python实现随机森林random forest的原理及方法

Kande ·

更新时间:2024-11-15

· 592 次阅读

引言

想通过随机森林来获取数据的主要特征

1、理论

随机森林是一个高度灵活的机器学习方法，拥有广泛的应用前景，从市场营销到医疗保健保险。既可以用来做市场营销模拟的建模，统计客户来源，保留和流失。也可用来预测疾病的风险和病患者的易感性。

根据个体学习器的生成方式，目前的集成学习方法大致可分为两大类，即个体学习器之间存在强依赖关系，必须串行生成的序列化方法，以及个体学习器间不存在强依赖关系，可同时生成的并行化方法；

前者的代表是Boosting，后者的代表是Bagging和“随机森林”（Random
Forest）

随机森林在以决策树为基学习器构建Bagging集成的基础上，进一步在决策树的训练过程中引入了随机属性选择（即引入随机特征选择）。

简单来说，随机森林就是对决策树的集成，但有两点不同：

（2）特征选取的差异性：每个决策树的n个分类特征是在所有特征中随机选择的（n是一个需要我们自己调整的参数）
随机森林，简单理解，比如预测salary，就是构建多个决策树job，age，house，然后根据要预测的量的各个特征（teacher，39，suburb）分别在对应决策树的目标值概率（salary<5000,salary>=5000），从而，确定预测量的发生概率（如，预测出P(salary<5000)=0.3）.

随机森林是一个可做能够回归和分类。它具备处理大数据的特性，而且它有助于估计或变量是非常重要的基础数据建模。

参数说明：

最主要的两个参数是n_estimators和max_features。

n_estimators：表示森林里树的个数。理论上是越大越好。但是伴随着就是计算时间的增长。但是并不是取得越大就会越好，预测效果最好的将会出现在合理的树个数。

max_features：随机选择特征集合的子集合，并用来分割节点。子集合的个数越少，方差就会减少的越快，但同时偏差就会增加的越快。根据较好的实践经验。如果是回归问题则：

max_features＝n_features，如果是分类问题则max_features＝sqrt(n_features)。

如果想获取较好的结果，必须将max_depth＝None,同时min_sample_split=1。
同时还要记得进行cross_validated（交叉验证），除此之外记得在random forest中，bootstrap=True。但在extra-trees中，bootstrap=False。

2、随机森林python实现

2.1Demo1

实现随机森林基本功能


#随机森林
from sklearn.tree import DecisionTreeRegressor 
from sklearn.ensemble import RandomForestRegressor 
import numpy as np 
from sklearn.datasets import load_iris 
iris=load_iris() 
#print iris#iris的４个属性是：萼片宽度　萼片长度　花瓣宽度　花瓣长度　标签是花的种类：setosa versicolour virginica 
print(iris['target'].shape)
rf=RandomForestRegressor()#这里使用了默认的参数设置 
rf.fit(iris.data[:150],iris.target[:150])#进行模型的训练 
#随机挑选两个预测不相同的样本 
instance=iris.data[[100,109]] 
print(instance)
rf.predict(instance[[0]])
print('instance 0 prediction；',rf.predict(instance[[0]]))
print( 'instance 1 prediction；',rf.predict(instance[[1]]))
print(iris.target[100],iris.target[109])

运行结果

(150,)
[[ 6.3 3.3 6. 2.5]
[ 7.2 3.6 6.1 2.5]]
instance 0 prediction； [ 2.]
instance 1 prediction； [ 2.]
2 2

2.2 Demo2

3种方法的比较


#random forest test
from sklearn.model_selection import cross_val_score
from sklearn.datasets import make_blobs
from sklearn.ensemble import RandomForestClassifier
from sklearn.ensemble import ExtraTreesClassifier
from sklearn.tree import DecisionTreeClassifier
X, y = make_blobs(n_samples=10000, n_features=10, centers=100,random_state=0)
clf = DecisionTreeClassifier(max_depth=None, min_samples_split=2,random_state=0)
scores = cross_val_score(clf, X, y)
print(scores.mean())    
clf = RandomForestClassifier(n_estimators=10, max_depth=None,min_samples_split=2, random_state=0)
scores = cross_val_score(clf, X, y)
print(scores.mean())    
clf = ExtraTreesClassifier(n_estimators=10, max_depth=None,min_samples_split=2, random_state=0)
scores = cross_val_score(clf, X, y)
print(scores.mean())

运行结果：

0.979408793821
0.999607843137
0.999898989899

2.3 Demo3-实现特征选择


#随机森林2
from sklearn.tree import DecisionTreeRegressor 
from sklearn.ensemble import RandomForestRegressor 
import numpy as np 
from sklearn.datasets import load_iris 
iris=load_iris() 
from sklearn.model_selection import cross_val_score, ShuffleSplit 
X = iris["data"] 
Y = iris["target"] 
names = iris["feature_names"] 
rf = RandomForestRegressor() 
scores = [] 
for i in range(X.shape[1]): 
 score = cross_val_score(rf, X[:, i:i+1], Y, scoring="r2", 
    cv=ShuffleSplit(len(X), 3, .3)) 
 scores.append((round(np.mean(score), 3), names[i])) 
print(sorted(scores, reverse=True))

运行结果：

[(0.89300000000000002, 'petal width (cm)'), (0.82099999999999995, 'petal length
(cm)'), (0.13, 'sepal length (cm)'), (-0.79100000000000004, 'sepal width (cm)')]

2.4 demo4-随机森林

本来想利用以下代码来构建随机随机森林决策树，但是，遇到的问题是，程序一直在运行，无法响应，还需要调试。


#随机森林4
#coding:utf-8 
import csv 
from random import seed 
from random import randrange 
from math import sqrt 
def loadCSV(filename):#加载数据，一行行的存入列表 
 dataSet = [] 
 with open(filename, 'r') as file: 
 csvReader = csv.reader(file) 
 for line in csvReader: 
  dataSet.append(line) 
 return dataSet 
# 除了标签列，其他列都转换为float类型 
def column_to_float(dataSet): 
 featLen = len(dataSet[0]) - 1 
 for data in dataSet: 
 for column in range(featLen): 
  data[column] = float(data[column].strip()) 
# 将数据集随机分成N块，方便交叉验证，其中一块是测试集，其他四块是训练集 
def spiltDataSet(dataSet, n_folds): 
 fold_size = int(len(dataSet) / n_folds) 
 dataSet_copy = list(dataSet) 
 dataSet_spilt = [] 
 for i in range(n_folds): 
 fold = [] 
 while len(fold) < fold_size: # 这里不能用if，if只是在第一次判断时起作用，while执行循环，直到条件不成立 
  index = randrange(len(dataSet_copy)) 
  fold.append(dataSet_copy.pop(index)) # pop() 函数用于移除列表中的一个元素（默认最后一个元素），并且返回该元素的值。 
 dataSet_spilt.append(fold) 
 return dataSet_spilt 
# 构造数据子集 
def get_subsample(dataSet, ratio): 
 subdataSet = [] 
 lenSubdata = round(len(dataSet) * ratio)#返回浮点数 
 while len(subdataSet) < lenSubdata: 
 index = randrange(len(dataSet) - 1) 
 subdataSet.append(dataSet[index]) 
 # print len(subdataSet) 
 return subdataSet 
# 分割数据集 
def data_spilt(dataSet, index, value): 
 left = [] 
 right = [] 
 for row in dataSet: 
 if row[index] < value: 
  left.append(row) 
 else: 
  right.append(row) 
 return left, right 
# 计算分割代价 
def spilt_loss(left, right, class_values): 
 loss = 0.0 
 for class_value in class_values: 
 left_size = len(left) 
 if left_size != 0: # 防止除数为零 
  prop = [row[-1] for row in left].count(class_value) / float(left_size) 
  loss += (prop * (1.0 - prop)) 
 right_size = len(right) 
 if right_size != 0: 
  prop = [row[-1] for row in right].count(class_value) / float(right_size) 
  loss += (prop * (1.0 - prop)) 
 return loss 
# 选取任意的n个特征，在这n个特征中，选取分割时的最优特征 
def get_best_spilt(dataSet, n_features): 
 features = [] 
 class_values = list(set(row[-1] for row in dataSet)) 
 b_index, b_value, b_loss, b_left, b_right = 999, 999, 999, None, None 
 while len(features) < n_features: 
 index = randrange(len(dataSet[0]) - 1) 
 if index not in features: 
  features.append(index) 
 # print 'features:',features 
 for index in features:#找到列的最适合做节点的索引，（损失最小） 
 for row in dataSet: 
  left, right = data_spilt(dataSet, index, row[index])#以它为节点的，左右分支 
  loss = spilt_loss(left, right, class_values) 
  if loss < b_loss:#寻找最小分割代价 
  b_index, b_value, b_loss, b_left, b_right = index, row[index], loss, left, right 
 # print b_loss 
 # print type(b_index) 
 return {'index': b_index, 'value': b_value, 'left': b_left, 'right': b_right} 
# 决定输出标签 
def decide_label(data): 
 output = [row[-1] for row in data] 
 return max(set(output), key=output.count) 
# 子分割，不断地构建叶节点的过程对对对 
def sub_spilt(root, n_features, max_depth, min_size, depth): 
 left = root['left'] 
 # print left 
 right = root['right'] 
 del (root['left']) 
 del (root['right']) 
 # print depth 
 if not left or not right: 
 root['left'] = root['right'] = decide_label(left + right) 
 # print 'testing' 
 return 
 if depth > max_depth: 
 root['left'] = decide_label(left) 
 root['right'] = decide_label(right) 
 return 
 if len(left) < min_size: 
 root['left'] = decide_label(left) 
 else: 
 root['left'] = get_best_spilt(left, n_features) 
 # print 'testing_left' 
 sub_spilt(root['left'], n_features, max_depth, min_size, depth + 1) 
 if len(right) < min_size: 
 root['right'] = decide_label(right) 
 else: 
 root['right'] = get_best_spilt(right, n_features) 
 # print 'testing_right' 
 sub_spilt(root['right'], n_features, max_depth, min_size, depth + 1) 
 # 构造决策树 
def build_tree(dataSet, n_features, max_depth, min_size): 
 root = get_best_spilt(dataSet, n_features) 
 sub_spilt(root, n_features, max_depth, min_size, 1) 
 return root 
# 预测测试集结果 
def predict(tree, row): 
 predictions = [] 
 if row[tree['index']] < tree['value']: 
 if isinstance(tree['left'], dict): 
  return predict(tree['left'], row) 
 else: 
  return tree['left'] 
 else: 
 if isinstance(tree['right'], dict): 
  return predict(tree['right'], row) 
 else: 
  return tree['right'] 
  # predictions=set(predictions) 
def bagging_predict(trees, row): 
 predictions = [predict(tree, row) for tree in trees] 
 return max(set(predictions), key=predictions.count) 
# 创建随机森林 
def random_forest(train, test, ratio, n_feature, max_depth, min_size, n_trees): 
 trees = [] 
 for i in range(n_trees): 
 train = get_subsample(train, ratio)#从切割的数据集中选取子集 
 tree = build_tree(train, n_features, max_depth, min_size) 
 # print 'tree %d: '%i,tree 
 trees.append(tree) 
 # predict_values = [predict(trees,row) for row in test] 
 predict_values = [bagging_predict(trees, row) for row in test] 
 return predict_values 
# 计算准确率 
def accuracy(predict_values, actual): 
 correct = 0 
 for i in range(len(actual)): 
 if actual[i] == predict_values[i]: 
  correct += 1 
 return correct / float(len(actual)) 
if __name__ == '__main__': 
 seed(1) 
 dataSet = loadCSV(r'G:\0研究生\tianchiCompetition\训练小样本2.csv') 
 column_to_float(dataSet) 
 n_folds = 5 
 max_depth = 15 
 min_size = 1 
 ratio = 1.0 
 # n_features=sqrt(len(dataSet)-1) 
 n_features = 15 
 n_trees = 10 
 folds = spiltDataSet(dataSet, n_folds)#先是切割数据集 
 scores = [] 
 for fold in folds: 
 train_set = folds[ 
   :] # 此处不能简单地用train_set=folds，这样用属于引用,那么当train_set的值改变的时候，folds的值也会改变，所以要用复制的形式。（L[:]）能够复制序列，D.copy() 能够复制字典，list能够生成拷贝 list(L) 
 train_set.remove(fold)#选好训练集 
 # print len(folds) 
 train_set = sum(train_set, []) # 将多个fold列表组合成一个train_set列表 
 # print len(train_set) 
 test_set = [] 
 for row in fold: 
  row_copy = list(row) 
  row_copy[-1] = None 
  test_set.append(row_copy) 
  # for row in test_set: 
  # print row[-1] 
 actual = [row[-1] for row in fold] 
 predict_values = random_forest(train_set, test_set, ratio, n_features, max_depth, min_size, n_trees) 
 accur = accuracy(predict_values, actual) 
 scores.append(accur) 
 print ('Trees is %d' % n_trees) 
 print ('scores:%s' % scores) 
 print ('mean score:%s' % (sum(scores) / float(len(scores))))

2.5 随机森林分类sonic data


# CART on the Bank Note dataset
from random import seed
from random import randrange
from csv import reader
# Load a CSV file
def load_csv(filename):
 file = open(filename, "r")
 lines = reader(file)
 dataset = list(lines)
 return dataset
# Convert string column to float
def str_column_to_float(dataset, column):
 for row in dataset:
 row[column] = float(row[column].strip())
# 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 data set 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, class_values):
 gini = 0.0
 for class_value in class_values:
 for group in groups:
  size = len(group)
  if size == 0:
  continue
  proportion = [row[-1] for row in group].count(class_value) / float(size)
  gini += (proportion * (1.0 - proportion))
 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:
  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
 print ({'index':b_index, 'value':b_value})
 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']
# Classification and Regression Tree Algorithm
def decision_tree(train, test, max_depth, min_size):
 tree = build_tree(train, max_depth, min_size)
 predictions = list()
 for row in test:
 prediction = predict(tree, row)
 predictions.append(prediction)
 return(predictions)
# Test CART on Bank Note dataset
seed(1)
# load and prepare data
filename = r'G:\0pythonstudy\决策树\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)
# evaluate algorithm
n_folds = 5
max_depth = 5
min_size = 10
scores = evaluate_algorithm(dataset, decision_tree, n_folds, max_depth, min_size)
print('Scores: %s' % scores)
print('Mean Accuracy: %.3f%%' % (sum(scores)/float(len(scores))))

运行结果：

{'index': 38, 'value': 0.0894}
{'index': 36, 'value': 0.8459}
{'index': 50, 'value': 0.0024}
{'index': 15, 'value': 0.0906}
{'index': 16, 'value': 0.9819}
{'index': 10, 'value': 0.0785}
{'index': 16, 'value': 0.0886}
{'index': 38, 'value': 0.0621}
{'index': 5, 'value': 0.0226}
{'index': 8, 'value': 0.0368}
{'index': 11, 'value': 0.0754}
{'index': 0, 'value': 0.0239}
{'index': 8, 'value': 0.0368}
{'index': 29, 'value': 0.1671}
{'index': 46, 'value': 0.0237}
{'index': 38, 'value': 0.0621}
{'index': 14, 'value': 0.0668}
{'index': 4, 'value': 0.0167}
{'index': 37, 'value': 0.0836}
{'index': 12, 'value': 0.0616}
{'index': 7, 'value': 0.0333}
{'index': 33, 'value': 0.8741}
{'index': 16, 'value': 0.0886}
{'index': 8, 'value': 0.0368}
{'index': 33, 'value': 0.0798}
{'index': 44, 'value': 0.0298}
Scores: [48.78048780487805, 70.73170731707317, 58.536585365853654, 51.2195121951
2195, 39.02439024390244]
Mean Accuracy: 53.659%
请按任意键继续. . .

知识点：

1.load CSV file


from csv import reader
# Load a CSV file
def load_csv(filename):
 file = open(filename, "r")
 lines = reader(file)
 dataset = list(lines)
 return dataset
filename = r'G:\0pythonstudy\决策树\sonar.all-data.csv'
dataset=load_csv(filename)
print(dataset)

2.把数据转化成float格式


# Convert string column to float
def str_column_to_float(dataset, column):
 for row in dataset:
 row[column] = float(row[column].strip())
 # print(row[column])
# convert string attributes to integers
for i in range(len(dataset[0])-1):
 str_column_to_float(dataset, i)

3.把最后一列的分类字符串转化成0、1整数


def str_column_to_int(dataset, column):
 class_values = [row[column] for row in dataset]#生成一个class label的list
 # print(class_values)
 unique = set(class_values)#set 获得list的不同元素
 print(unique)
 lookup = dict()#定义一个字典
 # print(enumerate(unique))
 for i, value in enumerate(unique):
 lookup[value] = i
 # print(lookup)
 for row in dataset:
 row[column] = lookup[row[column]]
 print(lookup['M'])

4、把数据集分割成K份


# Split a dataset into k folds
def cross_validation_split(dataset, n_folds):
 dataset_split = list()#生成空列表
 dataset_copy = list(dataset)
 print(len(dataset_copy))
 print(len(dataset))
 #print(dataset_copy)
 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))
  # print(index)
  fold.append(dataset_copy.pop(index))#使用.pop()把里边的元素都删除（相当于转移），这k份元素各不相同。
 dataset_split.append(fold)
 return dataset_split
n_folds=5 
folds = cross_validation_split(dataset, n_folds)#k份元素各不相同的训练集

5.计算正确率


# 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#这个是二值分类正确性的表达式

6.二分类每列


# Split a data set based on an attribute and an attribute value
def test_split(index, value, dataset):
 left, right = list(), list()#初始化两个空列表
 for row in dataset:
 if row[index] < value:
  left.append(row)
 else:
  right.append(row)
 return left, right #返回两个列表，每个列表以value为界限对指定行（index）进行二分类。

7.使用gini系数来获得最佳分割点


# Calculate the Gini index for a split dataset
def gini_index(groups, class_values):
 gini = 0.0
 for class_value in class_values:
 for group in groups:
  size = len(group)
  if size == 0:
  continue
  proportion = [row[-1] for row in group].count(class_value) / float(size)
  gini += (proportion * (1.0 - proportion))
 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:
  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
 # print(groups)
 print ({'index':b_index, 'value':b_value,'score':gini})
 return {'index':b_index, 'value':b_value, 'groups':b_groups}

这段代码，在求gini指数，直接应用定义式，不难理解。获得最佳分割点可能比较难看懂，这里用了两层迭代，一层是对不同列的迭代，一层是对不同行的迭代。并且，每次迭代，都对gini系数进行更新。

8、决策树生成


# 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)

这里使用了递归编程，不断生成左叉树和右叉树。

9.构建决策树


# 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 
tree=build_tree(train_set, max_depth, min_size)
print(tree)

10、预测test集


# Build a decision tree
def build_tree(train, max_depth, min_size):
 root = get_split(train)#获得最好的分割点,下标值，groups
 split(root, max_depth, min_size, 1)
 return root 
# tree=build_tree(train_set, max_depth, min_size)
# print(tree) 
# Make a prediction with a decision tree
def predict(node, row):
 print(row[node['index']])
 print(node['value'])
 if row[node['index']] < node['value']:#用测试集来代入训练的最好分割点，分割点有偏差时，通过搜索左右叉树来进一步比较。
 if isinstance(node['left'], dict):#如果是字典类型，执行操作
  return predict(node['left'], row)
 else:
  return node['left']
 else:
 if isinstance(node['right'], dict):
  return predict(node['right'], row)
 else:
  return node['right']
tree = build_tree(train_set, max_depth, min_size)
predictions = list()
for row in test_set:
 prediction = predict(tree, row)
 predictions.append(prediction)

11.评价决策树


# 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

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