Introduction to Naive Bayes
Naive Bayes is among one of the very simple and powerful algorithms for classification based on Bayes Theorem with an assumption of independence among the predictors. The Naive Bayes classifier assumes that the presence of a feature in a class is not related to any other feature. Naive Bayes is a classification algorithm for binary and multi-class classification problems.
Bayes Theorem
- Based on prior knowledge of conditions that may be related to an event, Bayes theorem describes the probability of the event
- conditional probability can be found this way
- Assume we have a Hypothesis(H) and evidence(E),
According to Bayes theorem, the relationship between the probability of Hypothesis before getting the evidence represented as P(H) and the probability of the hypothesis after getting the evidence represented as P(H|E) is:
P(H|E) = P(E|H)*P(H)/P(E)
- Prior probability = P(H) is the probability before getting the evidence
Posterior probability = P(H|E) is the probability after getting evidence - In general,
P(class|data) = (P(data|class) * P(class)) / P(data)
Bayes Theorem Example
Assume we have to find the probability of the randomly picked card to be king given that it is a face card.
There are 4 Kings in a Deck of Cards which implies that P(King) = 4/52
as all the Kings are face Cards so P(Face|King) = 1
there are 3 Face Cards in a Suit of 13 cards and there are 4 Suits in total so P(Face) = 12/52
Therefore,
P(King|face) = P(face|king)*P(king)/P(face) = 1/3
Code : Implementing Naive Bayes algorithm from scratch using Python
Python3
# Importing libraryimport mathimport randomimport csv# the categorical class names are changed to numberic data# eg: yes and no encoded to 1 and 0def encode_class(mydata): classes = [] for i in range(len(mydata)): if mydata[i][-1] not in classes: classes.append(mydata[i][-1]) for i in range(len(classes)): for j in range(len(mydata)): if mydata[j][-1] == classes[i]: mydata[j][-1] = i return mydata # Splitting the datadef splitting(mydata, ratio): train_num = int(len(mydata) * ratio) train = [] # initially testset will have all the dataset test = list(mydata) while len(train) < train_num: # index generated randomly from range 0 # to length of testset index = random.randrange(len(test)) # from testset, pop data rows and put it in train train.append(test.pop(index)) return train, test# Group the data rows under each class yes or# no in dictionary eg: dict[yes] and dict[no]def groupUnderClass(mydata): dict = {} for i in range(len(mydata)): if (mydata[i][-1] not in dict): dict[mydata[i][-1]] = [] dict[mydata[i][-1]].append(mydata[i]) return dict# Calculating Meandef mean(numbers): return sum(numbers) / float(len(numbers))# Calculating Standard Deviationdef std_dev(numbers): avg = mean(numbers) variance = sum([pow(x - avg, 2) for x in numbers]) / float(len(numbers) - 1) return math.sqrt(variance)def MeanAndStdDev(mydata): info = [(mean(attribute), std_dev(attribute)) for attribute in zip(*mydata)] # eg: list = [ [a, b, c], [m, n, o], [x, y, z]] # here mean of 1st attribute =(a + m+x), mean of 2nd attribute = (b + n+y)/3 # delete summaries of last class del info[-1] return info# find Mean and Standard Deviation under each classdef MeanAndStdDevForClass(mydata): info = {} dict = groupUnderClass(mydata) for classValue, instances in dict.items(): info[classValue] = MeanAndStdDev(instances) return info# Calculate Gaussian Probability Density Functiondef calculateGaussianProbability(x, mean, stdev): expo = math.exp(-(math.pow(x - mean, 2) / (2 * math.pow(stdev, 2)))) return (1 / (math.sqrt(2 * math.pi) * stdev)) * expo# Calculate Class Probabilitiesdef calculateClassProbabilities(info, test): probabilities = {} for classValue, classSummaries in info.items(): probabilities[classValue] = 1 for i in range(len(classSummaries)): mean, std_dev = classSummaries[i] x = test[i] probabilities[classValue] *= calculateGaussianProbability(x, mean, std_dev) return probabilities# Make prediction - highest probability is the predictiondef predict(info, test): probabilities = calculateClassProbabilities(info, test) bestLabel, bestProb = None, -1 for classValue, probability in probabilities.items(): if bestLabel is None or probability > bestProb: bestProb = probability bestLabel = classValue return bestLabel# returns predictions for a set of examplesdef getPredictions(info, test): predictions = [] for i in range(len(test)): result = predict(info, test[i]) predictions.append(result) return predictions# Accuracy scoredef accuracy_rate(test, predictions): correct = 0 for i in range(len(test)): if test[i][-1] == predictions[i]: correct += 1 return (correct / float(len(test))) * 100.0# driver code# add the data path in your systemfilename = r'E:\user\MACHINE LEARNING\machine learning algos\Naive bayes\filedata.csv'# load the file and store it in mydata listmydata = csv.reader(open(filename, "rt"))mydata = list(mydata)mydata = encode_class(mydata)for i in range(len(mydata)): mydata[i] = [float(x) for x in mydata[i]] # split ratio = 0.7# 70% of data is training data and 30% is test data used for testingratio = 0.7train_data, test_data = splitting(mydata, ratio)print('Total number of examples are: ', len(mydata))print('Out of these, training examples are: ', len(train_data))print("Test examples are: ", len(test_data))# prepare modelinfo = MeanAndStdDevForClass(train_data)# test modelpredictions = getPredictions(info, test_data)accuracy = accuracy_rate(test_data, predictions)print("Accuracy of your model is: ", accuracy) |
Output:
Total number of examples are: 200 Out of these, training examples are: 140 Test examples are: 60 Accuracy of your model is: 71.2376788
