Chapter 4-Classification, Q13

library(MASS)
names(Boston)

1. 'crim'
2. 'zn'
3. 'indus'
4. 'chas'
5. 'nox'
6. 'rm'
7. 'age'
8. 'dis'
9. 'rad'
10. 'tax'
11. 'ptratio'
12. 'black'
13. 'lstat'
14. 'medv'

crime_class = rep(0,length(Boston$crim))
crim_median = median(Boston$crim)
crime_class[Boston$crim>crim_median]=1
new_Boston = data.frame(Boston,crime_class)
names(new_Boston)

1. 'crim'
2. 'zn'
3. 'indus'
4. 'chas'
5. 'nox'
6. 'rm'
7. 'age'
8. 'dis'
9. 'rad'
10. 'tax'
11. 'ptratio'
12. 'black'
13. 'lstat'
14. 'medv'
15. 'crime_class'

cor(new_Boston)

	crim	zn	indus	chas	nox	rm	age	dis	rad	tax	ptratio	black	lstat	medv	crime_class
crim	1.00000000	-0.20046922	0.40658341	-0.055891582	0.42097171	-0.21924670	0.35273425	-0.37967009	0.625505145	0.58276431	0.2899456	-0.38506394	0.4556215	-0.3883046	0.40939545
zn	-0.20046922	1.00000000	-0.53382819	-0.042696719	-0.51660371	0.31199059	-0.56953734	0.66440822	-0.311947826	-0.31456332	-0.3916785	0.17552032	-0.4129946	0.3604453	-0.43615103
indus	0.40658341	-0.53382819	1.00000000	0.062938027	0.76365145	-0.39167585	0.64477851	-0.70802699	0.595129275	0.72076018	0.3832476	-0.35697654	0.6037997	-0.4837252	0.60326017
chas	-0.05589158	-0.04269672	0.06293803	1.000000000	0.09120281	0.09125123	0.08651777	-0.09917578	-0.007368241	-0.03558652	-0.1215152	0.04878848	-0.0539293	0.1752602	0.07009677
nox	0.42097171	-0.51660371	0.76365145	0.091202807	1.00000000	-0.30218819	0.73147010	-0.76923011	0.611440563	0.66802320	0.1889327	-0.38005064	0.5908789	-0.4273208	0.72323480
rm	-0.21924670	0.31199059	-0.39167585	0.091251225	-0.30218819	1.00000000	-0.24026493	0.20524621	-0.209846668	-0.29204783	-0.3555015	0.12806864	-0.6138083	0.6953599	-0.15637178
age	0.35273425	-0.56953734	0.64477851	0.086517774	0.73147010	-0.24026493	1.00000000	-0.74788054	0.456022452	0.50645559	0.2615150	-0.27353398	0.6023385	-0.3769546	0.61393992
dis	-0.37967009	0.66440822	-0.70802699	-0.099175780	-0.76923011	0.20524621	-0.74788054	1.00000000	-0.494587930	-0.53443158	-0.2324705	0.29151167	-0.4969958	0.2499287	-0.61634164
rad	0.62550515	-0.31194783	0.59512927	-0.007368241	0.61144056	-0.20984667	0.45602245	-0.49458793	1.000000000	0.91022819	0.4647412	-0.44441282	0.4886763	-0.3816262	0.61978625
tax	0.58276431	-0.31456332	0.72076018	-0.035586518	0.66802320	-0.29204783	0.50645559	-0.53443158	0.910228189	1.00000000	0.4608530	-0.44180801	0.5439934	-0.4685359	0.60874128
ptratio	0.28994558	-0.39167855	0.38324756	-0.121515174	0.18893268	-0.35550149	0.26151501	-0.23247054	0.464741179	0.46085304	1.0000000	-0.17738330	0.3740443	-0.5077867	0.25356836
black	-0.38506394	0.17552032	-0.35697654	0.048788485	-0.38005064	0.12806864	-0.27353398	0.29151167	-0.444412816	-0.44180801	-0.1773833	1.00000000	-0.3660869	0.3334608	-0.35121093
lstat	0.45562148	-0.41299457	0.60379972	-0.053929298	0.59087892	-0.61380827	0.60233853	-0.49699583	0.488676335	0.54399341	0.3740443	-0.36608690	1.0000000	-0.7376627	0.45326273
medv	-0.38830461	0.36044534	-0.48372516	0.175260177	-0.42732077	0.69535995	-0.37695457	0.24992873	-0.381626231	-0.46853593	-0.5077867	0.33346082	-0.7376627	1.0000000	-0.26301673
crime_class	0.40939545	-0.43615103	0.60326017	0.070096774	0.72323480	-0.15637178	0.61393992	-0.61634164	0.619786249	0.60874128	0.2535684	-0.35121093	0.4532627	-0.2630167	1.00000000

The predictors indus, nox, age, dis, rad and tax have a significant correlation with the crime_class

#creating training data and test data
nrow(new_Boston)

506

#new_Boston = new_Boston[sample(nrow(new_Boston)),]
training_data = new_Boston[1:406,]
test_data = new_Boston[407:nrow(new_Boston),]

LDA

lda.model = lda(crime_class~indus+nox+age+dis+rad+tax,data=training_data)
lda.pred = predict(lda.model,newdata=test_data)
table(lda.pred$class,test_data$crime_class)

   
     0  1
  0  2  1
  1 13 84

#error rate
mean(lda.pred$class!=test_data$crime_class)*100

14

#False positives is high. Increasing the cutoff probability
lda.pred2 = rep(0,length(test_data$crime_class))
lda.pred2[lda.pred$posterior[,2]>0.7]=1
table(lda.pred2,test_data$crime_class)

         
lda.pred2  0  1
        0 10  5
        1  5 80

#error rate
mean(lda.pred2!=test_data$crime_class)*100

10

Logistic Regression

lgs.model = glm(crime_class~indus+nox+rad+age+dis+tax,data=training_data,family=binomial)
summary(lgs.model)

Call:
glm(formula = crime_class ~ indus + nox + rad + age + dis + tax, 
    family = binomial, data = training_data)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.18209  -0.28262  -0.08559   0.04478   2.79128  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -28.284373   4.321520  -6.545 5.95e-11 ***
indus        -0.088696   0.049488  -1.792   0.0731 .  
nox          50.338126   8.474764   5.940 2.85e-09 ***
rad           0.557643   0.129025   4.322 1.55e-05 ***
age           0.003107   0.009210   0.337   0.7359    
dis           0.167513   0.151451   1.106   0.2687    
tax          -0.003928   0.003159  -1.243   0.2138    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 550.71  on 405  degrees of freedom
Residual deviance: 214.16  on 399  degrees of freedom
AIC: 228.16

Number of Fisher Scoring iterations: 8

lgs.probs = predict(lgs.model,newdata=test_data,type="response")
lgs.pred = rep(0,length(test_data$crime_class))
lgs.pred[lgs.probs>0.5]=1
#error rate
mean(lgs.pred!=test_data$crime_class)*100

5

table(lgs.pred,test_data$crime_class)

        
lgs.pred  0  1
       0 10  0
       1  5 85

KNN

library(class)
train.y = training_data$crime_class
test.y = test_data$crime_class
training_data_k = training_data[,-c(1,2,4,6)]
test_data_k = test_data[,-c(1,2,4,6)]
knn.pred = knn(training_data_k,test_data_k,train.y,k=3)

#error rate
mean(knn.pred!=test.y)*100

8

table(knn.pred,test.y)

        test.y
knn.pred  0  1
       0 10  3
       1  5 82