Logistic regression

binary classification:

• 0: Negative class
• 1: Positive class

logistic regression model:

Sigmoid function(Logistic function):

So,

Interpretations of Hypothesis output

• estimate probability that on input

Interpretations: Probability that , given , parameterized by

Example:
Interpretations: 70% chance of the tumor being malignant

Decision Boundary

• decision boundary is a property not of the training set but of the hypothesis and the parameters
• the training set is used to fit the parameters
• the parameters is used to define the decision boundary

• Suppose

predict if
predict if

image.png

we can see in the figure above:

• when then
• when then , and predict
• when then , and predict

An example of define decision boundary(parameters are certainly)

image.png

• if we has chose the parameters as follow:

• predict: when
• means , then we can get the purple line in the figure as above
• the decision boundary is

Cost function in logistic regression

• if , we can plot the figure as below:
  
  

  y1.png

1.
2. (we will penalized learning algorithm a very large cost)

• if , we can plot the figure as below:
  
  

  image.png

1.
2. (we will penalized learning algorithm a very large cost)

Simplified cost function and gradient descent

• Now we simplified the :

• So, the cost function is:

• Now, we use gradient descent to get :

Repeat

(Simultaneously update all )