4.6.1 The Stock Market Data
> library (ISLR) > names(Smarket ) [1] "Year" "Lag1" "Lag2" "Lag3" "Lag4" [6] "Lag5" "Volume " "Today" " Direction " > dim(Smarket ) [1] 1250 9
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. 这个还挺有意思的
> 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
4.6.2 Logistic Regression
The glm() function fits generalized glm() linear models, a class of models that includes logistic regression. The syntax
generalized of the glm() function is similar to that of lm(), except that we must pass in linear model the argument family=binomial in order to tell R to run a logistic regression rather than some other type of generalized linear model.
> 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
分析“
The 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.
”
看具体的参数
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.fit) (Intercept) Lag1 Lag2 Lag3 Lag4 -0.126000257 -0.073073746 -0.042301344 0.011085108 0.009358938 Lag5 Volume 0.010313068 0.135440659 > summary (glm.fit )$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 0.3924004 >
结果预测
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.
> attach(Smarket) >glm.probs= predict (glm.fit, type = "response") In 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. > contrasts (Direction ) Up Down 0 Up 1
之后
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 table() can be used to produce a confusion matrix in order to determine how many observations were correctly or incorrectly classified.
> glm.pred=rep ("Down " ,1250) > glm.pred[glm .probs >.5]=" Up"
> table(glm.pred ,Direction )
Direction
glm.pred Down Up
Up 457 507
Down 145 141
Cross validation create a held out data set of observations from 2005.
> train =(Year <2005) > Smarket.2005= Smarket [! train ,] > Direction.2005= Direction [! train]
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.fit=glm(Direction∼Lag1+Lag2+Lag3+Lag4+Lag5+Volume , data=Smarket ,family =binomial ,subset =train )
混乱，不继续这部分了。
4.6.3 Linear Discriminant Analysis
Now we will perform LDA on the Smarket data. In R, we fit a LDA model using the lda() function, which is part of the MASS library.
> 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
The LDA output indicates that ˆπ1 = 0.492 and ˆπ2 = 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 μk. 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 Lag1 and Lag2 that are used to form the LDA decision rule.
If −0.642×Lag1−0.514×Lag2 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 × Lag1 − 0.514 × Lag2 for each of the training observations. .
> lda.pred=predict (lda.fit , Smarket.2005) > names(lda.pred) [1] "class" "posterior" "x"
class, contains LDA’s predictions about the movement of the market.
The second element, posterior, is a matrix whose kth column contains the
posterior probability that the corresponding observation belongs to the kth
class, computed from (4.10). Finally, x contains the linear discriminants,
described earlier.
> lda.class =lda.pred$class > table(lda.class ,Direction.2005) Direction.2005 lda.class Down Up Down 35 35 Up 76 106
4.6.4 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.
> 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
The 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.
4.6.5 K-Nearest Neighbors
perform KNN using the knn() function, which is part of the class library.
The function requires four inputs.
1. A matrix containing the predictors associated with the training data, labeled train.X below.
2. A matrix containing the predictors associated with the data for which we wish to make predictions, labeled test.X below.
3. A vector containing the class labels for the training observations, labeled train.Direction (train.Y)below.
4. A value for K, the number of nearest neighbors to be used by the classifier.
We use the cbind() function, short for column bind, to bind the Lag1 and Lag2 variables together into two matrices, one for the training set and the other for the test set.
Seed
Now the knn() function can be used to predict the market’s movement for the dates in 2005. We set a random seed before we apply knn() because if several observations are tied as nearest neighbors, then R will randomly break the tie. Therefore, a seed must be set in order to ensure reproducibility of results.
> 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=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
results are bac, QDA is the best for this type of data
4.6.6 An Application to Caravan Insurance Data
Caravan data set includes 85 predictors that measure demographic characteristics for 5,822 individuals. The response variable is Purchase, which indicates whether or not a given individual purchases a caravan insurance policy. In this data set, only 6% of people purchased caravan insurance.
Limitations on KNN
Because the KNN classifier predicts the class of a given test observation by
identifying the observations that are nearest to it, the scale of the variables
matters. Any variables that are on a large scale will have a much larger
effect on the distance between the observations, and hence on the KNN
classifier, than variables that are on a small scale.
As far as KNN is concerned, a difference of $1,000
in salary is enormous compared to a difference of 50 years in age. Consequently,
salary will drive the KNN classification results, and age will have
almost no effect.
A good way to handle this problem is to standardize the data so that all variables are given a mean of zero and a standard deviation of one. we exclude column 86, because that is the qualitative Purchase variable.
standardized.X=scale(Caravan [,-86])
We now split the observations into a test set, containing the first 1,000
observations, and a training set, containing the remaining observations.
We fit a KNN model on the training data using K = 1, and evaluate its
performance on the test data.
> 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)