# 初学决策树

## 问题陈述

Since not all mushrooms are edible, you’d like to be able to tell whether a given mushroom is edible or poisonous based on it’s physical attributes You have some existing data that you can use for this task. Can you use the data to help you identify which mushrooms can be sold safely?

Note: The dataset used is for illustrative purposes only. It is not meant to be a guide on identifying edible mushrooms.

You have 10 examples of mushrooms. For each example, you have

• Three features
• Cap Color (Brown or Red),
• Stalk Shape (Tapering or Enlarging), and
• Solitary (Yes or No)
• Label
• Edible (1 indicating yes or 0 indicating poisonous)

## 建立决策树

### 示例数据

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  X_train = np.array( [[1, 1, 1], [1, 0, 1], [1, 0, 0], [1, 0, 0], [1, 1, 1], [0, 1, 1], [0, 0, 0], [1, 0, 1], [0, 1, 0], [1, 0, 0]]) y_train = np.array([1, 1, 0, 0, 1, 0, 0, 1, 1, 0]) # cat_data() def cat_data(): print("First few elements of X_train:\n", X_train[:3]) print("Type of X_train:", type(X_train)) print("First few elements of y_train:", y_train[:3]) print("Type of y_train:", type(y_train)) print('The shape of X_train is:', X_train.shape) print('The dim of X_train is:', X_train.ndim) print('The shape of y_train is: ', y_train.shape) print('The dim of y_train is: ', y_train.ndim) print('Number of training examples (m):', len(X_train)) 

### 计算熵

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33   def compute_entropy(y): """ Computes the entropy for Args: y (ndarray): Numpy array indicating whether each example at a node is edible (1) or poisonous (0) Returns: entropy (float): Entropy at that node """ # You need to return the following variables correctly entropy = 0. ### START CODE HERE ### if len(y) != 0: # Your code here to calculate the fraction of edible examples (i.e with value = 1 in y) p1 = len(y[y == 1]) / len(y) # For p1 = 0 and 1, set the entropy to 0 (to handle 0log0) if p1 != 0 and p1 != 1: # Your code here to calculate the entropy using the formula provided above entropy = -p1 * np.log2(p1) - (1 - p1) * np.log2(1 - p1) else: entropy = 0. ### END CODE HERE ### return entropy print("Entropy at root node: ", compute_entropy(y_train)) 

### Split the dataset

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43   def split_dataset(X, node_indices, feature): """ Splits the data at the given node into left and right branches Args: X (ndarray): Data matrix of shape(n_samples, n_features) node_indices (ndarray): List containing the active indices. I.e, the samples being considered at this step. feature (int): Index of feature to split on Returns: left_indices (ndarray): Indices with feature value == 1 right_indices (ndarray): Indices with feature value == 0 """ # You need to return the following variables correctly left_indices = [] right_indices = [] ### START CODE HERE ### for i in node_indices: if X[i][feature] == 1: # Your code here to check if the value of X at that index for the feature is 1 left_indices.append(i) else: right_indices.append(i) ### END CODE HERE ### return left_indices, right_indices root_indices = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] # Feel free to play around with these variables # The dataset only has three features, so this value can be 0 (Brown Cap), 1 (Tapering Stalk Shape) or 2 (Solitary) feature = 0 left_indices, right_indices = split_dataset(X_train, root_indices, feature) print("Left indices: ", left_indices) print("Right indices: ", right_indices) 

### 计算信息增益

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59   # UNQ_C3 # GRADED FUNCTION: compute_information_gain def compute_information_gain(X, y, node_indices, feature): """ Compute the information of splitting the node on a given feature Args: X (ndarray): Data matrix of shape(n_samples, n_features) y (array like): list or ndarray with n_samples containing the target variable node_indices (ndarray): List containing the active indices. I.e, the samples being considered in this step. Returns: cost (float): Cost computed """ # Split dataset left_indices, right_indices = split_dataset(X, node_indices, feature) # Some useful variables X_node, y_node = X[node_indices], y[node_indices] X_left, y_left = X[left_indices], y[left_indices] X_right, y_right = X[right_indices], y[right_indices] # You need to return the following variables correctly information_gain = 0 ### START CODE HERE ### # Your code here to compute the entropy at the node using compute_entropy() node_entropy = compute_entropy(y_node) left_entropy = compute_entropy(y_left) right_entropy = compute_entropy(y_right # Your code here to compute the proportion of examples at the left branch w_left = len(X_left) / len(X_node) w_right = len(X_right) / len(X_node) # Your code here to compute weighted entropy from the split using # w_left, w_right, left_entropy and right_entropy weighted_entropy = w_left * left_entropy + w_right * right_entropy # Your code here to compute the information gain as the entropy at the node # minus the weighted entropy information_gain = node_entropy - weighted_entropy ### END CODE HERE ### return information_gain info_gain0 = compute_information_gain(X_train, y_train, root_indices, feature=0) print("Information Gain from splitting the root on brown cap: ", info_gain0) info_gain1 = compute_information_gain(X_train, y_train, root_indices, feature=1) print("Information Gain from splitting the root on tapering stalk shape: ", info_gain1) info_gain2 = compute_information_gain(X_train, y_train, root_indices, feature=2) print("Information Gain from splitting the root on solitary: ", info_gain2) 

### Get best split

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  # UNQ_C4 # GRADED FUNCTION: get_best_split def get_best_split(X, y, node_indices): """ Returns the optimal feature and threshold value to split the node data Args: X (ndarray): Data matrix of shape(n_samples, n_features) y (array like): list or ndarray with n_samples containing the target variable node_indices (ndarray): List containing the active indices. I.e, the samples being considered in this step. Returns: best_feature (int): The index of the best feature to split """ # Some useful variables num_features = X.shape[1] # You need to return the following variables correctly best_feature = -1 ### START CODE HERE ### max_info_gain = 0 # Iterate through all features for feature in range(num_features): # Your code here to compute the information gain from splitting on this feature info_gain = compute_information_gain(X, y, node_indices, feature) # If the information gain is larger than the max seen so far if info_gain > max_info_gain: # Your code here to set the max_info_gain and best_feature max_info_gain = info_gain best_feature = feature ### END CODE HERE ## return best_feature best_feature = get_best_split(X_train, y_train, root_indices) print("Best feature to split on: %d" % best_feature) 

### Building the tree

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  # Not graded tree = [] def build_tree_recursive(X, y, node_indices, branch_name, max_depth, current_depth): """ Build a tree using the recursive algorithm that split the dataset into 2 subgroups at each node. This function just prints the tree. Args: X (ndarray): Data matrix of shape(n_samples, n_features) y (array like): list or ndarray with n_samples containing the target variable node_indices (ndarray): List containing the active indices. I.e, the samples being considered in this step. branch_name (string): Name of the branch. ['Root', 'Left', 'Right'] max_depth (int): Max depth of the resulting tree. current_depth (int): Current depth. Parameter used during recursive call. """ # Maximum depth reached - stop splitting if current_depth == max_depth: formatting = " "*current_depth + "-"*current_depth print(formatting, "%s leaf node with indices" % branch_name, node_indices) return # Otherwise, get best split and split the data # Get the best feature and threshold at this node best_feature = get_best_split(X, y, node_indices) tree.append((current_depth, branch_name, best_feature, node_indices)) formatting = "-"*current_depth print("%s Depth %d, %s: Split on feature: %d" % (formatting, current_depth, branch_name, best_feature)) # Split the dataset at the best feature left_indices, right_indices = split_dataset(X, node_indices, best_feature) # continue splitting the left and the right child. Increment current depth build_tree_recursive(X, y, left_indices, "Left", max_depth, current_depth+1) build_tree_recursive(X, y, right_indices, "Right", max_depth, current_depth+1) build_tree_recursive(X_train, y_train, root_indices, "Root", max_depth=2, current_depth=0) 

## code

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432   """ 问题陈述： Suppose you are starting a company that grows and sells wild mushrooms. Since not all mushrooms are edible, you'd like to be able to tell whether a given mushroom is edible or poisonous based on it's physical attributes You have some existing data that you can use for this task. Can you use the data to help you identify which mushrooms can be sold safely? Note: The dataset used is for illustrative purposes only. It is not meant to be a guide on identifying edible mushrooms. You have 10 examples of mushrooms. For each example, you have - Three features - Cap Color (Brown or Red), - Stalk Shape (Tapering or Enlarging), and - Solitary (Yes or No) - Label - Edible (1 indicating yes or 0 indicating poisonous) 回顾建立决策树的步骤： 1.Start with all examples at the root node 2.Calculate information gain for splitting on all possible features, and pick the one with the highest information gain 3.Split dataset according to the selected feature, and create left and right branches of the tree 4.Keep repeating splitting process until stopping criteria is met """ import numpy as np import matplotlib.pyplot as plt def compute_entropy_test(target): y = np.array([1] * 10) result = target(y) assert result == 0, "Entropy must be 0 with array of ones" y = np.array([0] * 10) result = target(y) assert result == 0, "Entropy must be 0 with array of zeros" y = np.array([0] * 12 + [1] * 12) result = target(y) assert result == 1, "Entropy must be 1 with same ammount of ones and zeros" y = np.array([1, 0, 1, 0, 1, 1, 1, 0, 1]) assert np.isclose(target(y), 0.918295, atol=1e-6), "Wrong value. Something between 0 and 1" assert np.isclose(target(-y + 1), target(y), atol=1e-6), "Wrong value" print("\033[92m All tests passed.") def split_dataset_test(target): X = np.array([[1, 0], [1, 0], [1, 1], [0, 0], [0, 1]]) X_t = np.array([[0, 1, 0, 1, 0]]) X = np.concatenate((X, X_t.T), axis=1) left, right = target(X, list(range(5)), 2) expected = {'left': np.array([1, 3]), 'right': np.array([0, 2, 4])} assert type(left) == list, f"Wrong type for left. Expected: list got: {type(left)}" assert type(right) == list, f"Wrong type for right. Expected: list got: {type(right)}" assert type(left[0]) == int, f"Wrong type for elements in the left list. Expected: int got: {type(left[0])}" assert type(right[0]) == int, f"Wrong type for elements in the right list. Expected: number got: {type(right[0])}" assert len(left) == 2, f"left must have 2 elements but got: {len(left)}" assert len(right) == 3, f"right must have 3 elements but got: {len(right)}" assert np.allclose(right, expected['right']), f"Wrong value for right. Expected: {expected['right']} \ngot: {right}" assert np.allclose(left, expected['left']), f"Wrong value for left. Expected: {expected['left']} \ngot: {left}" X = np.array([[0, 1], [1, 1], [1, 1], [0, 0], [1, 0]]) X_t = np.array([[0, 1, 0, 1, 0]]) X = np.concatenate((X_t.T, X), axis=1) left, right = target(X, list(range(5)), 0) expected = {'left': np.array([1, 3]), 'right': np.array([0, 2, 4])} assert np.allclose(right, expected['right']) and np.allclose(left, expected[ 'left']), f"Wrong value when target is at index 0." X = (np.random.rand(11, 3) > 0.5) * 1 # Just random binary numbers X_t = np.array([[0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0]]) X = np.concatenate((X, X_t.T), axis=1) left, right = target(X, [1, 2, 3, 6, 7, 9, 10], 3) expected = {'left': np.array([1, 3, 6]), 'right': np.array([2, 7, 9, 10])} assert np.allclose(right, expected['right']) and np.allclose(left, expected[ 'left']), f"Wrong value when target is at index 0. \nExpected: {expected} \ngot: \{left:{left}, 'right': {right}\}" print("\033[92m All tests passed.") def compute_information_gain_test(target): X = np.array([[1, 0], [1, 0], [1, 0], [0, 0], [0, 1]]) y = np.array([[0, 0, 0, 0, 0]]).T node_indexes = list(range(5)) result1 = target(X, y, node_indexes, 0) result2 = target(X, y, node_indexes, 0) assert result1 == 0 and result2 == 0, f"Information gain must be 0 when target variable is pure. Got {result1} and {result2}" y = np.array([[0, 1, 0, 1, 0]]).T node_indexes = list(range(5)) result = target(X, y, node_indexes, 0) assert np.isclose(result, 0.019973, atol=1e-6), f"Wrong information gain. Expected {0.019973} got: {result}" result = target(X, y, node_indexes, 1) assert np.isclose(result, 0.170951, atol=1e-6), f"Wrong information gain. Expected {0.170951} got: {result}" node_indexes = list(range(4)) result = target(X, y, node_indexes, 0) assert np.isclose(result, 0.311278, atol=1e-6), f"Wrong information gain. Expected {0.311278} got: {result}" result = target(X, y, node_indexes, 1) assert np.isclose(result, 0, atol=1e-6), f"Wrong information gain. Expected {0.0} got: {result}" print("\033[92m All tests passed.") def get_best_split_test(target): X = np.array([[1, 0], [1, 0], [1, 0], [0, 0], [0, 1]]) y = np.array([[0, 0, 0, 0, 0]]).T node_indexes = list(range(5)) result = target(X, y, node_indexes) assert result == -1, f"When the target variable is pure, there is no best split to do. Expected -1, got {result}" y = X[:, 0] result = target(X, y, node_indexes) assert result == 0, f"If the target is fully correlated with other feature, that feature must be the best split. Expected 0, got {result}" y = X[:, 1] result = target(X, y, node_indexes) assert result == 1, f"If the target is fully correlated with other feature, that feature must be the best split. Expected 1, got {result}" y = 1 - X[:, 0] result = target(X, y, node_indexes) assert result == 0, f"If the target is fully correlated with other feature, that feature must be the best split. Expected 0, got {result}" y = np.array([[0, 1, 0, 1, 0]]).T result = target(X, y, node_indexes) assert result == 1, f"Wrong result. Expected 1, got {result}" y = np.array([[0, 1, 0, 1, 0]]).T node_indexes = [2, 3, 4] result = target(X, y, node_indexes) assert result == 0, f"Wrong result. Expected 0, got {result}" n_samples = 100 X0 = np.array([[1] * n_samples]) X1 = np.array([[0] * n_samples]) X2 = (np.random.rand(1, 100) > 0.5) * 1 X3 = np.array([[1] * int(n_samples / 2) + [0] * int(n_samples / 2)]) y = X2.T node_indexes = list(range(20, 80)) X = np.array([X0, X1, X2, X3]).T.reshape(n_samples, 4) result = target(X, y, node_indexes) assert result == 2, f"Wrong result. Expected 2, got {result}" y = X0.T result = target(X, y, node_indexes) assert result == -1, f"When the target variable is pure, there is no best split to do. Expected -1, got {result}" print("\033[92m All tests passed.") def cat_data(): print("First few elements of X_train:\n", X_train[:3]) print("Type of X_train:", type(X_train)) print("First few elements of y_train:", y_train[:3]) print("Type of y_train:", type(y_train)) print('The shape of X_train is:', X_train.shape) print('The dim of X_train is:', X_train.ndim) print('The shape of y_train is: ', y_train.shape) print('The dim of y_train is: ', y_train.ndim) print('Number of training examples (m):', len(X_train)) # Calculate the entropy at a node # UNQ_C1 # GRADED FUNCTION: compute_entropy def compute_entropy(y): """ Computes the entropy for Args: y (ndarray): Numpy array indicating whether each example at a node is edible (1) or poisonous (0) Returns: entropy (float): Entropy at that node """ # You need to return the following variables correctly entropy = 0. ### START CODE HERE ### if len(y) != 0: # Your code here to calculate the fraction of edible examples (i.e with value = 1 in y) p1 = len(y[y == 1]) / len(y) # For p1 = 0 and 1, set the entropy to 0 (to handle 0log0) if p1 != 0 and p1 != 1: # Your code here to calculate the entropy using the formula provided above entropy = -p1 * np.log2(p1) - (1 - p1) * np.log2(1 - p1) else: entropy = 0. ### END CODE HERE ### return entropy # Split the dataset at a node into left and right branches based on a given feature # UNQ_C2 # GRADED FUNCTION: split_dataset def split_dataset(X, node_indices, feature): """ Splits the data at the given node into left and right branches Args: X (ndarray): Data matrix of shape(n_samples, n_features) node_indices (ndarray): List containing the active indices. I.e, the samples being considered at this step. feature (int): Index of feature to split on Returns: left_indices (ndarray): Indices with feature value == 1 right_indices (ndarray): Indices with feature value == 0 """ # You need to return the following variables correctly left_indices = [] right_indices = [] ### START CODE HERE ### for i in node_indices: if X[i][feature] == 1: # Your code here to check if the value of X at that index for the feature is 1 left_indices.append(i) else: right_indices.append(i) ### END CODE HERE ### return left_indices, right_indices # Calculate the information gain from splitting on a given feature # UNQ_C3 # GRADED FUNCTION: compute_information_gain def compute_information_gain(X, y, node_indices, feature): """ Compute the information of splitting the node on a given feature Args: X (ndarray): Data matrix of shape(n_samples, n_features) y (array like): list or ndarray with n_samples containing the target variable node_indices (ndarray): List containing the active indices. I.e, the samples being considered in this step. Returns: cost (float): Cost computed """ # Split dataset left_indices, right_indices = split_dataset(X, node_indices, feature) # Some useful variables X_node, y_node = X[node_indices], y[node_indices] X_left, y_left = X[left_indices], y[left_indices] X_right, y_right = X[right_indices], y[right_indices] # You need to return the following variables correctly information_gain = 0 ### START CODE HERE ### # Your code here to compute the entropy at the node using compute_entropy() node_entropy = compute_entropy(y_node) left_entropy = compute_entropy(y_left) right_entropy = compute_entropy(y_right) # Your code here to compute the proportion of examples at the left branch w_left = len(X_left) / len(X_node) w_right = len(X_right) / len(X_node) # Your code here to compute weighted entropy from the split using # w_left, w_right, left_entropy and right_entropy weighted_entropy = w_left * left_entropy + w_right * right_entropy # Your code here to compute the information gain as the entropy at the node # minus the weighted entropy information_gain = node_entropy - weighted_entropy ### END CODE HERE ### return information_gain # Choose the feature that maximizes information gain # UNQ_C4 # GRADED FUNCTION: get_best_split def get_best_split(X, y, node_indices): """ Returns the optimal feature and threshold value to split the node data Args: X (ndarray): Data matrix of shape(n_samples, n_features) y (array like): list or ndarray with n_samples containing the target variable node_indices (ndarray): List containing the active indices. I.e, the samples being considered in this step. Returns: best_feature (int): The index of the best feature to split """ # Some useful variables num_features = X.shape[1] # You need to return the following variables correctly best_feature = -1 ### START CODE HERE ### max_info_gain = 0 # Iterate through all features for feature in range(num_features): # Your code here to compute the information gain from splitting on this feature info_gain = compute_information_gain(X, y, node_indices, feature) # If the information gain is larger than the max seen so far if info_gain > max_info_gain: # Your code here to set the max_info_gain and best_feature max_info_gain = info_gain best_feature = feature ### END CODE HERE ## return best_feature # 创建树 # Not graded tree = [] def build_tree_recursive(X, y, node_indices, branch_name, max_depth, current_depth): """ Build a tree using the recursive algorithm that split the dataset into 2 subgroups at each node. This function just prints the tree. Args: X (ndarray): Data matrix of shape(n_samples, n_features) y (array like): list or ndarray with n_samples containing the target variable node_indices (ndarray): List containing the active indices. I.e, the samples being considered in this step. branch_name (string): Name of the branch. ['Root', 'Left', 'Right'] max_depth (int): Max depth of the resulting tree. current_depth (int): Current depth. Parameter used during recursive call. """ # Maximum depth reached - stop splitting if current_depth == max_depth: formatting = " " * current_depth + "-" * current_depth print(formatting, "%s leaf node with indices" % branch_name, node_indices) return # Otherwise, get best split and split the data # Get the best feature and threshold at this node best_feature = get_best_split(X, y, node_indices) tree.append((current_depth, branch_name, best_feature, node_indices)) formatting = "-" * current_depth print("%s Depth %d, %s: Split on feature: %d" % (formatting, current_depth, branch_name, best_feature)) # Split the dataset at the best feature left_indices, right_indices = split_dataset(X, node_indices, best_feature) # continue splitting the left and the right child. Increment current depth build_tree_recursive(X, y, left_indices, "Left", max_depth, current_depth + 1) build_tree_recursive(X, y, right_indices, "Right", max_depth, current_depth + 1) if __name__ == '__main__': X_train = np.array( [[1, 1, 1], [1, 0, 1], [1, 0, 0], [1, 0, 0], [1, 1, 1], [0, 1, 1], [0, 0, 0], [1, 0, 1], [0, 1, 0], [1, 0, 0]]) y_train = np.array([1, 1, 0, 0, 1, 0, 0, 1, 1, 0]) # cat_data() # print("Entropy at root node: ", compute_entropy(y_train)) root_indices = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] # Feel free to play around with these variables # The dataset only has three features, so this value can be 0 (Brown Cap), 1 (Tapering Stalk Shape) or 2 (Solitary) feature = 0 # left_indices, right_indices = split_dataset(X_train, root_indices, feature) # # print("Left indices: ", left_indices) # print("Right indices: ", right_indices) # build_tree_recursive(X_train, y_train, root_indices, "Root", max_depth=2, current_depth=0)