We will begin by examining some numerical and graphical summaries of the `Smarket`

data, which is part of the `ISLR2`

library. This data set consists of percentage returns for the S&P 500 stock index over \(1,250\)~days, from the beginning of 2001 until the end of 2005. For each date, we have recorded the percentage returns for each of the five previous trading days, `lagone`

through `lagfive`

. We have also recorded `volume`

(the number of shares traded on the previous day, in billions), `Today`

(the percentage return on the date in question) and `direction`

(whether the market was `Up`

or `Down`

on this date). Our goal is to predict `direction`

(a qualitative response) using the other features.

```
library(ISLR2)
names(Smarket)
```

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

`dim(Smarket)`

`## [1] 1250 9`

`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
##
##
##
##
```

`pairs(Smarket)`

The `cor()`

function produces a matrix that contains all of the pairwise correlations among the predictors in a data set. The first command below gives an error message because the `direction`

variable is qualitative.

`cor(Smarket)`

`## Error in cor(Smarket): 'x' must be numeric`

`cor(Smarket[, -9])`

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

As one would expect, the correlations between the lag variables and today’s returns are close to zero. In other words, there appears to be little correlation between today’s returns and previous days’ returns. The only substantial correlation is between `Year`

and `volume`

. By plotting the data, which is ordered chronologically, we see that `volume`

is increasing over time. In other words, the average number of shares traded daily increased from 2001 to 2005.

```
attach(Smarket)
plot(Volume)
```