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Figure 16.2 Three possible solutions to the data in Table 16.2. these solutions use backpropagation to train perceptron networks with two inputs, three hidden units, and one output unit. inputs re fully connected to the hidden and output layers, and hidden layer is fully connected to the output layer. all hidden and output units also have a bias input.

\ ) Zero Output I One Output

Test Case

The basic idea behind the financial ponzi (pyramid) scheme is that there are four types of people, as shown in Figure 16.3. Insiders with inside information cannot play. Risk-adverse individuals hold indices. Information traders use statistics to determine when to buy or sell. Three classes of noise traders provide liquidity and price changes used by the information traders to make money. Noise traders input information and make decisions that change prices. Information traders anticipate this behavior to make money. Predicting prices in this context is a problem of predicting human behavior.

If we accept this hypothesis, what we want to predict is the effects of the noise traders behavior. To predict the effects of their behavior, we need access to the same stimuli that they have. For many traders, the most common stimulus is some form of technical analysis. Both noise traders and information traders may use technical analysis. What distinguishes them is how they use it. Information traders collect statistics on



Figure 16.3 The Financial Ponzi Scheme. Markets are created as information-based traders interact with various types of noise traders. The kinds of information used by noise traders to make decisions

leads to changes in prices. these changes are anticipated by information-based traders and applied with profitable results.

historical relationships between technical indicators and future market conditions. They use this information to trade regardless of what they feel. Noise traders, to various degrees, are distinguished by "rules of thumb" that are inconsistently applied.

Lanes Stochastic

As an example of this, consider a popular indicator, Lanes Stochastic. {Note: Contrary to its use in mathematics, the "stochastic" in Lanes Stochastic has nothing to do with random numbers.) The formula for a stochastic is:

S(p) = (C - Lp) / (Hp - Lp) x 100

where

S(p) is the stochastic with period p (p > 1) is the closing price at the current time

Lp is the lowest low over p prior periods (between now t, and t - p + 1) Hp is the highest high over p prior periods (between now t, and t - p + l)3



There are many ways in which to use stochastics, and the rules for how to use them can become complex. Typically, they are combined in different ways to create a series of rules for when to go long, short, or exit a market. Noise traders often use simple rules of thumb to decide when to buy or sell based on stochastics. Furthermore, these rules of thumb are often violated in favor of "gut" feel, panic, or greed.

The following example illustrates how an information trader uses stochastics. It constructs a system for estimating the minimum, maximum, and expected return given the 5-period and 14-period stochastics for the S&P 500. The system uses several of these simplistic statistically based estimators as input to a network that decides the degree to which it can believe any one of them. Each of the estimators uses statistics from a different overlapping window of data or a different type of indicator. In this way, each estimator becomes specialized to a particular market. By deciding which estimator to believe, the network is deciding the type of the current market.

The inputs to this network may consist of technical or broad market indicators or other data such as the prediction error for each of the specialized feature extractors. The market selector network is a soft-max* backpropagation network trained on a 1-of-N code to select the best prediction of the future price change. This network learns the probability that a particular output is correct. These probabilities are multiplied by the market estimates to produce a single final output. The final output is used in a rule-based system to determine how to trade based on the estimated risk-reward and investors preferences.

The first step in building our stochastic trader is to collect some statistics. Table 16.3 shows statistics for the S&P 500 for March through December 1991 and 1992. First, compute the 5-day forward change in the S&P 500 for each trading day for the years 1991 through 1992. Compute the 5-day and 14-day stochastics for this same period. Divide each of the ranges for the 5-day and 14-day stochastics into uniform-size bins, five in this instance. For each day in the target range March through De-cember 1991, determine which "bin" that day belongs to based on the value of the 5-day and 14-day stochastics. Update the minimum, maximum, average, and count for that bin. When done, you will have a table similar to Table 16.3. The center point of each bin is used to label the rows and columns in the table.

The period over which data is collected must be sufficiently long to ensure that most of the bins have data in them. Using more bins requires more data. Fewer bins require less data. It may be necessary to experiment somewhat to find a balance between the number of bins and the amount of data.

*With regard to the nonlinear transformation occurring in a neural nets cells (neurons), a "hard-limit" transformation forces the cells output to be within a fixed range, typically from -1 to +1. A soft-max transformation has no fixed bounds, but makes large deviations from a mean value very difficult to attain. (Editor)

For many years, backpropagation was the most popular training algorithm applied to Perceptron based neural networks. Its mathematics is based on error reduction by gradient descent and is relatively simple to code. (Editor)



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