R Lab: Logistic Regression
The Stock Market Data
We will begin by examining some numerical and graphical summaries of the Smarket data, which is part of the ISLR 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, Lag1 through Lag5. 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).
library(ISLR)
str(`Smarket`)'data.frame': 1250 obs. of 9 variables:
$ Year : num 2001 2001 2001 2001 2001 ...
$ Lag1 : num 0.381 0.959 1.032 -0.623 0.614 ...
$ Lag2 : num -0.192 0.381 0.959 1.032 -0.623 ...
$ Lag3 : num -2.624 -0.192 0.381 0.959 1.032 ...
$ Lag4 : num -1.055 -2.624 -0.192 0.381 0.959 ...
$ Lag5 : num 5.01 -1.055 -2.624 -0.192 0.381 ...
$ Volume : num 1.19 1.3 1.41 1.28 1.21 ...
$ Today : num 0.959 1.032 -0.623 0.614 0.213 ...
$ Direction: Factor w/ 2 levels "Down","Up": 2 2 1 2 2 2 1 2 2 2 ...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 numericcor(`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 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 Lag1 through Lag5 and Volume. The glm() function fits generalized linear models, a class of models that includes 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)
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: 3The smallest p-value here is associated with Lag1. 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 Lag1 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
0.135440659 summary(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
0.3924004 The 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
0.5070841 0.4814679 0.4811388 0.5152224 0.5107812 0.5069565 0.4926509
8 9 10
0.5092292 0.5176135 0.4888378 contrasts(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
Up 457 507
Down 145 141(507+145) /1250[1] 0.5216mean(glm.pred==Direction )[1] 0.116The 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 ,]
Direction.2005= Direction[!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
Up 34 44
Down 77 97mean(glm.pred==Direction.2005)[1] 0.3055556mean(glm.pred!=Direction.2005)[1] 0.6944444The != 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 pvalues associated with all of the predictors, and that the smallest p-value, though not very small, corresponded to Lag1. 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 Lag1 and Lag2, 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
Up 76 106
Down 35 35mean(glm.pred==Direction.2005)[1] 0.1388889106/(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 na?ive 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 Lag1 and Lag2. In particular, we want to predict Direction on a day when Lag1 and Lag2 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 0.4960939