The Neural Network Market Model
Why should an artificial fragment of the brain do any better than the brain itself?
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Profiting with Artificial Brains?
Your only reason for playing the stock market is to make money -- else you should put your money in a bank and go play golf. The way to profit in the business is to buy equities at one price and sell them at a higher prices. The order in executing the transactions is an independent consideration.
Either way, long or short, to profit you have to be able to project future prices well enough and often enough to maintain a net gain. Or be very lucky.
To be lucky you just have to be lucky. But to be able to predict prices on a regular basis, you need a procedure that does the hard work for you. If you're lucky enough, you might have a productive formula, or model, as I discuss here.
Successful formulas -- indicators -- are hard to come by. Neural networks, too, are hard to come by, and neither do they offer a guarantee of success. After all, neural nets are really just artificial brain components. If we can't solve the trading problem using the brain itself, why should we expect a snippet of a fragment of a brain to be a successful substitute?
If we can find solace here, it lies in the fact that neural networks can be made to function sensibly on very specific problems. The network requires training, but it's possible that an artificial brain can be more effective than we ourselves can be, relative to a specific problem. (We make cars, don't we? And hammers and nails. And many other things that empower us.) It might be possible, too, to profit from a neural network model. Hurray for our side!
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The rationale for a neural network is hard to figure. On the one hand you can say it's simply a mechanical device. There's nothing hidden inside. Nor is it a mystery that it might be able to do something we can't do on our own. In that respect it's on a par with airplanes, or cars. Or bulldozers. Or computers. Like these things, the network is a mechanism designed to gain an advantage of some kind -- leverage -- and it's specifically intended to gain some ground on the pattern recognition problem as a sensor, which is to identify either spatial or temporal patterns. (The Cave has reading skills as its main topic.)
One way to view the network is to see it as a math function -- a mapping from a set of inputs to an output. Taken this way, each of the inputs becomes an independent variable, and the output is the value of the function, the dependent variable. There is a difficulty with this interpretation, though, one that has to do with the nature of the inputs. In fact there's more than one difficulty.
Treating the net as a function, the inputs: x1, x2, x3, and x4 become the independent variables, and the output becomes the dependent variable y(x). Now the first difficulty stems from the fact that each variable maps to each of the three nodes in the second layer. Since we like to think of functions as single-valued, does this multiplicity destroy the functional relationship?
Another difficulty is that it's commonly believed that a function should be expressible in closed form, using letter symbols -- like the single variable linear function:
y = 3x + 4,
for instance. But many networks can be constructed that don't have closed form descriptions. Does this mean that the function interpretation doesn't work?
Let's take the difficulties one at a time.
Where is the Function?
To make more sense of the functional relationship in the network, let's work back from the output and define a function f(x) across the network, assuming we have a function of the input variables, x (treated as a vector over the space of n inputs). Note there are three connections between the middle layer of nodes (the so-called "hidden" layer) and the output layer. These lines carry the output of the layer-2 nodes to the layer-3 node, so we can write a function f(y) to represent this transfer, where y represents the function at the second layer nodes. Normally, this function simply adds the y outputs to get the final output.
Now consider the connections between layer 1 and layer 2. Here we have weightings. The key to understanding neural networks lies in the weights assigned to each connection, i.e., to each of the input-output paths. Once the architecture is established, it's the weightings that identify the specific map defined by the net.
The upper node in layer 2 has three connections, involving the weights, w11, w21, and w31. So the function at that node is a function of the weighted inputs, x11w11, x21w21, and x31w31. We can write the function at the upper middle node as y(x, w). This can be defined as the sum of the weighted inputs. The function at the lower node is similar. The overall function is therefore:
f(y(x, w))
Is the Function Closed-Form?
To argue that the net defines a function doesn't mean you have to be able to write the function down in so many letter symbols, as a closed formula. They might rather be given as a table of independent and dependent values, or as a graph of values.
The finite, digital function given in the following table, for instance, can't be stated in closed form: It is a complete function, though, but only because I say it is.
|
x |
F(x) |
|
1 |
2 |
|
2 |
1 |
|
3 |
3 |
The function has the value zero for all other values of x.
To express the function, the best you can do is repeat the contents of the table:
f(x) = 2, for x = 1
f(x) = 1, for x = 2
f(x) = 3, for x = 3.
f(x) = 0, for x not equal to 1, 2, or 3.
Even so, it is still a function.
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The Significance of Weightings
The significance of the weightings is that they provide the locus for knowledge contained in the net. The knowledge can be narrow or wide, depending on the intensity of the "training" applied to the net. If the training is very "strict," so that only those patterns used in the training phase are identifiable in the test phase, then the knowledge in the net is very narrow and therefore highly specialized. In fact, the net becomes a memory device.
But if the training is less strict, so there is considerable flexibility in the interpretation of test patterns -- so that more than training patterns become identifiable in the test phase -- then the knowledge is broad and general, becoming more like a concept holder.
In the former case the net acts more as a memory device, remembering precisely the patterns used in the training. And in the latter case, the net is more of a generalizing device, being more loosely structured and able to identify as similar other test patterns that weren't among the training patterns.
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