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I/ take no trades in lo&back period if(clt[cbl c 910302) ( eqclsLcb1 = 0.0; continue; )
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To solve for the best parameters, brute force optimization would require that 2,041 tests be performed; in TradeStation, that works out to about 56 minutes of computing time, extrapolating from the earlier illustration in which a small subset of the current solution space was examined. Only 1 minute of running time was required by the genetic optimizer; in an attempt to put it at a significant disadvantage, it was prematurely stopped after performing only 133 tests. The output from the genetic optimizer appears in Table 3-2. In this table, PI represents the period of the faster moving average, P2 the period of the slower moving average, NETthe total net profit, NETLNG the net profit for long positions, NETSiS the net profit for short positions, PFAC the profit factor, ROA% the annualized rehm on account, DRAW the maximum drawdown, TRDS the number of trades taken by the system, WIN% the percentage of winning trades, AVGT the profit or loss resulting from the average trade, and FZTthe fitness of the solution (which, in this instance, is merely the total net p&it). As with the brute force data in Table 3-1, the genetic data have been sorted by net profit (fitness) and only the 25 best solutions were presented.
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Comparison of the brute force and genetic optimization results (Tables 3- 1 and 3-2, respectively) reveals that the genetic optimizer isolated a solution with a greater net profit ($172,725) than did the brute force optimizer ($145,125). This is no surprise since a larger solution space, not decimated by increments, was explored. The surprise is that the better solution was found so quickly, despite the handicap of a prematurely stopped evolutionary process. Results like these demonstrate the incredible effectiveness of genetic optimization.
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Optimizers based on annealing mimic the thermodynamic process by which liquids freeze and metals anneal. Starting out at a high temperature, the atoms of a liquid or molten metal bounce rapidly about in a random fashion. Slowly cooled, they mange themselves into an orderly configuration-a crystal-that represents a minimal energy state for the system. Simulated in software, this thermodynamic process readily solves large-scale optimization problems. As with genetic opimization, optimization by simulared annealing is a very powerful Stochastic technique, modeled upon a natural phenomenon, that can find globally optimal solutions and handle ill-behaved fitness functions. Simulated annealing has effectively solved significant combinatorial problems, including
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the famous traveling salesman problem, and the problem of how best to arrange the millions of circuit elements found on modem integrated circuit chips, such as those that power computers. Methods based on simulated annealing should not be construed as limited to combinatorial optimization; they can readily be adapted to the optimization of real-valued parameters. Consequently, optimizers based on simulated annealing are applicable to a wide variety of problems, including those faced by traders. Since genetic optimizers perform so well, we have experienced little need to explore optimizers based on simulated annealing. In addition, there have been a few reports suggesting that, in many cases, annealing algorithms do not perform as well as genetic algorithms. Because of these reasons, we have not provided examples of simulated annealing and have little more to say about the method.
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Analytic Optimizers
Analysis (as in real analysis or complex analysis ) is an extension of classical college calculus. Analytic optimizers involve the well-developed machinery of
analysis, specifically differential calculus and the study of analytic functions, in the solution of practical problems. In some instances, analytic methods can yield a direct (noniterative) solution to an optimization problem. This happens to be the case for multiple regression, where solutions can be obtained with a few matrix calculations. In multiple regression, the goal is to find a set of regression weights that minimize the sum of the squared prediction errors. In other cases, iterative techniques must be used. The connection weights in a neural network, for example, cannot be directly determined. They must be estimated using an iterative procedure, such as back-propagation. Many iterative techniques used to solve multivariate optimization problems (those involving several variables or parameters) employ some variation on the theme of steepest ascent. In its most basic form, optimization by steepest ascent works as follows: A point in the domain of the fitness function (that is, a set of parameter values) is chosen by some means. The gradient vector at that point is evaluated by computing the derivatives of the fitness function with respect to each of the variables or parameters; this defines the direction in ndimensional parameter space for which a fixed amount of movement will produce the greatest increase in fitness. A small step is taken up the hill in fitness space, along the direction of the gradient. The gradient is then recomputed at this new point, and another, perhaps smaller, step is taken. The process is repeated until convergence occurs. A real-world implementation of steepest ascent optimization has to specify how the step size will be determined at each iteration, and how the direction defined by the gradient will be adjusted for better overall convergence of the optimization process. Naive implementations assume that there is an analytic fitness
surface (one that can be approximated locally by a convergent power series) having hills that must be climbed. More sophisticated implementations go further, commonly assuming that the fitness function can be well approximated locally by a quadratic form. If a fitness function satisfies this assumption, then much faster convergence to a solution can be achieved. However, when the fitness surface has many irregularly shaped bills and valleys, quadratic forms often fail to provide a good approximation. In such cases, the more sophisticated methods break down entirely or their performance seriously degrades. Worse than degraded performance is the problem of local solutions. Almost all analytic methods, whether elementary or sophisticated, are easily trapped by local maxima: they generally fail to locate the globally best solution when there are many hills and valleys in the fitness surface. Least-squares, neural network predictive modeling gives rise to fitness surfaces that, although clearly analytic, are full of bumps, troughs, and other irregularities that lead standard analytic techniques (including back-propagation, a variant on steepest ascent) astray. Local maxima and other hazards that accompany such fitness surfaces can, however, be sidestepped by cleverly marrying a genetic algorithm with an analytic one. For titness surfaces amenable to analytic optimization, such a combined algorithm can provide the best of both worlds: fast, accurate solutions that are also likely to be globally optimal. Some fitness surfaces are simply not amenable to analytic optimization. More specifically, analytic methods cannot be used when the fitness surface has flat areas or discontinuities in the region of parameter space where a solution is to be sought. Flat areas imply null gradients, hence the absence of a preferred direction in which to take a step. At points of discontinuity, the gradient is not defined; again, a stepping direction cannot be determined. Even if a method does not explicitly use gradient information, such information is employed implicitly by the optimization algorithm. Unfortunately, many fitness functions of interest to traders-including, for instance, all functions that involve net profit, drawdown, percentage of winning trades, risk-to-reward ratios, and other like items-have plateaus and discontinuities. They are, therefore, not tractable using analytic methods. Although the discussion has centered on the maximization of fitness, everything said applies as well to the minimization of cost. Any maximization technique can be used for minimization, and vice versa: Multiply a fitness function by - 1 to obtain an equivalent cost function; multiply a cost function by - 1 and a fitness function is the result. If a minimization algorithm takes your fancy, but a maximization is required, use this trick to avoid having to recode the optimization algorithm. Linear Programming
The techniques of linear programming are designed for optimization problems involving linear cost or fitness functions, and linear constraints on the parameters
or input variables. Linear programming is typically used to solve resource allocation problems. In the world of trading, one use of linear programming might be to allocate capital among a set of investments to maximize net profit. If riskadjusted profit is to be optimized, linear programming methods cannot be used: Risk-adjusted profit is not a linear function of the amount of capital allocated to each of the investments; in such instances, other techniques (e.g., genetic algorithms) must be employed. Linear programming methods are rarely useful in the development of trading systems. They are mentioned here only to inform readers of their existence. HOW TO FAIL WITH OPTIMIZATION Most traders do not seek failure, at least not consciously. However, knowledge of the way failure is achieved can be of great benefit when seeking to avoid it. Failure with an optimizer is easy to accomplish by following a few key rules. First, be sure to use a small data sample when running sirindations: The smaller the sample, the greater the likelihood it will poorly represent the data on which the trading model will actually be traded. Next, make sure the trading system has a large number of parameters and rules to optimize: For a given data sample, the greater the number of variables that must be estimated, the easier it will be to obtain spurious results. It would also be beneficial to employ only a single sample on which to run tests; annoying out-of-sample data sets have no place in the rose-colored world of the ardent loser. Finally, do avoid the headache of inferential statistics. Follow these rules and failure is guaranteed. What shape will failure take Most likely, system performance will look great in tests, but terrible in real-time trading. Neural network developers call this phenomenon poor generalization ; traders are acquainted with it through the experience of margin calls and a serious loss of trading capital. One consequence of such a failure-laden outcome is the formation of a popular misconception: that all optimization is dangerous and to be feared. In actual fact, optimizers are not dangerous and not all optimization should be feared. Only bad optimization is dangerous and frightening. Optimization of large parameter sets on small samples, without out-of-sample tests or inferential statistics, is simply a bad practice that invites unhappy results for a variety of reasons. Small Samples Consider the impact of small samples on the optimization process. Small samples of market data are unlikely to be representative of the universe from which they are drawn: consequently, they will probably differ significantly from other samples obtained from the same universe. Applied to a small development sample, an optimizer will faithfully discover the best possible solution. The best solution for
the development sample, however, may turn out to be a dreadful solution for the later sample on which genuine trades will be taken. Failure ensues, not because optimization has found a bad solution, but because it has found a good solution to the wrong problem! Optimization on inadequate samples is also good at spawning solutions that represent only mathematical artifact. As the number of data points declines to the number of free (adjustable) parameters, most models (trading, regression, or otherwise) will attain a perfect tit to even random data. The principle involved is the same one responsible for the fact that a line, which is a two-parameter model, can always be drawn through any two distinct points, but cannot always be made to intersect three arbitrary points. In statistics, this is known as the degrees-of-freedom issue; there are as many degrees of freedom as there are data points beyond that which can be fitted perfectly for purely mathematical reasons. Even when there are enough data points to avoid a totally artifact-determined solution, some part of the model fitness obtained through optimization will be of an artifact-determined nature, a by-product of the process. For multiple regression models, a formula is available that can be used to estimate how much shrinkage would occur in the multiple correlation coefficient (a measure of model fitness) if the artifact-determined component were removed. The shrinkage correction formula, which shows the relationship between the number of parameters (regression coefficients) being optimized, sample size, and decreased levels of apparent fitness (correlation) in tests on new samples, is shown below in FORTRAN-style notation:
In this equation, N represents the number of data points, P the number of model parameters, R the multiple correlation coefficient determined for the sample by the regression (optimization) procedure, and RC the shrinkage-corrected multiple correlation coefficient. The inverse formula, one that estimates the optimizationinflated correlation (R) given the true correlation (RfJ existing in the population from which the data were sampled, appears below:
These formulas, although legitimate only for linear regression, are not bad for estimating how well a fully trained neural network model-which is nothing more than a particular kind of nonhnezu regression-will generalize. When working with neural networks, let P represent the total number of connection weights in the model. In addition, make sure that simple correlations are used when working with these formulas; if a neural network or regression package reports the squared multiple correlation, take the square root.
Large Parameter Sets An excessive number of free parameters or rules will impact an optimization effort in a manner similar to an insufficient number of data points. As the number of elements undergoing optimization rises, a model s ability to capitalize on idiosyncrasies in the development sample increases along with the proportion of the model s fitness that can be attributed to mathematical artifact. The result of optimizing a large number of variables-whether rules, parameters, or both-will be a model that performs well on the development data, but poorly on out-of-sample test data and in actual trading. It is not the absolute number of free parameters that should be of concern, but the number of parameters relative to the number of data points. The shrinkage formula discussed in the context of small samples is also heuristically relevant here: It illustrates how the relationship between the number of data points and the number of parameters affects the outcome. When there are too many parameters, given the number of data points, mathematical artifacts and capitalization on chance (curve-fitting, in the bad sense) become reasons for failure. No Verification One of the better ways to get into trouble is by failing to verify model performance using out-of-sample tests or inferential statistics. Without such tests, the spurious solutions resulting from small samples and large parameter sets, not to mention other less obvious causes, will go undetected. The trading system that appears to be ideal on the development sample will be put on-line, and devastating losses will follow. Developing systems without subjecting them to out-of-sample and statistical tests is like flying blind, without a safety belt, in an uninspected aircraft. HOW TO SUCCEED WITH OPTIMIZATION Four steps can be taken to avoid failure and increase the odds of achieving successful optimization. As a first step, optimize on the largest possible representative sample and make sure many simulated trades are available for analysis. The second step is to keep the number of free parameters or rules small, especially in relation to sample size. A third step involves running tests on out-of-sample data, that is, data not used or even seen during the optimization process. As a fourth and final step, it may be worthwhile to statistically assess the results. Large, Representative Samples As suggested earlier, failure is often a consequence of presenting an optimizer with the wrong problem to solve. Conversely, success is likely when the optimizer is
presented with the right problem. The conclusion is that trading models should be optimized on data from the near future, the data that will actually be traded; do that and watch the profits roll in. The catch is where to find tomorrow s data today. Since the future has not yet happened, it is impossible to present the optimizer with precisely the problem that needs to be solved. Consequently, it is necessary to attempt the next-best alternative: to present the optimizer with a broader problem, the solution to which should be as applicable as possible to the actual, but impossible-to-solve, problem. One way to accomplish this is with a data sample that, even though not drawn from the future, embodies many characteristics that might appear in future samples. Such a data sample should include bull and bear markets, trending and nontrending periods, and even crashes. In addition, the data in the sample should be as recent as possible so that it will reflect current patterns of market behavior. This is what is meant by a representative sample. As well as representative, the sample should be large. Large samples make it harder for optimizers to uncover spurious or artifact-determined solutions. Shrinkage, the expected decline in performance on unoptimized data, is reduced when large samples are employed in the optimization process. Sometimes, however, a trade-off must be made between the sample s size and the extent to which it is representative. As one goes farther back in history to bolster a sample, the data may become less representative of current market conditions. In some instances, there is a clear transition point beyond which the data become much less representative: For example, the S&P 500 futures began trading in 1983, effecting a structural change in the general market. Trade-offs become much less of an issue when working with intraday data on short time frames, where tens of thousands or even hundreds of thousands of bars of data can be gathered without going back beyond the recent past. Finally, when running simulations and optimizations, pay attention to the number of trades a system takes. Like large data samples, it is highly desirable that simulations and tests involve numerous trades. Chance or artifact can easily be responsible for any profits produced by a system that takes only a few trades, regardless of the number of data points used in the test! Few Rules and Parameters To achieve success, limit the number of free rules and parameters, especially when working with small data samples. For a given sample size, the fewer the rules or parameters to optimize, the greater the likelihood that a trading system will maintain its performance in out-of-sample tests and real-time trading. Although several dozen parameters may be acceptable when working with several thousand trades taken on 100,000 l-minute bars (about 1 year for the S&P 500 futures), even two or three parameters may be excessive when developing a system using a few years of end-of-day data. If a particular model requires many parameters, then
significant effort should be put into assembling a mammoth sample (the legendary Gann supposedly went back over 1,000 years in his study of wheat prices). An alternative that sometimes works is optimizing a trading model on a whole portfolio, using the same rules and parameters across all markets-a technique used extensively in this book. Verification of Results
After optimizing the rules and parameters of a trading system to obtain good behavior on the development or in-sample data, but before risking any real money, it is essential to verify the system s performance in some manner. Verification of system performance is important because it gives the trader a chance to veto faime and embrace success: Systems that fail the test of verification can be discarded, ones that pass can be traded with confidence. Verification is the single most critical step on the road to success with optimization or, in fact, with any other method of discovering a trading model that really works. To ensure success, verify any trading solution using out-of-sample tests or inferential statistics, preferably both. Discard any solution that fails to be profitable in an out-of-sample test: It is likely to fail again when the rubber hits the road. Compute inferential statistics on all tests, both in-sample and out-of-sample. These statistics reveal the probability that the performance observed in a sample reflects something real that will hold up in other samples and in real-time trading. Inferential statistics work by making probability inferences based on the distribution of profitability in a system s trades or returns. Be sure to use statistics that are corrected for multiple tests when analyzing in-sample optimization results. Out-of-sample tests should be analyzed with standard, uncorrected statistics. Such statistics appear in some of the performance reports that are displayed in the chapter on simulators. The use of statistics to evaluate trading systems is covered in depth in the following chapter. Develop a working knowledge of statistics; it will make you a better trader. Some suggest checking a model for sensitivity to small changes in parameter values. A model highly tolerant of such changes is more robust than a model not as tolerant, it is said. Do not pay too much attention to these claims. In truth, parameter tolerance cannot be relied upon as a gauge of model robustness. Many extremely robust models are highly sensitive to the values of certain parameters. The only true arbiters of system robustness are statistical and, especially, out-ofsample tests. ALTERNATIVES TO TRADITIONAL OPTIMIZATION
There are two major alternatives to traditional optimization: walk-forward optimization and self-adaptive systems. Both of these techniques have the advantage that any tests carried out are, from start to finish, effectively out-of-sample.
Examine the performance data, run some inferential statistics, plot the equity curve, and the system is ready to be traded. Everything is clean and mathematically unimpeachable. Corrections for shrinkage or multiple tests, worries over excessive curve-fitting, and many of the other concerns that plague traditional optimization methodologies can be forgotten. Moreover, with today s modem computer technology, walk-forward and self-adaptive models are practical and not even difficult to implement. The principle behind walk-forward optimization (also known as walk-forward testing) is to emulate the steps involved in actually trading a system that requires periodic optimization. It works like this: Optimize the system on the data points 1 through M. Then simulate trading on data points M + I through M + K. Reoptimize the system on data points K + 1 through K + M. Then simulate trading on points (K + M) + 1 through (K + M) + K. Advance through the data series in this fashion until no more data points are left to analyze. As should be evident, the system is optimized on a sample of historical data and then traded. After some period of time, the system is reoptimized and trading is resumed. The sequence of events guarantees that the data on which trades take place is always in the future relative to the optimization process; all trades occur on what is, essentially, out-ofsample data. In walk-forward testing, M is the look-back or optimization window and K the reoptimization interval. Self-adaptive systems work in a similar manner, except that the optimization or adaptive process is part of the system, rather than the test environment. As each bar or data point comes along, a self-adaptive system updates its internal state (its parameters or rules) and then makes decisions concerning actions required on the next bar or data point. When the next bar arrives, the decided-upon actions are carried out and the process repeats. Internal updates, which are how the system learns about or adapts to the market, need not occur on every single bar. They can be performed at fixed intervals or whenever deemed necessary by the model. The trader planning to work with self-adapting systems will need a powerful, component-based development platform that employs a strong language, such as Cf f, Object Pascal, or Visual Basic, and that provides good access to thirdparty libraries and software components. Components are designed to be incorporated into user-written software, including the special-purpose software that constitutes an adaptive system. The more components that are available, the less work there is to do. At the very least, a trader venturing into self-adaptive systems should have at hand genetic optimizer and trading simulator components that can be easily embedded within a trading model. Adaptive systems will be demonstrated in later chapters, showing how this technique works in practice. There is no doubt that walk-forward optimization and adaptive systems will become more popular over time as the markets become more efficient and difficult to trade, and as commercial software packages become available that place these techniques within reach of the average trader.
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