[人工智能]bagging

1.什么是 bootstraps?

bootstraps中文名称为自助法，是一种有放回的采样方法。其具体采样操作为：在m个样本组成的样本集合内，每次只抽取一个样本，有放回地抽取m次，这样就得到了m个样本组成的采样集合。

引理：D 中的某个样本在自助法采样中不被采到的概率约是 36.8%. 也就是说, 在 D 中大约有 36.8% 的样本未出现在采样数据集 D 中。
证明：对其中任意一个样本，它每次采样都不被抽中的概率为$p=\frac{m?1}{m}$，因此m次抽样它都不被抽中的概率为$(1?\frac{1}{m}{)}^m$,取极限得$1/e=36.8\%$。

自助法在数据集较小, 难以划分训练/验证集时很有用, 训练集大小仍是m. 然而, 自助法产生的数据集 改变了初始数据集分布, 会引入估计偏差. 因此, 当初始数据量足够时, 交叉验证和留出法更常用.

2. bootstraps与 bagging的联系

bootstraps采样是bagging其中必不可少的步骤。

3.什么是 bagging?

Bagging 的基本思路：一种朴素地增加基分类器多样性的方案是用训练集 D 的不同子集训练不同的个体学习器. 但是, 我们同时希望个体学习器不能太差. 如果每个子集互不相交, 则每个集学习器只用到了一小
部分训练数据, 甚至不足以有效学习. 因此, Bagging(boostrap aggregating) 从训练集 D 采样相互有交叠的采样子集。

Bagging 的具体过程包括如下三步.

1. 从训练集 D 中由自助法采样得到 T 个采样集$D_1,$
2. 基于自助采样集 $D_t$ 训练基学习器 $h_t$.
3. 使用相对多数投票 (分类任务) 或简单平均 (回归任务) 得到 $H$

4.随机森林与 bagging的联系与区别。

随机森林 (random forest)是在以决策树为集学习器构建 Bagging 集成 的基础上, 进一步在单变量决策树的训练过程中引入了随机属性选择.
具体地说, 经典单变量决策树是在选择划分属性时是在当前结点的属性集合 (假设有 d 个属性) 中选择一个属性, 而在随机森林中, 对基决策树的每个结点, 先从该结点的属性集合中随机选择一个包含$\hat{d}$ 个属性的子集, 然后再从这个子集中选择一个最优属性用于划分. 这里的参数 $\hat{d}$ 控制了随机性的引入程度:若 $\hat{d}=d$, 基决策树的构建和经典单变量决策树相同; 若$\hat{d}=1$, 则是随机选一个属性用于划分. 一般情况下, 推荐值 $\hat{d}=lgd$

5.使用偏差与方差理论阐述为什么 bagging能提升模型的预测精度

6.请尝试使用 bagging与基本分类模型或者回归模型做对比，观察 bagging是否相对于基础模型的精度有所提高？（必做）

7.假如让你来实现 bagging.你会使用 python+numpy+ sklearn的基础模型来实现 bagging吗？（拓展）

# Bagging Algorithm on the Sonar dataset
from random import seed
from random import randrange
from csv import reader
 
# Load a CSV file
def load_csv(filename):
	dataset = list()
	with open(filename, 'r') as file:
		csv_reader = reader(file)
		for row in csv_reader:
			if not row:
				continue
			dataset.append(row)
	return dataset
 
# Convert string column to float
def str_column_to_float(dataset, column):
	for row in dataset:
		row[column] = float(row[column].strip())
 
# Convert string column to integer
def str_column_to_int(dataset, column):
	class_values = [row[column] for row in dataset]
	unique = set(class_values)
	lookup = dict()
	for i, value in enumerate(unique):
		lookup[value] = i
	for row in dataset:
		row[column] = lookup[row[column]]
	return lookup
 
# Split a dataset into k folds
def cross_validation_split(dataset, n_folds):
	dataset_split = list()
	dataset_copy = list(dataset)
	fold_size = int(len(dataset) / n_folds)
	for i in range(n_folds):
		fold = list()
		while len(fold) < fold_size:
			index = randrange(len(dataset_copy))
			fold.append(dataset_copy.pop(index))
		dataset_split.append(fold)
	return dataset_split
 
# Calculate accuracy percentage
def accuracy_metric(actual, predicted):
	correct = 0
	for i in range(len(actual)):
		if actual[i] == predicted[i]:
			correct += 1
	return correct / float(len(actual)) * 100.0
 
# Evaluate an algorithm using a cross validation split
def evaluate_algorithm(dataset, algorithm, n_folds, *args):
	folds = cross_validation_split(dataset, n_folds)
	scores = list()
	for fold in folds:
		train_set = list(folds)
		train_set.remove(fold)
		train_set = sum(train_set, [])
		test_set = list()
		for row in fold:
			row_copy = list(row)
			test_set.append(row_copy)
			row_copy[-1] = None
		predicted = algorithm(train_set, test_set, *args)
		actual = [row[-1] for row in fold]
		accuracy = accuracy_metric(actual, predicted)
		scores.append(accuracy)
	return scores
 
# Split a dataset based on an attribute and an attribute value
def test_split(index, value, dataset):
	left, right = list(), list()
	for row in dataset:
		if row[index] < value:
			left.append(row)
		else:
			right.append(row)
	return left, right
 
# Calculate the Gini index for a split dataset
def gini_index(groups, classes):
	# count all samples at split point
	n_instances = float(sum([len(group) for group in groups]))
	# sum weighted Gini index for each group
	gini = 0.0
	for group in groups:
		size = float(len(group))
		# avoid divide by zero
		if size == 0:
			continue
		score = 0.0
		# score the group based on the score for each class
		for class_val in classes:
			p = [row[-1] for row in group].count(class_val) / size
			score += p * p
		# weight the group score by its relative size
		gini += (1.0 - score) * (size / n_instances)
	return gini
 
# Select the best split point for a dataset
def get_split(dataset):
	class_values = list(set(row[-1] for row in dataset))
	b_index, b_value, b_score, b_groups = 999, 999, 999, None
	for index in range(len(dataset[0])-1):
		for row in dataset:
		# for i in range(len(dataset)):
		# 	row = dataset[randrange(len(dataset))]
			groups = test_split(index, row[index], dataset)
			gini = gini_index(groups, class_values)
			if gini < b_score:
				b_index, b_value, b_score, b_groups = index, row[index], gini, groups
	return {'index':b_index, 'value':b_value, 'groups':b_groups}
 
# Create a terminal node value
def to_terminal(group):
	outcomes = [row[-1] for row in group]
	return max(set(outcomes), key=outcomes.count)
 
# Create child splits for a node or make terminal
def split(node, max_depth, min_size, depth):
	left, right = node['groups']
	del(node['groups'])
	# check for a no split
	if not left or not right:
		node['left'] = node['right'] = to_terminal(left + right)
		return
	# check for max depth
	if depth >= max_depth:
		node['left'], node['right'] = to_terminal(left), to_terminal(right)
		return
	# process left child
	if len(left) <= min_size:
		node['left'] = to_terminal(left)
	else:
		node['left'] = get_split(left)
		split(node['left'], max_depth, min_size, depth+1)
	# process right child
	if len(right) <= min_size:
		node['right'] = to_terminal(right)
	else:
		node['right'] = get_split(right)
		split(node['right'], max_depth, min_size, depth+1)
 
# Build a decision tree
def build_tree(train, max_depth, min_size):
	root = get_split(train)
	split(root, max_depth, min_size, 1)
	return root
 
# Make a prediction with a decision tree
def predict(node, row):
	if row[node['index']] < node['value']:
		if isinstance(node['left'], dict):
			return predict(node['left'], row)
		else:
			return node['left']
	else:
		if isinstance(node['right'], dict):
			return predict(node['right'], row)
		else:
			return node['right']
 
# Create a random subsample from the dataset with replacement
def subsample(dataset, ratio):
	sample = list()
	n_sample = round(len(dataset) * ratio)
	while len(sample) < n_sample:
		index = randrange(len(dataset))
		sample.append(dataset[index])
	return sample
 
# Make a prediction with a list of bagged trees
def bagging_predict(trees, row):
	predictions = [predict(tree, row) for tree in trees]
	return max(set(predictions), key=predictions.count)
 
# Bootstrap Aggregation Algorithm
def bagging(train, test, max_depth, min_size, sample_size, n_trees):
	trees = list()
	for i in range(n_trees):
		sample = subsample(train, sample_size)
		tree = build_tree(sample, max_depth, min_size)
		trees.append(tree)
	predictions = [bagging_predict(trees, row) for row in test]
	return(predictions)
 
# Test bagging on the sonar dataset
seed(1)
# load and prepare data
filename = 'sonar.all-data.csv'
dataset = load_csv(filename)
# convert string attributes to integers
for i in range(len(dataset[0])-1):
	str_column_to_float(dataset, i)
# convert class column to integers
str_column_to_int(dataset, len(dataset[0])-1)
# evaluate algorithm
n_folds = 5
max_depth = 6
min_size = 2
sample_size = 0.50
for n_trees in [1, 5, 10, 50]:
	scores = evaluate_algorithm(dataset, bagging, n_folds, max_depth, min_size, sample_size, n_trees)
	print('Trees: %d' % n_trees)
	print('Scores: %s' % scores)
	print('Mean Accuracy: %.3f%%' % (sum(scores)/float(len(scores))))