Lately, I've been thinking quite a bit about neural networks. In particular, I've been wondering whether it is actually possible to understand them. As a note, this posting assumes that the reader has some understanding of neural networks. Of course, we at Data Miners, heartily recommend our book

Data Mining Techniques for Marketing, Sales, and Customer Relationship Management for introducing neural networks (as well as a plethora of other data mining algorithms).

Let me start with a picture of a neural network. The following is a simple network that takes three inputs and has two nodes in the hidden layer:

Note that this structure of the network explains what is really happening. The "input layer" (the first layer connected to the inputs) standardizes the inputs. The "output layer" (connect to the output) is doing a regression or logistic regression, depending on whether the target is numeric or binary. The hidden layers are actually doing a mathematical operation as well. This could be the logistic function; more typically, though it is the hyperbolic tangent. All of the lines in the diagram have weights on them. Setting these weights -- plus a few others not shown -- is the process of training the neural network.

The topology of the neural network is specifically how SAS Enterprise Miner implements the network. Other tools have similar capabilities. Here, I am using SAS EM for three reasons. First, because we teach a class using this tool, I have pre-built neural network diagrams. Second, the neural network node allows me to score the hidden units. And third, the graphics provide a data-colored scatter plot, which I use to describe what's happening.

There are several ways to understand this neural network. The most basic way is "it's a black box and we don't need to understand it." In many respects, this is the standard data mining viewpoint. Neural networks often work well. However, if you want a technique that let's you undersand what it is doing, then choose another technique, such as regression or decision trees or nearest neighbor.

A related viewpoint is to write down the equation for what the network is doing. Then point out that this equation *is* the network. The problem is not that the network cannot explain what it is doing. The problem is that we human beings cannot understand what it is saying.

I am going to propose two other ways of looking at the network. One is geometrically. The inputs are projected onto the outputs of the hidden layer. The results of this projection are then combined to form the output. The other method is, for lack of a better term, "clustering". The hidden nodes actually identify patterns in the original data, and one hidden node usually dominates the output within a cluster.

Let me start with the geometric interpretation. For the network above, there are three dimensions of inputs and two hidden nodes. So, three dimensions are projected down to two dimensions.

I do need to emphasize that these projections are not the linear projections. This means that they are not described by simple matrices. These are non-linear projections. In particular, a given dimension could be stretched non-uniformly, which further complicates the situation.

I chose two nodes in the hidden layer on purpose, simply because two dimensions are pretty easy to visualize. Then I went and I tried it on a small neural network, using Enterprise Miner. The next couple of pictures are scatter plots made with EM. It has the nice feature that I can color the points based on data -- a feature sadly lacking from Excel.

The following scatter plot shows the original data points (about 2,700 of them). The positions are determined by the outputs of the hidden layers. The colors show the output of the network itself (blue being close to 0 and red being close to 1). The network is predicting a value of 0 or 1 based on a balanced training set and three inputs.

Hmm, the overall output is pretty much related to the H1 output rather than the H2 output. We see this becasuse the color changes primarily as we move horizontally across the scatter plot and not vertically. This is interesting. It means that H2 is contributing little to the network prediction. Under these particular circumstances, we can explain the output of the neural network by explaining what is happening at H1. And what is happening at H1 is a lot like a logistic regression, where we can determine the weights of different variables going in.

Note that this is an approximation, because H2 does make some contribution. But it is a close approximation, because for almost all input data points, H1 is the dominant node.

This pattern is a consequence of the distribution of the input data. Note that H2 is always negative and close to -1, whereas H1 varies from -1 to 1 (as we would expect, given the transfer function). This is because the inputs are always positive and in a particular range. The inputs do not result in the full range of values for each hidden node. This fact, in turn, provides a clue to what the neural network is doing. Also, this is close to a degenerate case because one hidden unit is almost always ignored. It does illustrate that looking at the outputs of the hidden layers are useful.

This suggests another approach. Imagine the space of H1 and H2 values, and further that any combination of them might exist (do remember that because of the transfer function, the values actually are limited to the range -1 to 1). Within this space, which node dominates the calculation of the output of the network?

To answer this question, I had to come up with some reasonable way to compare the following values:

- Network output: exp(bias + a1*H1 + a2*H2)
- H1 only: exp(bias + a1*H1)
- H2 only: exp(bias + a2*H2)

Let me give an example with numbers. For the network above, we have the following when H1 and H2 are both -1:

- Network output: 0.9994
- H1 only output: 0.9926
- H2 only output: 0.9749

To calculate the contribution of H1, I use the ratio of the sums of the squares of the differences, as in the following example for H1:

- H1 contribution: (0.9994 - 0.9926)^2 / ((0.9994 - 0.9926)^2 + (0.9994 - 0.9749)^2)

The following scatter plot shows the regions where H1 dominates the overall prediction of the network using this metric (red is H1 is dominant; blue is H2 is dominant):

There are four regions in this scatter plot, defined essentially by the intersection of two lines. In fact, each hidden node is going to add another line on this chart, generating more regions. Within each region, one node is going to dominate. The boundaries are fuzzy. Sometimes this makes no difference, because the output on either side is the same; sometimes it does make a difference.

Note that this scatter plot assumes that the inputs can generate all combinations of values from the hidden units. However, in practice, this is not true, as shown on the previous scatter plot, which essentially covers only the lowest eights of this one.

With the contribution metric, we can then say that for different regions in the hidden unit space, different hidden units dominate the output. This is essentially saying that in different areas, we only need one hidden unit to determine the outcome of the network. Within each region, then, we can identify the variables used by the hidden units and say that they are determining the outcome of the network.

This idea leads to a way to start to understand standard multilayer perceptron neural networks, at least in the space of the hidden units. We can identify the regions where particular hidden units dominate the output of the network. Within each region, we can identify which variables dominate the output of that hidden unit. Perhaps this explains what is happening in the network, because the input ranges limit the outputs only to one region.

More likely, we have to return to the original inputs to determine which hidden unit dominates for a given combination of inputs. I've only just started thinking about this idea, so perhaps I'll follow up in a later post.

--gordon

Labels: Data Mining, Enterprise Miner, gordon, Neural Networks