Weighted kNN, clustering, “early” trees and Bayesian Peter Fox Data Analytics – ITWS-4600/ITWS-6600/MATP-4450 Group 2 Module 6, February 12, 2018

Plot tools/ tips http://statmethods.net/advgraphs/layout.html http://flowingdata.com/2014/02/27/how-to-read-histograms-and-use-them-in-r/ pairs, gpairs, scatterplot.matrix, clustergram, etc. data() # precip, presidents, iris, swiss, sunspot.month (!), environmental, ethanol, ionosphere More script fragments in R are available on the web site (http://aquarius.tw.rpi.edu/html/DA )

And use a contingency table > data(Titanic) > mdl <- naiveBayes(Survived ~ ., data = Titanic) > mdl Naive Bayes Classifier for Discrete Predictors Call: naiveBayes.formula(formula = Survived ~ ., data = Titanic) A-priori probabilities: Survived No Yes 0.676965 0.323035 Conditional probabilities: Class Survived 1st 2nd 3rd Crew No 0.08187919 0.11208054 0.35436242 0.45167785 Yes 0.28551336 0.16596343 0.25035162 0.29817159 Sex Survived Male Female No 0.91543624 0.08456376 Yes 0.51617440 0.48382560 Age Survived Child Adult No 0.03489933 0.96510067 Yes 0.08016878 0.91983122 Lab5b_nbayes1.R

http://www. ugrad. stat. ubc. ca/R/library/mlbench/html/HouseVotes84 http://www.ugrad.stat.ubc.ca/R/library/mlbench/html/HouseVotes84.html require(mlbench) data(HouseVotes84) model <- naiveBayes(Class ~ ., data = HouseVotes84) predict(model, HouseVotes84[1:10,-1]) predict(model, HouseVotes84[1:10,-1], type = "raw") pred <- predict(model, HouseVotes84[,-1]) table(pred, HouseVotes84$Class)

Exercise for you > data(HairEyeColor) > mosaicplot(HairEyeColor) > margin.table(HairEyeColor,3) Sex Male Female 279 313 > margin.table(HairEyeColor,c(1,3)) Hair Male Female Black 56 52 Brown 143 143 Red 34 37 Blond 46 81 How would you construct a naïve Bayes classifier and test it?

Cars?

Linear regression? Or?

Ionosphere: group2/lab2_kknn2.R require(kknn) data(ionosphere) ionosphere.learn <- ionosphere[1:200,] ionosphere.valid <- ionosphere[-c(1:200),] fit.kknn <- kknn(class ~ ., ionosphere.learn, ionosphere.valid) table(ionosphere.valid$class, fit.kknn$fit) # vary kernel (fit.train1 <- train.kknn(class ~ ., ionosphere.learn, kmax = 15, kernel = c("triangular", "rectangular", "epanechnikov", "optimal"), distance = 1)) table(predict(fit.train1, ionosphere.valid), ionosphere.valid$class) #alter distance (fit.train2 <- train.kknn(class ~ ., ionosphere.learn, kmax = 15, kernel = c("triangular", "rectangular", "epanechnikov", "optimal"), distance = 2)) table(predict(fit.train2, ionosphere.valid), ionosphere.valid$class)

Results ionosphere.learn <- ionosphere[1:200,] # convenience samping!!!! ionosphere.valid <- ionosphere[-c(1:200),] fit.kknn <- kknn(class ~ ., ionosphere.learn, ionosphere.valid) table(ionosphere.valid$class, fit.kknn$fit) b g b 19 8 g 2 122

(fit. train1 <- train. kknn(class ~. , ionosphere (fit.train1 <- train.kknn(class ~ ., ionosphere.learn, kmax = 15, + kernel = c("triangular", "rectangular", "epanechnikov", "optimal"), distance = 1)) Call: train.kknn(formula = class ~ ., data = ionosphere.learn, kmax = 15, distance = 1, kernel = c("triangular", "rectangular", "epanechnikov", "optimal")) Type of response variable: nominal Minimal misclassification: 0.12 Best kernel: rectangular Best k: 2 table(predict(fit.train1, ionosphere.valid), ionosphere.valid$class) b g b 25 4 g 2 120

(fit. train2 <- train. kknn(class ~. , ionosphere (fit.train2 <- train.kknn(class ~ ., ionosphere.learn, kmax = 15, + kernel = c("triangular", "rectangular", "epanechnikov", "optimal"), distance = 2)) Call: train.kknn(formula = class ~ ., data = ionosphere.learn, kmax = 15, distance = 2, kernel = c("triangular", "rectangular", "epanechnikov", "optimal")) Type of response variable: nominal Minimal misclassification: 0.12 Best kernel: rectangular Best k: 2 table(predict(fit.train2, ionosphere.valid), ionosphere.valid$class) b g b 20 5 g 7 119

However… there is more

Naïve Bayes – what is it? Example: testing for a specific item of knowledge that 1% of the population has been informed of (don’t ask how). An imperfect test: 99% of knowledgeable people test positive 99% of ignorant people test negative If a person tests positive – what is the probability that they know the fact?

Naïve approach… We have 10,000 representative people 100 know the fact/item, 9,900 do not We test them all: Get 99 knowing people testing knowing Get 99 not knowing people testing not knowing But 99 not knowing people testing as knowing Testing positive (knowing) – equally likely to know or not = 50%

Tree diagram 10000 ppl 1% know (100ppl) 99% test to know (99ppl) 1% test not to know (1per) 99% do not know (9900ppl) 1% test to know (99ppl) 99% test not to know (9801ppl)

Relation between probabilities For outcomes x and y there are probabilities of p(x) and p (y) that either happened If there’s a connection, then the joint probability = that both happen = p(x,y) Or x happens given y happens = p(x|y) or vice versa then: p(x|y)*p(y)=p(x,y)=p(y|x)*p(x) So p(y|x)=p(x|y)*p(y)/p(x) (Bayes’ Law) E.g. p(know|+ve)=p(+ve|know)*p(know)/p(+ve)= (.99*.01)/(.99*.01+.01*.99) = 0.5

How do you use it? If the population contains x what is the chance that y is true? p(SPAM|word)=p(word|SPAM)*p(SPAM)/p(word) Base this on data: p(spam) counts proportion of spam versus not p(word|spam) counts prevalence of spam containing the ‘word’ p(word|!spam) counts prevalence of non-spam containing the ‘word’

Or.. What is the probability that you are in one class (i) over another class (j) given another factor (X)? Invoke Bayes: Maximize p(X|Ci)p(Ci)/p(X) (p(X)~constant and p(Ci) are equal if not known) So: conditional indep -

P(xk | Ci) is estimated from the training samples Categorical: Estimate P(xk | Ci) as percentage of samples of class i with value xk Training involves counting percentage of occurrence of each possible value for each class Numeric: Actual form of density function is generally not known, so “normal” density (i.e. distribution) is often assumed

Digging into iris classifier<-naiveBayes(iris[,1:4], iris[,5]) table(predict(classifier, iris[,-5]), iris[,5], dnn=list('predicted','actual')) classifier$apriori classifier$tables$Petal.Length plot(function(x) dnorm(x, 1.462, 0.1736640), 0, 8, col="red", main="Petal length distribution for the 3 different species") curve(dnorm(x, 4.260, 0.4699110), add=TRUE, col="blue") curve(dnorm(x, 5.552, 0.5518947 ), add=TRUE, col = "green")

Bayes > cl <- kmeans(iris[,1:4], 3) > table(cl$cluster, iris[,5]) setosa versicolor virginica 2 0 2 36 1 0 48 14 3 50 0 0 # > m <- naiveBayes(iris[,1:4], iris[,5]) > table(predict(m, iris[,1:4]), iris[,5]) setosa 50 0 0 versicolor 0 47 3 virginica 0 3 47 pairs(iris[1:4],main="Iris Data (red=setosa,green=versicolor,blue=virginica)", pch=21, bg=c("red","green3","blue")[unclass(iris$Species)])

Ex: Classification Bayes Retrieve the abalone.csv dataset Predicting the age of abalone from physical measurements. Perform naivebayes classification to get predictors for Age (Rings). Interpret. Discuss in lab or on LMS.

Using a contingency table > data(Titanic) > mdl <- naiveBayes(Survived ~ ., data = Titanic) > mdl Naive Bayes Classifier for Discrete Predictors Call: naiveBayes.formula(formula = Survived ~ ., data = Titanic) A-priori probabilities: Survived No Yes 0.676965 0.323035 Conditional probabilities: Class Survived 1st 2nd 3rd Crew No 0.08187919 0.11208054 0.35436242 0.45167785 Yes 0.28551336 0.16596343 0.25035162 0.29817159 Sex Survived Male Female No 0.91543624 0.08456376 Yes 0.51617440 0.48382560 Age Survived Child Adult No 0.03489933 0.96510067 Yes 0.08016878 0.91983122

Using a contingency table > predict(mdl, as.data.frame(Titanic)[,1:3]) [1] Yes No No No Yes Yes Yes Yes No No No No Yes Yes Yes Yes Yes No No No Yes Yes Yes Yes No [26] No No No Yes Yes Yes Yes Levels: No Yes

At this point… You may realize the inter-relation among classifications and clustering methods, at an absolute and relative level (i.e. hierarchical -> trees…) is COMPLEX… Trees are interesting from a decision perspective: if this or that, then this…. Beyond just distance measures: clustering (kmeans) to probabilities (Bayesian) And, so many ways to visualize them…