朴素贝叶斯算法简介及python代码实现分析

Posted 2021-01-11 虚生

概念：

 

　　贝叶斯定理：贝叶斯理论是以18世纪的一位神学家托马斯.贝叶斯(Thomas Bayes)命名。通常，事件A在事件B（发生）的条件下的概率，与事件B在事件A（发生）的条件下的概率是不一样的；然而，这两者是有确定的关系的，贝叶斯定理就是这种关系的陈述

　　朴素贝叶斯：朴素贝叶斯方法是基于贝叶斯定理和特征条件独立假设的分类方法。对于给定的训练数据集，首先基于特征条件独立假设学习输入/输出的联合概率分布；然后基于此模型，对给定的输入x，利用贝叶斯定理求出后验概率(Maximum A Posteriori)最大的输出y。

通俗的来讲，在给定数据集的前提下，对于一个新样本（未分类），在数据集中找到和新样本特征相同的样本，最后根据这些样本算出每个类的概率，概率最高的类即为新样本的类。

 

运算公式：

 

P( h | d) = P ( d | h ) * P( h) / P(d)

这里：
P ( h | d )：是因子h基于数据d的假设概率,叫做后验概率
P ( d | h ) : 是假设h为真条件下的数据d的概率
P( h)　: 是假设条件h为真的时候的概率（和数据无关），它叫做h的先验概率
P(d)　: 数据d的概率，和先验条件无关．

 

算法实现分解：

 

１　数据处理：加载数据并把他们分成训练数据和测试数据
２　汇总数据：汇总训练数据的概率以便后续计算概率和做预测
３　结果预测：　通过给定的测试数据和汇总的训练数据做预测
４　评估准确性：使用测试数据来评估预测的准确性

 

代码实现：

  1 # Example of Naive Bayes implemented from Scratch in Python
  2 import csv
  3 import random
  4 import math
  5 
  6 def loadCsv(filename):
  7         lines = csv.reader(open(filename, "rb"))
  8         dataset = list(lines)
  9         for i in range(len(dataset)):
 10                 dataset[i] = [float(x) for x in dataset[i]]
 11         return dataset
 12 
 13 def splitDataset(dataset, splitRatio):
 14         trainSize = int(len(dataset) * splitRatio)
 15         trainSet = []
 16         copy = list(dataset)
 17         while len(trainSet) < trainSize:
 18                 index = random.randrange(len(copy))
 19                 trainSet.append(copy.pop(index))
 20         return [trainSet, copy]
 21 
 22 def separateByClass(dataset):
 23         separated = {}
 24         for i in range(len(dataset)):
 25                 vector = dataset[i]
 26                 if (vector[-1] not in separated):
 27                         separated[vector[-1]] = []
 28                 separated[vector[-1]].append(vector)
 29         return separated
 30 
 31 def mean(numbers):
 32         return sum(numbers)/float(len(numbers))
 33 
 34 def stdev(numbers):
 35         avg = mean(numbers)
 36         variance = sum([pow(x-avg,2) for x in numbers])/float(len(numbers)-1)
 37         return math.sqrt(variance)
 38 
 39 def summarize(dataset):
 40         summaries = [(mean(attribute), stdev(attribute)) for attribute in zip(*dataset)]
 41         del summaries[-1]
 42         return summaries
 43 
 44 def summarizeByClass(dataset):
 45         separated = separateByClass(dataset)
 46         summaries = {}
 47         for classValue, instances in separated.iteritems():
 48                 summaries[classValue] = summarize(instances)
 49         return summaries
 50 
 51 def calculateProbability(x, mean, stdev):
 52         exponent = math.exp(-(math.pow(x-mean,2)/(2*math.pow(stdev,2))))
 53         return (1 / (math.sqrt(2*math.pi) * stdev)) * exponent
 54 
 55 def calculateClassProbabilities(summaries, inputVector):
 56         probabilities = {}
 57         for classValue, classSummaries in summaries.iteritems():
 58                 probabilities[classValue] = 1
 59                 for i in range(len(classSummaries)):
 60                         mean, stdev = classSummaries[i]
 61                         x = inputVector[i]
 62                         probabilities[classValue] *= calculateProbability(x, mean, stdev)
 63         return probabilities
 64 
 65 def predict(summaries, inputVector):
 66         probabilities = calculateClassProbabilities(summaries, inputVector)
 67         bestLabel, bestProb = None, -1
 68         for classValue, probability in probabilities.iteritems():
 69                 if bestLabel is None or probability > bestProb:
 70                         bestProb = probability
 71                         bestLabel = classValue
 72         return bestLabel
 73 
 74 def getPredictions(summaries, testSet):
 75         predictions = []
 76         for i in range(len(testSet)):
 77                 result = predict(summaries, testSet[i])
 78                 predictions.append(result)
 79         return predictions
 80 
 81 def getAccuracy(testSet, predictions):
 82         correct = 0
 83         for i in range(len(testSet)):
 84                 if testSet[i][-1] == predictions[i]:
 85                         correct += 1
 86         return (correct/float(len(testSet))) * 100.0
 87 
 88 def main():
 89         filename = ‘pima-indians-diabetes.data.csv‘
 90         splitRatio = 0.67
 91         dataset = loadCsv(filename)
 92         trainingSet, testSet = splitDataset(dataset, splitRatio)
 93         print(‘Split {0} rows into train={1} and test={2} rows‘).format(len(dataset), len(trainingSet), len(testSet))
 94         # prepare model
 95         summaries = summarizeByClass(trainingSet)
 96         # test model
 97         predictions = getPredictions(summaries, testSet)
 98         accuracy = getAccuracy(testSet, predictions)
 99         print(‘Accuracy: {0}%‘).format(accuracy)
100 
101 main()

 

pima-indians-diabetes.data.csv的下载地址：

https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv 

 

参考文档：

１　https://en.wikipedia.org/wiki/Naive_Bayes_classifier

２　https://machinelearningmastery.com/naive-bayes-classifier-scratch-python/

３　https://machinelearningmastery.com/naive-bayes-for-machine-learning/