Laboratorio 4

Lab 4: Logistic Regression, LDA, QDA, and KNN

The Stock Market Data

library(ISLR)
names(Smarket)

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

dim(Smarket)

[1] 1250    9

summary(Smarket)

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

pairs(Smarket)

# producira error por tener columna no numerica
# cor(Smarket)
cor(Smarket[,-9])

             Year         Lag1         Lag2
Year   1.00000000  0.029699649  0.030596422
Lag1   0.02969965  1.000000000 -0.026294328
Lag2   0.03059642 -0.026294328  1.000000000
Lag3   0.03319458 -0.010803402 -0.025896670
Lag4   0.03568872 -0.002985911 -0.010853533
Lag5   0.02978799 -0.005674606 -0.003557949
Volume 0.53900647  0.040909908 -0.043383215
Today  0.03009523 -0.026155045 -0.010250033
               Lag3         Lag4         Lag5
Year    0.033194581  0.035688718  0.029787995
Lag1   -0.010803402 -0.002985911 -0.005674606
Lag2   -0.025896670 -0.010853533 -0.003557949
Lag3    1.000000000 -0.024051036 -0.018808338
Lag4   -0.024051036  1.000000000 -0.027083641
Lag5   -0.018808338 -0.027083641  1.000000000
Volume -0.041823686 -0.048414246 -0.022002315
Today  -0.002447647 -0.006899527 -0.034860083
            Volume        Today
Year    0.53900647  0.030095229
Lag1    0.04090991 -0.026155045
Lag2   -0.04338321 -0.010250033
Lag3   -0.04182369 -0.002447647
Lag4   -0.04841425 -0.006899527
Lag5   -0.02200231 -0.034860083
Volume  1.00000000  0.014591823
Today   0.01459182  1.000000000

attach(Smarket)
plot(Volume)

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)

Deviance Residuals: 
   Min      1Q  Median      3Q     Max  
-1.446  -1.203   1.065   1.145   1.326  

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

coef(glm.fit)

 (Intercept)         Lag1         Lag2 
-0.126000257 -0.073073746 -0.042301344 
        Lag3         Lag4         Lag5 
 0.011085108  0.009358938  0.010313068 
      Volume 
 0.135440659 

summary(glm.fit)$coef

                Estimate Std. Error    z value
(Intercept) -0.126000257 0.24073574 -0.5233966
Lag1        -0.073073746 0.05016739 -1.4565986
Lag2        -0.042301344 0.05008605 -0.8445733
Lag3         0.011085108 0.04993854  0.2219750
Lag4         0.009358938 0.04997413  0.1872757
Lag5         0.010313068 0.04951146  0.2082966
Volume       0.135440659 0.15835970  0.8552723
             Pr(>|z|)
(Intercept) 0.6006983
Lag1        0.1452272
Lag2        0.3983491
Lag3        0.8243333
Lag4        0.8514445
Lag5        0.8349974
Volume      0.3924004

glm.probs = predict(glm.fit, type = "response")
glm.probs[1:10]

        1         2         3         4         5 
0.5070841 0.4814679 0.4811388 0.5152224 0.5107812 
        6         7         8         9        10 
0.5069565 0.4926509 0.5092292 0.5176135 0.4888378 

contrasts(Direction)

     Up
Down  0
Up    1

glm.pred = rep("Down", 1250)
glm.pred[glm.probs > 0.5] = "Up"

table(glm.pred, Direction)

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

(507+145)/1250

[1] 0.5216

mean(glm.pred == Direction)

[1] 0.5216

train = (Year < 2005)
Smarket.2005 = Smarket[!train,]
dim(Smarket.2005)

[1] 252   9

Direction.2005 = Direction[!train]

glm.fit = glm(Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 + Volume, data = Smarket, family = binomial, subset = train)
glm.probs = predict(glm.fit, Smarket.2005, type = "response")

glm.pred = rep("Down", 252)
glm.pred[glm.probs > 0.5] = "Up"
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

mean(glm.pred != Direction.2005)

[1] 0.5198413

glm.fit = glm(Direction ~ Lag1 + Lag2, data = Smarket, family = binomial, subset = train)
glm.probs = predict(glm.fit, Smarket.2005, type = "response")
glm.pred = rep("Down", 252)
glm.pred[glm.probs > 0.5] = "Up"
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

106/(106+76)

[1] 0.5824176

predict(glm.fit, newdata = data.frame(Lag1 = c(1.2, 1.5), Lag2 = c(1.1, -0.8)), type = "response")

        1         2 
0.4791462 0.4960939 

Linear Discriminant Analysis

library(MASS)
lda.fit = lda(Direction ~ Lag1 + Lag2, data = Smarket, subset = train)
lda.fit

Call:
lda(Direction ~ Lag1 + Lag2, data = Smarket, subset = train)

Prior probabilities of groups:
    Down       Up 
0.491984 0.508016 

Group means:
            Lag1        Lag2
Down  0.04279022  0.03389409
Up   -0.03954635 -0.03132544

Coefficients of linear discriminants:
            LD1
Lag1 -0.6420190
Lag2 -0.5135293

plot(lda.fit)

lda.pred = predict(lda.fit, Smarket.2005)
names(lda.pred)

[1] "class"     "posterior" "x"        

lda.class = lda.pred$class
table(lda.class, Direction.2005)

         Direction.2005
lda.class Down  Up
     Down   35  35
     Up     76 106

mean(lda.class == Direction.2005)

[1] 0.5595238

sum(lda.pred$posterior[,1] >= 0.5)

[1] 70

sum(lda.pred$posterior[,1] < 0.5)

[1] 182

lda.pred$posterior[1:20, 1]

      999      1000      1001      1002      1003 
0.4901792 0.4792185 0.4668185 0.4740011 0.4927877 
     1004      1005      1006      1007      1008 
0.4938562 0.4951016 0.4872861 0.4907013 0.4844026 
     1009      1010      1011      1012      1013 
0.4906963 0.5119988 0.4895152 0.4706761 0.4744593 
     1014      1015      1016      1017      1018 
0.4799583 0.4935775 0.5030894 0.4978806 0.4886331 

lda.class[1:20]

 [1] Up   Up   Up   Up   Up   Up   Up   Up   Up  
[10] Up   Up   Down Up   Up   Up   Up   Up   Down
[19] Up   Up  
Levels: Down Up

sum(lda.pred$posterior[,1] > 0.9)

[1] 0

Quadratic Discriminant Analysis

qda.fit = qda(Direction ~ Lag1 + Lag2, data = Smarket, subset = train)
qda.fit

Call:
qda(Direction ~ Lag1 + Lag2, data = Smarket, subset = train)

Prior probabilities of groups:
    Down       Up 
0.491984 0.508016 

Group means:
            Lag1        Lag2
Down  0.04279022  0.03389409
Up   -0.03954635 -0.03132544

qda.class = predict(qda.fit, Smarket.2005)$class
table(qda.class, Direction.2005)

         Direction.2005
qda.class Down  Up
     Down   30  20
     Up     81 121

mean(qda.class == Direction.2005)

[1] 0.5992063

K-Nearest Neighbors

library(class)
train.X = cbind(Lag1, Lag2)[train,]
test.X = cbind(Lag1, Lag2)[!train,]
train.Direction = Direction[train]

set.seed(1)
knn.pred = knn(train.X, test.X, train.Direction, k = 1)
table(knn.pred, Direction.2005)

        Direction.2005
knn.pred Down Up
    Down   43 58
    Up     68 83

(83+43)/252

[1] 0.5

knn.pred = knn(train.X, test.X, train.Direction, k = 3)
table(knn.pred, Direction.2005)

        Direction.2005
knn.pred Down Up
    Down   48 54
    Up     63 87

mean(knn.pred == Direction.2005)

[1] 0.5357143

An Application to Caravan Insurance Data

dim(Caravan)

[1] 5822   86

attach(Caravan)
summary(Purchase)

  No  Yes 
5474  348 

348/5822

[1] 0.05977327

standardized.X = scale(Caravan[,-86])
var(Caravan[,1])

[1] 165.0378

var(Caravan[,2])

[1] 0.1647078

var(standardized.X[,1])

[1] 1

var(standardized.X[,2])

[1] 1

test = 1:1000
train.X = standardized.X[-test,]
test.X = standardized.X[test,]
train.Y = Purchase[-test]
test.Y = Purchase[test]
set.seed(1)
knn.pred = knn(train.X, test.X, train.Y, k = 1)
mean(test.Y != knn.pred)

[1] 0.118

mean(test.Y != "No")

[1] 0.059

table(knn.pred, test.Y)

        test.Y
knn.pred  No Yes
     No  873  50
     Yes  68   9

9/(68+9)

[1] 0.1168831

knn.pred = knn(train.X, test.X, train.Y, k = 3)
table(knn.pred, test.Y)

        test.Y
knn.pred  No Yes
     No  920  54
     Yes  21   5

5/26

[1] 0.1923077

knn.pred = knn(train.X, test.X, train.Y, k = 5)
table(knn.pred, test.Y)

        test.Y
knn.pred  No Yes
     No  930  55
     Yes  11   4

4/15

[1] 0.2666667

glm.fit = glm(Purchase ~ ., data = Caravan, family = binomial, subset = -test)

glm.fit: fitted probabilities numerically 0 or 1 occurred

glm.probs = predict(glm.fit, Caravan[test,], type = "response")
glm.pred = rep("No", 1000)
glm.pred[glm.probs > 0.5] = "Yes"
table(glm.pred, test.Y)

        test.Y
glm.pred  No Yes
     No  934  59
     Yes   7   0

glm.pred = rep("No", 1000)
glm.pred[glm.probs > 0.25] = "Yes"
table(glm.pred, test.Y)

        test.Y
glm.pred  No Yes
     No  919  48
     Yes  22  11

11/(22+11)

[1] 0.3333333