This lab on Logistic Regression is a Python adaptation of p. 161-163 of "Introduction to Statistical Learning with Applications in R" by Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani. Adapted by R. Jordan Crouser at Smith College for SDS293: Machine Learning (Spring 2016).
Want to follow along on your own machine? Download the .py or Jupyter Notebook version.
import pandas as pd
import numpy as np
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis, QuadraticDiscriminantAnalysis
from sklearn.metrics import confusion_matrix, classification_report, precision_score
%matplotlib inline
Let's return to the Smarket data from ISLR.
df = pd.read_csv('Smarket.csv', usecols=range(1,10), index_col=0, parse_dates=True)
df.head()
Now we will perform LDA on the Smarket data from the ISLR package. In Python, we can fit a LDA model using the LinearDiscriminantAnalysis() function, which is part of the discriminant_analysis module of the sklearn library. As we did with logistic regression and KNN, we'll fit the model using only the observations before 2005, and then test the model on the data from 2005.
X_train = df[:'2004'][['Lag1','Lag2']]
y_train = df[:'2004']['Direction']
X_test = df['2005':][['Lag1','Lag2']]
y_test = df['2005':]['Direction']
lda = LinearDiscriminantAnalysis()
model = lda.fit(X_train, y_train)
print(model.priors_)
The LDA output indicates prior probabilities of ${\hat{\pi}}_1 = 0.492$ and ${\hat{\pi}}_2 = 0.508$; in other words, 49.2% of the training observations correspond to days during which the market went down.
print(model.means_)
The above 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.
print(model.coef_)
The coefficients of linear discriminants output provides the linear
combination of Lag1 and Lag2 that are used to form the LDA decision rule.
If $−0.0554\times{\tt Lag1}−0.0443\times{\tt Lag2}$ 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. Note: these coefficients differ from those produced by R.
The predict() function returns a list of LDA’s predictions about the movement of the market on the test data:
pred=model.predict(X_test)
print(np.unique(pred, return_counts=True))
The model assigned 70 observations to the "Down" class, and 182 observations to the "Up" class. Let's check out the confusion matrix to see how this model is doing. We'll want to compare the predicted class (which we can find in pred) to the true class (found in `y_test})$.
print(confusion_matrix(pred, y_test))
print(classification_report(y_test, pred, digits=3))
We will now fit a QDA model to the Smarket data. QDA is implemented
in sklearn using the QuadraticDiscriminantAnalysis() function, which is again part of the discriminant_analysis module. The
syntax is identical to that of LinearDiscriminantAnalysis().
qda = QuadraticDiscriminantAnalysis()
model2 = qda.fit(X_train, y_train)
print(model2.priors_)
print(model2.means_)
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.
pred2=model2.predict(X_test)
print(np.unique(pred2, return_counts=True))
print(confusion_matrix(pred2, y_test))
print(classification_report(y_test, pred2, digits=3))
Interestingly, 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!
Let's see how the LDA/QDA approach performs on the Carseats data set, which is
included with ISLR.
Recall: this is a simulated data set containing sales of child car seats at 400 different stores.
df2 = pd.read_csv('Carseats.csv')
df2.head()
See if you can build a model that predicts ShelveLoc, the shelf location (Bad, Good, or Medium) of the product at each store. Don't forget to hold out some of the data for testing!
# Your code here
To get credit for this lab, please post your answers to the prompt in #lab5.