# Understanding This Graphical Depiction of a Radial-Basis Function Network

Let $f$ be a Radial-Basis Function Network:

$$f(X) = \sum_{i=1}^N a_i p( \lvert \lvert b_i X - c_i \lvert \lvert)$$

From Artificial Intelligence for Human Beings, the following depicts $f$:

In the concrete example of this diagram, it seems that $N = 3$ (i.e., there are $3$ RBFs).

Question 1: How is it possible that there are multiple outputs on the right of this diagram coming from multiple summations? In particular, there are 3 summations producing 3 outputs. But shouldn't there just be 1 summation producing 1 output? Are the second and third summations coming from a different set of RBFs than the first summation?

The textbook continues:

Because multiple summations exist, you can see the development of a classification problem. The highest summation specifies the predicted class. A regression problem indicates that the model will output a single numeric value.

Question 2: Here, does "the highest summation" refer to the very top "Output 1" in the diagram? Or does it refer to the largest summation in the numerical sense?

Question 3: Where in the equation for $f$ above is the $Bias$ (depicted in the diagram as $Bias (1)$) represented?

• "How is it possible that..." - It'd help if you told us why you think it's not possible. This might give us more to work with, to explain better what's going on. Also, please ask only one question per post. If you have multiple questions, I suggest you ask one first, wait to get an answer, and then if you still have other questions, post another one separately. (That way, if the answer to the first answers you question you don't need to ask the others.) But in any case, we want you to ask only one question per post -- the site format doesn't work as well with multiple questions. Thanks! – D.W. Jul 11 '16 at 8:25
• Also, when you find something you don't understand, a very good practice is to try to find another different explanation of the topic and see if that resolves it. Have you looked for any other explanations of RBF networks? – D.W. Jul 11 '16 at 8:26