Definition and interpretation of z in logistic regression

Course Q&A Machine Learning Specialization Supervised ML: Regression and Classification
1

Prof Ng defines z as z = w.x + b; however he doesn’t give the interpretation of this variable z.
In linear regression it is pretty intuitive that we are taking a training example as an input and projecting it’s output on a graph using our pre-defined linear/polynomial regression model ( f(x) =w.x + b ).
In logistic regression, I don’t understand what z is, intuitively and graphically. z is in fact a scalar, but does this scalar represent ?
Thanks in advance.

2

Hello @renesultan

Welcome to the community.

Here is the reference to an article that explains what “z” is. Do let me know if this gives you the clarity.

3

I think he did explain Z’s intuition with respect to logistic regression when he was explaining decision boundaries. The line drawn using Z acts as the decision boundary. If Z is linear, you will have linear decision boundary and if it contains power of x, you will get non linear decision boundary.

The sigmoid function of Z becomes greater than 0.5 if Z is >=0. So, if you use a threshold 0.5 to separate classes, it will mean whatever is on positive side of line defined by Z is one class and the other class is one opposite side.

4

Great sharing! @shanup presented the convenience of having z in calculus; @Shivanshu_Singh told us Professor Andrew Ng’s geometric interpretation of the polynomial function z as a decision boundary (course 1 week 3 Video: Decision Boundary, and a relevant discussion); my last point is, mathematically, z is also called a logits in our context (logistic regression), so your wx+b projects x to logits z and your sigmoid function projects z to a probability. logits is a name you will see again in course 2 week 2.

5

Following on from this point, and from the conversation referred to in @shanup’s post, I am unclear whether z is a scalar or a vector/array.
As z = (dot product of w and x) + b, I would have thought this would produce a scalar as @renesultan states. However, in the final C1 W3 Logistic Regression Programming Assignment, it states (in relation to z in the Sigmoid function):

Blockquote

Note that

  • z is not always a single number, but can also be an array of numbers.
  • If the input is an array of numbers, we’d like to apply the sigmoid function to each value in the input array.

Blockquote

Please could you clarify?

Jem

6

Hello Jem,

If we have one sample, then we have one z and z is a scalar.

If we have two samples, then we have two z and z is a vector/array.

For example,

import numpy as np

x = np.array([
    [1., 2., 3., ], # sample 1
    [3., 4., 5., ], # sample 2
])

w = np.array([
    [6., ],
    [7., ],
    [8., ],
])

sigmoid = lambda z: 1/(1+np.exp(-z))

z = x @ w
a = sigmoid(z)

print(z)
print(a)

Both z and a are arrays.

Cheers,
Raymond

7

Great, thanks Raymond.

Jem