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Another issue found in trading system development is optimization, i.e., improving the performance of a system by adjusting its parameters until the system performs its best on what the developer hopes is a representative sample. When the system fails to hold up in the future (or on out-of-sample data), the optimization process is pejoratively called curve-fitting. However, there is good curve-fitting and bad curve-fitting. Good curve-fitting is when a model can be fit to the entire relevant population (or, at least, to a sufficiently large sample thereof), suggesting that valid characteristics of the entire population have been captured in the model. Bad curve-@zing occurs when the system only fits chance characteristics, those that are not necessarily representative of the population from which the sample was drawn. Developers are correct to fear bad curve-fitting, i.e., the situation in which parameter values are adapted to the particular sample on which the system was optimized, not to the population as a whole. If the sample was small or was not representative of the population from which it was drawn, it is likely that the system will look good on that one sample but fail miserably on another, or worse, lose money in real-time trading. However, as the sample gets larger, the chance of this happening becomes smaller: Bad curve-fitting declines and good curve-fitting increases. All the statistics discussed reflect this, even the ones that specifically concern optimization. It is true that the more combinations of things optimized, the greater the likelihood good performance may be obtained by chance alone.
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However, if the statistical result was sufficiently good, or the sample on which it was based large enough to reduce the probability that the outcome was due to chance, the result might still be very real and significant, even if many parameters were optimized. Some have argued that size does not matter, i.e., that sample size and the number of trades studied have little or nothing to do with the risk of overoptimization, and that a large sample does not mitigate curve-fitting. This is patently untrue, both intuitively and mathematically. Anyone would have less confidence in a system that took only three or four trades over a lo-year period than in one that took over 1,000 reasonably profitable trades. Think of a linear regression model in which a straight line is being fit to a number of points. If there are only two points, it is easy to fit the line perfectly every time, regardless of where the points are located. If there are three points, it is harder. If there is a scatterplot of points, it is going to be harder still, unless those points reveal some real characteristic of the population that involves a linear relationship. The linear regression example demonstrates that bad curve-fitting does become more difficult as the sample size gets larger. Consider two trading systems: One system had a profit per trade of $100, it took 2 trades, and the standard deviation was $100 per trade: the other system took 1,000 trades, with similar means and standard deviations. When evaluated statistically, the system with 1,000 trades will be a lot more statistically significant than the one with the 2 trades. In multiple linear regression models, as the number of regression parameters (beta weights) being estimated is increased relative to the sample size, the amount of curve-fitting increases and statistical significance lessens for the same degree of model fit. In other words, the greater the degree of curve-fitting, the harder it is to get statistical significance. The exception is if the improvement in fit when adding regressors is sufficient to compensate for the loss in significance due to the additional parameters being estimated. In fact, an estimate of shrinkage (the degree to which the multiple correlation can be expected to shrink when computed using out-of-sample data) can even be calculated given sample size and number of regressors: Shrinkage increases with regressors and decreases with sample size. In short, there is mathematical evidence that curve-fitting to chance characteristics of a sample, with concomitant poor generalization, is more likely if the sample is small relative to the number of parameters being fit by the model. In fact, as n (the sample size) goes to infinity, the probability that the curve-fitting (achieved by optimizing a set of parameters) is nonrepresentative of the population goes to zero. The larger the number of parameters being optimized, the larger the sample required. In the language of statistics, the parameters being estimated use up the available degrees of freedom. All this leads to the conclusion that the larger the sample, the more likely its curves are representative of characteristics of the market as a whole. A small
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sample almost certainly will be nonrepresentative of the market: It is unlikely that its curves will reflect those of the entire market that persist over time. Any model built using a small sample will be capitalizing purely on the chance of sampling. Whether curve-fitting is good or bad depends on if it was done to chance or to real market patterns, which, in turn, largely depends on the size and representativeness of the sample. Statistics are useful because they make it possible to take curve-fitting into account when evaluating a system. When dealing with neural networks, concerns about overtraining or generalization are tantamount to concerns about bad curve-fitting. If the sample is large enough and representative, curve-fitting some real characteristic of the market is more likely, which may be good because the model should fit the market. On the other hand, if the sample is small, the model will almost certainly be fit to peculiar characteristics of the sample and not to the behavior of the market generally. In neural networks, the concern about whether the neural network will generalize is the same as the concern about whether other kinds of systems will hold up in the future. To a great extent, generalization depends on the size of the sample on which the neural network is trained. The larger the sample, or the smaller the number of connection weights (parameters) being estimated, the more likely the network will generalize. Again, this can be demonstrated mathematically by examining simple cases. As was the case with regression, au estimate of shrinkage (the opposite of generalization) may be computed when developing neural networks. In a very real sense, a neural network is actually a multiple regression, albeit, nonlinear, and the correlation of a neural net s output with the target may be construed as a multiple correlation coefficient. The multiple correlation obtained between a net s output and the target may be corrected for shrinkage to obtain some idea of how the net might perform on out-of-sample data. Such shrinkage-corrected multiple correlations should routinely be computed as a means of determining whether a network has merely curve-fit the data or has discovered something useful. The formula for correcting a multiple correlation for shrinkage is as follows:
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A FORTRAN-style expression was used for reasons of typsetting. In this formula, SQRT represents the square root operator; N is the number of data points or, in the case of neural networks, facts; P is the number of regression coefticients or, in the case of neural networks, connection weights; R represents the uncorrected multiple correlation; and RC is the multiple correlation corrected for shrinkage. Although this formula is strictly applicable only to linear multiple regression (for which it was originally developed), it works well with neural networks and may be used to estimate how much performance was inflated on the in-sample data due to curve-fitting. The formula expresses a relationship between sample size, number of parameters, and deterioration of results. The
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statistical correction embodied in the shrinkage formula is used in the chapter on neural network entry models.
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