-world objects are more likely to be described as fractional, orhaving a fractal dimension. The classic example of this is the algebra of coastline dimen-
° An excellent discussion of this topic can be found in Edgar Peters, Chaos and Order in the Capital MarketsJohn Wiley & Sons, New York, 1991). The coastline example is originally credited to Benoit Mandelbrot.FRACTALS AND CHWS 49)
 
sion. We will see that the questions “How long is the coastline?” and “How far did pricesmove?” are very similar.
The answer to both these questions is “That depends on how it is measured.” Considerthe problem of measuring the coastline of Australia using a large wall map. If we take a 12-inch ruler, we might find that the coastline is about 10 feet (perhaps 10,000 miles accord-ing to the scale). Had we taken a slightly smaller ruler we would have been more accurateand perhaps have found the coastline to be 11,000 miles; even a smaller ruler would havefollowed the contours better and found 12,000 miles of coast. As the ruler gets smaller thecoastline appears to get longer. If we had an infinitely smaller ruler, the coast would be infi-nitely long. There is really no correct answer to the question, “How long is the coastline?”
Fractal dimension is the degree of roughness or irregularity of a structure or system.In many chaotic systems, there is a constant fractal dimension; that is, the interval used formeasuring will have a predictable impact on the resulting values in a manner similar to anormal distribution. Therefore, using a 12-inch ruler, we measure the coastline and get10,000 miles:
24-inch naler 5,000-mile coastline- 12-inch naler 10,000-mile coastline6-inch ruler 20,000-mile coastline
Note that the large 24-inch ruler returns a value that is actually smaller than what webelieve is a reasonable answer. This is because, when you place a long ruler from one pointto another on the map, it cuts across part of the land mass.
Using Fractal Efficiency
In Chapter 17 (“Adaptive Techniques”), there is a discussion of Kaufman’s efficiency ratio.This ratio is formed by dividing the net change in price movement over 7 periods by thesum of all component moves, taken as positive numbers, over the same 7 periods. If theratio approaches the value 1.0, then the movement is smooth (not chaotic); if the ratioapproaches 0, then there is great inefficiency or chaos. This same measurement has morerecently been called fractal efficiency.
Fractal efficiency measures the amount of chaotic movement in prices; this can also beconsidered similar to market noise. In Chapter 17, Kaufman related this to trending andnontrending patterns, when the ratio approached 1.0 and 0, respectively. While each mar-ket has its unique underlying level of noise, the measurement of fractal efficiency is con-sistent over all markets. Markets may vary in volatility, although their chaotic behavior istechnically the same; therefore, the characteristics of one market may be compared withothers by matching both the fractal efficiency and volatility,
Interpreting fractal efficiency as noise allows other trading rules to develop. For exam-ple, a market with less noise should be entered quickly using a trending system, while itmay be best to wait for a better price if the market is classified as having high noise. A noisymarket is one that continues to change direction, while an efficient market is smooth.These characteristics-are important in choosing trading rules and in turning a theoreticalmodel into a profitable trading system.
Chaotic Patterns and Market Behavior
Chaotic patterns are easy to imagine in the behavior of prices, but very difficult to measure.There would be no problem in predicting price direction if every participant reacted in thesame way to the same event, much the way a single planet would smoothly orbit a singlesun. In the real world nothing is quite as simple. Consider the pattern of prices repre-sented as planets that are affected equally by two events, El and E2. We will get a wobblypattern whenever the planet passes across the midpoint where one attractor is stronger492 ADVANCED TECHNIQUES
 
than the other, shown in Figure 20-7a. At point a, the object is most affected by the near-est attractor, E1, but as it circles it becomes closer to E2 and tries to form an orbit aroundit. The possible patterns are too complex and they vary based on the distance between Eland E2 and the size of E2 compared with E1. If attractor E2 is much larger than E1, therewill simply be a distortion in the orbit around E2; if El and E2 are the same size, objects aand & will switch orbits, forming figure eights.
As complex as these patterns might get, they are simple when compared with reality.Each day brings events of various importance into the market, acting as attractors. Eachattractor has an initial importance that loses value over time. To make matters worse, wecannot predict when a new attractor, or news event, will appear. This makes the chaoticpattern very similar to raindrops falling on a pond. Each new drop, equivalent to a newevent, hits at an unpredictable time and place, with variable size, and forms circular rip-ples. These ripples dissipate as they get further away from the point of contact in the sameway that the importance of an event fades away over time. The interesting aspect of theraindrop analogy is that, while we cannot predict where the next raindrop will fall, once ithas landed we can completely determine its effects—until the next drop hits. This isremarkably similar to the market. Less often there is a significant event, a price shock, thatoverwhelms the smaller events, the market noise, for a short time.
NEURAL NETWORKS"® '
Another area of analysis that has continued to grow rapidly is neural networks. During thepast 5 years there has been a deluge of material on the advantages of using neural net-
*° Parts of this section are based on Perry Kaufman, Smarter Trading (McGraw-Hill, 1995, pp 164-171).
 
FIGURE 20-7 (a) Equal attractors cause a symmetric pattern that switches between El andE2. (b) Very unequal attractors show only a smail disturbance in a regular orbit.
 
(-Qy )
    
Greater GreaterGreater attraction Greater attractionattraction to E2 attraction toE2to E1 to E1
Size of orbit shows reflective effect of events,
{a) (6)NEURAL NETWORK 493
 
works as a tool for uncovering market relationships and finding better systems. Most ofwhat has been said is true. This technique offers unparalleled power for discovering non-linear relationships between any combination of fundamental information, technical indi-cators, and price data. Its disadvantage is that it is potentially so powerful that, withoutproper control, it will find relationships that exist only by chance."
Although the idea and words for the computerized neural network are based on thebiological functions of the human brain, an artificial neural network is not a model of abrain, nor does it learn in the human sense. It is simply very good at finding patterns,whether they are continuous or appear at different times.
The operation of an artificial neural network can be thought of as a feedback process,similar to Pavlov's approach to training a dog:
1. A bell rings.
2. The dog runs to one of three bowls.
3. If right, the dog gets a treat; if wrong, the dog gets a shock.4, If trained, stop; if not trained, continue at step 1.
Terminology of Neural Networks
The terminology used in the computerized neural network is drawn from the human bio-logical counterpart, shown in Figure 20-8. The principal elements are:Neurons, the cells that compose the brain, process and store information.Networks, groups of neurons.Dendrites, receivers of information, passing it directly to the neurons.Axons, which come out of the neuron and allow information to pass from one neuronto another. : -
Synapses, which exist on the path between neurons and may inhibit or enhance theflow of information between neurons. They can be considered selectors.
™ Readers are referred to E. Michael Azoff, Neural Network Time Series Forecasting of Financial Markets (JohnWiley & Sons, 1994), for a thorough treatment of this subject, and Edward Gately, Neural Networks for FinancialForecasting (John Wiley & Sons, 1996), for a more introductory approach.
 
FIGURE 20-8 A biological neural network. Information is received through dendrites andpassed to a neuron for storage. Data are shared by other cells by moving through the outputconnector, called an axon.A synapse may be located on the path between some individual neu-rons or neural networks; it selects the relevant data by inhibiting or enhancing the flow.
 
Neuron
{information =Storage Cell) To OtherLb Neurons
(a
mY
     
 
= Axon Synapse. (Output (Optionalrenites ) Connector) Inhibitor andeceivers) Enhancer)
 
Source: Perry Kaufman, Smarter Trading (McGraw-Hill, 1995, p. 165).494 ADVANCED TECHNIQUES
 
Artificial Neural Networks
Using essentially the same structure as a biological neural network, the computerized, orartificial neural network (ANN), can generate a decision on the direction of the stock mar-ket. It relies heavily on the synapses, which are interpreted as weighting factors in thisprocess. To achieve its result, it will also combine inputs that interact with one another intoa single, more complex piece of data using layers of neurons, as shown in Figure 20-9, aclassic 3-layer neural network.
To be most efficient, the inputs to the ANN should be those factors considered mostrelevant to the direction of stock prices. The five items chosen here were the Gross Domes-tic Product, unemployment, inventories, interest rates, and the value of the U.S. dollar—all readily available data, Each of these items is input and stored in separate neurons.Changing values may have a positive or negative effect on the final output, which is thedirection of stocks. An improved GDP lower unemployment, higher inventories, and alower U.S. dollar are all good signs for the economy and result in the possibility of higherinterest rates (defensive action by the Federal Reserve to prevent inflation) and a likelydecline in stock prices. Interest rates themselves have a direct effect on stock prices,improving profits as rates decline.
Each neuron in layer 1, which receives the data from the dendrites, is connected to asecond layer of neurons through a synapse. Each synapse can be used to filter or enhanceinformation by assigning a weighting factor. For example, if changes in unemploymenthave a greater impact than changes in the GDP then it may receive a weight of 1.5 com-pared with .9 for the GDP If very small changes in any of the data is considered unimpor-tant to the result, then the synapse: can act as a threshold and only allows data to pass if itexceeds a minimum value,
The second layer of neurons is used to combine initial data into significant subgroups.In Figure 20-9, the GDP, unemployment, and inventories are combined into a single item
 
FIGURE 20-9 A 3-layer artificial neural network to determine the direction of stock prices.
 
Inputs ~ Data Selection Combined Selection Combined ~ OutputStored in by Valves by Valves result \level 1 Weighting Storedin Weighting Stored inNeurons " Level 2 #2 Level 3
Neurons Neurons
   
   
InterestRates.
   
MarketReaction
= Output
     
  
AnticipatedForward FXRates
    
 
Source: Perry Kaufman, Smarter Trading (McGraw-Hill, 1995, p. 168).NEURAL NETWORKS 495
 
called Domestic Economic Health. The synapses allow each element to be assigned a spe-cific level of importance using a weighting factor. Also note that this neuron is altered byanticipated interest rates, which is the result of data flowing to another neuron in layer 2.Finally, the three neurons in layer 2 are combined according to importance, giving the netstock market reaction to the input data.
The human brain works in a way very similar to the artificial neural network shown inFigure 20-9. It groups and weighs the data, combining them into subgroups and finallyproducing a decision. The human process of weighting the data is complex and not nec-essarily transparent; that is, we may. never know the precise flow of data, how many layersexist, and how weights are assigned and reassigned before the final decision.
Acomputerized neural network is not as complex. Because it cannot know whether itsanswer is correct, you must tell it. This is done by giving the computer the historic data andthe corresponding answers. By giving the artificial neural network a long history of infor-mation, it can determine, using feedback, the weighting factors that would have given thecorrect results most often. The more history that is given to the computer, the more likelyit will find a robust answer. Figure 20-10 shows the results of using five different inputs topredict the direction of the stock market. Two of the inputs, the Academy Award Winnersand the Number of NASA Launches are not likely to be useful in the long term, but mayappear to provide valuable information for short intervals. By using enough comparisons,the weighting factors are found to show that unemployment has a strong negative effect onprices, the GDP a strong positive effect, and inventories have a weak positive effect. Theother items had no consistent predictive ability and received a weight of zero. This feed-back process is called training.
Selecting and Preprocessing the Inputs
There is considerable debate over the inputs needed to find a successful solution using aneural network; however, everyone agrees that the selection of inputs is critical. Theseinputs must be presented in the most direct form because, unlike an expert system, the
 
FIGURE 20-10 Learning by feedback.
    
FeedbackLoop
                    
  
Inputs Weighting CombinedFactors OutputMutated
  
Gross DomesticProductAcademy AwardWinners.
Number ofNASA SpaceLaunches
     
DomesticHealth
     
MatchesHistory,?
    
 
       
 
Source: Perry Kaufman, Smarter Trading (McGraw-Hill, 1995, p. 166).496 ADVANCED TECHNIQUES
 
neural network will not be able to change them. This step is called preprocessing. We mustdecide what factors affect the direction of stocks and the ability to anticipate thay direction,then prepare data that contains information with these qualities. For example, we maywant to know the short-term and long-term trends; the direction of interest rates and theDow Utility Index; the ratio of interest rates to gold; economic data such as the GDP andbalance of trade; technical indicators such as the RSI, stochastic, and ADX; and a 20-daymoving correlation between the U.S. stock market and other major markets. There arecountless factors that might influence the direction of stocks; the more you choose, theslower the solution and the greater the chance of a less robust model. If you choose toofew, they may not contain all of the information necessary; therefore, the preprocessingproblem requires practice. You may also construct a number of systems that produce prof.its and include their basic components as inputs to the neural network. You might createa performance series for a specific system that has only values -1, 0, and 1, representingshort, neutral, and long market positions. In that way the neural net may be used toenhance an existing trading strategy.
Selecting the Output
In a manner similar to evaluating optimization results, it is necessary to select the successcriteria. This should be based on a combination of frequency of trading, the size of theprofits per trade, a reward/risk criterion, and a frequency of profitability. According to Rug-giero,” neural networks can be used to’predict price direction, such as the percentagechange 5 days into the future; however, they are much better at predicting forward-shiftedtechnical indicators, because these tend to have smoother results.
Most technical indicators, such as trend and momentum calculations, smooth theinput data, which is most often the price. The longer the time period used in the calcula-tion, the greater the smoothing. Ruggiero suggests an output function such as
@Average((close[-5] ~ @Lowest(Close[-5],5))/(@Highest(close(-5],5) -@Lowest(close [-5],5)),5)
which is similar to a smoothed 5-period stochastic shifted forward by 5 periods. In this
notation, the bracketed value [-5] represents 5 periods into the future.\
The Training Process
At the heart of the neural network approach is the feedback process used for training,shown in Figure 20-10. To observe sound statistical practice, it is advised that only about70% of the data be used for training. As the neural net refines the weighting factors for eachof the inputs and combinations of inputs (in the layers between the input and output), itwill need more data to test these results. Of the remaining 30% of the data, 20% should bedesignated for testing. The success or failure of the method to find a solution is based onthe performance of the test data. The remaining 10% is saved for out-of-sample validation.
Weighting factors are found using a method called a genetic algorithm, discussed inthe next section. For now, we need to know that the training process begins with an ar-bitrary or random value assigned to the weighting factors of each input. As the trainingproceeds, these weighting factors are raridomly mutated, or changed, until the best com-bination is found. The genetic algorithm changes and combines weighting factors in amanner referred to as “survival of the fittest,” giving preference to the best and discarding
the worst. -
" Murray A. Ruggiero, Jr, “Build a real neural net,” Futures June 1996), the first of a series of excellent articleson this subject.NEURAL NETWORKS 497
 
Testing is completed when the results of the test data converge to a single value. Thatis, after a number of feedback loops, the test data is used with the new weighting factors.If the results are improving the process continues. If the results are improving at a veryslow rate, or have become stable at one level, the neural network process is consideredcompleted. Sometimes the results get worse, rather than better. This can be fixed by begin-ning again using new random weighting factors. If this doesn’t work, then the inputs mustbe reevaluated for relevance.
Because the genetic algorithm is a trial-and-error process, rather than an analyticapproach, the best results could be found by coincidence, rather than by cause and effect.With enough data series it is always possible that two series of events will appear related,although they are not. It is necessary to review the results to avoid simple mistakes.
ATraining Example
We would like to train a neural net to tell us whether we should buy or sell stocks. Asinputs, we select what we believe to be the five most relevant fundamental factors: GNPunemployment, inventories, U.S. dollar index, and short-term interest rates. This test doesnot use any preprocessed data, such as trends or indicators. To simplify the process, thefollowing approach is taken:
1. Each input is normalized so that it has values between +100 and —100, indicatingstrength to weakness, with 0 as neutral.
2. When the combined values of the five indicators exceed +125, we will enter a longposition; when the combined value is below -125, we will enter a short.
3. Combined values between +125 and ~-125 are considered neutral.
To show how the training process works, two events are shown in Tables 20-4 andTable 20-5 as Case 1 and Case 2. Table 20-4 is the initial state of the neural network, wherewe have chosen to set all of the weighting factors to 1.0. In actual training, the networkmight require that the sum of ail weights total to 1.0. The actual values of the normalizedinputs are shown in the columns marked “relative value,” and the correct historic answeris at the bottom, marked as “strong” and “weak” stock market reactions to these values. Forthe neural network to return the correct answers, it must produce a result greater than+125 for Case 1 and below —125 for Case 2. By assigning initial weights of 1.0 to all inputs,the value of Case 1 is +75 and the value of Case 2 is +40; both fail and testing continuessearching for better weighting factors.
 
TABLE 20-4 Two Training Cases (Initial State)
 
 
 
Case | Case 2 -
Input Relative Value Weight Net _ Relative Value Weight NetGNP - Strong 50 1.0 50. Weak ~60 1.0 ~60Unemployment Low ~25 1.0 ~25 High 40 1.0 40Inventories Low —50 1.0 -50 Neutral 15 10 ISUS. dollar Vry Strg 75 1.0 75 Neutral. 0 1.0 0Interest rates Falling ~25 1.0 725 Rising 45 1.0 45Total value 75 40Threshold- +/~125 +1-125Current response None None .Actual market reaction Strong . Weak
neSource: Perry Kaufman, Smarter Trading (McGraw-Hill, 1995)498 ADVANCED TECHNIQUES
ee
TABLE 20-5 Two Training Cases (After Mutated Weighting Factors)
 
 
 
 
Case t Case 2
Input Relative Value Weight Net Relative Value Weight NetGNP Strong 50 1.0 50 Weak 60 1.0 ~60Unemployment Low ~25 -1.0 25 High 40 -10 —40Inventories Low ~50 1.0 -50 Neutral IS 4.0 15US, dollar Vry Strg 75 1.0 75 Neutrat 0 1.0 0interest rates Falling ~25 -10 25 Rising 45 -1.0 45Total value 125 —i30Threshold +/-125 +H~125Current response None NoneActual market reaction Strong Weak
 
Source: Perry Kaufman, Smarter Trading (McGraw-Hill, 1995),
In Table 20-5, the weighting factors have undergone mutation using a genetic algo-rithm. This example attempts to use only weighting factors of +1.0, 0, and —1.0. By revers-ing the effect of unemployment and interest rates on the direction of the stock market, andby random selection of weighting factors, the results now match the ‘historic pattern ofstock movement. Using fractional values for weighting factors, and many more trainingcases, the ANN method should find the current underlying relationship between theseinputs and stock market movement.
Reducing the Number of Decision Levels and Neurons
The robustness of a neural network solution is directly related to the number of decisionlevels, the number of neurons in each level, and the total number of inputs. Fewer ele-ments produce more generalized and, therefore, more robust solutions. When there aremany decision layers and many neurons, then the inputs can be combined and recom-bined in many different ways allowing very specific patterns to be found. The more spe-cific, the greater the chance that the final solution will be overfitted, or that the neuralnetwork program will fail because it cannot converge on a satisfactory result. ‘
The trade-offs using a neural network are the same as most other optimization meth-ods. Too many inputs and combinations require lengthy testing and a greater chance ofasolution that is overfit. Too few values can produce a result that is too general and not prac-tical. It is best to begin with the most general and proceed in clear steps toward a morespecific solution. In this way you will understand the process better, ultimately_save time,and be able to stop when you have reached the practical limitations of this method.
GENETIC ALGORITHMS -
The concept of a genetic algorithm is based on Darwin’s theory of survival of the fittest. Inthe real world, a mutation with traits that improve its ability to survive will continue to pro-create. This has been applied to the process of finding the best system parameters.Although a genetic algorithm” is actually a sophisticated search method that replaces thestandard optimization, it uses a technique that parallels the survival of the fittest. It is par-ticularly valuable when the number of viables is so large that a test of all combinations isimpractical. Instead of the typical sequential search, it is a process of random seeding,
'S This technique is attributed to John H, Holland, Adaptations in Natural Language and Artificial Systems (Uni-versity of Michigan Press, 1975).GENETIC ALGORITHMS 499
 
selection, and combination to find the best set of trading rules. Standard statistical criteriaare used in the selection process to qualify the results.
Representation of a Genetic Algorithm
Using the words common to this methodology, the most basic component of a geneticalgorithm is a gene; a number of genes will comprise an individual, and a combination ofindividuals (and therefore genes) is a chromosome. Each chromosome represents a poten-tial solution, a set of trading rules in which the genes are the specific values and calculationexpressions, and these form individuals that represent rules and ultimately form a tradingstrategy. For example, Chromosome 1 might be a rule to buy on strength:
If a 10-day moving average is less than yesterday’s close and a 5-day stochastic isgreater than 50, then buy.
Chromosome 2 could be a rule that buys on weakness:
If a 20-day exponential is less than yesterday's low and a 10-day RSI is less than 50,then buy.
If we rewrite these two chromosomes in a notational form, the genes and individuals in itsstructure become more apparent:
Chromosome l: MA, 10, <, C, [0], & Stoch, 5, >, 50, 1Chromosome 2: Exp, 20, <, L, [1], & RSI, 10, <, 50, 1
Each of these chromosomes has 11 genes, any one of which can be changed. In addition,each chromosome has two individuals, separated by an “&” operator. In Table 20-6, thedescription of the genes indicates other values that can replace the current ones. Table20-6 is a way to represent the:chromosomes and individuals in a general form. It is easy tosee that each gene can be changed, and each change will represent a new trading rule. Acombination of trading rules, or chromosomes, will create a trading strategy. Before contin-uing, the following steps will be needed to use the genetic algorithm to find the best results:
1. A clear way of representing the chromosomes, or individuals
2. Acriterion to decide that one chromosome is better than another
3. A selection procedure that determines which chromosomes will survive, and inwhat manner -
4. A process for mutation (introducing new characteristics) and crossover (combininggenes) to procreate chromosomes with greater potential
nlTABLE 20-6 Functional Description of the Genes in Chromosomes | and 2
 
 
Gene Chr I Chr 2 Function' MA Exp A trend calculation (moving average, linear regression, breakout)2 10 20 —The calculation period for the trend calculation3 < < Relational operacor (<,<=,>)4 c L Price used in trend calculation ((H + L + C)/3, indexed value)5° 0 1 Reference data ({0] = current day, [1] = previous day)6 & & Method of combining individuals {and, or)7 Stoch RSI Indicator calculation8 5 10 Indicator calculation period9 > <8 Relational operator0 50 50 Comparisen value for relational operatorI
1 1 Market action (1 = Buy. —! = Self)500 ADVANCEDTECHNIQUES
 
Fitness .
Having represented the chromosome in Table 20-6, we next must define a fitness criterion,which ranks the results. Because fitness will lead to survival, it is very important to decidewhich chromosomes should be discarded and which should be used further. A fitness crite-rion must combine the most important features associated with a successful trading strategy:
1. Net profits, or profits per trade2. The number of trades, or an error criteria3. The smoothness of the results
Ideally, we prefer systems that have large profits, lots of trades, and very consistent perfor-mance. To measure that result, the following might be used:
Rank = Profit per trade * (1 — (1/sqrt(number of trades) * (gross profits/gross losses)
If the profits per trade are larger, the rank is also higher; if there are more trades, the rankis higher, and if ratio of gross profits to losses is greater, the rank is higher. These factorsare not weighted, but they serve as a simple criterion for comparing or measuring the fit-ness of a chromosome. For example, a trading method that returned $500 per trade for 10trades, with a gross profitfoss ratio of .5 would have a rank of 170.75. Another system thatreturned only $250 per trade over 100 trades with a profit/loss ratio of 1.4 would rank315.00. Therefore, the system with the smaller returns has a much more agreeable tradingprofile according to the fitness criteria. Each analyst must create a criterion that allowsthose strategies with a personally desirable profile to survive.
Mutation and Crossover
Remembering that a genetic algorithm is a sophisticated search method, it needs a way tointroduce new mules (genes), combine genes into individuals that have passed the fitnesscriteria, and discard genes and individuals that are less promising. This is done throughmutation and crossover. Mutation is the process of introducing new genes, or combina-tions of genes, from a gene pool, which we have defined as a set of rules and relationaloperators, similar to those in Table 20-6. For example, the first gene, which represents atrend-following calculation, could be a moving average, exponential smoothing, linearregression, or breakout. In mutating the gene, one of these four techniques is chosen ran-domly. Similarly, the calculation period and the way in which rules are combined ‘areselected randomly. Typically, only one of the individuals in the chromosome is mutated ineach step; therefore, in Chromosome 1,
Chromosome 1: MA, 10, <, C, [0], & Stoch, 5, >, 50, 1the genes in the second individual (Stoch, 5,>, 50, 1) might be mutated to (RSI, 10, >, 40, 1).
Crossover is a way of combining two chromosomes by exchanging their individuals.The result is called an offspring. Using chromosomes 1 and 2, we can switch the first indi-viduals to get
Offspring 1: Exp, 20, <,L, [1], & Stoch, 5,>,50,1 _
Offspring 2: MA, 10, <, C, [0], & RSI, 10, <, 50, 1The two methods of mutation and crossover provide the only tools needed to introduce
new features to the optimization and to combine features in all ways. If you could continuethis process indefinitely, you can study every possible mutation and combination.
Selection
The process of natural selection is to allow only the best individuals to survive. A strongselection criterion chooses only those individuals with the highest ranking, as determinedGENETIC ALGORITHMS 501
 
by the fitness test. A weak criterion allows lower rankings to survive. The implementationof the selection process is also drawn from evolution. When an individual has a high fitnessscore, it becomes a larger part of the