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Recitation 3

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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

Linear Classification

Usually we talk about linear regression: inferring a continuous value as a linear function of input variables. In this homework, we’re using linear regression to predict a category variable. Let’s see what this looks like on a simple test dataset:

data = np.concatenate((np.random.normal(size=(n, 2)), np.random.normal(size=(n, 2)) + 2.5))
cls = np.array([-1]*n + [1]*n)

plt.scatter(data[cls==-1,0], data[cls==-1,1])
plt.scatter(data[cls== 1,0], data[cls== 1,1])

We fit a simple linear regression model to this data and examine the output. Notice that the output is continuous, not discrete.

model = LinearRegression()
model.fit(data, cls)
pred = model.predict(data)


Let’s draw the scatter data and highlight misclassified points:

plt.scatter(data[pred<0,0], data[pred<0,1])
plt.scatter(data[pred>0,0], data[pred>0,1])

# Plot misclassified points:
errs = ((cls==-1) & (pred>0)) | ((cls==1) & (pred<0)) 
plt.scatter(data[errs,0], data[errs,1])

Lets visualize the classifier using regularly-spaced points:

xx, yy = np.meshgrid(np.linspace(-3, 6, 101), np.linspace(-3, 6, 101))
grid = np.vstack([xx.ravel(), yy.ravel()]).T
ccls = model.predict(grid)
plt.scatter(grid[ccls<0,0], grid[ccls<0,1])
plt.scatter(grid[ccls>0,0], grid[ccls>0,1])

# Post the original data division as well:
plt.scatter(data[cls==-1,0], data[cls==-1,1])
plt.scatter(data[cls== 1,0], data[cls== 1,1])

Notice that the class boundary is a line; this is characteristic of a linear classifier. We cannot separate classes like these using a linear classifier:

plt.subplot(1, 2, 1)
plt.scatter([-1, 1], [0, 0])
plt.scatter([0, 0], [-1, 1])

plt.subplot(1, 2, 2)
plt.scatter([-1, 1], [-1, 1])
plt.scatter([0], [0])

F1 vs Accuracy

In hw3_text, we introduce the $F_1$-score to measure the performance of a binary classifier. We define this in terms of the number of true and false positives and negatives in our classifier.

  Predicted Actual
true positive T T
false positive T F
false negative F T
true negative F F

To calculate the $F_1$ score, we can calculate:


We care about this because it works when you want to measure classifier performance on rare events.

For example, lets look at a pair of ficticious medical test in a population where 1% of people have some disorder.

When Zico’s Magic Classifier (ZMC) and the much simpler Always False Classifier (AFC) are run on 1000 people, they both obtain 99% accuracy. Here are the confusion matrices for both:

Always False Classifier Actual True Actual False
Predicted True 0 0
Predicted False 10 990
Zico’s Magic Classifier Actual True Actual False
Predicted True 10 10
Predicted False 0 980

The accuracy for both classifiers is 99%, despite AFC not depending on the data (and consequently being useless). In this example, accuracy does not tell us which classifier is better.

Let’s calculate the $F_1$ score for both classifiers:

  Precision Recall $F_1$ score
ATC $\frac{0}{0}$ $\frac{0}{10}$ $0$ or NaN
ZMC $\frac{10}{20}$ $\frac{10}{10}$ $\frac{2}{3}$

The $F_1$ score is good when you want measure your classifier’s performance on catching rare events.