+\\epsilon) - g(x) - h(x)}{\\epsilon} && \\quad \\text{using }f(x) = g(x) + h(x) \\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon) - g(x) + h(x+\\epsilon) - h(x)}{\\epsilon} && \\quad \\text{just moving terms around}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon} + \\underset{\\epsilon \\to 0}\\lim\\dfrac{h(x+\\epsilon) - h(x)}{\\epsilon} && \\quad \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"&& = g'(x) + h'(x) && \\quad \\text{using the definitions of }g'(x) \\text{ and } h'(x)\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon) + h(x+\\epsilon) - g(x) - h(x)}{\\epsilon} && \\quad \\text{using }f(x) = g(x) + h(x) \\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon) - g(x) + h(x+\\epsilon) - h(x)}{\\epsilon} && \\quad \\text{just moving terms around}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon} + \\underset{\\epsilon \\to 0}\\lim\\dfrac{h(x+\\epsilon) - h(x)}{\\epsilon} && \\quad \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"& = g'(x) + h'(x) && \\quad \\text{using the definitions of }g'(x) \\text{ and } h'(x)\n",
"\\end{align*}\n",
"$"
]
},
@ -1305,12 +1305,12 @@
"## Constant: $f(x)=c$\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{c - c}{\\epsilon} && \\quad \\text{using }f(x) = c \\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim 0 && \\quad \\text{since }c - c = 0\\\\\n",
"&& = 0 && \\quad \\text{since the limit of a constant is that constant}\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{c - c}{\\epsilon} && \\quad \\text{using }f(x) = c \\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim 0 && \\quad \\text{since }c - c = 0\\\\\n",
"& = 0 && \\quad \\text{since the limit of a constant is that constant}\n",
"\\end{align*}\n",
"$\n"
]
},
@ -1324,18 +1324,18 @@
"## Product rule: $f(x)=g(x)h(x)$\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon)h(x+\\epsilon) - g(x)h(x)}{\\epsilon} && \\quad \\text{using }f(x) = g(x)h(x) \\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon)h(x+\\epsilon) - g(x)h(x+\\epsilon) + g(x)h(x + \\epsilon) - g(x)h(x)}{\\epsilon} && \\quad \\text{subtracting and adding }g(x)h(x + \\epsilon)\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon)h(x+\\epsilon) - g(x)h(x+\\epsilon)}{\\epsilon} + \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x)h(x + \\epsilon) - g(x)h(x)}{\\epsilon} && \\quad \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}h(x+\\epsilon)\\right]} \\,+\\, \\underset{\\epsilon \\to 0}\\lim{\\left[g(x)\\dfrac{h(x + \\epsilon) - h(x)}{\\epsilon}\\right]} && \\quad \\text{factorizing }h(x+\\epsilon) \\text{ and } g(x)\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}h(x+\\epsilon)\\right]} \\,+\\, g(x)\\underset{\\epsilon \\to 0}\\lim{\\dfrac{h(x + \\epsilon) - h(x)}{\\epsilon}} && \\quad \\text{taking } g(x) \\text{ out of the limit since it does not depend on }\\epsilon\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}h(x+\\epsilon)\\right]} \\,+\\, g(x)h'(x) && \\quad \\text{using the definition of h'(x)}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}\\right]}\\underset{\\epsilon \\to 0}\\lim{h(x+\\epsilon)} + g(x)h'(x) && \\quad \\text{since the limit of a product is the product of the limits}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}\\right]}h(x) + h(x)g'(x) && \\quad \\text{since } h(x) \\text{ is continuous}\\\\\n",
"&& = g'(x)h(x) + g(x)h'(x) && \\quad \\text{using the definition of }g'(x)\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon)h(x+\\epsilon) - g(x)h(x)}{\\epsilon} && \\quad \\text{using }f(x) = g(x)h(x) \\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon)h(x+\\epsilon) - g(x)h(x+\\epsilon) + g(x)h(x + \\epsilon) - g(x)h(x)}{\\epsilon} && \\quad \\text{subtracting and adding }g(x)h(x + \\epsilon)\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x+\\epsilon)h(x+\\epsilon) - g(x)h(x+\\epsilon)}{\\epsilon} + \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(x)h(x + \\epsilon) - g(x)h(x)}{\\epsilon} && \\quad \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}h(x+\\epsilon)\\right]} \\,+\\, \\underset{\\epsilon \\to 0}\\lim{\\left[g(x)\\dfrac{h(x + \\epsilon) - h(x)}{\\epsilon}\\right]} && \\quad \\text{factorizing }h(x+\\epsilon) \\text{ and } g(x)\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}h(x+\\epsilon)\\right]} \\,+\\, g(x)\\underset{\\epsilon \\to 0}\\lim{\\dfrac{h(x + \\epsilon) - h(x)}{\\epsilon}} && \\quad \\text{taking } g(x) \\text{ out of the limit since it does not depend on }\\epsilon\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}h(x+\\epsilon)\\right]} \\,+\\, g(x)h'(x) && \\quad \\text{using the definition of h'(x)}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}\\right]}\\underset{\\epsilon \\to 0}\\lim{h(x+\\epsilon)} + g(x)h'(x) && \\quad \\text{since the limit of a product is the product of the limits}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(x+\\epsilon) - g(x)}{\\epsilon}\\right]}h(x) + h(x)g'(x) && \\quad \\text{since } h(x) \\text{ is continuous}\\\\\n",
"& = g'(x)h(x) + g(x)h'(x) && \\quad \\text{using the definition of }g'(x)\n",
"\\end{align*}\n",
"$\n",
"\n",
"Note that if $g(x)=c$ (a constant), then $g'(x)=0$, so the equation simplifies to:\n",
@ -1353,18 +1353,18 @@
"## Chain rule: $f(x)=g(h(x))$\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{\\epsilon} && \\quad \\text{using }f(x) = g(h(x))\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{h(x+\\epsilon)-h(x)}{h(x+\\epsilon)-h(x)}\\,\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{\\epsilon}\\right]} && \\quad \\text{multiplying and dividing by }h(x+\\epsilon) - h(x)\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{h(x+\\epsilon)-h(x)}{\\epsilon}\\,\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{h(x+\\epsilon)-h(x)}\\right]} && \\quad \\text{swapping the denominators}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{h(x+\\epsilon)-h(x)}{\\epsilon}\\right]} \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{h(x+\\epsilon)-h(x)}\\right]} && \\quad \\text{the limit of a product is the product of the limits}\\\\\n",
"&& = h'(x) \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{h(x+\\epsilon)-h(x)}\\right]} && \\quad \\text{using the definition of }h'(x)\\\\\n",
"&& = h'(x) \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(u) - g(v)}{u-v}\\right]} && \\quad \\text{using }u=h(x+\\epsilon) \\text{ and } v=h(x)\\\\\n",
"&& = h'(x) \\underset{u \\to v}\\lim{\\left[\\dfrac{g(u) - g(v)}{u-v}\\right]} && \\quad \\text{ since } h \\text{ is continuous, so } \\underset{\\epsilon \\to 0}\\lim{u}=v\\\\\n",
"&& = h'(x)g'(v) && \\quad \\text{ using the definition of } g'(v)\\\\\n",
"&& = h'(x)g'(h(x)) && \\quad \\text{ since } v = h(x)\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{\\epsilon} && \\quad \\text{using }f(x) = g(h(x))\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{h(x+\\epsilon)-h(x)}{h(x+\\epsilon)-h(x)}\\,\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{\\epsilon}\\right]} && \\quad \\text{multiplying and dividing by }h(x+\\epsilon) - h(x)\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{h(x+\\epsilon)-h(x)}{\\epsilon}\\,\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{h(x+\\epsilon)-h(x)}\\right]} && \\quad \\text{swapping the denominators}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{h(x+\\epsilon)-h(x)}{\\epsilon}\\right]} \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{h(x+\\epsilon)-h(x)}\\right]} && \\quad \\text{the limit of a product is the product of the limits}\\\\\n",
"& = h'(x) \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(h(x+\\epsilon)) - g(h(x))}{h(x+\\epsilon)-h(x)}\\right]} && \\quad \\text{using the definition of }h'(x)\\\\\n",
"& = h'(x) \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{g(u) - g(v)}{u-v}\\right]} && \\quad \\text{using }u=h(x+\\epsilon) \\text{ and } v=h(x)\\\\\n",
"& = h'(x) \\underset{u \\to v}\\lim{\\left[\\dfrac{g(u) - g(v)}{u-v}\\right]} && \\quad \\text{ since } h \\text{ is continuous, so } \\underset{\\epsilon \\to 0}\\lim{u}=v\\\\\n",
"& = h'(x)g'(v) && \\quad \\text{ using the definition of } g'(v)\\\\\n",
"& = h'(x)g'(h(x)) && \\quad \\text{ since } v = h(x)\n",
"\\end{align*}\n",
"$"
]
},
@ -1380,15 +1380,15 @@
"There are several equivalent definitions of the number $e$. One of them states that $e$ is the unique positive number for which $\\underset{\\epsilon \\to 0}\\lim{\\dfrac{e^\\epsilon - 1}{\\epsilon}}=1$. We will use this in this proof:\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{e^{x+\\epsilon} - e^x}{\\epsilon} && \\quad \\text{using }f(x) = e^x\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{e^x e^\\epsilon - e^x}{\\epsilon} && \\quad \\text{using the fact that } x^{a+b}=x^a x^b\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[e^x\\dfrac{e^\\epsilon - 1}{\\epsilon}\\right]} && \\quad \\text{factoring out }e^x\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{e^x} \\, \\underset{\\epsilon \\to 0}\\lim{\\dfrac{e^\\epsilon - 1}{\\epsilon}} && \\quad \\text{the limit of a product is the product of the limits}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{e^x} && \\quad \\text{since }\\underset{\\epsilon \\to 0}\\lim{\\dfrac{e^\\epsilon - 1}{\\epsilon}}=1\\\\\n",
"&& = e^x && \\quad \\text{since } e^x \\text{ does not depend on }\\epsilon\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{e^{x+\\epsilon} - e^x}{\\epsilon} && \\quad \\text{using }f(x) = e^x\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{e^x e^\\epsilon - e^x}{\\epsilon} && \\quad \\text{using the fact that } x^{a+b}=x^a x^b\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[e^x\\dfrac{e^\\epsilon - 1}{\\epsilon}\\right]} && \\quad \\text{factoring out }e^x\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{e^x} \\, \\underset{\\epsilon \\to 0}\\lim{\\dfrac{e^\\epsilon - 1}{\\epsilon}} && \\quad \\text{the limit of a product is the product of the limits}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{e^x} && \\quad \\text{since }\\underset{\\epsilon \\to 0}\\lim{\\dfrac{e^\\epsilon - 1}{\\epsilon}}=1\\\\\n",
"& = e^x && \\quad \\text{since } e^x \\text{ does not depend on }\\epsilon\n",
"\\end{align*}\n",
"$\n"
]
},
@ -1412,19 +1412,19 @@
"This will come in handy in a second:\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{\\ln(x+\\epsilon) - \\ln(x)}{\\epsilon} && \\quad \\text{using }f(x) = \\ln(x)\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{\\ln\\left(\\dfrac{x+\\epsilon}{x}\\right)}{\\epsilon} && \\quad \\text{since }\\ln(a)-\\ln(b)=\\ln\\left(\\dfrac{a}{b}\\right)\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{1}{\\epsilon} \\, \\ln\\left(1 + \\dfrac{\\epsilon}{x}\\right)\\right]} && \\quad \\text{just moving things around a bit}\\\\\n",
"&& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{1}{xu} \\, \\ln\\left(1 + u\\right)\\right]} && \\quad \\text{defining }u=\\dfrac{\\epsilon}{x} \\text{ and thus } \\epsilon=xu\\\\\n",
"&& = \\underset{u \\to 0}\\lim{\\left[\\dfrac{1}{xu} \\, \\ln\\left(1 + u\\right)\\right]} && \\quad \\text{replacing } \\underset{\\epsilon \\to 0}\\lim \\text{ with } \\underset{u \\to 0}\\lim \\text{ since }\\underset{\\epsilon \\to 0}\\lim u=0\\\\\n",
"&& = \\underset{u \\to 0}\\lim{\\left[\\dfrac{1}{x} \\, \\ln\\left((1 + u)^{1/u}\\right)\\right]} && \\quad \\text{since }a\\ln(b)=\\ln(a^b)\\\\\n",
"&& = \\dfrac{1}{x}\\underset{u \\to 0}\\lim{\\left[\\ln\\left((1 + u)^{1/u}\\right)\\right]} && \\quad \\text{taking }\\dfrac{1}{x} \\text{ out since it does not depend on }\\epsilon\\\\\n",
"&& = \\dfrac{1}{x}\\ln\\left(\\underset{u \\to 0}\\lim{(1 + u)^{1/u}}\\right) && \\quad \\text{taking }\\ln\\text{ out since it is a continuous function}\\\\\n",
"&& = \\dfrac{1}{x}\\ln(e) && \\quad \\text{since }e=\\underset{u \\to 0}\\lim{(1 + u)^{1/u}}\\\\\n",
"&& = \\dfrac{1}{x} && \\quad \\text{since }\\ln(e)=1\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\epsilon \\to 0}\\lim\\dfrac{f(x+\\epsilon) - f(x)}{\\epsilon} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{\\ln(x+\\epsilon) - \\ln(x)}{\\epsilon} && \\quad \\text{using }f(x) = \\ln(x)\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim\\dfrac{\\ln\\left(\\dfrac{x+\\epsilon}{x}\\right)}{\\epsilon} && \\quad \\text{since }\\ln(a)-\\ln(b)=\\ln\\left(\\dfrac{a}{b}\\right)\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{1}{\\epsilon} \\, \\ln\\left(1 + \\dfrac{\\epsilon}{x}\\right)\\right]} && \\quad \\text{just moving things around a bit}\\\\\n",
"& = \\underset{\\epsilon \\to 0}\\lim{\\left[\\dfrac{1}{xu} \\, \\ln\\left(1 + u\\right)\\right]} && \\quad \\text{defining }u=\\dfrac{\\epsilon}{x} \\text{ and thus } \\epsilon=xu\\\\\n",
"& = \\underset{u \\to 0}\\lim{\\left[\\dfrac{1}{xu} \\, \\ln\\left(1 + u\\right)\\right]} && \\quad \\text{replacing } \\underset{\\epsilon \\to 0}\\lim \\text{ with } \\underset{u \\to 0}\\lim \\text{ since }\\underset{\\epsilon \\to 0}\\lim u=0\\\\\n",
"& = \\underset{u \\to 0}\\lim{\\left[\\dfrac{1}{x} \\, \\ln\\left((1 + u)^{1/u}\\right)\\right]} && \\quad \\text{since }a\\ln(b)=\\ln(a^b)\\\\\n",
"& = \\dfrac{1}{x}\\underset{u \\to 0}\\lim{\\left[\\ln\\left((1 + u)^{1/u}\\right)\\right]} && \\quad \\text{taking }\\dfrac{1}{x} \\text{ out since it does not depend on }\\epsilon\\\\\n",
"& = \\dfrac{1}{x}\\ln\\left(\\underset{u \\to 0}\\lim{(1 + u)^{1/u}}\\right) && \\quad \\text{taking }\\ln\\text{ out since it is a continuous function}\\\\\n",
"& = \\dfrac{1}{x}\\ln(e) && \\quad \\text{since }e=\\underset{u \\to 0}\\lim{(1 + u)^{1/u}}\\\\\n",
"& = \\dfrac{1}{x} && \\quad \\text{since }\\ln(e)=1\n",
"\\end{align*}\n",
"$\n"
]
},
@ -1655,16 +1655,16 @@
"Now the second thing we need to prove before we can tackle the derivative of the $\\sin$ function is the fact that $\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta}=0$. Here we go:\n",
"\n",
"$\n",
"\\begin{split}\n",
"\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta} && = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta}\\frac{\\cos(\\theta) + 1}{\\cos(\\theta) + 1} && \\quad \\text{ multiplying and dividing by }\\cos(\\theta)+1\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos^2(\\theta) - 1}{\\theta(\\cos(\\theta) + 1)} && \\quad \\text{ since }(a-1)(a+1)=a^2-1\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin^2(\\theta)}{\\theta(\\cos(\\theta) + 1)} && \\quad \\text{ since }\\cos^2(\\theta) - 1 = \\sin^2(\\theta)\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta}\\dfrac{\\sin(\\theta)}{\\cos(\\theta) + 1} && \\quad \\text{ just rearranging the terms}\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta} \\, \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\cos(\\theta) + 1} && \\quad \\text{ since the limit of a product is the product of the limits}\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\cos(\\theta) + 1} && \\quad \\text{ since } \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta}=1\\\\\n",
"&& = \\dfrac{0}{1+1} && \\quad \\text{ since } \\underset{\\theta \\to 0}\\lim\\sin(\\theta)=0 \\text{ and } \\underset{\\theta \\to 0}\\lim\\cos(\\theta)=1\\\\\n",
"&& = 0\\\\\n",
"\\end{split}\n",
"\\begin{align*}\n",
"\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta} & = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta}\\frac{\\cos(\\theta) + 1}{\\cos(\\theta) + 1} && \\quad \\text{ multiplying and dividing by }\\cos(\\theta)+1\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos^2(\\theta) - 1}{\\theta(\\cos(\\theta) + 1)} && \\quad \\text{ since }(a-1)(a+1)=a^2-1\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin^2(\\theta)}{\\theta(\\cos(\\theta) + 1)} && \\quad \\text{ since }\\cos^2(\\theta) - 1 = \\sin^2(\\theta)\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta}\\dfrac{\\sin(\\theta)}{\\cos(\\theta) + 1} && \\quad \\text{ just rearranging the terms}\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta} \\, \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\cos(\\theta) + 1} && \\quad \\text{ since the limit of a product is the product of the limits}\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\cos(\\theta) + 1} && \\quad \\text{ since } \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta}=1\\\\\n",
"& = \\dfrac{0}{1+1} && \\quad \\text{ since } \\underset{\\theta \\to 0}\\lim\\sin(\\theta)=0 \\text{ and } \\underset{\\theta \\to 0}\\lim\\cos(\\theta)=1\\\\\n",
"& = 0\\\\\n",
"\\end{align*}\n",
"$\n",
"\n",
"<hr />\n",
@ -1694,15 +1694,15 @@
},
"source": [
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\theta \\to 0}\\lim\\dfrac{f(x+\\theta) - f(x)}{\\theta} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(x+\\theta) - \\sin(x)}{\\theta} && \\quad \\text{using }f(x) = \\sin(x)\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(x)\\sin(\\theta) + \\sin(x)\\cos(\\theta) - \\sin(x)}{\\theta} && \\quad \\text{since } cos(a+b)=\\cos(a)\\sin(b)+\\sin(a)\\cos(b)\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(x)\\sin(\\theta)}{\\theta} + \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(x)\\cos(\\theta) - \\sin(x)}{\\theta} && \\quad \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"&& = \\cos(x)\\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta} + \\sin(x)\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta} && \\quad \\text{bringing out } \\cos(x) \\text{ and } \\sin(x) \\text{ since they don't depend on }\\theta\\\\\n",
"&& = \\cos(x)\\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta} && \\quad \\text{since }\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta}=0\\\\\n",
"&& = \\cos(x) && \\quad \\text{since }\\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta}=1\\\\\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\theta \\to 0}\\lim\\dfrac{f(x+\\theta) - f(x)}{\\theta} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(x+\\theta) - \\sin(x)}{\\theta} && \\quad \\text{using }f(x) = \\sin(x)\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(x)\\sin(\\theta) + \\sin(x)\\cos(\\theta) - \\sin(x)}{\\theta} && \\quad \\text{since } cos(a+b)=\\cos(a)\\sin(b)+\\sin(a)\\cos(b)\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(x)\\sin(\\theta)}{\\theta} + \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(x)\\cos(\\theta) - \\sin(x)}{\\theta} && \\quad \\text{since the limit of a sum is the sum of the limits}\\\\\n",
"& = \\cos(x)\\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta} + \\sin(x)\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta} && \\quad \\text{bringing out } \\cos(x) \\text{ and } \\sin(x) \\text{ since they don't depend on }\\theta\\\\\n",
"& = \\cos(x)\\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta} && \\quad \\text{since }\\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(\\theta) - 1}{\\theta}=0\\\\\n",
"& = \\cos(x) && \\quad \\text{since }\\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(\\theta)}{\\theta}=1\\\\\n",
"\\end{align*}\n",
"$\n"
]
},
@ -1718,16 +1718,16 @@
"Since we have proven that $\\sin'(x)=\\cos(x)$, proving that $\\cos'(x)=-\\sin(x)$ will be much easier.\n",
"\n",
"$\n",
"\\begin{split}\n",
"f'(x) && = \\underset{\\theta \\to 0}\\lim\\dfrac{f(x+\\theta) - f(x)}{\\theta} && \\quad\\text{by definition}\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(x+\\theta) - \\cos(x)}{\\theta} && \\quad \\text{using }f(x) = \\cos(x)\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin\\left(x+\\dfrac{\\pi}{2}+\\theta\\right) - \\sin\\left(x+\\dfrac{\\pi}{2}\\right)}{\\theta} && \\quad \\text{since }\\cos(x) = \\sin\\left(x+\\dfrac{\\pi}{2}\\right)\\\\\n",
"&& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(z+\\theta) - \\sin(z)}{\\theta} && \\quad \\text{using }z = x + \\dfrac{\\pi}{2}\\\\\n",
"&& = \\sin'(z) && \\quad \\text{using the definition of }\\sin'(z)\\\\\n",
"&& = \\cos(z) && \\quad \\text{since we proved that }\\sin'(z)=\\cos(z)\\\\\n",
"&& = \\cos\\left(x + \\dfrac{\\pi}{2}\\right) && \\quad \\text{using the definition of }z\\\\\n",
"&& = -\\sin(x) && \\quad \\text{using this well-known rule of trigonometry}\n",
"\\end{split}\n",
"\\begin{align*}\n",
"f'(x) & = \\underset{\\theta \\to 0}\\lim\\dfrac{f(x+\\theta) - f(x)}{\\theta} && \\quad\\text{by definition}\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\cos(x+\\theta) - \\cos(x)}{\\theta} && \\quad \\text{using }f(x) = \\cos(x)\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin\\left(x+\\dfrac{\\pi}{2}+\\theta\\right) - \\sin\\left(x+\\dfrac{\\pi}{2}\\right)}{\\theta} && \\quad \\text{since }\\cos(x) = \\sin\\left(x+\\dfrac{\\pi}{2}\\right)\\\\\n",
"& = \\underset{\\theta \\to 0}\\lim\\dfrac{\\sin(z+\\theta) - \\sin(z)}{\\theta} && \\quad \\text{using }z = x + \\dfrac{\\pi}{2}\\\\\n",
"& = \\sin'(z) && \\quad \\text{using the definition of }\\sin'(z)\\\\\n",
"& = \\cos(z) && \\quad \\text{since we proved that }\\sin'(z)=\\cos(z)\\\\\n",
"& = \\cos\\left(x + \\dfrac{\\pi}{2}\\right) && \\quad \\text{using the definition of }z\\\\\n",
"& = -\\sin(x) && \\quad \\text{using this well-known rule of trigonometry}\n",
"\\end{align*}\n",
"$\n"
]
},

6
requirements.txt
View File

@ -49,14 +49,16 @@ tensorflow-addons==0.12.1
# There are a few dependencies you need to install first, check out:
# https://github.com/openai/gym#installing-everything
gym[atari,Box2D]==0.18.0
gym[Box2D]==0.18.0
atari_py==0.2.5
# On Windows, install atari_py using:
# pip install --no-index -f https://github.com/Kojoley/atari-py/releases atari_py
tf-agents==0.7.1
##### Image manipulation
Pillow==8.2.0
Pillow==8.3.2
graphviz==0.16
opencv-python==4.5.1.48
pyglet==1.5.0