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There are some classical problems that linear classifiers can’t solve in their basic form, but I will show you how to modify logistic regression in order to do so. The outputs are defined as follows: 0 XOR 0 = 0 1 XOR 0 = 1 0 XOR 1 = 1 1 XOR 1 = 0 As you can see, there is no line that separates the two classes. So what do we do in these two cases? For the XOR problem, create a third dimension that is derived from the first two, x3 = x1x2. Try to draw a 3-D plot to see how a plane could separate the two classes now.
You might want to check this course out if you found the material in this book too challenging. com/data-science-logistic-regression-in-python This next course was the basis for Chapter 3 in this book on linear regression.
We would have: a1 = w1Tx a0 = w0Tx Recall that a0 and a1 can be either negative or positive. Since probabilities must be 0 or positive, we can enforce positivity by exponentiating a. How do we ensure these sum to 1? Simply divide by exp(a1) + exp(a0). So now: p(y=1 | x) = exp(a1) / [ exp(a1) + exp(a0) ] p(y=0 | x) = exp(a0) / [ exp(a1) + exp(a0) ] You can see that it would be very easy to extend this to any number of classes. Appropriately, when you hook up a bunch of neurons / logistic units together, you get a neural network.