Assignment 7

Author

Yi Wang

require(ISLR)

Loading required package: ISLR

Warning: package 'ISLR' was built under R version 4.3.3

# Check dataset Smarket
?Smarket

starting httpd help server ...

 done

names(Smarket)

[1] "Year"      "Lag1"      "Lag2"      "Lag3"      "Lag4"      "Lag5"     
[7] "Volume"    "Today"     "Direction"

summary(Smarket)

      Year           Lag1                Lag2                Lag3          
 Min.   :2001   Min.   :-4.922000   Min.   :-4.922000   Min.   :-4.922000  
 1st Qu.:2002   1st Qu.:-0.639500   1st Qu.:-0.639500   1st Qu.:-0.640000  
 Median :2003   Median : 0.039000   Median : 0.039000   Median : 0.038500  
 Mean   :2003   Mean   : 0.003834   Mean   : 0.003919   Mean   : 0.001716  
 3rd Qu.:2004   3rd Qu.: 0.596750   3rd Qu.: 0.596750   3rd Qu.: 0.596750  
 Max.   :2005   Max.   : 5.733000   Max.   : 5.733000   Max.   : 5.733000  
      Lag4                Lag5              Volume           Today          
 Min.   :-4.922000   Min.   :-4.92200   Min.   :0.3561   Min.   :-4.922000  
 1st Qu.:-0.640000   1st Qu.:-0.64000   1st Qu.:1.2574   1st Qu.:-0.639500  
 Median : 0.038500   Median : 0.03850   Median :1.4229   Median : 0.038500  
 Mean   : 0.001636   Mean   : 0.00561   Mean   :1.4783   Mean   : 0.003138  
 3rd Qu.: 0.596750   3rd Qu.: 0.59700   3rd Qu.:1.6417   3rd Qu.: 0.596750  
 Max.   : 5.733000   Max.   : 5.73300   Max.   :3.1525   Max.   : 5.733000  
 Direction 
 Down:602  
 Up  :648

# Create a dataframe for data browsing
sm=Smarket

# Bivariate Plot of inter-lag correlations
pairs(Smarket,col=Smarket$Direction,cex=.5, pch=20)

# Logistic regression
glm.fit=glm(Direction~Lag1+Lag2+Lag3+Lag4+Lag5+Volume,
            data=Smarket,family=binomial)
summary(glm.fit)


Call:
glm(formula = Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 + 
    Volume, family = binomial, data = Smarket)

Coefficients:
             Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.126000   0.240736  -0.523    0.601
Lag1        -0.073074   0.050167  -1.457    0.145
Lag2        -0.042301   0.050086  -0.845    0.398
Lag3         0.011085   0.049939   0.222    0.824
Lag4         0.009359   0.049974   0.187    0.851
Lag5         0.010313   0.049511   0.208    0.835
Volume       0.135441   0.158360   0.855    0.392

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1731.2  on 1249  degrees of freedom
Residual deviance: 1727.6  on 1243  degrees of freedom
AIC: 1741.6

Number of Fisher Scoring iterations: 3

glm.probs=predict(glm.fit,type="response") 
glm.probs[1:5]

        1         2         3         4         5 
0.5070841 0.4814679 0.4811388 0.5152224 0.5107812

glm.pred=ifelse(glm.probs>0.5,"Up","Down")
attach(Smarket)
table(glm.pred,Direction)

        Direction
glm.pred Down  Up
    Down  145 141
    Up    457 507

mean(glm.pred==Direction)

[1] 0.5216

# Make training and test set for prediction
train = Year<2005
glm.fit=glm(Direction~Lag1+Lag2+Lag3+Lag4+Lag5+Volume,
            data=Smarket,family=binomial, subset=train)
glm.probs=predict(glm.fit,newdata=Smarket[!train,],type="response") 
glm.pred=ifelse(glm.probs >0.5,"Up","Down")
Direction.2005=Smarket$Direction[!train]
table(glm.pred,Direction.2005)

        Direction.2005
glm.pred Down Up
    Down   77 97
    Up     34 44

mean(glm.pred==Direction.2005)

[1] 0.4801587

#Fit smaller model
glm.fit=glm(Direction~Lag1+Lag2,
            data=Smarket,family=binomial, subset=train)
glm.probs=predict(glm.fit,newdata=Smarket[!train,],type="response") 
glm.pred=ifelse(glm.probs >0.5,"Up","Down")
table(glm.pred,Direction.2005)

        Direction.2005
glm.pred Down  Up
    Down   35  35
    Up     76 106

mean(glm.pred==Direction.2005)

[1] 0.5595238

# Check accuracy rate
106/(76+106)

[1] 0.5824176

# Can you interpret the results?

Answer:

Result Interpretation

2 a. In LDA, predictor variables are assumed to be normally distributed with equal variance, and the response variable must be categorical, as LDA is tailored for classification tasks.

2 b. LDA is suitable for multiclass classification, while logistic regression is typically used for binary classification. Additionally, LDA imposes distributional assumptions on the data, unlike logistic regression.

2 c. The Receiver Operating Characteristic (ROC) curve assesses the performance of a classifier model.

2 d. Sensitivity measures the true positive rate, while specificity measures the true negative rate. Sensitivity is often prioritized due to the severe consequences of false positives, such as wrongful convictions. Both type I and II errors are undesirable, but high sensitivity is crucial.

2 e. Based on the chart, sensitivity is deemed more critical for prediction.

3.Calculate the prediction error: prediction_error <- (252+23)/10000 = 0.0275 = 2.75%