Link to the previous post : https://test.statinfer.com/204-5-1-neural-networks-a-recap-of-logistic-regression/

In the last session we recapped logistic regression. There is something more to understand before we move further which is a Decision Boundary. Once we get decision boundary right we can move further to Neural networks.

Decision Boundary – Logistic Regression

The line or margin that separates the classes.
Classification, algorithms are all about finding the decision boundaries.
It need not be straight line always.
The final function of our decision boundary looks like,
- Y=1 if $w^Tx+w_0>0$ ; else Y=0

In logistic regression, it can be derived from the logistic regression coefficients and the threshold.
- Imagine the logistic regression line p(y)=$e^(b_0+b_1x_1+b_2x_2)/1+exp^(b_0+b_1x_1+b_2x_2)$
- Suppose if p(y)>0.5 then class-1 or else class-0
  - $log(y/1-y)=b_0+b_1x_1+b_2x_2$
  - $Log(0.5/0.5)=b_0+b_1x_1+b_2x_2$
  - $0=b_0+b_1x_1+b_2x_2$
  - $b_0+b_1x_1+b_2x_2=0 is the line$
- Rewriting it in mx+c form
  - $X_2=(-b_1/b_2)X_1+(-b_0/b_2)$
- Anything above this line is class-1, below this line is class-0
  - $X_2>(-b_1/b_2)X_1+(-b_0/b_2)$ is class-1
  - $X_2<(-b_1/b_2)X_1+(-b_0/b_2)$ is class-0
  - $X_2=(-b_1/b_2)X_1+(-b_0/b_2)$ tie probability of 0.5
- We can change the decision boundary by changing the threshold value(here 0.5)

Practice : Decision Boundary

Draw a scatter plot that shows Age on X axis and Experience on Y-axis. Try, to distinguish the two classes with colors or shapes (visualizing the classes)
Build a logistic regression model to predict Productivity using age and experience.
Finally, draw the decision boundary for this logistic regression model.
Create, the confusion matrix.
Calculate, the accuracy and error rates.

Solution : We have covered all these tasks in previous post. However, we will again plot the decision boundary.

In [11]:

import matplotlib.pyplot as plt

fig = plt.figure()
ax1 = fig.add_subplot(111)

ax1.scatter(Emp_Productivity1.Age[Emp_Productivity1.Productivity==0],Emp_Productivity1.Experience[Emp_Productivity1.Productivity==0], s=10, c='b', marker="o", label='Productivity 0')
ax1.scatter(Emp_Productivity1.Age[Emp_Productivity1.Productivity==1],Emp_Productivity1.Experience[Emp_Productivity1.Productivity==1], s=10, c='r', marker="+", label='Productivity 1')
plt.legend(loc='upper left');

x_min, x_max = ax1.get_xlim()
ax1.plot([0, x_max], [intercept1, x_max*slope1+intercept1])
ax1.set_xlim([15,35])
ax1.set_ylim([0,10])
plt.show()

We did cover this part in our last post too, but we will move further for one or two more posts to understand different kind of decision boundaries.

New Representation for Logistic Regression

$y = \frac{e^{(} b_{0} + b_{1} x_{1} + b_{2} x_{2})}{1 + e^{(} b_{0} + b_{1} x_{1} + b_{2} x_{2})}$

$y = \frac{1}{1 + e^{-} (b_{0} + b_{1} x_{1} + b_{2} x_{2})}$

$y = g (w_{0} + w_{1} x_{1} + w_{2} x_{2}) w h e r e g (x) = \frac{1}{1 + e^{-} (x)}$

$y = g (\sum w_{k} x_{k})$

Finding the weights in logistic regression

out(x) = $y = g (\sum w_{k} x_{k})$ The above output is a non linear function of linear combination of inputs – A typical multiple logistic regression line

We find w to minimize $\sum_{i = 1}^{n} [y_{i} - g (\sum w_{k} x_{k})]^{2}$

The next post is a practice session on non-linear decision boundary.

Link to the next post : https://test.statinfer.com/204-5-3-practice-non-linear-decision-boundary/

9th April 2019

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204.5.2 Decision Boundary – Logistic Regression

Few more things before getting into Neural Networks.