A Work Point Count System Coupled with Back-Propagation for Solving Double Dummy Bridge Problem

R Amalraj

Associate Professor in ComputerScience

Department of Mathematics andComputer Science

Sri Vasavi College, Erode 638 316,Bharathiyar University Coimbatore

Tamil Nadu, South India. r_amalraj05@yahoo.co.in

M DharmalingamPh.D Research Scholar

Department of Computer ScienceSri Vasavi College, Erode 638 316,Bharathiyar University Coimbatore

Tamil Nadu, South India. emdharma@gmail.com

ABSTRACT- The game ‘Contract Bridge’ is one of the most widely known card games comprising manyfascinating aspects, such as bidding, playing and winning the trick including estimation of hand strength,the additional data based on the human knowledge of the game to improve the quality of results. Thegame classified under a game of imperfect information is to be equally well-defined, since the decisionmade on any stage of the game is purely based on the decision that was made on the immediatepreceding stage. The incompleteness of information, the real spirit of the card game in proceedingfurther deals of the game are taking into many forms especially during the distribution of cards for thenext deal. One among the architectures of the artificial neural network is considered by training onsample deals and used to estimate the number of tricks taken by one pair of bridge players is the keyidea behind the Double Dummy Bridge Problem, implemented in this paper. The Cascade CorrelationNeural Network architecture with supervised learning implemented to train data and hence to test it iscoupled along with Work Point Count System.

Keywords: Artificial Neural Networks, Cascade-Correlation Neural Network, BackPropagation Algorithm, Sigmoid functions, Contract Bridge, Double Dummy BridgeProblem, Work Point Count System.

1. Introduction

Contract bridge is a trick-taking card game where, on each ofseveral deals, the opposing side first competes in a biddingauction for the right to establish the contract for that deal,the side winning the auction being known as the declaring side.The contract is an exchange of the right to establish which suit,if any, is a trump for an undertaking to win at least the numberof tricks specified by the highest bid [1]. After the contracthas been established, the play of the cards proceeds as in mosttrick-taking card games until all thirteen tricks have beenplayed. One side may claim a stated number of the remainingtricks and concede the balance if any, during the play1. The

11www.wikihow.com/play-Bridge

declaring side will have either succeeded or failed in fulfillingthe contract is based on the actual number of tricks taken. Ifsuccessful, the declaring side scores the points, otherwise thedefending side scores the points.

The overriding objective is to win the contest by accumulatingmore points than the opponent. Although each variant of bridgehas its own particular scheme for awarding and accumulatingpoints, all are based upon whether or not the contract for eachdeal was made or defeated and the total number of tricks playedin the game. It can sometimes be advantageous to bid a contractthat one does not expect to make and to be defeated, thus losingsome points, rather than allow the opposing side to bid and makea contract which would score them still greater number of points.In the standard 52-card deck used in bridge, the ace is rankedthe highest followed by the king, queen, and jack and the spot-cards from ten down through two. The suit denominations also havea rank order with no-trump being the highest followed by spades,hearts, diamonds and clubs.

A key feature of bridge is the concept of vulnerability, since oneach deal, each side is said to be either vulnerable or notvulnerable depending upon whether or not it has won a game. Thescoring points that are won on a deal as a result of making acontract, and the points which are lost when failing to make acontract, are both significantly increased for the side that isvulnerable. Accordingly, whether one's side is vulnerable affectsits strategy for both bidding and play [2]. In certainsituations, if one pair of players has made the most recent bid,in turn the other pair of players may double the stakes. i.e., ifthe said pair makes a contract at this stage, then they windouble the number of points, but the risk is also correspondinglygreater. Any pair between the two can then redoubles and sincethree passes end every auction, it is quite possible for thefinal contract to be doubled or redoubled, increasing the score.

Artificial Neural Networks (ANN) are based on non-linearactivation function approximations which make them suitable formost of the applications especially games, since the outcome ofthe game could only be foreseen, but can’t be stated earlier at

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any stage. The focus of the paper is mainly on verification ofANN’s abilities to learn the evaluation function to solve theDouble Dummy Bridge Problems (DDBP) rather than finding thesolution to the problem. There are many Feed Forward NeuralNetworks (FFNN) available which are trained in bridge game [3, 4,5] and have been formalized in the best defense model, which alsopresents the strongest possible assumptions about the opponent[6, 7]. This is used by human players because modeling thestrongest possible opponents provides a lower bound on the payoff that can be expected when the opponents are less informed. Ofthe two methods known as Work Point Count Method (WPCM) [8] [13]and Distributional Point Method (DPM) [9][13] to evaluate thehand strength during the game, is an exclusive and a popularsystem used to bid a final contract in bridge game. The knowledgeof the game of bridge acquired by an individual player over aperiod of time is also considered and it is supplied to theneural network architecture in the form of input neurons bymultiplying the card values by the appropriate numbers defined inWPCM. The sum of these values is given in the network modelthereby making it to imbibe the expertise of the players alongwith the usual definition in the input process.

The paper is organized as follows. Section 2 provides a briefdescription of the game of bridge followed by a definition of thedouble dummy bridge problem and the nature of the source data andhow they are mapped to each card. In Section 3 the literaturesurvey pertaining to previous efforts of applying neural networksin game of bridge are mentioned along with the data used in thegame obtained from GIB library which includes possiblecombinations of deals for each of the tricks and the number oftricks taken by the appropriate pair during the game. Section 4details about cascade-correlation architecture and the way inwhich the neurons are added in the hidden layer one after anotherbased on the convergence of error both in the training phase witha diagrammatical representation. Section 5 summarizes the back-propagation algorithm implemented in the cascade-correlationarchitecture. Though many methods are explained to assign valuesto hand strength, a factor which reflects the individual players’expertise in the game of bridge is assigned through work point

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count method and it is illustrated in Section 6. The bridge game,implemented through cascade-correlation network model with numberof input neurons equivalent to the number of cards used in thegame say 52, the required number of hidden neurons and one outputneuron is explained in Section 7. The representation of resultswith the defined architecture is discussed with a sample of datain Section 8. Also, the architecture illustrated with sample datais compared with state of the art architecture with a completedifferent algorithm to establish its accuracy with respect toconvergence of results.

2. Problem description

The game of bridge is one of the well known card games playedworldwide with randomly dealt cards, which makes it also a gameof chance, or more exactly, a tactical game with inbuiltrandomness, imperfect information and restricted communication.The randomness and imperfect information built-in the game isdrawing attention of many of the researchers and ComputationalIntelligence (CI) methods are applied to focus mainly on theaspect of learning in the game playing systems [10]. The bridgeis a partnership game requiring four players, each player sitsopposite to his partner and it is traditional to refer to theplayers according to their position at the table as North, East,South and West, so North and South are partners playing againstEast and West. It is played with a standard deck of 52 playingcards, where one of the players deals all of the cards, 13 toeach player, in clockwise rotation, beginning with the player tothe left of the dealer. In bridge games, basic representationincludes value of each card as (Ace (A), King (K), Queen (Q),Jack (J ), 10, 9, 8, 7, 6, 5, 4, 3, 2) and suit as ( (Spades),♠the highest, (Hearts), (Diamonds), (Clubs), the lowest) for♥ ♦ ♣assignment of cards into particular hands and into public orhidden subsets, depending on the game rules. The ranking is for‘bidding’ purposes only and in ‘play’ all suits are equal, unlessone suit has been named as ‘trumps’, then it beats all the othercards. The cards are shuffled by the player to dealer's left andcut by the player to dealer's right.

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The first card played to each trick is called the ‘lead’ and onemay play any card in hand on ‘lead’ as the next three players havean obligation to follow the suit and if it is not possible, theymay play any card in hand. After four cards have been played, thetrick is complete and the winner of a trick is determined. Thebidding determines the declarer, which suit will be trumps andthe number of tricks declarer must win. The dealer is the one whodeals the cards and any of the players may become the declarer.The player to the left of declarer makes the first lead, which iscalled the opening lead. The hand held by declarer’s partner isthen displayed face up for all to see called the dummy and theplayer who held it does not participate in the play. The declarermust play both the dummy and his own hand, although each inproper turn. After the opening lead, the hand that wins eachtrick must lead to the next trick. If a trick contains a trumpcard, it is won by the highest trump played; otherwise it is wonby the highest card of the suit led. After each trick, one playerof the side that wins it should collect the cards, arrange themneatly so that the number of tricks won can be counted easily andthe play continues the same way for all 13 tricks [11]. Theconcept of exposing one of the hands for all to see is thehallmark of bridge. The dummy should be arranged neatly,separated into suits and the cards in each suit should be inorder of rank and overlapped, with the rank of each card clearlyvisible. If there is a trump suit, it is placed on dummy’s rightside, viewed by declarer, trumps are on the left. The purpose ofbidding is just to determine which player will be the declarerand how many tricks must be won with the chosen suit as trumps.

2.1. The game of contract bridge

In Contract bridge, the four players in two fixed partnershipsas pairs facing each other [12] and referred according to theirposition at the table as North (N), East (E), South (S) andWest (W), so N and S are partners playing against E and W isillustrated as below in Fig. 1 as an arrangement in the game. Theteam who made the final bid will at the moment try to make thecontract. The first player of this group who mentioned the valueof the contract becomes the declarer. The declarer’s partner is

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well-known as the dummy.

Fig. 1 The arrangement inBridge Game

The player to the left of thedeclarer leads to the firsttrick and instantly afterthis opening lead, thedummy’s cards are shown. The play proceeds clockwise and eachplayer must, if potential, play a card of the suit led. A trickconsists of four cards and is won by the maximum trump in it orif no trumps were played by the maximum card of the suit led.The champion of a trick leads to the next stage and the aim ofthe declarer is to take at least the number of tricksannounced during the bidding phase when the opponents try toprevent from doing it. In bridge, special focus in gamerepresentation is on the fact that players cooperate in pairs,thus sharing potentials of their hands [13].

2.2. The bidding and playing phases

The game then proceeds through bidding and playing phases and thepurpose of the biding phase is the identification of trumps anddeclarer of the contract. The playing phase consists of 13 trickswith each player contributing one card to each trick in aclockwise fashion with another level bid to decide who will be

6

32 543 A432 KQ J 7

QJ 10 9 8

J 7 6K J 910 2

7 65 410 98

Q10 A 98 3

AK AKQ2 8765 654

the declarer. The side which bids the highest will try to win atleast that number of tricks bid, with the specified suit astrumps. There are 5 possible trump suits: spades (♠), hearts(♥), diamonds (♦), clubs (♣) and “no-trump” which is the termfor contracts played without a trump. After three successivepasses, the last bid becomes the contract.

The bidding phase is a conversation between two cooperating teammembers against an opposing partnership which aims to decide whowill be the declarer. Each partnership uses an establishedbidding system to exchange information, interpret the partner'sbidding sequence as each player has knowledge of his own hand andan interesting aspect of the bidding phase is the cooperation ofplayers in North with South and West with East.

The play phase seems to be much less interesting than the biddingphase. The player to the left of the declarer leads to the firsttrick and may play any card and instantly after this openinglead, the dummy's cards are exposed. The play proceeds clockwiseand each of the other three players in turn must, if foundpotential, play a card of the same suit that the person in-chargeplayed. A player with no card of the suit may play any card ofhis selection. A trick consists of four cards, one from eachplayer, declared won by the maximum trump in it, or if no trumpsare played by the maximum card of the suit. The winner of a trickleads subsequently with any card as the dummy takes no activerole in the play and not permitted to offer any advice orobservation. Finally, the scoring depends on the number of trickstaken by the declarer team and the contract [14, 15, 16].

2.3 No-trump & Trump-suit

A trick contains four cards one contributed by each player andthe first player starts with the most important card, placing itface up on the table. In a clockwise direction, each player hasto track suit, by playing a card of the similar suit as the oneled. If a ‘heart’ is lead, for instance, each player must play a‘heart’ if potential and only if a player doesn’t have a ‘heart’ itcan be discarded. The maximum card in the suit wins the trick for

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the player who played it, known as playing in no-trump. The no-trump contracts seem to be potentially simpler than the suits ,because it is not possible to ruff a card of a high rank with atrump card [17]. Though it simplifies the rules, it doesn’tsimplify the strategy as there is no guarantee that a card willtake a trick, because even Aces are useless in tricks of othersuits in no-trump contracts. The success of a contract oftenlies in the hand making the opening lead and the knowledge ofthe location of cards may sometimes be insufficient to indicatecards that will take tricks [18]. A card that belongs to thesuit has been chosen to have the highest value in a particulargame, since a trump can be any of the cards belonging to any oneof the players in the pair. The rule of the game stillnecessitates that if a player can track suit, the player must doso, otherwise a player can no longer go behind suit, however, atrump can be played, and the trump is higher and more influentialthan any card in the suit led [19].

2.4 Work Point Count System

The work point count method which scores 4 points for Ace, 3points for King, 2 points for Queen and 1point for a Jack isfollowed in which no points are counted for 10 and below. Duringthe bidding phase of contract bridge, when a team reaches thecombined score of 26 points, they should use WPCM for gettingfinal contract and out of thirteen tricks in contract bridge,there is a possibility to make use of eight tricks by using WPCM.

3. The relevance of game of bridge to Artificial Intelligence

Though the game of bridge has not attracted much attention of theresearchers of Soft Computing, there are many interesting nuancesof the game of bridge from the point of view of ArtificialIntelligence (AI), since many factors of the game of bridgesatisfy the very definition of it such as the imprecise data,imperfect information through which the fact is to ascertainedetc. The two completely different phases of the game- the biddingand the play, both should be played optimally well to gain thebest possible result can be matched with that of the training and

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testing phases of AI, since the best possible training of theneurons can alone reflect the level of accuracy of inference inthe testing phase of the network.

During 1982, an article regarding a game was reported in the formof a computer program playing game of bridge [20]. A bridgeprogram capable of learning, hence generalize and able to inferdata was published in 1971. Also, the solution of no-trumpsinstances of the double dummy bridge problem was reported [15]. Aprogram for both bidding and playing which used the knowledgefrom the bidding phase was developed after few years [14]. Acomputer bridge program that would be able to defeat the besthuman bridge players was developed [21] and slowly theresearchers started utilizing their skills towards theapplication of games especially the game of bridge. The ideapresented by Levy was followed and Monte Carlo algorithm with avery fast DDBP solver in a computer bridge program was developed[22].

The data used in the game of DDBP was taken from the Ginsberg’sIntelligent Bridge (GIB) Library [23], which includes 7,171,02deals and for each of the tricks, it provides the number oftricks to be taken by N S pair for each combination of thetrump suit and the hand which makes the opening lead. There are20 numbers of each deal i.e. 5 trump suits by 4 sides as No-trumps, spades, Hearts, Diamonds and Clubs. The term ‘No-trump’is used for contracts played without trump in the four sidesWest, North, East and South.

4. Cascade-correlation neural network architecture

The cascade-correlation architecture was introduced by [24]defined with number of input neurons, output neurons representedin the input layer and output layer respectively and hiddenneurons are added to the network depends on the necessity of theaccuracy of the results. The Cascade-correlation begins with aminimal network, then mechanically trains and adds new hiddenunits one by one, creating a multi-layer configuration. Once anew hidden unit has been added to the network, its input-side

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weights are frozen. The new hidden neuron is added in eachtraining set and weights are adjusted to minimize the magnitudeof the correlation between the new hidden neuron output and theresidual error signal on the network output that has to beeliminated. The cascade-correlation architecture has many rewardsover its counterpart, as it learns at a faster rate, the networkdetermines its own dimension and topology, it retains thestructures it had built, still if the preparation set changes,and it requires no back-propagation of error signals through theassociations of the network.

During the learning process, new neurons are added to the networkone by one as in Fig.2 and each one of them is placed into a newhidden layer and connected to all the preceding input and hiddenneurons. Once a neuron is added to the network and activated, itsinput connections become frozen and do not change anymore.

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Input n

Input 1

Bias

1 Hidden

2 Hidden

1 Outpu

t

Fig. 2 The architecture of Cascade-Correlation Neural Network(CCNN)

The neuron to be added to the existing network can be made in thefollowing two steps: (i) The candidate neuron is connected to allthe input and hidden neurons by trainable input connections, butits output is not connected to the network. Then the weights ofthe candidate neuron can be trained while all the other weightsin the network are frozen. (ii) The candidate is connected to theoutput neurons and then all the output connections are trained.The whole process is repeated until the desired network accuracyis obtained. In equation (1), the correlation parameter ‘S’defined as below is to be maximized.

where ‘O’ is the number of network outputs, ‘P’ is the number oftraining patterns, ‘Vp’ is output on the new hidden neuron and‘Epo’ is the error on the network output. In the equation (2) theweight adjustment for the new neuron can be found by gradientdescent rule as

The output neurons are trained using the generalized deltalearning rule for faster convergence in Back -Propagationalgorithm. Each hidden neuron is trained just once and then itsweights are frozen. The network’s learning process is completedwhen satisfied results are obtained. The cascade-correlationarchitecture needs only a forward sweep to compute the networkoutput and then this information can be used to train thecandidate neurons.

5. Back-Propagation training algorithm

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The cascade correlation neural network is a widely used type ofarchitecture consisting of an input layer, a hidden layer, anoutput layer and two levels of adaptive connections [25]. It isalso fully interconnected, i.e. each neuron is connected to allthe neurons in the next level. The overall idea behind backpropagation is to make large change to a particular weight, ‘w’,the change leads to a large reduction in the errors observed atthe output nodes. In equation (3), let ‘y’ be a smooth function ofseveral variables xi, and it is required to know how to makeincremental changes to initial values of each xi, so as toincrease the value of y as fast as possible. The change to eachinitial xi value should be in proportion to the partialderivative of ‘y’ with respect to that particular ‘xi’. Supposethat ‘y’ is a function of a several intermediate variables ‘xi’ andthat each ‘xi’ is a function of one variable ‘z’ and we want to knowthe derivative of ‘y’ with respect to ‘z’, then using the chainrule.

The standard way of measuring performance is to pick a particularsample input and then sum up the squared error at each of theoutputs. We sum over all sample inputs and add a minus sign foran overall measurement of performance that peaks at o.

Where ‘P’ is the measured performance, S is an index that rangesover all sample inputs, Z is an index that ranges overall outputnodes, dsz is the desired output for sample input 's' at the zth

node, osz is the actual output for sample input 's' at the zth

node. The performance measure P is a function of the weights andthe idea of gradient ascent can be deployed if one can calculatethe partial derivative of performance with respect to each digit.

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With these partial derivatives in hand, one can climb theperformance hill most rapidly by altering all weights inproportion to the corresponding partial derivative. Theperformance is given as a sum over all sample inputs. We cancompute the partial derivative of performance with respect to aparticular weight by adding up the partial derivative ofperformance for each sample input considered separately. Theequation (6) each weight will be adjusted by summing theadjustments derived from each sample input. Consider the partialderivative

where the weight is a weight connecting ith layer of nodesto jth layer of nodes. The equation (7) our goal is to find anefficient way to compute the partial derivative of P with withrespect to . The effect of on value P, is through theintermediate variable oj, the output of the jth node and using thechain rule, it is express as

Determine oj by adding up all the inputs to node 'j' and passingthe results through a function.

Hence,where is a threshold function. Let

We can apply the chain rule again.

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Substituting Equation (8) in Equation (5), we have

Thus, the two important consequences of the above equations are,1) The partial derivative of performance with respect to a weightdepends on the partial derivative of performance with respect tothe following output. 2) The partial derivative of performancewith respect to one output depends on the partial derivative ofperformance with respect to the outputs in the next layer. Thesystem error will be reduced if the error for each trainingpattern is reduced. The equation (14) and (15) thus, at step’s+1'of the training process, the weight adjustment should beproportional to the derivative of the error measure computed oniteration’s’. This can be written as

where is a constant learning coefficient, and there isanother possible way to improve the rate of convergence by addingsome inertia or momentum to the gradient expression, accomplishedby adding a fraction of the previous weight change with currentweight change. The addition of such term helps to smooth out thedescent path by preventing extreme changes in the gradient due tolocal anomalies. Hence, the partial derivatives of the errors

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must be accumulated for all training patterns. This indicatesthat the weights are updated only after the presentation of allof the training patterns.

6. Work point count method

In bridge games, though the basic representation includes valueof each card as (Ace (A), King (K), Queen (Q), Jack (J ),10, 9, 8, 7, 6, 5, 4, 3, 2) for assignment of cards intoparticular hands and into public or hidden subsets, a uniformlinear transformation in the range 0.10 through 0.90 where 0.10is assigned to the smallest card value 2 with an increment of0.067 to the next card value i.e., 3 and so on till 0.90 for thehighest card value A is assigned as represented in Table 1.Though the ranking is for ‘bidding’ purposes only, with respectto ‘play’ all suits are considered equal, unless one suit hasbeen named as ‘trumps’, then it beats all the other cards.

Table1. The range of values assigned to each card of the deck

S.No.

RankCard

Rank CardValue

1 2 0.102 3 0.173 4 0.234 5 0.305 6 0.376 7 0.437 8 0.508 9 0.579 10 0.6310 J 0.7011 Q 0.7712 K 0.8313 A 0.90

Also, suit cards such as ( (Spades), the highest, (Hearts), ♠ ♥ ♦(Diamonds), (Clubs), the lowest)♣ are assigned a real numberusing the following mapping: Spades (0.3), Hearts (0.5), Diamonds

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(0.7) and Clubs (0.9). There are 52 input values and each valuerepresents one card from the deck and the positions of cards inthe input layer are fixed, i.e. from the leftmost input neuronsto the rightmost, the following cards are represented as 2♠, 3♠,K♠, A♠, 2♥, A♥, 2♦, A♦, 2♣,. .. , A . ♣ A value presented to thisneuron determined the hand to which the respective card belongsto i.e. 1.0 for North, 0.8 for South, −1.0 for West, and −0.8 forEast. Apart from the usual card values as input to the neurons inthe input layer which are multiplied with respective weights oftheir connections to the hidden neurons and hence from hiddenlayer to the output layer, the human knowledge is represented byvarious numerical estimators of hand’s strength used byexperienced human bridge players in order to declare the optimalpossible contract. The human estimators of hand strength can bedivided into two categories such as point count methods anddistributional point methods.

The human point count methods are based on calculating thestrength of a hand as a sum of single cards’ strength and thevalue of each card depends only on card’s rank. Though there aremany human point count methods such as Bamberger point count,Collet point count, Four aces points, Polish points etc., areavailable, work point count method is employed in our discussion,because it is the most widely used points counting system, whichscores 4 points for an Ace, 3 points for a King, 2 points for aQueen and 1 point for a Jack. The other category of human hand’sstrength estimators contains distributional points, in which thepatterns are scored based on its existence in a set of cardsassigned to one hand. The most important patterns are suits’lengths and existence of groups of honors in one suit. Anotherimportant pattern is a group of honors in one suit located in thecards of both players in a pair, since having a group of tophonors in a suit allows predicting more precisely the number oftricks available in this suit.

7. Implementation of CCNN architecture

Though there are several neural network architectures have beenused to solve the Double Dummy Bridge Problem, in this paper,

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Cascade-Correlation Neural Network (CCNN) architecture with 52,(13x4) input neurons for solving the DDBP is attempted and theresults are discussed. The 52 input card representation deals areimplemented in CCNN architecture as shown in Fig.3

Fig. 3 CCNN architecture with 13x4 input pattern

Layers are fully connected, i.e., in the 52 – 32 − 1 network 52input neurons are connected to all 32 hidden neurons and allhidden neurons are connected to a single output neuron.

7.1 Input layer

In the representation of bridge game as a neural network model, 52neurons are used in the input layer, since 52 cards are used intotal as each player has received 13 cards with all possiblecombinations of card values and combinations of suits among thecards. The rank of the card gets transformed using a uniformlinear transformation in the range of 0.10 through 0.90 withlowest values to biggest values of the cards respectively.

7.2 Hidden layer

Though the number of hidden neurons to a particular problem isstill decided by a rule of thumb, when the number of neurons is

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minimum, the model may take too much of time to learn or may notbe able to learn at all resulting in a poor performance during thetraining session. On the other hand if the number of neurons inthe hidden layer is equivalent to the input neurons, then theobjective of the training phase itself may become obsolete andinstead of learning during training session, the network mightmemorize the patterns, which will result, very badly in thetesting phase of the network. Thus, it is decided to have halfthe size of the input neurons as a rule of thumb and in theimplementation phase after a trial with 25 neurons, 26 neurons, 32neurons, it is concluded to stick with 32 neurons since it isslightly more than half the size of the input neurons.

7.3 Output layer

Thus there are two ways of transforming the networks’ outputs intothe number of tricks and in the first case one output neuron isused to get the result. The decision boundaries are defined in therange 0.1 through 0.9, denoting particular number of tricks isdiscussed in this paper and the final number of tricks iscalculated by dividing the output interval into 14 subintervals ofpair wise equal lengths of 0.067. For training and learning thedata, two activation functions viz., log sigmoid transfer functionand hyperbolic tangent sigmoid functions are used. The backpropagation algorithm is used for training and testing throughMATLAB 2008a.

Though there are 7,171,02 data in the GIB Library, for theconvenience to deal with the algorithm, first 5000 data areconsidered for training and 2500 alternate values among them areconsidered for testing and are furnished in Table 2 as originaldata selected at random from GIB Library used for training, inTable 3 as data after uniform linear transformation and in Table 4as the calculated value using two activate functions of thenetwork architecture with the original data respectively inAppendix A. From the table, it is inferred that hyperbolic tangentsigmoid function produces better results when compared with logsigmoid transfer function.

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The other way of transforming the networks’ outputs into thenumber of tricks relied on using 14 output neurons, with each ofthe output neurons represented one target number of tricks. In thetraining phase exactly one out of 14 output values is set to anon-zero value , usually 1.0 and in the testing phase, the outputneuron with the highest value defined the final prediction.Although the second method seems to be more suitable, in most ofthe experiments the networks using 14 output neurons achievedworse results than the corresponding networks having only oneoutput neuron [13]. Hence, the graphical demonstration of themethodology is furnished in Fig. 4 as below.

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Original data from GIB Library

Data converted by uniform linear transformation

Input to CCNN Model

2♠, A♠, 2♥, A♥, 2♦, A♦, 2♣,. .. , A♣

Spades Hearts Diamonds Clubs

Output of the ANN Model

Calculation of Mean Squared Error

32

52

1

Application of Back-Propagation

Reached Convergenc

e

STOPYesNo

Fig. 4 Graphical representation of the methodology8. Representation of results

A total number of five thousand deals from GIB library fortraining and two thousand five hundred among the trained data areused for testing on CCNN with fifty two input neurons, thirty twohidden neurons and only one output neuron (52-32-1). There are20 numbers for each deal i.e. 5 trump suits classified as no-trumps, spades, hearts, diamonds and clubs by 4 sides. The meansquared error during the training phase and testing phase usinglog sigmoid as the activation function is illustrated in Fig.5 ,while the mean squared error during the training phase andtesting phase using hyperbolic sigmoid as the activation functionis illustrated in Fig.6. From the table, it is inferred thathyperbolic tangent sigmoid function produces better results whencompared to log sigmoid transfer function.

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Fig.5 Mean Squared Error (MSE) during training and testing phaseof Log Sigmoid function

Fig.6 Mean Squared Error (MSE) during training and testing phaseof Hyperbolic Sigmoid functionThe result also revealed that, log sigmoid transfer function andhyperbolic tangent sigmoid functions are compared with each otherand it is inferred that the hyperbolic tangent sigmoid functionprovides significantly better results than log sigmoid transferfunction. A sample deal is also illustrated in Fig.7 with gamearrangement for reference in Section 8.1.

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8.1. Sample deals 7♠ with 25 points using WPCS

In the first example, since A K Q represented in ♠ amounts tothe maximum points in North side, hence found in NS pair, it isaimed to score 25 points, which is possible with 8 tricks to allplayers, irrespective of their level of mastery of bridge game.While incorporating work point count method, it is possible tofind the missing trick i.e., ♦ also, using the neural networkmodel discussed in this paper.

Fig. 7 A Sample deal (NS pair in ♠ contract with South openinglead)

8.2 Comparison to a proven architecture

Thus to validate the convergence of the algorithm in the Cascade-Correlation Neural Network (CCNN) architecture with 52, (13x4)input neurons for solving the DDBP, the problem is attempted withResilient back-propagation (Rprop), considered being the bestalgorithm, measured in terms of convergence speed, accuracy androbustness with respect to training parameters. Rprop takes intoaccount only the sign of the partial derivative over allpatterns, not the magnitude and acts independently on each‘weight’. The bias is updated based on the direction of thepartial derivative, and not on the magnitude [26]. The size of

22

the weight update is increased if the direction remains unchangedfor consecutive iterations, hence, adaptation of the weight-stepis not “blurred” by gradient behavior, instead, each weight hasan individual evolving update-value, and the weight-step is onlydetermined by its update-value and the sign of the gradient. Theperformance during training and testing phases of the same datafrom GIB Library used in this architecture is illustrated for thepurpose of comparison in Fig 8 and Fig 9 respectively.

When the performance of the algorithm is measured, all humanpoint count and distributional point methods achieved onlyslightly better results. Also results of these networks for no-trump contract are comparable, but for spades, contracts usingdistributional point methods turned out to be definitely moreeffective [27].

It is observed from paper [3] that extending the dealrepresentation by adding human estimation didn’t improve the bestoverall result accomplished by pure 52x4 in the case of suitcontracts and only slight improvement was notified in the case ofnon-trump contracts. This observation suggested that therelevance of additional information related to suit lengths andpoint distribution in particular hands has been autonomouslydiscovered by the best 52x4 networks during the training process.

Also, humans are visibly better at solving the no-trump contractsthan the suit ones and the opposite conclusion is also valid inthe case of neural networks. The study reported in [1]demonstrates that a neural network can be trained to capture theimplicit reasoning used for bidding a hand in bridge.

23

Fig. 8 Mean Squared Error (MSE) during training phase ofRprop Algorithm

Fig. 9 Mean Squared Error (MSE) during testing phase ofRprop Algorithm

9. Conclusion

24

The results revealed that, the data tested through abovearchitecture show significantly better performance and the timetaken for training and testing are relatively minimum which isconverging towards the possible minimum error during theiterations. The mean squared error between the actual valueobtained from GIB and the corresponding calculated value with logsigmoid function as the activation function during the trainingand testing phase is illustrated in Fig 5. Similarly, the meansquared error between the actual value obtained from GIB and thecorresponding calculated value with hyperbolic tangent functionas the activation function during the training and testing phaseis illustrated in Fig 6.

In Cascade-Correlation neural network, during training processnew hidden nodes are added to the network one by one. For eachnew hidden node, the correlation magnitude between the new nodeoutput and the residual error signal is maximized. When the nodeis being added to the network, the input weights of hidden nodesare frozen, and only the output connections are trainedrepeatedly.

In CCNN model, log sigmoid transfer function and hyperbolictangent sigmoid functions are used and the hyperbolic tangentsigmoid function produced better result when compared to logsigmoid transfer function. Thus during a game, a trick normallyfailed to be utilized by a player, irrespective of his expertisein the game of bridge at tournaments is easily overcome by theimplementation of cascade-correlation neural network model alongwith work point count method. The problem of demonstrating aspecific knowledge gained through the learning process isextremely specialized and it is inferred that work point countmethod is a good information system that provides some new ideasto the bridge players and helpful for beginners and semiprofessional players as well in improving their bridge skills.

References

[1] B.Yegnanarayana,D. Khemani,M. Sarkar, Neural Networkfor contract bridge bidding. 21(3), 1996, pp 395-413.

25

[2] A.Amit,S.Markovitch, Learning to bid in bridge. MachineLearning, 63(3), 2006, pp. 287-327.

[3] J.Mandziuk,K.Mossakowski, Neural Networks compete withexpert human players in solving the double dummy bridgeproblem. In Proceedings of the Conference onComputational Intelligence and games, 5,(2009), pp 117-124.

[4] K.Mossakowski,J. Mandziuk,Artificial Neural Networksfor solving double dummy bridge problems. In L.Rutkowski, J. H. Siekmann, R. Tadeusiewicz, and L. A.Zadeh,(Eds.), Lecture Notes in ArtificialIntelligence,LNAI:vol,3070. Artificial Intelligence andSoft Computing ICAISC, 2004, pp.915-921.

[5] M.Sarkar,B. Yegnanarayana,D. Khemani,Application ofneural network in contract bridge bidding. In Proceedingof National Conference on Neural Networks and FuzzySystems, Madres, 1995, pp144-151.

[6] M.Dharmalingam,R. Amalraj, Neural Network Architecturesfor Solving the Double Dummy Bridge Problem in ContractBridge. In Proceeding of the PSG-ACM National Conferenceon Intelligent Computing.1, (2013), pp 31-37.

[7] M.Dharmalingam, R.Amalraj, Back-Propagation NeuralNetwork Architecture for Solving the Double Dummy BridgeProblem in Contract Bridge, IEEE International Conferenceon Intelligent Computing Applications, 2014, pp 454-461.

[8] H.Francis, A.Truscott, D. Francis, The OfficialEncyclopedia of Bridge, American Contract Bridge League.(6th) Edition, 2001.

[9] B. Seifert, Encyclopedia of Bridge, Warsaw: PolishScientific Publishers PWN, 1996.

[10] J.Mandziuk, Knowledge-free and Learning – Based methodsin Intelligent Game Playing. Springer, 2010.

[11] M.Dharmalingam, R.Amalraj, Supervised Learning inImperfect Information Game. International Journal of

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Advanced Research in Computer Science,4(2), 2013, pp 195-200.

[12] M.Dharmalingam, R.Amalraj, A Solution to the DoubleDummy Bridge Problem in Contract Bridge Influenced bySupervised learning module Adapted by Artificial NeuralNetwork, ICTACT Journal on Soft Computing: Special Issueon Distributed Intelligent System and Applications,05(01), 2014, pp 836-843.

[13] K. Mossakowski,J Mandziuk, Learning without humanexpertise: A case study of Double Dummy Bridge Problem.IEEE Transactions on Neural Networks, 20(2), 2009, pp278-299.

[14] S.J.J.Smith,D.S Nau,T.A Throop, Computer Bridge - A BigWin for AI planning. Artificial Intelligence Magazine,19(2),1998, pp 93-106.

[15] I.Frank, D.A Basin, A Theoretical and EmpiricalInvestigation of Search in Imperfect Information Game.Theoretical Computer Science, 252(1), 2001, pp 217-256.

[16] T.Ando,Y. Sekiya,T. Uehara, Partnership bidding forcomputer bridge. Systems and Computers in Japan, 31(2),2010, pp 72-82.

[17] M Dharmalingam and R Amalraj, Supervised Elman NeuralNetwork Architecture for solving the Double Dummy BridgeProblem in Contract Bridge, International Journal ofScience and Research, 3(6), 2014, pp 2745-2750.

[18] W.Jamroga, Modeling Artificial Intelligence on a caseof bridge card play bidding. In Proceedings ofInternational Workshop on Intelligent Information System.(1999), pp 276-277.

[19] J.Mandziuk, Some thoughts on using ComputationalIntelligence methods in classical mind board games. InProceedings of the International Joint Conference onNeural Networks.(2008),pp 4001-4007.

27

[20] H.W.Root, The ABCs of Bridge, Three RiversPress,1998.

[21] S.J.J Smith,D.S Nau,T.A Throop, Success in Spades:Using AI Planning Techniques to Win the WorldChampionship of Computer Bridge. In Proceeding of theNational Conference on Artifitical Intelligence, (1998),pp1079-1086.

[22] D.N.Levy, The million pound bridge program, inHeuristic programming in Artificial Intelligence,(1989),pp 95-103.

[23] M.L Ginsberg GIB: Imperfect Information in aComputationally Challenging Game Journal of ArtificialIntelligence Research, vol. 14, (2001), pp 303-358.

[24] S.E Fahlman, V Lebiere, The cascade-correlationlearning architecture. Advances in Neural InformationProcessing. (1990), pp 524-532.

[25] D.E Rumelhart, G.E Hinton, R.J Williams, Learninginternal representations by error back propagation,Parallel Distributed Processing: Explorations in theMicrostructure of Cognition, (l), (1986), pp 533-536.

[26] C.S Chen, S.L Su, Resilient Back propagation NeuralNetwork for Approximation 2-D GDOP, Proceedings of theIntl. Multi Conference of Engineers and ComputerScientists, vol. 2, (2010), pp 17-19.

[27] J. Mandziuk and K. Mossakowski, Example – basedestimation of hands strength in the game of bridge withor without using explicit human knowledge, In Proceedingof the IEEE Symposium Computational Intelligence in Datamining, (2007), pp.413-420.

28

Dr. R. Amalraj is an Associated Professor inComputer Science in the Department of ComputerScience, Sri Vasavi College, Erode. He obtained hisPhD in Computer Science from PSG College ofTechnology, affiliated to Bharathiar University,Coimbatore in 2003. He is a Life Member of ComputerSociety of India. He has been a Principal

Investigator for a second Minor Research Project sponsoredby University Grants Commission, New Delhi. He has publishedseveral research papers in reputed National andInternational journals, with specific interest in ArtificialIntelligence, Image Processing and Soft Computing.

Mr. M Dharmalingam received his Under-Graduate,Post-Graduate and Master of Philosophy degrees fromBharathiar University, Coimbatore in the years2000, 2004 and 2008 respectively. He is a Part timeResearch Scholar in Computer Science at Sri VasaviCollege, Erode. He has published research papers in

reputed National and International journals and the broadfield of his research interest is Soft Computing.

29

Appendix- A(a) Table 2- An Extract of original Ginsberg’s Intelligent Bridge (GIB) Library data

JT852.93.KQ7.J82 AQ97.JT654.T6.A5 43.AK8.A542.7643 K6.Q72.J983.KQT9:88887777A9A977778888765.T87.Q953.K98 T4.KJ63.KT76.AJ5 KQ83.A94.J.QT432 AJ92.Q52.A842.76:AAA89998A9A9AAAA777732.972.QJ6543.K6 95.K86.AK.Q98743 JT86.JT4.T972.T5 AKQ74.AQ53.8.AJ2:CCCCBBCBCCCC8888CCCCA743.JT72.Q64.QJ J5.Q63.T985.K986 62.K984.J2.T7542 KQT98.A5.AK73.A3:AAA9AAA97777BBBA8888KQ97.63.T84.KT84 A543.AQ7.Q5.A752 J82.JT854.K92.Q3 T6.K92.AJ763.J96:99997777989899999898QJ742.A2.QT8.Q82 AT5.63.J7632.A73 K9.KJ84.K54.J964 863.QT975.A9.KT5:555554546676666655558754.QJ752.QJ83. JT.T98.AT9762.62 AKQ962.A3.5.KQT3 3.K64.K4.AJ98754:4444222233337777878782.AK764.A4.9432 A964.JT82.K3.AK8 QJ7.93.QT9875.J6 KT53.Q5.J62.QT75:99999999999966669999T5.32.KJ42.AKJ52 KJ6.65.A85.QT743 A98432.AK74.93.8 Q7.QJT98.QT76.96:65653333656566665555AK9.KQ854.QT6.74 Q83.J9.AK42.KJ63 742.T7632.J3.T85 JT65.A.9875.AQ92:878799996565AAAAAAAA4.AJ752.86.K9765 KT8532.96.Q2.T42 A97.KQT.KJ94.AQ3 QJ6.843.AT753.J8:21217777111144441111J.J863.AT8742.86 Q543.K5.KQ.J9742 8762.T72.J63.KT3 AKT9.AQ94.95.AQ5:CCDCCCDCBBBB7777CCDC2.J863.QT3.JT753 T763.T52.KJ752.K A54.AKQ.A64.A962 KQJ98.974.98.Q84:5353777733335555222253.T8.A765.KJT92 Q72.632.J94.8765 KT6.AKQJ754.K3.Q AJ984.9.QT82.A43:32326676223265652232854.QT65432.5.AT AT3.K9.JT92.5432 KQJ76.J7.Q864.J8 92.A8.AK73.KQ976:BABA87876565BABABBBBAJ8.AK85.Q742.J9 942.T4.KT9.KT632 QT6.J962.AJ86.85 K753.Q73.53.AQ74:54545555333333336666J9732.73.A6.AK85 86.J65.95432.J32 AQT4.AT942.J7.94 K5.KQ8.KQT8.QT76:55553333444476766565T3.AT762.T63.973 AQJ976.85.Q92.A2 K.KJ93.K874.KQT8 8542.Q4.AJ5.J654:9898AAAA4454777777769743.8.KQ865.J97 AJ.QT9.J9732.863 Q8.K7642.A4.AKT5 KT652.AJ53.T.Q42:7777878788886666656585.K85.KJ53.QT42 T94.QT63.QT.KJ75 AKJ762.7.A4.A963 Q3.AJ942.98762.8:22222121777744442222T832.92.864.T954 KQJ97.4.A72.QJ72 A4.KQJT86.KT3.K6 65.A753.QJ95.A83:9797A9A96666BABAAAAA

301www.wikihow.com/play-Bridge

Q75.K7432.KQT7.2 KT943.QT.A6.9543 A.J9.J98543.K876 J862.A865.2.AQJT:A7A7BABA77774343AAAAT85.A865.6.87654 62.QT93.Q83.AT93 AQ974.4.KJT742.Q KJ3.KJ72.A95.KJ2:AAAA4454AAA96565A9A998.J98632.AJT.AT K753.A5.973.8432 QJT6.Q4.KQ42.QJ9 A42.KT7.865.K765:64645455444444447676AQ4.KT5.J65.KJ82 862.J94.9843.653 J973.8732.T72.QT KT5.AQ6.AKQ.A974:7777777776777777777763.AK98.AKQ9542. .QJT632.T76.T976 AQT84.74.J3.J532 KJ9752.5.8.AKQ84:33536566666622229888AKQT7.JT9652.2.T 84.K8.KJ.Q985432 J632.AQ3.AQ953.J 95.74.T8764.AK76:70701010101044447777AJ.K64.Q8542.652 842.AJT8732.63.8 KQ965.5.JT7.KQJT T73.Q9.AK9.A9743:A8A87474AAAA55557676AT5.96.AKQ96.AK3 9432.KT85.T4.J85 KJ87.AQ7.873.T76 Q6.J432.J52.Q942:00000000545400003232AQ764.J984.J8.52 5.AK7653.Q54.K76 JT83.T.AK62.AQT8 K92.Q2.T973.J943:444422226666555455549.T852.QT64.KT85 AT3.K963.AK52.A9 QJ8765.Q74.7.Q63 K42.AJ.J983.J742:AAAA8888A9A9BBBBA9A97.QJT2.K97.QJ642 QJ952.K8.Q8.9753 K864.653.AJ62.T8 AT3.A974.T543.AK:8888AAAA777788887777A8.AQ2.K82.QJ982 Q3.KJ84.AQT53.53 542.T763.64.AK74 KJT976.95.J97.T6:656598987666989855558.A984.A83.JT963 JT94.KJT.T7.KQ74 Q.Q7532.Q9542.85 AK76532.6.KJ6.A2:BBBBCBCB666566668888AK6.AT654.AT.KQ3 84.Q9.J954.A7652 J3.K732.KQ2.JT98 QT9752.J8.8763.4:11115555212133432222K2.86.QJT65.J753 9654.KQJ54.842.4 AJ7.A92.A.AQ9862 QT83.T73.K973.KT:43437676555533332222QJ5.AT98.3.AKQT3 K98763.65.KJ5.64 T.KQ73.QT92.J752 A42.J42.A8764.98:44447777222266662222J8.AKJ8.AT652.T8 AT65.QT754.K4.54 7.93.J9873.AKQ96 KQ9432.62.Q.J732:77778888756522223333J6543.AKJ9.Q753. 9872.4.AJT862.KQ AK.QT863.9.AT962 QT.752.K4.J87543:44444444111176766565T9862.4.K432.KJ2 AQ75.AJ932..AQ98 K43.KQ8.AQ9.T543 J.T765.JT8765.76:65656555BBBA87878787AQ8542.J75.J64.A KT6.QT83.A.JT752 73.9642.Q972.986 J9.AK.KT853.KQ43:BABA7777BABAA9A9CBCBA843.AT7.J82.AJ6 65.K8.AK75.KQT54 QJT92.QJ4.Q4.982 K7.96532.T963.73:74445454866698988777654.KQ9.T762.J52 K932.JT84.QJ9.98 JT7.653.A4.AKQT6 AQ8.A72.K853.743:75758888777787875555T4.AJ7652.AK76.K A972.83.JT95.A54 KQ6.Q4.83.QT8763 J853.KT9.Q42.J92:55556666334354553333K75.KQT3.K86.Q63 T6.AJ.Q954.J9854 AJ92.42.AJ732.AK Q843.98765.T.T72:11112222333311113333T92.74.JT9.QJ943 753.T9653.Q7.765 864.J8.A54.AKT82 AKQJ.AKQ2.K8632.:9797CACACCCCBABA6666QT954.AKQT7.87.J 73.92.AK2.875432 6.J8653.953.AKT6 AKJ82.4.QJT64.Q9:75757777313188888787A6.7652.AQ93.KT6 QJ743.QJ8.JT842. KT2.AT.K7.QJ9853 985.K943.65.A742:41417676544465652121KT64.93.KJ4.K953 J932.K85.QT632.4 85.T64.A9875.AQT AQ7.AQJ72..J8762:66668888A9A966667777KQ6.AJT9.876.974 T842.Q7.AJ52.KT8 953.K865.QT94.AQ AJ7.432.K3.J6532:55556665434344446666

311www.wikihow.com/play-Bridge

732.K72.J7542.93 T4.QJ4.QT9.KQ842 AQ965.965.A86.J5 KJ8.AT83.K3.AT76:AAAA8888AAAA8888AAAAK862.843.64.T843 A953.JT92.AK9.J6 7.K65.JT73.AKQ92 QJT4.AQ7.Q852.75:C8C8CBCBCBCBBABA6666AK94.K652.986.J6 J8765.AJ9.QT3.54 Q3.QT873.AK42.AQ T2.4.J75.KT98732:33335454223232327777AQJ5.8642..Q9862 972.Q3.KQ632.K73 KT86.AJ97.985.JT 43.KT5.AJT74.A54:888844444454AAAA6565J4.T643.AK.KT752 987.Q87.Q9654.QJ AQT2.2.8732.9643 K653.AKJ95.JT.A8:76777677878876763333Q6532.JT6.K.Q954 AJT98.K.9754.A73 K7.98432.QJ86.JT 4.AQ75.AT32.K862:A9A9A9A99999BABAAAAAJ72.932.K762.Q62 KQT65.QJ85.53.T9 A8.AK.AJT4.KJ875 943.T764.Q98.A43:52526666666621212222AK86.AKJ9.Q4.Q83 T.876543.T32.AK6 Q9432.Q2.J75.J94 J75.T.AK986.T752:44444343666688887676K6532.64.A74.Q84 J7.QT8.QJ62.K753 AQT8.A5.K95.AJ96 94.KJ9732.T83.T2:11111111666644442221AK85.KT73.753.K2 64.AQ854.J64.Q86 92.J962.KQ2.JT43 QJT73..AT98.A975:777788885555888877774.AQJT65.KQ8.752 QJ93.8743.6.AKQ6 A2.9.AJ97532.JT8 KT8765.K2.T4.943:40409999303030307676JT962.A.AJ42.974 8543.T4.65.T8632 KQ.KQJ2.KT8.KQJ5 A7.987653.Q973.A:22222222444433333333975.KT97.J984.65 Q82.QJ65.Q3.T972 AKJ6.4.T7652.Q83 T43.A832.AK.AKJ4:88886565999855558898JT32.KQ752.654.8 Q9.T86.AKJ72.AQ9 K875.A.983.T7432 A64.J943.QT.KJ65:BABA66668887BABAAAA9Q954.J2.AT3.KT84 AT32.K53.762.A63 KJ76.Q9874.K.Q97 8.AT6.QJ9854.J52:85854444656588886666KQ9875.3.KJ982.9 .J985.AQ7.KQT765 AJT6.KQ4.T54.842 432.AT762.63.AJ3:97975454CCCC5555CCCCAK5.943.A85.7542 8762.QJ87.Q.QJT3 T.AT652.KT962.96 QJ943.K.J743.AK8:76769999545454547777A74.T2.T73.KQ754 3.J63.QJ842.AJT3 QJ85.98754.K65.9 KT962.AKQ.A9.862:999998989898AAAA9898T4.T2.AQ97.KQJ86 AQ75.KJ6.KJT5.93 J92.AQ73.642.T75 K863.9854.83.A42:76769998988876766666J853.KJ632.64.Q8 .Q874.JT7.AKT743 A764.AT5.A52.952 KQT92.9.KQ983.J6:888877776666AAAA9999.QJ9.KQ752.KJT94 AJ32.AT73.64.863 T976.K5.AJ983.75 KQ854.8642.T.AQ2:77778888989822225252843.984.A.KJT974 QJT6.JT7.92.A532 A97.KQ2.KQT73.86 K52.A653.J8654.Q:64647676777777774343AK76.Q.AK842.K43 QJ32.AKJ872.963. T84.653.T.JT8762 95.T94.QJ75.AQ95:98987565989877774333QT2.KQ5.QJ85.K43 KJ6.432.A64.J872 A75.J976.KT3.QT5 9843.AT8.972.A96:54445555444444445555T852.74.643.KT53 K94.QJT6.972.J97 QJ6.A82.AQT.Q864 A73.K953.KJ85.A2:88887777A9A988986666A4.AK8765.T8.764 J92.J4.K74.AQ982 KQ.T2.AJ652.KJT5 T87653.Q93.Q93.3:53527777333343434444AK9.Q975.T3.AQ98 QJ642.K.J96.7543 T.84.AK8752.KJ62 8753.AJT632.Q4.T:222277776565222221215.KQT8.J32.KJ952 A983.AJ753.A.Q87 KQ76.64.KT9764.3 JT42.92.Q85.AT64:76769888877755558788Q75.K97.K9.KT987 J8643.Q.AJ652.A4 AKT2.JT643.T7.Q3 9.A852.Q843.J652:777776766464A9A97575

32

T6.JT93.T632.J96 K84.A84.Q87.A873 A7.K72.AKJ95.QT4 QJ9532.Q65.4.K52:98969999656555559898KJ4.KQ43.T64.QJ5 T6.J92.AKJ.AKT32 953.T85.Q98.9876 AQ872.A76.7532.4:9999AAAA9999AAAA999943.J9.KT98.AKJ72 QJ2.AK85.A4.T865 KT765.64.QJ75.94 A98.QT732.632.Q3:88887676999955556666J72.843.9872.KT9 T6.KQJT2.A4.Q642 K953.A95.T6.J753 AQ84.76.KQJ53.A8:CBCCA9A9CBCCBBBBA9A99832.874.J2.QJ65 KT54.KQJT5.AT.T8 AQJ7.A9.973.AK97 6.632.KQ8654.432:6464444498988888333363.K8.AQ98.KJ875 K52.AT4.K5432.Q6 AQJT87.J972.J6.A 94.Q653.T7.T9432:33323332534444444444K.KJT654.Q8542.8 AJT6.A9.7.AKT542 Q98.82.AKJ9.9763 75432.Q73.T63.QJ:8888BBBB43443333BABAT642.Q9.7643.QJ7 A93.T75.QJT8.862 KJ.AKJ864.K9.AT9 Q875.32.A52.K543:55657777444477776666QT75.5.A852.AQ83 2.963.KQ74.KJ652 K3.AKQJ87.6.T974 AJ9864.T42.JT93.:43436464424286964444T82.652.Q7.QJ752 9753.K874.KJT2.3 AJ4.Q93.9864.964 KQ6.AJT.A53.AKT8:CCCCCCCCCCCCCCCCAAAA953.KQ63.KJ963.T AJ7.T974.T875.A8 K62.J2.AQ.Q97652 QT84.A85.42.KJ43:76767777666655556565KJ.5.J632.AQJ642 T9654.QT73.T9.93 A32.KJ8.AK75.K75 Q87.A9642.Q84.T8:11115555555521211111JT76.542.954.A97 KQ42.A963.Q6.JT2 A8..AKT87.KQ6543 953.KQJT87.J32.8:64647676888821212222KQ4.KJ732.A3.942 J53.T84.K98754.Q A82.A965.Q2.AKT5 T976.Q.JT6.J8763:21113333111166663333K862.6.AT85.AJ86 JT43.Q72.QJ963.7 A75.98543.42.952 Q9.AKJT.K7.KQT43:98997676888898987777QJ3.82.JT76.AK43 K984.AQJ5.K95.J2 AT65.KT963.Q.T85 72.74.A8432.Q976:65655454555577775555T.AKQT54.Q97.764 AK53.63.832.Q853 QJ87642.872.A5.J 9.J9.KJT64.AKT92:7676444443439898A9A9T975.94.AJT42.65 83.T763.Q6.Q9872 AJ.KQJ.K975.KT43 KQ642.A852.83.AJ:44546666878733436666932.J653.AJT.A86 64.AKT82.943.732 T87.Q9.KQ765.KJ4 AKQJ5.74.82.QT95:76769898888865658787J9.J975.863.AT76 A83.AQ63.KT52.85 QT752.K.AJ74.432 K64.T842.Q9.KQJ9:99998888AAAA88989898.74.AT97653.8732 QJ973.A952.KQ.Q5 AT5.JT83.J2.A964 K8642.KQ6.84.KJT:8585AAAA9798545466669732.AK2.J7.AT52 A54.T983.A632.K7 8.QJ754.T95.9643 KQJT6.6.KQ84.QJ8:9797BABA6666BBBB6565AQJT.K5.J984.KJ4 54.QT962.QT.T932 987632.843.K72.A K.AJ7.A653.Q8765:63633333777744448888K6.KJT52.JT7643. Q54.Q63.K.Q96532 T92.974.A985.JT8 AJ873.A8.Q2.AK74:B9B9BABA44444444BABAA874.K2.K632.AQ9 KJT6.QJT543.AJ.K Q3.A96.Q874.8532 952.87.T95.JT764:43435454767633334444AQ8654.Q62.AT3.A JT3.JT98.982.862 K9.K43.QJ754.K53 72.A75.K6.QJT974:10101010333310105555A.K42.K6543.K863 KT98.A3.Q98.AQ54 Q2.QJ98.AJT.JT72 J76543.T765.72.9:7474A999544453433333AK763.T.K962.AT4 QT5.AK976.AJ8.32 J94.J85.Q74.J975 82.Q432.T53.KQ86:878855559999666676764.Q53.T8643.T962 K853.KJ92.Q9.K43 Q9762.T874.AK75. AJT.A6.J2.AQJ875:D8D89999A9A96565DBDB

33

QT74.AK3.T653.Q5 J985.752.987.JT2 AK6.QJ864.AK42.6 32.T9.QJ.AK98743:71712222111110107777J.QT7.AQT762.742 QT874.K865.98.AJ AK9.AJ93.543.QT5 6532.42.KJ.K9863:21216666313131314444864.K.AJ976.AK95 K.Q9853.85432.J3 AT73.AJT6.T.8642 QJ952.742.KQ.QT7:43435454444443432222A95.T653.KJ3.KQ5 KQJ.K87.QT5.JT76 32.QJ942.A97.842 T8764.A.8642.A93:77779899555587777777AT532.AQ5.T6.754 74.KJT973.KJ84.A 8.8642.Q97.T9862 KQJ96..A532.KQJ3:AAAAAAAA9898BBBB9999

The data continues………….

(b) Table 3- An Extract of data after Uniform Linear Transformation0.70 0.63 0.50 0.30 0.10 0.57 0.17 0.83 0.77 0.43 0.70 0.50 0.10 0.90 0.77 0.57 0.43 0.70 0.63 0.37 0.30 0.23 0.63 0.37 0.90 0.30 0.23 0.17 0.90 0.83 0.50 0.90 0.30 0.23 0.10 0.43 0.37 0.23 0.17 0.83 0.37 0.77 0.43 0.10 0.70 0.57 0.50 0.17 0.83 0.77 0.63 0.57 8 7 9 7 8 0.43 0.37 0.30 0.63 0.50 0.43 0.77 0.57 0.30 0.17 0.83 0.57 0.50 0.63 0.23 0.83 0.70 0.37 0.17 0.83 0.63 0.43 0.37 0.90 0.70 0.30 0.83 0.77 0.50 0.17 0.90 0.57 0.23 0.70 0.77 0.63 0.23 0.17 0.10 0.90 0.70 0.57 0.10 0.77 0.30 0.10 0.90 0.50 0.23 0.10 0.43 0.37 8 8 9 A 7 0.17 0.10 0.57 0.43 0.10 0.77 0.70 0.37 0.30 0.23 0.17 0.83 0.37 0.57 0.30 0.83 0.50 0.37 0.90 0.83 0.77 0.57 0.50 0.43 0.23 0.17 0.70 0.63 0.50 0.37 0.70 0.63 0.23 0.63 0.57 0.43 0.10 0.63 0.30 0.90 0.83 0.77 0.43 0.23 0.90 0.77 0.30 0.17 0.50 0.90 0.70 0.10 C B C 8 C 0.90 0.43 0.23 0.17 0.70 0.63 0.43 0.10 0.77 0.37 0.23 0.77 0.70 0.70 0.30 0.77 0.37 0.17 0.63 0.57 0.50 0.30 0.83 0.57 0.50 0.37 0.37 0.10 0.83 0.57 0.50 0.23 0.70 0.10 0.63 0.43 0.30 0.23 0.10 0.83 0.77 0.63 0.57 0.50 0.90 0.30 0.90 0.83 0.43 0.17 0.90 0.17 9 9 7 A 8 0.83 0.77 0.57 0.43 0.37 0.17 0.63 0.50 0.23 0.83 0.63 0.50 0.23 0.90 0.30 0.23 0.17 0.90 0.77 0.43 0.77 0.30 0.90 0.43 0.30 0.10 0.70 0.50 0.10 0.70 0.63 0.50 0.30 0.23 0.83 0.57 0.10 0.77 0.17 0.63 0.37 0.83 0.57 0.10 0.90 0.70 0.43 0.37 0.17 0.70 0.57 0.37 9 7 8 9 8 0.77 0.70 0.43 0.23 0.10 0.90 0.10 0.77 0.63 0.50 0.77 0.50 0.10 0.90 0.63 0.30 0.37 0.17 0.70 0.43 0.37 0.17 0.10 0.90 0.43 0.17 0.83 0.57 0.83 0.70 0.50 0.23 0.83 0.30 0.23 0.70 0.57 0.37 0.23 0.50 0.37 0.17 0.77 0.63 0.57 0.43 0.30 0.90 0.57 0.83 0.63 0.30 5 4 6 6 5 0.50 0.43 0.30 0.23 0.77 0.70 0.43 0.30 0.10 0.77 0.70 0.50 0.17 0.70 0.63 0.63 0.57 0.50 0.90 0.63 0.57 0.43 0.37 0.10 0.37 0.10 0.90 0.83 0.77 0.57 0.37 0.10 0.90 0.17 0.30 0.83 0.77 0.63 0.17 0.17 0.83 0.37 0.23 0.83 0.23 0.90 0.70 0.57 0.50 0.43 0.30 0.23 4 2 3 7 7 0.50 0.10 0.90 0.83 0.43 0.37 0.23 0.90 0.23 0.57 0.23 0.17 0.10 0.90 0.57 0.37 0.23 0.70 0.63 0.50 0.10 0.83 0.17 0.90 0.83 0.50 0.77 0.70 0.43 0.57 0.17 0.77 0.63 0.57 0.50 0.43 0.30 0.70 0.37 0.83 0.63 0.30 0.17 0.77 0.30 0.70 0.37 0.10 0.77 0.63 0.43 0.30 9 9 9 6 9 0.63 0.30 0.17 0.10 0.83 0.70 0.23 0.10 0.90 0.83 0.70 0.30 0.10 0.83 0.70 0.37 0.37 0.30 0.90 0.50 0.30 0.77 0.63 0.43 0.23 0.17 0.90 0.57 0.50 0.23 0.17 0.10 0.90 0.83 0.43 0.23 0.57 0.17 0.50 0.77 0.43 0.77 0.70 0.63 0.57 0.50 0.77 0.63 0.43 0.37 0.57 0.37 5 3 5 6 5

34

0.90 0.83 0.57 0.83 0.77 0.50 0.30 0.23 0.77 0.63 0.37 0.43 0.23 0.77 0.50 0.17 0.70 0.57 0.90 0.83 0.23 0.10 0.83 0.70 0.37 0.17 0.43 0.23 0.10 0.63 0.43 0.37 0.17 0.10 0.70 0.17 0.63 0.50 0.30 0.70 0.63 0.37 0.30 0.90 0.57 0.50 0.43 0.30 0.90 0.77 0.57 0.10 7 9 5 A A 0.23 0.90 0.70 0.43 0.30 0.10 0.50 0.37 0.83 0.57 0.43 0.37 0.30 0.83 0.63 0.50 0.30 0.17 0.10 0.57 0.37 0.77 0.10 0.63 0.23 0.10 0.90 0.57 0.43 0.83 0.77 0.63 0.83 0.70 0.57 0.23 0.90 0.77 0.17 0.77 0.70 0.37 0.50 0.23 0.17 0.90 0.63 0.43 0.30 0.17 0.70 0.50 1 7 1 4 1 0.70 0.70 0.50 0.37 0.17 0.90 0.63 0.50 0.43 0.23 0.10 0.50 0.37 0.77 0.30 0.23 0.17 0.83 0.30 0.83 0.77 0.70 0.57 0.43 0.23 0.10 0.50 0.43 0.37 0.10 0.63 0.43 0.10 0.70 0.37 0.17 0.83 0.63 0.17 0.90 0.83 0.63 0.57 0.90 0.77 0.57 0.23 0.57 0.30 0.90 0.77 0.30 C C B 7 C 0.10 0.70 0.50 0.37 0.17 0.77 0.63 0.17 0.70 0.63 0.43 0.30 0.17 0.63 0.43 0.37 0.17 0.63 0.30 0.10 0.83 0.70 0.43 0.30 0.10 0.83 0.90 0.30 0.23 0.90 0.83 0.77 0.90 0.37 0.23 0.90 0.57 0.37 0.10 0.83 0.77 0.70 0.57 0.50 0.57 0.43 0.23 0.57 0.50 0.77 0.50 0.23 3 7 3 5 2 0.30 0.17 0.63 0.50 0.90 0.43 0.37 0.30 0.83 0.70 0.63 0.57 0.10 0.77 0.43 0.10 0.37 0.17 0.10 0.70 0.57 0.23 0.50 0.43 0.37 0.30 0.83 0.63 0.37 0.90 0.83 0.77 0.70 0.43 0.30 0.23 0.83 0.17 0.77 0.90 0.70 0.57 0.50 0.23 0.57 0.77 0.63 0.50 0.10 0.90 0.23 0.17 2 6 2 5 2 0.50 0.30 0.23 0.77 0.63 0.37 0.30 0.23 0.17 0.10 0.30 0.90 0.63 0.90 0.63 0.17 0.83 0.57 0.70 0.63 0.57 0.10 0.30 0.23 0.17 0.10 0.83 0.77 0.70 0.43 0.37 0.70 0.43 0.77 0.50 0.37 0.23 0.70 0.50 0.57 0.10 0.90 0.50 0.90 0.83 0.43 0.17 0.83 0.77 0.57 0.43 0.37 A 7 5 A B 0.90 0.70 0.50 0.90 0.83 0.50 0.30 0.77 0.43 0.23 0.10 0.70 0.57 0.57 0.23 0.10 0.63 0.23 0.83 0.63 0.57 0.83 0.63 0.37 0.17 0.10 0.77 0.63 0.37 0.70 0.57 0.37 0.10 0.90 0.70 0.50 0.37 0.50 0.30 0.83 0.43 0.30 0.17 0.77 0.43 0.17 0.30 0.17 0.90 0.77 0.43 0.23 4 5 3 3 6 0.70 0.57 0.43 0.17 0.10 0.43 0.17 0.90 0.37 0.90 0.83 0.50 0.30 0.50 0.37 0.70 0.37 0.30 0.57 0.30 0.23 0.17 0.10 0.70 0.17 0.10 0.90 0.77 0.63 0.23 0.90 0.63 0.57 0.23 0.10 0.70 0.43 0.57 0.23 0.83 0.30 0.83 0.77 0.50 0.83 0.77 0.63 0.50 0.77 0.63 0.43 0.37 5 3 4 6 5 0.63 0.17 0.90 0.63 0.43 0.37 0.10 0.63 0.37 0.17 0.57 0.43 0.17 0.90 0.77 0.70 0.57 0.43 0.37 0.50 0.30 0.77 0.57 0.10 0.90 0.10 0.83 0.83 0.70 0.57 0.17 0.83 0.50 0.43 0.23 0.83 0.77 0.63 0.50 0.50 0.30 0.23 0.10 0.77 0.23 0.90 0.70 0.30 0.70 0.37 0.30 0.23 8 A 4 7 6 0.57 0.43 0.23 0.17 0.50 0.83 0.77 0.50 0.37 0.30 0.70 0.57 0.43 0.90 0.70 0.77 0.63 0.57 0.70 0.57 0.43 0.17 0.10 0.50 0.37 0.17 0.77 0.50 0.83 0.43 0.37 0.23 0.10 0.90 0.23 0.90 0.83 0.63 0.30 0.83 0.63 0.37 0.30 0.10 0.90 0.70 0.30 0.17 0.63 0.77 0.23 0.10 7 7 8 6 5 0.50 0.30 0.83 0.50 0.30 0.83 0.70 0.30 0.17 0.77 0.63 0.23 0.10 0.63 0.57 0.23 0.77 0.63 0.37 0.17 0.77 0.63 0.83 0.70 0.43 0.30 0.90 0.83 0.70 0.43 0.37 0.10 0.43 0.90 0.23 0.90 0.57 0.37 0.17 0.77 0.17 0.90 0.70 0.57 0.23 0.10 0.57 0.50 0.43 0.37 0.10 0.50 2 1 7 4 2 0.63 0.50 0.17 0.10 0.57 0.10 0.50 0.37 0.23 0.63 0.57 0.30 0.23 0.83 0.77 0.70 0.57 0.43 0.23 0.90 0.43 0.10 0.77 0.70 0.43 0.10 0.90 0.23 0.83 0.77 0.70 0.63 0.50 0.37 0.83 0.63 0.17 0.83 0.37 0.37 0.30 0.90 0.43 0.30 0.17 0.77 0.70 0.57 0.30 0.90 0.50 0.17 7 9 6 A A 0.77 0.43 0.30 0.83 0.43 0.23 0.17 0.10 0.83 0.77 0.63 0.43 0.10 0.83 0.63 0.57 0.23 0.17 0.77 0.63 0.90 0.37 0.57 0.30 0.23 0.17 0.90 0.70 0.57 0.70 0.57 0.50 0.30 0.23 0.17 0.83 0.50 0.43 0.37 0.70 0.50 0.37 0.10 0.90 0.50 0.37 0.30 0.10 0.90 0.77 0.70 0.63 7 A 7 3 A 0.63 0.50 0.30 0.90 0.50 0.37 0.30 0.37 0.50 0.43 0.37 0.30 0.23 0.37 0.10 0.77 0.63 0.57 0.17 0.77 0.50 0.17 0.90 0.63 0.57 0.17 0.90 0.77 0.57 0.43 0.23 0.23 0.83 0.70 0.63 0.43 0.23 0.10 0.77 0.83 0.70 0.17 0.83 0.70 0.43 0.10 0.90 0.57 0.30 0.83 0.70 0.10 A 4 9 5 9 0.57 0.50 0.70 0.57 0.50 0.37 0.17 0.10 0.90 0.70 0.63 0.90 0.63 0.83 0.43 0.30 0.17 0.90 0.30 0.57 0.43 0.17 0.50 0.23 0.17 0.10 0.77 0.70 0.63 0.37 0.77 0.23 0.83 0.77 0.23 0.10 0.77 0.70 0.57 0.90 0.23 0.10 0.83 0.63 0.43 0.50 0.37 0.30 0.83 0.43 0.37 0.30 4 4 4 4 6 0.90 0.77 0.23 0.83 0.63 0.30 0.70 0.37 0.30 0.83 0.70 0.50 0.10 0.50 0.37 0.10 0.70 0.57 0.23 0.57 0.50 0.23 0.17 0.37 0.30 0.17 0.70 0.57 0.43 0.17 0.50 0.43 0.17 0.10 0.63 0.43 0.10 0.77 0.63 0.83 0.63 0.30 0.90 0.77 0.37 0.90 0.83 0.77 0.90 0.57 0.43 0.23 7 7 6 7 7 0.37 0.17 0.90 0.83 0.57 0.50 0.90 0.83 0.77 0.57 0.30 0.23 0.10 0.77 0.70 0.63 0.37 0.17 0.10 0.63 0.43 0.37 0.63 0.57 0.43 0.37 0.90 0.77 0.63 0.50 0.23 0.43 0.23 0.70 0.17 0.70 0.30 0.17 0.10 0.83 0.70 0.57 0.43 0.30 0.10 0.30 0.50 0.90 0.83 0.77 0.50 0.23 3 5 6 2 8 0.90 0.83 0.77 0.63 0.43 0.70 0.63 0.57 0.37 0.30 0.10 0.10 0.63 0.50 0.23 0.83 0.50 0.83 0.70 0.77 0.57 0.50 0.30 0.23 0.17 0.10 0.70 0.37 0.17 0.10 0.90 0.77 0.17 0.90 0.77 0.57 0.30 0.17 0.70 0.57 0.30 0.43 0.23 0.63 0.50 0.43 0.37 0.23 0.90 0.83 0.43 0.37 0 0 0 4 7 0.90 0.70 0.83 0.37 0.23 0.77 0.50 0.30 0.23 0.10 0.37 0.30 0.10 0.50 0.23 0.10 0.90 0.70 0.63 0.50 0.43 0.17 0.10 0.37 0.17 0.50 0.83 0.77 0.57 0.37 0.30 0.30 0.70 0.63 0.43 0.83 0.77 0.70 0.63 0.63 0.43 0.17 0.77 0.57 0.90 0.83 0.57 0.90 0.57 0.43 0.23 0.17 8 4 A 5 6 0.90 0.63 0.30 0.57 0.37 0.90 0.83 0.77 0.57 0.37 0.90 0.83 0.17 0.57 0.23 0.17 0.10 0.83 0.63 0.50 0.30 0.63 0.23 0.70 0.50 0.30 0.83 0.70 0.50 0.43 0.90 0.77 0.43 0.50 0.43 0.17 0.63 0.43 0.37 0.77 0.37 0.70 0.23 0.17 0.10 0.70 0.30 0.10 0.77 0.57 0.23 0.10 0 0 4 0 2 0.90 0.77 0.43 0.37 0.23 0.70 0.57 0.50 0.23 0.70 0.50 0.30 0.10 0.30 0.90 0.83 0.43 0.37 0.30 0.17 0.77 0.30 0.23 0.83 0.43 0.37 0.70 0.63 0.50 0.17 0.63 0.90 0.83 0.37 0.10 0.90 0.77 0.63 0.50 0.83 0.57 0.10 0.77 0.10 0.63 0.57 0.43 0.17 0.70 0.57 0.23 0.17 4 2 6 4 4 0.57 0.63 0.50 0.30 0.10 0.77 0.63 0.37 0.23 0.83 0.63 0.50 0.30 0.90 0.63 0.17 0.83 0.57 0.37 0.17 0.90 0.83 0.30 0.10 0.90 0.57 0.77 0.70 0.50 0.43 0.37 0.30 0.77 0.43 0.23 0.43 0.77 0.37 0.17 0.83 0.23 0.10 0.90 0.70 0.70 0.57 0.50 0.17 0.70 0.43 0.23 0.10 A 8 9 B 9

35

0.43 0.77 0.70 0.63 0.10 0.83 0.57 0.43 0.77 0.70 0.37 0.23 0.10 0.77 0.70 0.57 0.30 0.10 0.83 0.50 0.77 0.50 0.57 0.43 0.30 0.17 0.83 0.50 0.37 0.23 0.37 0.30 0.17 0.90 0.70 0.37 0.10 0.63 0.50 0.90 0.63 0.17 0.90 0.57 0.43 0.23 0.63 0.30 0.23 0.17 0.90 0.83 8 A 7 8 7 0.90 0.50 0.90 0.77 0.10 0.83 0.50 0.10 0.77 0.70 0.57 0.50 0.10 0.77 0.17 0.83 0.70 0.50 0.23 0.90 0.77 0.63 0.30 0.17 0.30 0.17 0.30 0.23 0.10 0.63 0.43 0.37 0.17 0.37 0.23 0.90 0.83 0.43 0.23 0.83 0.70 0.63 0.57 0.43 0.37 0.57 0.30 0.70 0.57 0.43 0.63 0.37 5 8 6 8 5 0.50 0.90 0.57 0.50 0.23 0.90 0.50 0.17 0.70 0.63 0.57 0.37 0.17 0.70 0.63 0.57 0.23 0.83 0.70 0.63 0.63 0.43 0.83 0.77 0.43 0.23 0.77 0.77 0.43 0.30 0.17 0.10 0.77 0.57 0.30 0.23 0.10 0.50 0.30 0.90 0.83 0.43 0.37 0.30 0.17 0.10 0.37 0.83 0.70 0.37 0.90 0.10 B B 5 6 8 0.90 0.83 0.37 0.90 0.63 0.37 0.30 0.23 0.90 0.63 0.83 0.77 0.17 0.50 0.23 0.77 0.57 0.70 0.57 0.30 0.23 0.90 0.43 0.37 0.30 0.10 0.70 0.17 0.83 0.43 0.17 0.10 0.83 0.77 0.10 0.70 0.63 0.57 0.50 0.77 0.63 0.57 0.43 0.30 0.10 0.70 0.50 0.50 0.43 0.37 0.17 0.23 1 5 1 3 2 0.83 0.10 0.50 0.37 0.77 0.70 0.63 0.37 0.30 0.70 0.43 0.30 0.17 0.57 0.37 0.30 0.23 0.83 0.77 0.70 0.30 0.23 0.50 0.23 0.10 0.23 0.90 0.70 0.43 0.90 0.57 0.10 0.90 0.90 0.77 0.57 0.50 0.37 0.10 0.77 0.63 0.50 0.17 0.63 0.43 0.17 0.83 0.57 0.43 0.17 0.83 0.63 3 6 5 3 2 0.77 0.70 0.30 0.90 0.63 0.57 0.50 0.17 0.90 0.83 0.77 0.63 0.17 0.83 0.57 0.50 0.43 0.37 0.17 0.37 0.30 0.83 0.70 0.30 0.37 0.23 0.63 0.83 0.77 0.43 0.17 0.77 0.63 0.57 0.10 0.70 0.43 0.30 0.10 0.90 0.23 0.10 0.70 0.23 0.10 0.90 0.50 0.43 0.37 0.23 0.57 0.50 4 7 2 6 2 0.70 0.50 0.90 0.83 0.70 0.50 0.90 0.63 0.37 0.30 0.10 0.63 0.50 0.90 0.63 0.37 0.30 0.77 0.63 0.43 0.30 0.23 0.83 0.23 0.30 0.23 0.43 0.57 0.17 0.70 0.57 0.50 0.43 0.17 0.90 0.83 0.77 0.57 0.37 0.83 0.77 0.57 0.23 0.17 0.10 0.37 0.10 0.77 0.70 0.43 0.17 0.10 7 8 5 2 3 0.70 0.37 0.30 0.23 0.17 0.90 0.83 0.70 0.57 0.77 0.43 0.30 0.17 0.57 0.50 0.43 0.10 0.23 0.90 0.70 0.63 0.50 0.37 0.10 0.83 0.77 0.90 0.83 0.77 0.63 0.50 0.37 0.17 0.57 0.90 0.63 0.57 0.37 0.10 0.77 0.63 0.43 0.30 0.10 0.83 0.23 0.70 0.50 0.43 0.30 0.23 0.17 4 4 1 6 5 0.63 0.57 0.50 0.37 0.10 0.23 0.83 0.23 0.17 0.10 0.83 0.70 0.10 0.90 0.77 0.43 0.30 0.90 0.70 0.57 0.17 0.10 0.90 0.77 0.57 0.50 0.83 0.23 0.17 0.83 0.77 0.50 0.90 0.77 0.57 0.63 0.30 0.23 0.17 0.70 0.63 0.43 0.37 0.30 0.70 0.63 0.50 0.43 0.37 0.30 0.43 0.37 5 5 A 7 7 0.90 0.77 0.50 0.30 0.23 0.10 0.70 0.43 0.30 0.70 0.37 0.23 0.90 0.83 0.63 0.37 0.77 0.63 0.50 0.17 0.90 0.70 0.63 0.43 0.30 0.10 0.43 0.17 0.57 0.37 0.23 0.10 0.77 0.57 0.43 0.10 0.57 0.50 0.37 0.70 0.57 0.90 0.83 0.83 0.63 0.50 0.30 0.17 0.83 0.77 0.23 0.17 A 7 A 9 B 0.90 0.50 0.23 0.17 0.90 0.63 0.43 0.70 0.50 0.10 0.90 0.70 0.37 0.37 0.30 0.83 0.50 0.90 0.83 0.43 0.30 0.83 0.77 0.63 0.30 0.23 0.77 0.70 0.63 0.57 0.10 0.77 0.70 0.23 0.77 0.23 0.57 0.50 0.10 0.83 0.43 0.57 0.37 0.30 0.17 0.10 0.63 0.57 0.37 0.17 0.43 0.17 4 4 6 8 7 0.37 0.30 0.23 0.83 0.77 0.57 0.63 0.43 0.37 0.10 0.70 0.30 0.10 0.83 0.57 0.17 0.10 0.70 0.63 0.50 0.23 0.77 0.70 0.57 0.57 0.50 0.70 0.63 0.43 0.37 0.30 0.17 0.90 0.23 0.90 0.83 0.77 0.63 0.37 0.90 0.77 0.50 0.90 0.43 0.10 0.83 0.50 0.30 0.17 0.43 0.23 0.17 5 8 7 7 5 0.63 0.23 0.90 0.70 0.43 0.37 0.30 0.10 0.90 0.83 0.43 0.37 0.83 0.90 0.57 0.43 0.10 0.50 0.17 0.70 0.63 0.57 0.30 0.90 0.30 0.23 0.83 0.77 0.37 0.77 0.23 0.50 0.17 0.77 0.63 0.50 0.43 0.37 0.17 0.70 0.50 0.30 0.17 0.83 0.63 0.57 0.77 0.23 0.10 0.70 0.57 0.10 5 6 3 4 3 0.83 0.43 0.30 0.83 0.77 0.63 0.17 0.83 0.50 0.37 0.77 0.37 0.17 0.63 0.37 0.90 0.70 0.77 0.57 0.30 0.23 0.70 0.57 0.50 0.30 0.23 0.90 0.70 0.57 0.10 0.23 0.10 0.90 0.70 0.43 0.17 0.10 0.90 0.83 0.77 0.50 0.23 0.17 0.57 0.50 0.43 0.37 0.30 0.63 0.63 0.43 0.10 1 2 3 1 3 0.63 0.57 0.10 0.43 0.23 0.70 0.63 0.57 0.77 0.70 0.57 0.23 0.17 0.43 0.30 0.17 0.63 0.57 0.37 0.30 0.17 0.77 0.43 0.43 0.37 0.30 0.50 0.37 0.23 0.70 0.50 0.90 0.30 0.23 0.90 0.83 0.63 0.50 0.10 0.90 0.83 0.77 0.70 0.90 0.83 0.77 0.10 0.83 0.50 0.37 0.17 0.10 7 A C A 6 0.77 0.63 0.57 0.30 0.23 0.90 0.83 0.77 0.63 0.43 0.50 0.43 0.70 0.43 0.17 0.57 0.10 0.90 0.83 0.10 0.50 0.43 0.30 0.23 0.17 0.10 0.37 0.70 0.50 0.37 0.30 0.17 0.57 0.30 0.17 0.90 0.83 0.63 0.37 0.90 0.83 0.70 0.50 0.10 0.23 0.77 0.70 0.63 0.37 0.23 0.77 0.57 5 7 1 8 7 0.90 0.37 0.43 0.37 0.30 0.10 0.90 0.77 0.57 0.17 0.83 0.63 0.37 0.77 0.70 0.43 0.23 0.17 0.77 0.70 0.50 0.70 0.63 0.50 0.23 0.10 0.83 0.63 0.10 0.90 0.63 0.83 0.43 0.77 0.70 0.57 0.50 0.30 0.17 0.57 0.50 0.30 0.83 0.57 0.23 0.17 0.37 0.30 0.90 0.43 0.23 0.10 1 6 4 5 1 0.83 0.63 0.37 0.23 0.57 0.17 0.83 0.70 0.23 0.83 0.57 0.30 0.17 0.70 0.57 0.17 0.10 0.83 0.50 0.30 0.77 0.63 0.37 0.17 0.10 0.23 0.50 0.30 0.63 0.37 0.23 0.90 0.57 0.50 0.43 0.30 0.90 0.77 0.63 0.90 0.77 0.43 0.90 0.77 0.70 0.43 0.10 0.70 0.50 0.43 0.37 0.10 6 8 9 6 7 0.83 0.77 0.37 0.90 0.70 0.63 0.57 0.50 0.43 0.37 0.57 0.43 0.23 0.63 0.50 0.23 0.10 0.77 0.43 0.90 0.70 0.30 0.10 0.83 0.63 0.50 0.57 0.30 0.17 0.83 0.50 0.37 0.30 0.77 0.63 0.57 0.23 0.90 0.77 0.90 0.70 0.43 0.23 0.17 0.10 0.83 0.17 0.70 0.37 0.30 0.17 0.10 5 5 3 4 6 0.43 0.17 0.10 0.83 0.43 0.10 0.70 0.43 0.30 0.23 0.10 0.57 0.17 0.63 0.23 0.77 0.70 0.23 0.77 0.63 0.57 0.83 0.77 0.50 0.23 0.10 0.90 0.77 0.57 0.37 0.30 0.57 0.37 0.30 0.90 0.50 0.37 0.70 0.30 0.83 0.70 0.50 0.90 0.63 0.50 0.17 0.83 0.17 0.90 0.63 0.43 0.37 A 8 A 8 A 0.83 0.50 0.37 0.10 0.50 0.23 0.17 0.37 0.23 0.63 0.50 0.23 0.17 0.90 0.57 0.30 0.17 0.70 0.63 0.57 0.10 0.90 0.83 0.57 0.70 0.37 0.43 0.83 0.37 0.30 0.70 0.63 0.43 0.17 0.90 0.83 0.77 0.57 0.10 0.77 0.70 0.63 0.23 0.90 0.77 0.43 0.77 0.50 0.30 0.10 0.43 0.30 8 B B A 6 0.90 0.83 0.57 0.23 0.83 0.37 0.30 0.10 0.57 0.50 0.37 0.70 0.37 0.70 0.50 0.43 0.37 0.30 0.90 0.70 0.57 0.77 0.63 0.17 0.30 0.23 0.77 0.17 0.77 0.63 0.50 0.43 0.17 0.90 0.83 0.23 0.10 0.90 0.77 0.63 0.10 0.23 0.70 0.43 0.30 0.83 0.63 0.57 0.50 0.43 0.17 0.10 3 4 2 2 7

36

0.90 0.77 0.70 0.30 0.50 0.37 0.23 0.10 0.77 0.57 0.50 0.37 0.10 0.57 0.43 0.10 0.77 0.17 0.83 0.77 0.37 0.17 0.10 0.83 0.43 0.17 0.83 0.63 0.50 0.37 0.90 0.70 0.57 0.43 0.57 0.50 0.30 0.70 0.63 0.23 0.17 0.83 0.63 0.30 0.90 0.70 0.63 0.43 0.23 0.90 0.30 0.23 8 4 4 A 5 0.70 0.23 0.63 0.37 0.23 0.17 0.90 0.83 0.83 0.63 0.43 0.30 0.10 0.57 0.50 0.43 0.77 0.50 0.43 0.77 0.57 0.37 0.30 0.23 0.77 0.70 0.90 0.77 0.63 0.10 0.10 0.50 0.43 0.17 0.10 0.57 0.37 0.23 0.17 0.83 0.37 0.30 0.17 0.90 0.83 0.70 0.57 0.30 0.70 0.63 0.90 0.50 6 6 7 6 3 0.77 0.37 0.30 0.17 0.10 0.70 0.63 0.37 0.83 0.77 0.57 0.30 0.23 0.90 0.70 0.63 0.57 0.50 0.83 0.57 0.43 0.30 0.23 0.90 0.43 0.17 0.83 0.43 0.57 0.50 0.23 0.17 0.10 0.77 0.70 0.50 0.37 0.70 0.63 0.23 0.90 0.77 0.43 0.30 0.90 0.63 0.17 0.10 0.83 0.50 0.37 0.10 9 9 9 A A 0.70 0.43 0.10 0.57 0.17 0.10 0.83 0.43 0.37 0.10 0.77 0.37 0.10 0.83 0.77 0.63 0.37 0.30 0.77 0.70 0.50 0.30 0.30 0.17 0.63 0.57 0.90 0.50 0.90 0.83 0.90 0.70 0.63 0.23 0.83 0.70 0.50 0.43 0.30 0.57 0.23 0.17 0.63 0.43 0.37 0.23 0.77 0.57 0.50 0.90 0.23 0.17 2 6 6 1 2 0.90 0.83 0.50 0.37 0.90 0.83 0.70 0.57 0.77 0.23 0.77 0.50 0.17 0.63 0.50 0.43 0.37 0.30 0.23 0.17 0.63 0.17 0.10 0.90 0.83 0.37 0.77 0.57 0.23 0.17 0.10 0.77 0.10 0.70 0.43 0.30 0.70 0.57 0.23 0.70 0.43 0.30 0.63 0.90 0.83 0.57 0.50 0.37 0.63 0.43 0.30 0.10 4 3 6 8 6 0.83 0.37 0.30 0.17 0.10 0.37 0.23 0.90 0.43 0.23 0.77 0.50 0.23 0.70 0.43 0.77 0.63 0.50 0.77 0.70 0.37 0.10 0.83 0.43 0.30 0.17 0.90 0.77 0.63 0.50 0.90 0.30 0.83 0.57 0.30 0.90 0.70 0.57 0.37 0.57 0.23 0.83 0.70 0.57 0.43 0.17 0.10 0.63 0.50 0.17 0.63 0.10 1 1 6 4 1 0.90 0.83 0.50 0.30 0.83 0.63 0.43 0.17 0.43 0.30 0.17 0.83 0.10 0.37 0.23 0.90 0.77 0.50 0.30 0.23 0.70 0.37 0.23 0.77 0.50 0.37 0.57 0.10 0.70 0.57 0.37 0.10 0.83 0.77 0.10 0.70 0.63 0.23 0.17 0.77 0.70 0.63 0.43 0.17 0.90 0.63 0.57 0.50 0.90 0.57 0.43 0.30 7 8 5 8 7 0.23 0.90 0.77 0.70 0.63 0.37 0.30 0.83 0.77 0.50 0.43 0.30 0.10 0.77 0.70 0.57 0.17 0.50 0.43 0.23 0.17 0.37 0.90 0.83 0.77 0.37 0.90 0.10 0.57 0.90 0.70 0.57 0.43 0.30 0.17 0.10 0.70 0.63 0.50 0.83 0.63 0.50 0.43 0.37 0.30 0.83 0.10 0.63 0.23 0.57 0.23 0.17 0 9 0 0 6 0.70 0.63 0.57 0.37 0.10 0.90 0.90 0.70 0.23 0.10 0.57 0.43 0.23 0.50 0.30 0.23 0.17 0.63 0.23 0.37 0.30 0.63 0.50 0.37 0.17 0.10 0.83 0.77 0.83 0.77 0.70 0.10 0.83 0.63 0.50 0.83 0.77 0.70 0.30 0.90 0.43 0.57 0.50 0.43 0.37 0.30 0.17 0.77 0.57 0.43 0.17 0.90 2 2 4 3 3 0.57 0.43 0.30 0.83 0.63 0.57 0.43 0.70 0.57 0.50 0.23 0.37 0.30 0.77 0.50 0.10 0.77 0.70 0.37 0.30 0.77 0.17 0.63 0.57 0.43 0.10 0.90 0.83 0.70 0.37 0.23 0.63 0.43 0.37 0.30 0.10 0.77 0.50 0.17 0.63 0.23 0.17 0.90 0.50 0.17 0.10 0.90 0.83 0.90 0.83 0.70 0.23 8 5 8 5 8 0.70 0.63 0.17 0.10 0.83 0.77 0.43 0.30 0.10 0.37 0.30 0.23 0.50 0.77 0.57 0.63 0.50 0.37 0.90 0.83 0.70 0.43 0.10 0.90 0.77 0.57 0.83 0.50 0.43 0.30 0.90 0.57 0.50 0.17 0.63 0.43 0.23 0.17 0.10 0.90 0.37 0.23 0.70 0.57 0.23 0.17 0.77 0.63 0.83 0.70 0.37 0.30 A 6 7 A 9 0.77 0.57 0.30 0.23 0.70 0.10 0.90 0.63 0.17 0.83 0.63 0.50 0.23 0.90 0.63 0.17 0.10 0.83 0.30 0.17 0.43 0.37 0.10 0.90 0.37 0.17 0.83 0.70 0.43 0.37 0.77 0.57 0.50 0.43 0.23 0.83 0.77 0.57 0.43 0.50 0.90 0.63 0.37 0.77 0.70 0.57 0.50 0.30 0.23 0.70 0.30 0.10 5 4 5 8 6 0.83 0.77 0.57 0.50 0.43 0.30 0.17 0.83 0.70 0.57 0.50 0.10 0.57 0.70 0.57 0.50 0.30 0.90 0.77 0.43 0.83 0.77 0.63 0.43 0.37 0.30 0.90 0.70 0.63 0.37 0.83 0.77 0.23 0.63 0.30 0.23 0.50 0.23 0.10 0.23 0.17 0.10 0.90 0.63 0.43 0.37 0.10 0.37 0.17 0.90 0.70 0.17 7 4 C 5 C 0.90 0.83 0.30 0.57 0.23 0.17 0.90 0.50 0.30 0.43 0.30 0.23 0.10 0.50 0.43 0.37 0.10 0.77 0.70 0.50 0.43 0.77 0.77 0.70 0.63 0.17 0.63 0.90 0.63 0.37 0.30 0.10 0.83 0.63 0.57 0.37 0.10 0.57 0.37 0.77 0.70 0.57 0.23 0.17 0.83 0.70 0.43 0.23 0.17 0.90 0.83 0.50 6 9 4 4 7 0.90 0.43 0.23 0.63 0.10 0.63 0.43 0.17 0.83 0.77 0.43 0.30 0.23 0.17 0.70 0.37 0.17 0.77 0.70 0.50 0.23 0.10 0.90 0.70 0.63 0.17 0.77 0.70 0.50 0.30 0.57 0.50 0.43 0.30 0.23 0.83 0.37 0.30 0.57 0.83 0.63 0.57 0.37 0.10 0.90 0.83 0.77 0.90 0.57 0.50 0.37 0.10 9 8 8 A 8 0.63 0.23 0.63 0.10 0.90 0.77 0.57 0.43 0.83 0.77 0.70 0.50 0.37 0.90 0.77 0.43 0.30 0.83 0.70 0.37 0.83 0.70 0.63 0.30 0.57 0.17 0.70 0.57 0.10 0.90 0.77 0.43 0.17 0.37 0.23 0.10 0.63 0.43 0.30 0.83 0.50 0.37 0.17 0.57 0.50 0.30 0.23 0.50 0.17 0.90 0.23 0.10 6 8 8 6 6 0.70 0.50 0.30 0.17 0.83 0.70 0.37 0.17 0.10 0.37 0.23 0.77 0.50 0.77 0.50 0.43 0.23 0.70 0.63 0.43 0.90 0.83 0.63 0.43 0.23 0.17 0.90 0.43 0.37 0.23 0.90 0.63 0.30 0.90 0.30 0.10 0.57 0.30 0.10 0.83 0.77 0.63 0.57 0.10 0.57 0.83 0.77 0.57 0.50 0.17 0.70 0.37 8 7 6 A 9 0.77 0.70 0.57 0.83 0.77 0.43 0.30 0.10 0.83 0.70 0.63 0.57 0.23 0.90 0.70 0.17 0.10 0.90 0.63 0.43 0.17 0.37 0.23 0.50 0.37 0.17 0.63 0.57 0.43 0.37 0.83 0.30 0.90 0.70 0.57 0.50 0.17 0.43 0.30 0.83 0.77 0.50 0.30 0.23 0.50 0.37 0.23 0.10 0.63 0.90 0.77 0.10 7 8 8 2 2 0.50 0.23 0.17 0.57 0.50 0.23 0.90 0.83 0.70 0.63 0.57 0.43 0.23 0.77 0.70 0.63 0.37 0.70 0.63 0.43 0.57 0.10 0.90 0.30 0.17 0.10 0.90 0.57 0.43 0.83 0.77 0.10 0.83 0.77 0.63 0.43 0.17 0.50 0.37 0.83 0.30 0.10 0.90 0.37 0.30 0.17 0.70 0.50 0.37 0.30 0.23 0.77 4 6 7 7 3 0.90 0.83 0.43 0.37 0.77 0.90 0.83 0.50 0.23 0.10 0.83 0.23 0.17 0.77 0.70 0.17 0.10 0.90 0.83 0.70 0.50 0.43 0.10 0.57 0.37 0.17 0.63 0.50 0.23 0.37 0.30 0.17 0.63 0.70 0.63 0.50 0.43 0.37 0.10 0.57 0.30 0.63 0.57 0.23 0.77 0.70 0.43 0.30 0.90 0.77 0.57 0.30 8 5 8 7 3 0.77 0.63 0.10 0.83 0.77 0.30 0.77 0.70 0.50 0.30 0.83 0.23 0.17 0.83 0.70 0.37 0.23 0.17 0.10 0.90 0.37 0.23 0.70 0.50 0.43 0.10 0.90 0.43 0.30 0.70 0.57 0.43 0.37 0.83 0.63 0.17 0.77 0.63 0.30 0.57 0.50 0.23 0.17 0.90 0.63 0.50 0.57 0.43 0.10 0.90 0.57 0.37 4 5 4 4 5 0.63 0.50 0.30 0.10 0.43 0.23 0.37 0.23 0.17 0.83 0.63 0.30 0.17 0.83 0.57 0.23 0.77 0.70 0.63 0.37 0.57 0.43 0.10 0.70 0.57 0.43 0.77 0.70 0.37 0.90 0.50 0.10 0.90 0.77 0.63 0.77 0.50 0.37 0.23 0.90 0.43 0.17 0.83 0.57 0.30 0.17 0.83 0.70 0.50 0.30 0.90 0.10 8 7 9 8 6

37

0.90 0.23 0.90 0.83 0.50 0.43 0.37 0.30 0.63 0.50 0.43 0.37 0.23 0.70 0.57 0.10 0.70 0.23 0.83 0.43 0.23 0.90 0.77 0.57 0.50 0.10 0.83 0.77 0.63 0.10 0.90 0.70 0.37 0.30 0.10 0.83 0.70 0.63 0.30 0.63 0.50 0.43 0.37 0.30 0.17 0.77 0.57 0.17 0.77 0.57 0.17 0.17 2 7 3 3 4 0.90 0.83 0.57 0.77 0.57 0.43 0.30 0.63 0.17 0.90 0.77 0.57 0.50 0.77 0.70 0.37 0.23 0.10 0.83 0.70 0.57 0.37 0.43 0.30 0.23 0.17 0.63 0.50 0.23 0.90 0.83 0.50 0.43 0.30 0.10 0.83 0.70 0.37 0.10 0.50 0.43 0.30 0.17 0.90 0.70 0.63 0.37 0.17 0.10 0.77 0.23 0.63 2 7 5 2 1 0.30 0.83 0.77 0.63 0.50 0.70 0.17 0.10 0.83 0.70 0.57 0.30 0.10 0.90 0.57 0.50 0.17 0.90 0.70 0.43 0.30 0.17 0.90 0.77 0.50 0.43 0.83 0.77 0.43 0.37 0.37 0.23 0.83 0.63 0.57 0.43 0.37 0.23 0.17 0.70 0.63 0.23 0.10 0.57 0.10 0.77 0.50 0.30 0.90 0.63 0.37 0.23 6 8 7 5 7 0.77 0.43 0.30 0.83 0.57 0.43 0.83 0.57 0.83 0.63 0.57 0.50 0.43 0.70 0.50 0.37 0.23 0.17 0.77 0.90 0.70 0.37 0.30 0.10 0.90 0.23 0.90 0.83 0.63 0.10 0.70 0.63 0.37 0.23 0.17 0.63 0.43 0.77 0.17 0.57 0.90 0.50 0.30 0.10 0.77 0.50 0.23 0.17 0.70 0.37 0.30 0.10 7 6 4 9 5 0.63 0.37 0.70 0.63 0.57 0.17 0.63 0.37 0.17 0.10 0.70 0.57 0.37 0.83 0.50 0.23 0.90 0.50 0.23 0.77 0.50 0.43 0.90 0.50 0.43 0.17 0.90 0.43 0.83 0.43 0.10 0.90 0.83 0.70 0.57 0.30 0.77 0.63 0.23 0.77 0.70 0.57 0.30 0.17 0.10 0.77 0.37 0.30 0.23 0.83 0.30 0.10 6 9 5 5 8 0.83 0.70 0.23 0.83 0.77 0.23 0.17 0.63 0.37 0.23 0.77 0.70 0.30 0.63 0.37 0.70 0.57 0.10 0.90 0.83 0.70 0.90 0.83 0.63 0.17 0.10 0.57 0.30 0.17 0.63 0.50 0.30 0.77 0.57 0.50 0.57 0.50 0.43 0.37 0.90 0.77 0.50 0.43 0.10 0.90 0.43 0.37 0.43 0.30 0.17 0.10 0.23 9 A 9 A 9 0.23 0.17 0.70 0.57 0.83 0.63 0.57 0.50 0.90 0.83 0.70 0.43 0.10 0.77 0.70 0.10 0.90 0.83 0.50 0.30 0.90 0.23 0.63 0.50 0.37 0.30 0.83 0.63 0.43 0.37 0.30 0.37 0.23 0.77 0.70 0.43 0.30 0.57 0.23 0.90 0.57 0.50 0.77 0.63 0.43 0.17 0.10 0.37 0.17 0.10 0.77 0.17 8 6 9 5 6 0.70 0.43 0.10 0.50 0.23 0.17 0.57 0.50 0.43 0.10 0.83 0.63 0.57 0.63 0.37 0.83 0.77 0.70 0.63 0.10 0.90 0.23 0.77 0.37 0.23 0.10 0.83 0.57 0.30 0.17 0.90 0.57 0.30 0.63 0.37 0.70 0.43 0.30 0.17 0.90 0.77 0.50 0.23 0.43 0.37 0.83 0.77 0.70 0.30 0.17 0.90 0.50 B 9 B B 9 0.57 0.50 0.17 0.10 0.50 0.43 0.23 0.70 0.10 0.77 0.70 0.37 0.30 0.83 0.63 0.30 0.23 0.83 0.77 0.70 0.63 0.30 0.90 0.63 0.63 0.50 0.90 0.77 0.70 0.43 0.90 0.57 0.57 0.43 0.17 0.90 0.83 0.57 0.43 0.37 0.37 0.17 0.10 0.83 0.77 0.50 0.37 0.30 0.23 0.23 0.17 0.10 4 4 8 8 3 0.37 0.17 0.83 0.50 0.90 0.77 0.57 0.50 0.83 0.70 0.50 0.43 0.30 0.83 0.30 0.10 0.90 0.63 0.23 0.83 0.30 0.23 0.17 0.10 0.77 0.37 0.90 0.77 0.70 0.63 0.50 0.43 0.70 0.57 0.43 0.10 0.70 0.37 0.90 0.57 0.23 0.77 0.37 0.30 0.17 0.63 0.43 0.63 0.57 0.23 0.17 0.10 2 2 3 4 4 0.83 0.83 0.70 0.63 0.37 0.30 0.23 0.77 0.50 0.30 0.23 0.10 0.50 0.90 0.70 0.63 0.37 0.90 0.57 0.43 0.90 0.83 0.63 0.30 0.23 0.10 0.77 0.57 0.50 0.50 0.10 0.90 0.83 0.70 0.57 0.57 0.43 0.37 0.17 0.43 0.30 0.23 0.17 0.10 0.77 0.43 0.17 0.63 0.37 0.17 0.77 0.70 8 B 3 3 A 0.63 0.37 0.23 0.10 0.77 0.57 0.43 0.37 0.23 0.17 0.77 0.70 0.43 0.90 0.57 0.17 0.63 0.43 0.30 0.77 0.70 0.63 0.50 0.50 0.37 0.10 0.83 0.70 0.90 0.83 0.70 0.50 0.37 0.23 0.83 0.57 0.90 0.63 0.57 0.77 0.50 0.43 0.30 0.17 0.10 0.90 0.30 0.10 0.83 0.30 0.23 0.17 5 7 4 7 6 0.77 0.63 0.43 0.30 0.30 0.90 0.50 0.30 0.10 0.90 0.77 0.50 0.17 0.10 0.57 0.37 0.17 0.83 0.77 0.43 0.23 0.83 0.70 0.37 0.30 0.10 0.83 0.17 0.90 0.83 0.77 0.70 0.50 0.43 0.37 0.63 0.57 0.43 0.23 0.90 0.70 0.57 0.50 0.37 0.23 0.63 0.23 0.10 0.70 0.63 0.57 0.17 3 4 2 6 4 0.63 0.50 0.10 0.37 0.30 0.10 0.77 0.43 0.77 0.70 0.43 0.30 0.10 0.57 0.43 0.30 0.17 0.83 0.50 0.43 0.23 0.83 0.70 0.63 0.10 0.17 0.90 0.70 0.23 0.77 0.57 0.17 0.57 0.50 0.37 0.23 0.57 0.37 0.23 0.83 0.77 0.37 0.90 0.70 0.63 0.90 0.30 0.17 0.90 0.83 0.63 0.50 C C C C A 0.57 0.30 0.17 0.83 0.77 0.37 0.17 0.83 0.70 0.57 0.37 0.17 0.63 0.90 0.70 0.43 0.63 0.57 0.43 0.23 0.63 0.50 0.43 0.30 0.90 0.50 0.83 0.37 0.10 0.70 0.10 0.90 0.77 0.77 0.57 0.43 0.37 0.30 0.10 0.77 0.63 0.50 0.23 0.90 0.50 0.30 0.23 0.10 0.83 0.70 0.23 0.17 6 7 6 5 5 0.83 0.70 0.30 0.70 0.37 0.17 0.10 0.90 0.77 0.70 0.37 0.23 0.10 0.63 0.57 0.37 0.30 0.23 0.77 0.63 0.43 0.17 0.63 0.57 0.57 0.17 0.90 0.17 0.10 0.83 0.70 0.50 0.90 0.83 0.43 0.30 0.83 0.43 0.30 0.77 0.50 0.43 0.90 0.57 0.37 0.23 0.10 0.77 0.50 0.23 0.63 0.50 1 5 5 1 1 0.70 0.63 0.43 0.37 0.30 0.23 0.10 0.57 0.30 0.23 0.90 0.57 0.43 0.83 0.77 0.23 0.10 0.90 0.57 0.37 0.17 0.77 0.37 0.70 0.63 0.10 0.90 0.50 0.90 0.83 0.63 0.50 0.43 0.83 0.77 0.37 0.30 0.23 0.17 0.57 0.30 0.17 0.83 0.77 0.70 0.63 0.50 0.43 0.70 0.17 0.10 0.50 4 6 8 1 2 0.83 0.77 0.23 0.83 0.70 0.43 0.17 0.10 0.90 0.17 0.57 0.23 0.10 0.70 0.30 0.17 0.63 0.50 0.23 0.83 0.57 0.50 0.43 0.30 0.23 0.77 0.90 0.50 0.10 0.90 0.57 0.37 0.30 0.77 0.10 0.90 0.83 0.63 0.30 0.63 0.57 0.43 0.37 0.77 0.70 0.63 0.37 0.70 0.50 0.43 0.37 0.17 1 3 1 6 3 0.83 0.50 0.37 0.10 0.37 0.90 0.63 0.50 0.30 0.90 0.70 0.50 0.37 0.70 0.63 0.23 0.17 0.77 0.43 0.10 0.77 0.70 0.57 0.37 0.17 0.43 0.90 0.43 0.30 0.57 0.50 0.30 0.23 0.17 0.23 0.10 0.57 0.30 0.10 0.77 0.57 0.90 0.83 0.70 0.63 0.83 0.43 0.83 0.77 0.63 0.23 0.17 8 6 8 8 7 0.77 0.70 0.17 0.50 0.10 0.70 0.63 0.43 0.37 0.90 0.83 0.23 0.17 0.83 0.57 0.50 0.23 0.90 0.77 0.70 0.30 0.83 0.57 0.30 0.70 0.10 0.90 0.63 0.37 0.30 0.83 0.63 0.57 0.37 0.17 0.77 0.63 0.50 0.30 0.43 0.10 0.43 0.23 0.90 0.50 0.23 0.17 0.10 0.77 0.57 0.43 0.37 5 4 5 7 5 0.63 0.90 0.83 0.77 0.63 0.30 0.23 0.77 0.57 0.43 0.43 0.37 0.23 0.90 0.83 0.30 0.17 0.37 0.17 0.50 0.17 0.10 0.77 0.50 0.30 0.17 0.77 0.70 0.50 0.43 0.37 0.23 0.10 0.50 0.43 0.10 0.90 0.30 0.70 0.57 0.70 0.57 0.83 0.70 0.63 0.37 0.23 0.90 0.83 0.63 0.57 0.10 6 4 3 8 9 0.63 0.57 0.43 0.30 0.57 0.23 0.90 0.70 0.63 0.23 0.10 0.37 0.30 0.50 0.17 0.63 0.43 0.37 0.17 0.77 0.37 0.77 0.57 0.50 0.43 0.10 0.90 0.70 0.83 0.77 0.70 0.83 0.57 0.43 0.30 0.83 0.63 0.23 0.17 0.83 0.77 0.37 0.23 0.10 0.90 0.50 0.30 0.10 0.50 0.17 0.90 0.70 4 6 7 3 6

38

The data continues……

39

( c) Table 4-An Extract of actual value compared with values obtained from two activation functions of cascade-correlation networks.

S.NoActual value in

GIBCalculated valuein Log Sigmoid

function

Calculated value inHyperbolic Sigmoid

function1 0.69000 0.83335 0.637652 0.76000 0.28511 0.729583 0.92000 0.97990 0.904124 0.76000 0.97068 0.876775 0.69000 0.82824 0.914996 0.46000 0.32592 0.443117 0.53000 0.83548 0.742498 0.69000 0.62037 0.406849 0.46000 0.73385 0.5439410 0.76000 0.64635 0.6053311 0.53000 0.47536 0.7491012 0.92000 0.90222 0.7069513 0.53000 0.76448 0.5737214 0.46000 0.30870 0.4874015 0.76000 0.44391 0.6627316 0.46000 0.58714 0.3753517 0.46000 0.40213 0.6258118 0.76000 0.40387 0.3922319 0.61000 0.89480 0.6311820 0.53000 0.69884 0.4173721 0.76000 0.37357 0.5302022 0.76000 0.38286 0.7133423 0.76000 0.51486 0.7308224 0.46000 0.31351 0.6396825 0.53000 0.49473 0.7597226 0.61000 0.54258 0.4157127 0.53000 0.51326 0.4495328 0.76000 0.78753 0.6616729 0.30000 0.20979 0.3445930 0.46000 0.50345 0.6126331 0.84000 0.94730 0.6622232 0.76000 0.95802 0.65976

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33 0.61000 0.86574 0.5420634 0.84000 0.94381 0.7431835 0.38000 0.58444 0.5150636 0.46000 0.72572 0.5572037 0.53000 0.56715 0.6420238 0.61000 0.59770 0.5594639 0.46000 0.86681 0.4887240 0.76000 0.63020 0.7795441 0.84000 0.57169 0.7180942 0.61000 0.77635 0.8920543 0.61000 0.76880 0.5991044 0.46000 0.47845 0.5811845 0.23000 0.32951 0.5968346 0.92000 0.66339 0.8167747 0.61000 0.94995 0.7217248 0.46000 0.36253 0.5238149 0.69000 0.83971 0.6366150 0.46000 0.45303 0.4818451 0.76000 0.61212 0.8431652 0.84000 0.63145 0.8007353 0.53000 0.76219 0.4119254 0.46000 0.40733 0.4355455 0.53000 0.89537 0.5913756 0.76000 0.89543 0.9237357 0.46000 0.33528 0.5987758 0.61000 0.39483 0.6749059 0.46000 0.25362 0.4208760 0.61000 0.89595 0.7839561 0.69000 0.71038 0.8193462 0.30000 0.74626 0.4895763 0.61000 0.42135 0.5288764 0.76000 0.75359 0.7053765 0.61000 0.69023 0.7866166 0.92000 0.51392 0.6069867 0.69000 0.75857 0.6509868 0.76000 0.77227 0.7666269 0.61000 0.67357 0.7299070 0.76000 0.90963 0.6793171 0.61000 0.41452 0.6387072 0.53000 0.74739 0.6425373 0.61000 0.61081 0.66776

41

74 0.38000 0.43853 0.4257375 0.69000 0.94996 0.6635576 0.53000 0.45178 0.3109877 0.53000 0.58841 0.4428378 0.61000 0.43248 0.8172179 0.69000 0.68011 0.3861280 0.69000 0.37439 0.5681781 0.76000 0.94242 0.7948282 0.69000 0.91246 0.7548883 0.84000 0.82203 0.9080884 0.61000 0.49005 0.5705385 0.30000 0.31854 0.2174286 0.84000 0.72787 0.8869487 0.53000 0.45357 0.5923188 0.46000 0.87370 0.4970989 0.92000 0.58892 0.8893790 0.53000 0.43798 0.5124291 0.38000 0.61122 0.4189992 0.61000 0.53027 0.5661293 0.46000 0.52042 0.39828

The data continues…

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