Linear Classification
This lab is a minor modification of the lab from `An Introduction to Statistical Learning with Applications in R’, Chapter 4.
The Stock Market Data
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)## Warning: 패키지 'ISLR2'는 R 버전 4.3.2에서 작성되었습니다names(Smarket)## [1] "Year" "Lag1" "Lag2" "Lag3" "Lag4" "Lag5"
## [7] "Volume" "Today" "Direction"dim(Smarket)## [1] 1250 9summary(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'는 반드시 수치형이어야 합니다cor(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 1.000000000As 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)
Logistic Regression
Next, we will fit a logistic regression model in order to predict
direction using lagone through
lagfive and volume. The glm()
function can be used to fit many types of generalized linear models,
including logistic regression. The syntax of the glm()
function is similar to that of lm(), except that we must
pass in the argument family = binomial in order to tell
R to run a logistic regression rather than some other type
of generalized linear model.
glm.fits <- glm(
Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 + Volume,
data = Smarket, family = binomial
)
summary(glm.fits)##
## 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: 3The smallest \(p\)-value here is
associated with lagone. The negative coefficient for this
predictor suggests that if the market had a positive return yesterday,
then it is less likely to go up today. However, at a value of \(0.15\), the \(p\)-value is still relatively large, and so
there is no clear evidence of a real association between
lagone and direction.
We use the coef() function in order to access just the
coefficients for this fitted model. We can also use the
summary() function to access particular aspects of the
fitted model, such as the \(p\)-values
for the coefficients.
coef(glm.fits)## (Intercept) Lag1 Lag2 Lag3 Lag4 Lag5
## -0.126000257 -0.073073746 -0.042301344 0.011085108 0.009358938 0.010313068
## Volume
## 0.135440659summary(glm.fits)$coef## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.126000257 0.24073574 -0.5233966 0.6006983
## Lag1 -0.073073746 0.05016739 -1.4565986 0.1452272
## Lag2 -0.042301344 0.05008605 -0.8445733 0.3983491
## Lag3 0.011085108 0.04993854 0.2219750 0.8243333
## Lag4 0.009358938 0.04997413 0.1872757 0.8514445
## Lag5 0.010313068 0.04951146 0.2082966 0.8349974
## Volume 0.135440659 0.15835970 0.8552723 0.3924004summary(glm.fits)$coef[, 4]## (Intercept) Lag1 Lag2 Lag3 Lag4 Lag5
## 0.6006983 0.1452272 0.3983491 0.8243333 0.8514445 0.8349974
## Volume
## 0.3924004The predict() function can be used to predict the
probability that the market will go up, given values of the predictors.
The type = "response" option tells R to output
probabilities of the form \(P(Y=1|X)\),
as opposed to other information such as the logit. If no data set is
supplied to the predict() function, then the probabilities
are computed for the training data that was used to fit the logistic
regression model. Here we have printed only the first ten probabilities.
We know that these values correspond to the probability of the market
going up, rather than down, because the contrasts()
function indicates that R has created a dummy variable with
a 1 for Up.
glm.probs <- predict(glm.fits, type = "response")
glm.probs[1:10]## 1 2 3 4 5 6 7 8
## 0.5070841 0.4814679 0.4811388 0.5152224 0.5107812 0.5069565 0.4926509 0.5092292
## 9 10
## 0.5176135 0.4888378contrasts(Direction)## Up
## Down 0
## Up 1In order to make a prediction as to whether the market will go up or
down on a particular day, we must convert these predicted probabilities
into class labels, Up or Down. The following
two commands create a vector of class predictions based on whether the
predicted probability of a market increase is greater than or less than
\(0.5\).
glm.pred <- rep("Down", 1250)
glm.pred[glm.probs > .5] = "Up"The first command creates a vector of 1,250 Down
elements. The second line transforms to Up all of the
elements for which the predicted probability of a market increase
exceeds \(0.5\). Given these
predictions, the table() function can be used to produce a
confusion matrix in order to determine how many observations were
correctly or incorrectly classified.
table(glm.pred, Direction)## Direction
## glm.pred Down Up
## Down 145 141
## Up 457 507(507 + 145) / 1250## [1] 0.5216mean(glm.pred == Direction)## [1] 0.5216The diagonal elements of the confusion matrix indicate correct
predictions, while the off-diagonals represent incorrect predictions.
Hence our model correctly predicted that the market would go up on \(507\) days and that it would go down on
\(145\) days, for a total of \(507+145 = 652\) correct predictions. The
mean() function can be used to compute the fraction of days
for which the prediction was correct. In this case, logistic regression
correctly predicted the movement of the market \(52.2\) % of the time.
At first glance, it appears that the logistic regression model is working a little better than random guessing. However, this result is misleading because we trained and tested the model on the same set of \(1,250\) observations. In other words, \(100\%-52.2\%=47.8\%\), is the training error rate. As we have seen previously, the training error rate is often overly optimistic—it tends to underestimate the test error rate. In order to better assess the accuracy of the logistic regression model in this setting, we can fit the model using part of the data, and then examine how well it predicts the held out data. This will yield a more realistic error rate, in the sense that in practice we will be interested in our model’s performance not on the data that we used to fit the model, but rather on days in the future for which the market’s movements are unknown.
To implement this strategy, we will first create a vector corresponding to the observations from 2001 through 2004. We will then use this vector to create a held out data set of observations from 2005.
train <- (Year < 2005)
Smarket.2005 <- Smarket[!train, ]
dim(Smarket.2005)## [1] 252 9Direction.2005 <- Direction[!train]The object train is a vector of \(1{,}250\) elements, corresponding to the
observations in our data set. The elements of the vector that correspond
to observations that occurred before 2005 are set to TRUE,
whereas those that correspond to observations in 2005 are set to
FALSE. The object train is a Boolean
vector, since its elements are TRUE and FALSE.
Boolean vectors can be used to obtain a subset of the rows or columns of
a matrix. For instance, the command Smarket[train, ] would
pick out a submatrix of the stock market data set, corresponding only to
the dates before 2005, since those are the ones for which the elements
of train are TRUE. The ! symbol
can be used to reverse all of the elements of a Boolean vector. That is,
!train is a vector similar to train, except
that the elements that are TRUE in train get
swapped to FALSE in !train, and the elements
that are FALSE in train get swapped to
TRUE in !train. Therefore,
Smarket[!train, ] yields a submatrix of the stock market
data containing only the observations for which train is
FALSE—that is, the observations with dates in 2005. The
output above indicates that there are 252 such observations.
We now fit a logistic regression model using only the subset of the
observations that correspond to dates before 2005, using the
subset argument. We then obtain predicted probabilities of
the stock market going up for each of the days in our test set—that is,
for the days in 2005.
glm.fits <- glm(
Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 + Volume,
data = Smarket, family = binomial, subset = train
)
glm.probs <- predict(glm.fits, Smarket.2005,
type = "response")Notice that we have trained and tested our model on two completely separate data sets: training was performed using only the dates before 2005, and testing was performed using only the dates in 2005. Finally, we compute the predictions for 2005 and compare them to the actual movements of the market over that time period.
glm.pred <- rep("Down", 252)
glm.pred[glm.probs > .5] <- "Up"
table(glm.pred, Direction.2005)## Direction.2005
## glm.pred Down Up
## Down 77 97
## Up 34 44mean(glm.pred == Direction.2005)## [1] 0.4801587mean(glm.pred != Direction.2005)## [1] 0.5198413The != notation means not equal to, and so the
last command computes the test set error rate. The results are rather
disappointing: the test error rate is \(52\) %, which is worse than random
guessing! Of course this result is not all that surprising, given that
one would not generally expect to be able to use previous days’ returns
to predict future market performance. (After all, if it were possible to
do so, then the authors of this book would be out striking it rich
rather than writing a statistics textbook.)
We recall that the logistic regression model had very underwhelming
\(p\)-values associated with all of the
predictors, and that the smallest \(p\)-value, though not very small,
corresponded to lagone. Perhaps by removing the variables
that appear not to be helpful in predicting direction, we
can obtain a more effective model. After all, using predictors that have
no relationship with the response tends to cause a deterioration in the
test error rate (since such predictors cause an increase in variance
without a corresponding decrease in bias), and so removing such
predictors may in turn yield an improvement. Below we have refit the
logistic regression using just lagone and
lagtwo, which seemed to have the highest predictive power
in the original logistic regression model.
glm.fits <- glm(Direction ~ Lag1 + Lag2, data = Smarket,
family = binomial, subset = train)
glm.probs <- predict(glm.fits, Smarket.2005,
type = "response")
glm.pred <- rep("Down", 252)
glm.pred[glm.probs > .5] <- "Up"
table(glm.pred, Direction.2005)## Direction.2005
## glm.pred Down Up
## Down 35 35
## Up 76 106mean(glm.pred == Direction.2005)## [1] 0.5595238106 / (106 + 76)## [1] 0.5824176Now the results appear to be a little better: \(56\%\) of the daily movements have been correctly predicted. It is worth noting that in this case, a much simpler strategy of predicting that the market will increase every day will also be correct \(56\%\) of the time! Hence, in terms of overall error rate, the logistic regression method is no better than the naive approach. However, the confusion matrix shows that on days when logistic regression predicts an increase in the market, it has a \(58\%\) accuracy rate. This suggests a possible trading strategy of buying on days when the model predicts an increasing market, and avoiding trades on days when a decrease is predicted. Of course one would need to investigate more carefully whether this small improvement was real or just due to random chance.
Suppose that we want to predict the returns associated with
particular values of lagone and lagtwo. In
particular, we want to predict direction on a day when
lagone and lagtwo equal 1.2 and~1.1,
respectively, and on a day when they equal 1.5 and $-$0.8. We do this
using the predict() function.
predict(glm.fits,
newdata =
data.frame(Lag1 = c(1.2, 1.5), Lag2 = c(1.1, -0.8)),
type = "response"
)## 1 2
## 0.4791462 0.4960939Now, we plot the decision boundary of the logistic classifier overlayed with data. We do this by set a grid and predict classification values on the grid. We first set up the grid.
color <- c('#e66101','#5e3c99','#fdb863','#b2abd2')
grid_n <- 400
grid_L1 <- seq(from = min(Smarket[["Lag1"]]), to = max(Smarket[["Lag1"]]),
length.out = grid_n)
grid_L2 <- seq(from = min(Smarket[["Lag2"]]), to = max(Smarket[["Lag2"]]),
length.out = grid_n)
grid_L <- expand.grid(Lag1 = grid_L1, Lag2 = grid_L2)Then, we predict classification values on the grid, which will
determine the decision boundary. Then we plot the decision boundary
using the .filled.contour() function and overlay data on
it.
p_grid_glm <- matrix(
predict(glm.fits, newdata = grid_L, type = "response") > 0.5,
nrow = grid_n, ncol = grid_n)
plot(NA, main = "Logistic Classifier", xlab = "Lag1", ylab = "Lag2",
xlim = range(grid_L1), ylim = range(grid_L2))
.filled.contour(x = grid_L1, y = grid_L2, z = p_grid_glm, levels = c(0, 0.5, 1),
col = color[3:4])
points(Smarket[["Lag1"]], Smarket[["Lag2"]],
col = color[Smarket[["Direction"]]], pch = 16)
legend("topright", legend = c("Down", "Up"), col = color[1:2], pch = 16)
You can see that the decision boundary is linear.
Linear Discriminant Analysis
Now we will perform LDA on the Smarket data. In
R, we fit an LDA model using the lda()
function, which is part of the MASS library. Notice that
the syntax for the lda() function is identical to that of
lm(), and to that of glm() except for the
absence of the family option. We fit the model using only
the observations before 2005.
library(MASS)##
## 다음의 패키지를 부착합니다: 'MASS'## The following object is masked from 'package:ISLR2':
##
## Bostonlda.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 -0.5135293plot(lda.fit)
The LDA output indicates that \(\hat\pi_1=0.492\) and \(\hat\pi_2=0.508\); in other words, \(49.2\) % of the training observations
correspond to days during which the market went down. It also provides
the group means; these are the average of each predictor within each
class, and are used by LDA as estimates of \(\mu_k\). These suggest that there is a
tendency for the previous 2~days’ returns to be negative on days when
the market increases, and a tendency for the previous days’ returns to
be positive on days when the market declines. The coefficients of
linear discriminants output provides the linear combination of
lagone and lagtwo that are used to form the
LDA decision rule. In other words, these are the multipliers of the
elements of \(X=x\) in (4.24). If \(-0.642 \times \text{lagone} - 0.514 \times
\text{lagtwo}\) is large, then the LDA classifier will predict a
market increase, and if it is small, then the LDA classifier will
predict a market decline.
The plot() function produces plots of the linear
discriminants, obtained by computing \(-0.642 \times \text{lagone} - 0.514 \times
\text{lagtwo}\) for each of the training observations. The
Up and Down observations are displayed
separately.
The predict() function returns a list with three
elements. The first element, class, contains LDA’s
predictions about the movement of the market. The second element,
posterior, is a matrix whose \(k\)th column contains the posterior
probability that the corresponding observation belongs to the \(k\)th class, computed from (4.15). Finally,
x contains the linear discriminants, described earlier.
lda.pred <- predict(lda.fit, Smarket.2005)
names(lda.pred)## [1] "class" "posterior" "x"As we observed in Section 4.5, the LDA and logistic regression predictions are almost identical.
lda.class <- lda.pred$class
table(lda.class, Direction.2005)## Direction.2005
## lda.class Down Up
## Down 35 35
## Up 76 106mean(lda.class == Direction.2005)## [1] 0.5595238Applying a \(50\) % threshold to the
posterior probabilities allows us to recreate the predictions contained
in lda.pred$class.
sum(lda.pred$posterior[, 1] >= .5)## [1] 70sum(lda.pred$posterior[, 1] < .5)## [1] 182Notice that the posterior probability output by the model corresponds to the probability that the market will decrease:
lda.pred$posterior[1:20, 1]## 999 1000 1001 1002 1003 1004 1005 1006
## 0.4901792 0.4792185 0.4668185 0.4740011 0.4927877 0.4938562 0.4951016 0.4872861
## 1007 1008 1009 1010 1011 1012 1013 1014
## 0.4907013 0.4844026 0.4906963 0.5119988 0.4895152 0.4706761 0.4744593 0.4799583
## 1015 1016 1017 1018
## 0.4935775 0.5030894 0.4978806 0.4886331lda.class[1:20]## [1] Up Up Up Up Up Up Up Up Up Up Up Down Up Up Up
## [16] Up Up Down Up Up
## Levels: Down UpIf we wanted to use a posterior probability threshold other than \(50\) % in order to make predictions, then we could easily do so. For instance, suppose that we wish to predict a market decrease only if we are very certain that the market will indeed decrease on that day—say, if the posterior probability is at least \(90\) %.
sum(lda.pred$posterior[, 1] > .9)## [1] 0No days in 2005 meet that threshold! In fact, the greatest posterior probability of decrease in all of 2005 was \(52.02\) %.
We again plot the decision boundary of the logistic classifier
overlayed with data. We do this by set a grid and predict classification
values on the grid. We use the grid computed before. Then, we predict
classification values on the grid, which will determine the decision
boundary. Then we plot the decision boundary using the
.filled.contour() function and overlay data on it.
p_grid_lda <- matrix(as.numeric(predict(lda.fit, newdata = grid_L)[["class"]]),
nrow = grid_n, ncol = grid_n)
plot(NA, main = "Linear Discriminant Analysis", xlab = "Lag1", ylab = "Lag2",
xlim = range(grid_L1), ylim = range(grid_L2))
.filled.contour(x = grid_L1, y = grid_L2, z = p_grid_lda, levels = c(1, 1.5, 2),
col = color[3:4])
points(Smarket[["Lag1"]], Smarket[["Lag2"]],
col = color[Smarket[["Direction"]]], pch = 16)
points(lda.fit[["means"]], pch = "+", cex = 3, col = color[3:4])
legend("topright", legend = c("Down", "Up"), col = color[1:2], pch = 16)
You can see that the decision boundary is linear. Two crosses are the estimated centers for Gaussian distributions for each class. Note that the decision boundary need not pass between the centers.
Quadratic Discriminant Analysis
We will now fit a QDA model to the Smarket data. QDA is
implemented in R using the qda() function,
which is also part of the MASS library. The syntax is
identical to that of lda().
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 -0.03132544The output contains the group means. But it does not contain the
coefficients of the linear discriminants, because the QDA classifier
involves a quadratic, rather than a linear, function of the predictors.
The predict() function works in exactly the same fashion as
for LDA.
qda.class <- predict(qda.fit, Smarket.2005)$class
table(qda.class, Direction.2005)## Direction.2005
## qda.class Down Up
## Down 30 20
## Up 81 121mean(qda.class == Direction.2005)## [1] 0.5992063Interestingly, the QDA predictions are accurate almost \(60\) % of the time, even though the 2005 data was not used to fit the model. This level of accuracy is quite impressive for stock market data, which is known to be quite hard to model accurately. This suggests that the quadratic form assumed by QDA may capture the true relationship more accurately than the linear forms assumed by LDA and logistic regression. However, we recommend evaluating this method’s performance on a larger test set before betting that this approach will consistently beat the market!
We again plot the decision boundary of the quadratic discriminant
analysis overlayed with data. We do this by set a grid and predict
classification values on the grid. We use the grid computed before.
Then, we predict classification values on the grid, which will determine
the decision boundary. Then we plot the decision boundary using the
.filled.contour() function and overlay data on it.
p_grid_qda <- matrix(as.numeric(predict(qda.fit, newdata = grid_L)[["class"]]),
nrow = grid_n, ncol = grid_n)
plot(NA, main = "Quadratic Discriminant Analysis", xlab = "Lag1", ylab = "Lag2",
xlim = range(grid_L1), ylim = range(grid_L2))
.filled.contour(x = grid_L1, y = grid_L2, z = p_grid_qda, levels = c(1, 1.5, 2),
col = color[3:4])
points(Smarket[["Lag1"]], Smarket[["Lag2"]],
col = color[Smarket[["Direction"]]], pch = 16)
points(qda.fit[["means"]], pch = "+", cex = 3, col = color[3:4])
legend("topright", legend = c("Down", "Up"), col = color[1:2], pch = 16)
You can see that the decision boundary is quadratic. Two crosses are the estimated centers for Gaussian distributions for each class. Note that the decision boundary need not pass between the centers.