Research Article Predictive Modeling in Race Walkingdownloads.hindawi.com › journals › cin › 2015 › 735060.pdfResearch Article Predictive Modeling in Race Walking KrzysztofWiktorowicz,

Research ArticlePredictive Modeling in Race Walking

Krzysztof Wiktorowicz1 Krzysztof Przednowek2

LesBaw Lassota2 and Tomasz Krzeszowski1

1Faculty of Electrical and Computer Engineering Rzeszow University of Technology 35-959 Rzeszow Poland2Faculty of Physical Education University of Rzeszow 35-959 Rzeszow Poland

Correspondence should be addressed to Krzysztof Wiktorowicz kwiktorprzedupl

Received 15 December 2014 Accepted 18 June 2015

Academic Editor Okyay Kaynak

Copyright copy 2015 Krzysztof Wiktorowicz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents the use of linear and nonlinear multivariable models as tools to support training process of race walkers Thesemodels are calculated using data collected from race walkersrsquo training events and they are used to predict the result over a 3 km racebased on training loads The material consists of 122 training plans for 21 athletes In order to choose the best model leave-one-outcross-validation method is usedThemain contribution of the paper is to propose the nonlinear modifications for linear models inorder to achieve smaller prediction error It is shown that the best model is a modified LASSO regression with quadratic terms inthe nonlinear part This model has the smallest prediction error and simplified structure by eliminating some of the predictors

1 Introduction

The level of todayrsquos high-performance sport is very high andvery even Both coaches and competitors are forced to searchfor and use newer and sometimes innovative solutions inthe process of sports training [1] A solution supporting thisprocess may be the application of various types of regressionmodels

Prediction in sport concerns many aspects includingthe prediction of performance results [2 3] or predictingsporting talent [4 5] Models predicting results in sporttaking into account the seasonal statistics of each team werealso constructed [6] The application of predictive modelsin athletics was described by Maszczyk et al [2] where theregressionwas used to predict results in a javelin throwThesemodels were applied to support the choice and selection ofprospective javelin throwers

Prediction of sports results using linear regression wasalso presented in the work by Przednowek and Wiktorow-icz [7] A linear predictive model implemented by ridgeregression was applied to predict the outcomes of a walkingrace after the immediate preparation phase As input for themodel the basic somatic features (height and weight) andtraining loads (training components) for each day of training

were provided and the output was the result expected overa distance of 5 km In addition to linear models artificialneural networks whose parameters were specified in cross-validation were also used to implement this task

In the paper by Drake and James [8] the regressionsestimating the results over distances of 5 10 20 and 50 kmand the levels of the selected physiological parameters (egVO2max) were presented The regressions applied were theclassical linear models and the 119877

2 criterion was chosen forthe quality evaluation This study included 23 women and 45men The amount of collected data was different dependingon the task and ranged from 21 to 68 records

A nonlinear regression equation to predict the maximumaerobic capacity of footballers was proposed by Chatterjee etal [9] The data came from 35 young players aged from 14to 16 The experiment was to verify the use of the 20m MST(Multistage Shuttle Run Test) in evaluating the performanceof VO2maxThe talent of young hockey players was identifiedby Roczniok et al [5] using a regression equation Theresearch involved 60 boys aged between 15 and 16 whoattended selection camps The applied regression modelclassified candidates for future training based on selectedparameters of the players The logistic regression was used inthe model as the classification method

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 735060 9 pageshttpdxdoiorg1011552015735060

2 Computational Intelligence and Neuroscience

The nonlinear predictive models used in sport are alsobased on the selected methods of ldquodata miningrdquo [10] Amongthem an important role is played by fuzzy logic expertsystems Papic et al [4] described practical application of sucha system The proposed system was based on knowledge ofexperts in the field of sport as well as the data obtained as aresult of motor tests The model suggested the most suitablesport and it was designed to search for prospective sportstalents

The application of fuzzy modeling techniques in sportsprediction was also presented by Męzyk and Unold [11] Thegoal of their paper was to find the rules that can expressswimmerrsquos feelings the day after in-water training The datawas collected for two months among competitors practicingswimmingThe swimmers were characterized by a good levelof sports attainment (2nd sport class) The material obtainedconsisted of 12 attributes and the total number of modelswas 480 out of which 136 were used in the final stageThe authors proved that their method was characterizedby better predictive ability than the traditional methods ofclassification

Other papers also concern the use of artificial neuralnetworks in sports prediction [6] Neural models are usedto analyze the effectiveness of the training of swimmers toidentify handball playersrsquo tactics or to predict sporting talent[12] Many studies present the application of neural networksin various aspects of sports training [13ndash15] These modelssupport the planning of training loads practice control orthe selection of sports

An approach developed by the authors is the constructionof models performing the task of predicting the resultsachieved by a competitor in the proposed sports trainingThis allows for the proper selection of training componentsand thus supports the achievement of the desired result Theaim of this study is to determine the effectiveness of selectedlinear and nonlinear models in predicting the outcome ina 3-kilometer walking race for the proposed training Theresearch hypothesis of the paper is stated as follows theprediction error of 3 kilometersrsquo result in race walking fornonlinear models can be smaller than for linear models

The paper is organized as follows In Section 2 the train-ing data of the race walkers recorded during annual trainingcycle is described Section 3 contains the methods used tobuild the linear and nonlinear predictive models includingordinary least squares regression regularized methods thatis ridge LASSO and elastic net regressions nonlinearleast squares regression and artificial neural networks asmultilayer perceptron and radial basis function network InSection 3 the criterion used to evaluate the performance ofthemodels calculated usingmean square error in the processof cross-validation is also defined Section 4 describes theprocedures used for building models and their evaluation in119877 language and STATISTICA software The obtained resultsare analyzed and discussed in Section 5 Finally in Section 6the performed work is concluded

2 Material

The predictive models were built using the training dataof athletes practising race walking The analysis involved

a group of colts and juniors from Poland Among thecompetitors were the finalists in the Polish Junior IndoorChampionships and the Polish Junior Championships Thedata of race walkers was recorded during the 2011-2012 seasonin the form of trainingmeans and training loadsThe trainingmean is the type of work performed while the training load isthe amount ofwork at a particular intensity done by an athleteduring exercise [1] In the material which has been collected11 means of training were distinguished The material wasdrawn from the annual training cycle for the following fourphases transition general preparation special preparationand starting phase The training data has the form of sums oftraining loads completed in onemonth of the chosen trainingphaseThematerial included 122 training patternsmade by 21race walkers

Control of the training process in race walking requiresdifferent tests of physical fitness at every training levelBecause this research concerns the competitors in colt andjunior categories thus in order to determine a unifiedcriterion of the level of training a result for 3000m racewalking was used The choice of the distance of 3000m isvalid because this is the indoor walking competition

The description of the variables under consideration andtheir basic statistics are presented in Table 1 The variablesare as follows arithmetic mean of 119909 minimum value 119909minmaximum value 119909max standard deviation SD and coefficientof variation 119881 = SD119909 sdot 100 The qualitative variablesare 1198831 1198832 1198833 1198834 which take their values from the set0 1Theother variables that is1198835 11988318 are quantitativevariables If the value at inputs 1198831 1198832 1198833 is 0 it meansthat the transitional period is considered Setting the value1 on one of the inputs 1198831 1198832 1198833 it means the trainingperiod is selected The variable 1198834 represents the gender ofthe competitor where the value 0 denotes a female while thevalue 1 denotes amale and the age is represented by1198835 Basicsomatic features of race walkers such as weight and height arepresented in the form of BMI (1198836) expressed by the formula

BMI = 119872

1198672 [kgm2] (1)

where 119872 is the body weight [kg] and 119867 is the body height[m] The variable 1198837 denotes the current result over 3 km inseconds Training loads are characterized by the followingvariables running exercises (1198838) walking with differentlevels of intensity (1198839 11988310 11988311) exercises forming differ-ent types of endurance (11988312 11988313 11988314) exercises formingtechniques (11988315) exercises forming muscle strength (11988316)exercises forming general fitness (11988317) and warming upexercises (11988318)

An example of data used for building the model has theform

x5 = [0 1 0 0

23 2209 800 32 400 112 20 16 324 48 8

280 640 400]

1199105= 800

(2)

Computational Intelligence and Neuroscience 3

Table 1 The variables and their basic statistics

Variable Description 119909 119909min 119909max SD 119881 []119884 Result over 3 km [s] 9369 780 1155 784 84

1198831General preparationphase

mdash mdash mdash mdash mdash

1198832Special preparationphase


1198833 Starting phase mdash mdash mdash mdash mdash1198834 Competitorrsquos gender mdash mdash mdash mdash mdash

1198835Competitorrsquos age[years]

189 14 24 30 156

1198836BMI (body massindex) [kgm2]

193 164 221 17 87

1198837Current result over 3km [s]

9626 795 1210 877 91

1198838Overall runningendurance [km]

309 0 56 106 344

1198839

Overall walkingendurance in the 1stintensity range [km]

2246 57 440 961 428

11988310

Overall walkingendurance in the 2ndintensity range [km]

532 0 120 346 651

11988311

Overall walkingendurance in the 3rdintensity range [km]

79 0 30 94 1197

11988312Short tempoendurance [km]

89 0 24 5 560

11988313Medium tempoendurance [km]

83 0 324 86 1032

11988314Long tempoendurance [km]

129 0 56 161 1250

11988315

Exercises formingtechnique (rhythm) ofwalking [km]

44 0 12 42 960

11988316Exercises formingmuscle strength [min]

902 0 360 1048 1163

11988317Exercises forminggeneral fitness [min]

5220 120 720 1099 210

11988318Universal exercises(warm up) [min]

3173 150 420 725 228

The vector x5 represents a 23-year-old race walker withBMI = 2209 kgm2 who completes training in the specialpreparation phase The result both before and after thetraining was the same and is equal to 800 s

3 Methods

In this study two approaches were considered The firstapproach was based on white boxmodels realized bymodernregularized methodsThese models are interpretable becausetheir structure and parameters are known The secondapproach was based on black boxmodels realized by artificialneural networks

X1

Xp

YY = f(X1 Xp)

Figure 1 A diagram of a system with multiple inputs and oneoutput

31 Constructing Regression Models Consider a multivari-able regression model with the inputs (predictors or regres-sors) 119883

119895 119895 = 1 119901 and one output (response) 119884 shown in

Figure 1 We assume that the model is linear and has the form

= 1199080 +11988311199081 + sdot sdot sdot +119883119901119908119901

= 1199080 +

119901

sum

119895=1119883119895119908119895

(3)

where is the estimated response and 1199080 119908119895 are unknownweights of the model The weight 1199080 is called constantterm or intercept Furthermore we assume that the data isstandardized and centered and the model can be simplifiedto the form (see eg [16])

= 11988311199081 + sdot sdot sdot +119883119901119908119901

=

119901

sum

119895=1119883119895119908119895

(4)

Observations are written as pairs (x119894 119910119894) where x

119894=

[1199091198941 119909119894119901] 119894 = 1 119899 119909

119894119895is the value of the 119895th predictor

in the 119894th observation and 119910119894is the value of the response in

the 119894th observation Based on formula (4) the 119894th observationcan be expressed as

119910119894= 11990911989411199081 + sdot sdot sdot + 119909

119894119901119908119901

=

119901

sum

119895=1119909119894119895119908119895= x119894w

(5)

where w = [1199081 119908119901]119879 Introducing matrix X in the form

of

X =

[[[[[[[

[

11990911 11990912 sdot sdot sdot 1199091119901

11990921 11990922 sdot sdot sdot 1199092119901

d

1199091198991 1199091198992 sdot sdot sdot 119909

119899119901

]]]]]]]

]

(6)

formula (5) can be written as

y = Xw (7)

where y = [1199101 119910119899]119879


In order to construct regression models an error (resid-ual) is introduced as the difference between the real value 119910

119894

and the estimated value 119910119894in the form of

119890119894= 119910119894minus119910119894= 119910119894minus

119901

sum

119895=1119909119894119895119908119895= 119910119894minus x119894w (8)

Using matrix (6) the error can be written as

e = y minus y = y minusXw (9)

where e = [1198901 119890119899]119879 and y = [1199101 119910119899]

119879Denoting by 119869(w sdot) the cost function the problem of find-

ing the optimal estimator can be formulated as to minimizethe function 119869(w sdot) which means solving the problem

w = argminw

(119869 (w sdot)) (10)

where w is the vector of solutionsDepending on the function 119869(w sdot) different regression

models can be obtained In this paper the following modelsare considered ordinary least squares regression (OLS) ridgeregression LASSO (least absolute shrinkage and selectionoperator) elastic net regression (ENET) and nonlinear leastsquares regression (NLS)

32 Linear Regressions In OLS regression (see eg [16ndash18])the model is calculated by minimizing the sum of squarederrors

119869 (w) = e119879e

= (y minusXw)119879

(y minusXw)

=1003817100381710038171003817y minusXw1003817100381710038171003817

22

(11)

where sdot2 denotes the Euclidean norm (1198712) Minimizing thecost function (11) which is the quadratic function ofw we getthe following solution

w = (X119879X)minus1X119879y (12)

It should be noted that solution (12) does not exist if thematrix X119879X is singular (due to correlated predictors or if119901 gt 119899) In this case different methods of regularizationincluding the previouslymentioned ridge LASSO and elasticnet regressions can be used

In ridge regression by Hoerl and Kennard [19] the costfunction includes a penalty and has the form

119869 (w 120582) = e119879e+120582w119879w

= (y minusXw)119879

(y minusXw) + 120582w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w

22

(13)

The parameter 120582 ge 0 determines the size of the penalty for120582 gt 0 the model is penalized for 120582 = 0 ridge regression

reduces to OLS regression Solving problem (10) for ridgeregression we get

w = (X119879X+120582I)minus1X119879y (14)

where I is the identity matrix with the size of 119901 times 119901 Becausethe diagonal of the matrix X119879X is increased by a positiveconstant the matrix X119879X + 120582I is invertible and the problembecomes nonsingular

In LASSO regression by Tibshirani [20] similarly to ridgeregression the penalty is added to the cost function wherethe 1198711-norm (the sum of absolute values) is used

119869 (w 120582) = e119879e+120582z119879w

= (y minusXw)119879

(y minusXw) + 120582z119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w1

(15)

where z = [1199111 119911119901]119879 119911119895= sgn(119908

119895) and sdot 1 denotes the

Manhattan norm (1198711) Because problem (10) is not linear inrelation to y (due to the use of 1198711-norm) the solution cannotbe obtained in the compact form as in ridge regressionThe most popular algorithm used in this case is the LARSalgorithm (least angle regression) by Efron et al [21]

In elastic net regression by Zou and Hastie [22] thefeatures of ridge and LASSO regressions are combined Thecost function in the so-called naive elastic net has the form of

119869 (w 1205821 1205822) = e119879e+1205821z119879w +1205822w

119879w

= (y minusXw)119879

(y minusXw) + 1205821z119879w

+1205822w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +1205821 w1 +1205822 w

22

(16)

To solve the problem Zou and Hastie [22] proposed theLARS-EN algorithm which was based on the LARS algo-rithm developed for LASSO regression They used the factthat elastic net regression reduces to LASSO regression forthe augmented data set (Xlowast ylowast)

33 Nonlinear Regressions To take into account the non-linearity in the models we can apply the transformation ofpredictors or use nonlinear regression In this paper the lattersolution is applied

In OLS regression the model is described by formula (5)while in more general nonlinear regression the relationshipbetween the output and the predictors is expressed by acertain nonlinear function 119891(sdot) in the form of

119910119894= 119891 (x

119894w) (17)

In this case the cost function 119869(w) is formulated as

119869 (w) =

119899

sum

119894=11198902119894=

119899

sum

119894=1(119910119894minus119910119894)2

=

119899

sum

119894=1(119910119894minus119891 (x

119894w))

2

(18)


Since the minimization of function (18) is associated withsolving nonlinear equations numerical optimization is usedin this case The main problem connected with the construc-tion of nonlinear models is the choice of the appropriatefunction 119891(sdot)

34 Artificial Neural Networks Artificial neural networks(ANNs) were also used for building predictive models Twotypes of ANNs were implemented a multilayer perceptron(MLP) and networks with radial basis function (RBF) [18]

The MLP network is the most common type of neuralmodels The calculation of the output in 3-layer multiple-input-one-output network is performed in feed-forwardarchitecture In the first step 119898 linear combinations or theso-called activations of the input variables are constructed as

119886119896=

119901

sum

119895=1119909119895119908(1)119896119895

(19)

where 119896 = 1 119898 and 119908(1)119896119895

denotes the weights for the firstlayer From the activations 119886

119896 using a nonlinear activation

function ℎ(sdot) hidden variables 119911119896are calculated as

119911119896= ℎ (119886

119896) (20)

The function ℎ(sdot) is usually chosen as logistic or ldquotanhrdquofunction The hidden variables are used next to calculate theoutput activation

119886 =

119898

sum

119896=1119911119896119908(2)119896

(21)

where119908(2)119896

are weights for the second layer Finally the outputof the network is calculated using an activation function 120590(sdot)

in the form of

119910 = 120590 (119886) (22)

For regression problems the function 120590(sdot) is chosen asidentity function and so we obtain 119910 = 119886 The MLP networkutilizes iterative supervised learning known as error back-propagation for training the weightsThis method is based ongradient descent applied to the sum of squares function Toavoid the problemwith overtraining the network the number119898 of hidden neurons which is a free parameter should bedetermined to give the best predictive performance

In the RBF network the concept of radial basis function isused Linear regression (5) is extended by linear combinationsof nonlinear functions of the inputs in the form of

119910119894=

119901

sum

119895=1120601119895(119909119894119895)119908119895= 120593 (x

119894)w (23)

where 120593 = [1206011 120601119901]119879 is a vector of basis functions Using

nonlinear basis functions we get a nonlinear model whichis however a linear function of parameters 119908

119895 In the RBF

network the hidden neurons perform a radial basis functionwhose value depends on the distance from selected center 119888

119895

120593 (x119894 c) = 120593 (

1003817100381710038171003817x119894 minus c1003817100381710038171003817) (24)

where c = [1198881 119888119901] and sdot is usually the Euclidean normThere are many possible choices for the basis functions butthe most popular is Gaussian function It is known that RBFnetwork can exactly interpolate any continuous functionthat is the function passes exactly through every data pointIn this case the number of hidden neurons is equal to thenumber of observations and the values of coefficients 119908

119895

are found by simple standard inversion technique Such anetwork matches the data exactly but it has poor predictiveability because the network is overtrained

35 Choosing the Model In this paper the best predic-tive model is chosen using leave-one-out cross-validation(LOOCV)method [23] in which the number of tests is equalto the number of data 119899 and one pair (x

119894 119910119894) creates a testing

set The quality of the model is evaluated by means of thesquare root of the mean square error (RMSECV) defined as

MSECV =1119899

119899

sum

119894=1(119910119894minus119910minus119894)2

RMSECV = radicMSECV

(25)

where 119910minus119894denotes the output of themodel built in the 119894th step

of validation process using a data set containing no testingpair (x

119894 119910119894) and MSECV is the mean square error

In order to describe the measure to which the modelfits the training data the root mean square error of training(RMSET) is considered This error is defined as

MSET =1119899

119899

sum

119894=1(119910119894minus119910119894)2

RMSET = radicMSET

(26)

where 119910119894denotes the output of the model built in the 119894th step

using the full data set and MSET is the mean square error oftraining

4 Implementation of the Predictive Models

All the regression models were calculated using 119877 languagewith additional packages [24]

The lmridge function from ldquoMASSrdquo package [25] wasused for calculating OLS regression (where 120582 = 0) and ridgeregression (where 120582 gt 0) With the function enet includedin the package ldquoelastic netrdquo [26] LASSO regression and elasticnet regression were calculated The parameters of the enetfunction are 119904 isin [0 1] and 120582 ge 0 where 119904 is a fractionof the 1198711 norm whereas 120582 denotes 1205822 in formula (16) Theparameterization of elastic net regression using the pair (120582 119904)instead of (1205821 1205822) in formula (16) is possible because elasticnet regression can be treated as LASSO regression for anaugmented data set (Xlowast ylowast) [22] Assuming that 120582 = 0 weget LASSO regression with one parameter 119904 for the originaldata (X y)

All the nonlinear regressionmodels were calculated usingthe nls function coming from the ldquostatsrdquo package [27] It


calculates the parameters of the model using the nonlinearleast squares method One of the parameters of the nlsfunction is a formula that specifies the function119891(sdot) in model(18) To calculate the weights Gauss-Newtonrsquos algorithm wasused which was selected by default in the nls function In allthe calculations it was assumed that the initial values of theweights are equal to zero

For the implementation of artificial neural networksStatSoft STATISTICA [28] software was usedThe learning ofMLP networks was implemented using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm [18]While calculatingthe RBF network the parameters of the basis functions wereautomatically set by the learning procedure

The parameters in all models were selected using leave-one-out cross-validation In the case of regularized regres-sions the penalty coefficients were calculated while in thecase of neural networks the number of neurons in the hiddenlayer was calculated The primary performance criterion ofthe model was RMSECV error Cross-validation functions inthe STATISTICA program were implemented using VisualBasic language

5 Results and Discussion

From a coachrsquos point of view the prediction of results is veryimportant in the process of sport training A coach using themodel which was constructed earlier can predict how thetraining loadswill influence the sport outcomeThepresentedmodels can be used for predictions based on the proposedmonthly training introduced as the sum of the training loadsof each type implemented in a given month

The results of the research are presented in Table 2 thedescription of the selected regressions will be presented inthe next paragraphs Linear models such as OLS ridge andLASSO regressions have been calculated by the authors inwork [3] They will be briefly described here The nonlinearmodels implemented using nonlinear regression and artificialneural networks will be discussed in greater detail All themethods will be compared taking into account the accuracyof the prediction

51 Linear Regressions The regression model calculated bythe OLS method generated the prediction error RMSECV =

2690 s and the training error RMSET = 2270 s (Table 2)In the second column of Table 2 the weights 1199080 and 119908

119895are

presentedThe search for the ridge regression model is based on

finding the parameter 120582 for which the model achieves thesmallest prediction error In this paper ridge regressionmodels for 120582 changing from 0 to 2 with step of 01 wereanalyzed Based on the results it was found that the bestridge model is achieved for 120582opt = 1 The prediction errorRMSECV = 2676 s was smaller than in the OLS model whilethe training error RMSET = 2282 s was greater (Table 2)Theobtained ridge regression improved the predictive ability ofthe model It is seen from Table 2 that as in the case of OLSregression all weights are nonzero and all the input variablesare used in computing the output of the model

Table 2 Coefficients of linear models and linear part of nonlinearmodel NLS1 and error results

Regression OLS RIDGE LASSO ENET NLS11199080 2372 3257 2966 20051199081 4567 3467 3275 41241199082 9061 7484 7191 77121199083 3970 2749 2445 minus34391199084 minus2838 2424 15451199085 minus09755 minus1770 minus1416 minus22441199086 1072 05391 minus24711199087 07331 06805 07069 minus17821199088 minus02779 minus03589 minus03410 minus15001199089 minus01428 minus01420 minus01364 minus0096611990810 minus01579 minus00948 minus00200 0741711990811 07472 04352 00618 0693311990812 04845 03852 01793 minus0672611990813 01216 01454 01183 minus0093611990814 minus01510 minus00270 223111990815 minus05125 minus03070 0734911990816 minus00601 minus00571 minus00652 minus0268511990817 minus00153 minus00071 0035811990818 minus00115 minus00403 minus00220 minus00662RMSECV [s] 2690 2676 2620 2883RMSET [s] 2270 2282 2289 2021

The LASSO regression model was calculated using theLARS-EN algorithm in which the penalty is associated withthe parameter 119904 changing from 0 to 1 with step of 001 Itwas found that the optimal LASSO regression is calculatedfor 119904opt = 078 The best LASSO model generates the errorRMSECV = 2620 s which improves the results of OLS andridge models However it should be noted that this model ischaracterized by the worst data fit with the greatest trainingerror RMSET = 2289 s The LASSO method is also used forcalculating an optimal set of input variables It can be seenin the fourth column of Table 2 that the LASSO regressioneliminated the five input variables (1198834 1198836 11988314 11988315 and11988317) which made the model simpler than for OLS and ridgeregression

The use of elastic net regression model has not improvedthe value of the prediction errorThe bestmodel was obtainedfor a pair of parameters 119904opt = 078 and 120582opt = 0 Becausethe parameter 120582 is zero the model is identical to the LASSOregression (fourth column of Table 2)

52 Nonlinear Regressions Nonlinear regression modelswere obtained using various functions 119891(sdot) in formula (18)It was assumed that the function 119891(sdot) consists of two com-ponents the linear part in which the weights are calculatedas in OLS regression and the nonlinear part containingexpressions of higher orders in the form of a quadraticfunction of selected predictors

119891 (x119894w k) =

119901

sum

119895=1119909119894119895119908119895+

119901

sum

119895=11199092119894119895V119895 (27)


Table 3 Coefficients of nonlinear part of nonlinear models anderror results (all coefficients have to be multiplied by 10minus2)

Regression NLS1 NLS2 NLS3 NLS4V5 5335 minus03751 minus03686 minus06995V6 5943 minus10454 minus13869V7 01218 00003 00004 00001V8 1880 00710 00372 minus00172V9 minus00016 00093 00093 00085V10 minus06646 minus00577 minus00701 minus01326V11 minus30394 minus03608 00116 08915V12 48741 03807 04170 10628V13 04897 minus02496 minus02379 minus01391V14 minus47399 minus01141 minus01362V15 minus136418 13387 08183V16 00335 minus00015 minus00003 minus00004V17 minus00033 minus00006 minus00006V18 00054 00012 00013 minus00002RMSECV [s] 2883 2524 2534 2460RMSET [s] 2021 2263 2274 2279

where w = [1199081 119908119901]119879 is the vector of the weights of the

linear part and k = [V1 V119901]119879 is the vector of the weights

of the nonlinear part The following cases of nonlinearregression were considered (Table 3) wherein each of thefollowing models does not take into account the squares ofqualitative variables 1198831 1198832 1198833 and 1198834 (V1 = V2 = V3 =

V4 = 0)

(i) NLS1 both the weights of the linear part and theweights V5 V18 of the nonlinear part are calcu-lated

(ii) NLS2 the weights of the linear part are constantand their values come from the OLS regression (thesecond column of Table 2) the weights V5 V18 ofthe nonlinear part are calculated (the third column ofTable 3)

(iii) NLS3 the weights of the linear part are constantand their values come from the ridge regression (thethird column of Table 2) the weights V5 V18 of thenonlinear part are calculated (the fourth column ofTable 3)

(iv) NLS4 the weights of the linear part are constantand their values come from the LASSO regres-sion (the fourth column of Table 2) the weightsV5 V7 V13 V16 V18 of the nonlinear part are calcu-lated (the fifth column of Table 3)

Based on the results shown in Table 3 the best nonlinearregression model is the NLS4 model that is the modifiedLASSO regression This model is characterized by the small-est prediction error and the reduced number of predictors

0 2 4 6 8 100

50

100

150

Number of neurons in hidden layer

RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]

Figure 2 Cross-validation error (RMSECV) and training error(RMSET) for MLP(tanh) neural network vertical line drawn for119898 = 1 signifies the number of hidden neurons chosen in cross-validation

0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]

Figure 3 Cross-validation error (RMSECV) and training error(RMSET) for MLP(exp) neural network vertical line drawn for119898 =

1 signifies the number of hidden neurons chosen in cross-validation

53 Neural Networks In order to select the best structureof a neural network the number of neurons 119898 isin [1 10]in the hidden layer was analyzed In Figures 2 3 and4 the relationships between cross-validation error and thenumber of hidden neurons are presentedThe smallest cross-validation errors for the MLP(tanh) and MLP(exp) networkswere obtained for one hidden neuron (18-1-1 architecture)and they were respectively 2989 s and 3002 s (Table 4) Forthe RBF network the best architecture was the one withfour neurons in the hidden layer (18-4-1) and cross-validationerror in this case was 5571 s Comparing the results it is seenthat the best model is the MLP(tanh) network with the 18-1-1 architecture However it is worse than the best regressionmodel NLS4 (Table 3) by more than 5 seconds


1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]

Figure 4 Cross-validation error (RMSECV) and training error(RMSET) for RBF neural network vertical line drawn for 119898 = 4signifies the number of hidden neurons chosen in cross-validation

Table 4 The number of hidden neurons and error results for thebest neural nets

ANN MLP(tanh) MLP(exp) RBFm 1 1 4RMSECV [s] 2989 3002 5571RMSET [s] 2519 2517 5263

6 Conclusions

This paper presents linear and nonlinear models used topredict sports results for race walkers Introducing a monthlytraining schedule for a selected phase in the annual cyclea decline in physical performance may be predicted basedon the generated results Thanks to that it is possible totake into account earlier changes in the scheduled trainingThe novelty of this research is the use of nonlinear modelsincluding modifications of linear regressions and artificialneural networks in order to reduce the prediction errorgenerated by linearmodelsThe bestmodel was the nonlinearmodification of LASSO regression for which the error was246 seconds In addition the method has simplified thestructure of the model by eliminating 9 out of 32 predictorsThe research hypothesis was confirmed Comparing withother results is difficult because there is a lack of publicationsconcerning predictive models in race walking

Experts in the fields of sports theory and training wereconsulted during the construction of the models in orderto maintain the theoretical and practical principles of sporttraining The importance of the work is that practitioners(coaches) can use predictive models for planning of trainingloads in race walking

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgment

Thiswork has been partially supported by the PolishMinistryof Science and Higher Education under Grant U-649DSM

References

[1] T O Bompa and G Haff Periodization Theory and Methodol-ogy of Training Human Kinetics Champaign Ill USA 1999

[2] A Maszczyk A Zając and I Ryguła ldquoA neural network modelapproach to athlete selectionrdquo Sports Engineering vol 13 no 2pp 83ndash93 2011

[3] K Przednowek and K Wiktorowicz ldquoPrediction of the resultin race walking using regularized regression modelsrdquo Journal ofTheoretical and Applied Computer Science vol 7 no 2 pp 45ndash58 2013

[4] V Papic N Rogulj and V Plestina ldquoIdentification of sport tal-ents using a web-oriented expert system with a fuzzy modulerdquoExpert Systems with Applications vol 36 no 5 pp 8830ndash88382009

[5] R Roczniok A Maszczyk A Stanula et al ldquoPhysiological andphysical profiles and on-ice performance approach to predicttalent in male youth ice hockey players during draft to hockeyteamrdquo Isokinetics and Exercise Science vol 21 no 2 pp 121ndash1272013

[6] M Haghighat H Rastegari N Nourafza N Branch andI Esfahan ldquoA review of data mining techniques for resultprediction in sportsrdquo Advances in Computer Science vol 2 no5 pp 7ndash12 2013

[7] K Przednowek and K Wiktorowicz ldquoNeural system of sportresult optimization of athletes doing race walkingrdquo MetodyInformatyki Stosowanej vol 29 no 4 pp 189ndash200 2011 (Polish)

[8] ADrake andR James ldquoPrediction of racewalking performancevia laboratory and field testsrdquo New Studies in Athletics vol 23no 4 pp 35ndash41 2009

[9] P Chatterjee A K Banerjee P Dasb and P Debnath ldquoA regres-sion equation to predict VO

2max of young football players of

Nepalrdquo International Journal of Applied Sports Sciences vol 21no 2 pp 113ndash121 2009

[10] B Ofoghi J Zeleznikow C MacMahon and M Raab ldquoDatamining in elite sports a review and a frameworkrdquoMeasurementin Physical Education and Exercise Science vol 17 no 3 pp 171ndash186 2013

[11] E Męzyk and O Unold ldquoMachine learning approach to modelsport trainingrdquoComputers in Human Behavior vol 27 no 5 pp1499ndash1506 2011

[12] M Pfeiffer and A Hohmann ldquoApplications of neural networksin training sciencerdquoHumanMovement Science vol 31 no 2 pp344ndash359 2012

[13] I Ryguła ldquoArtificial neural networks as a tool of modeling oftraining loadsrdquo in Proceedings of the 27th Annual InternationalConference of the Engineering in Medicine and Biology Society(IEEE-EMBS rsquo05) pp 2985ndash2988 IEEE September 2005

[14] A J Silva A M Costa P M Oliveira et al ldquoThe use of neuralnetwork technology to model swimming performancerdquo Journalof Sports Science and Medicine vol 6 no 1 pp 117ndash125 2007

[15] A Maszczyk R Roczniok Z Waskiewicz et al ldquoApplication ofregression and neural models to predict competitive swimmingperformancerdquo Perceptual and Motor Skills vol 114 no 2 pp610ndash626 2012


[16] T Hastie R Tibshirani and J Friedman The Elements ofStatistical Learning Data Mining Inference and PredictionSpringer New York NY USA 2009

[17] G S Maddala Introduction to Econometrics Wiley ChichesterUK 2001

[18] C M Bishop Pattern Recognition and Machine LearningSpringer New York NY USA 2006

[19] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquo Technometrics vol 12no 1 pp 55ndash67 1970

[20] R Tibshirani ldquoRegression shrinkage and selection via theLassordquo Journal of the Royal Statistical Society Series B Method-ological vol 58 no 1 pp 267ndash288 1996

[21] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004

[22] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical SocietymdashSeries BStatistical Methodology vol 67 no 2 pp 301ndash320 2005

[23] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010

[24] R Development Core Team R A Language and Environmentfor Statistical Computing R Development Core Team ViennaAustria 2011

[25] B Ripley B Venables K Hornik A Gebhardt and D FirthPackage ldquoMASSrdquo Version 73ndash20 CRAN 2012

[26] H Zou and T Hastie Package ldquoelasticnetrdquo version 11 CRAN2012

[27] R Development Core Team and Contributors Worldwide TheR ldquoStatsrdquo Package R Development Core Team Vienna Austria2011

[28] StatSoft Statistica (Data Analysis Software System) Version 10StatSoft 2011

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



The nonlinear predictive models used in sport are alsobased on the selected methods of ldquodata miningrdquo [10] Amongthem an important role is played by fuzzy logic expertsystems Papic et al [4] described practical application of sucha system The proposed system was based on knowledge ofexperts in the field of sport as well as the data obtained as aresult of motor tests The model suggested the most suitablesport and it was designed to search for prospective sportstalents

The application of fuzzy modeling techniques in sportsprediction was also presented by Męzyk and Unold [11] Thegoal of their paper was to find the rules that can expressswimmerrsquos feelings the day after in-water training The datawas collected for two months among competitors practicingswimmingThe swimmers were characterized by a good levelof sports attainment (2nd sport class) The material obtainedconsisted of 12 attributes and the total number of modelswas 480 out of which 136 were used in the final stageThe authors proved that their method was characterizedby better predictive ability than the traditional methods ofclassification

Other papers also concern the use of artificial neuralnetworks in sports prediction [6] Neural models are usedto analyze the effectiveness of the training of swimmers toidentify handball playersrsquo tactics or to predict sporting talent[12] Many studies present the application of neural networksin various aspects of sports training [13ndash15] These modelssupport the planning of training loads practice control orthe selection of sports

An approach developed by the authors is the constructionof models performing the task of predicting the resultsachieved by a competitor in the proposed sports trainingThis allows for the proper selection of training componentsand thus supports the achievement of the desired result Theaim of this study is to determine the effectiveness of selectedlinear and nonlinear models in predicting the outcome ina 3-kilometer walking race for the proposed training Theresearch hypothesis of the paper is stated as follows theprediction error of 3 kilometersrsquo result in race walking fornonlinear models can be smaller than for linear models

The paper is organized as follows In Section 2 the train-ing data of the race walkers recorded during annual trainingcycle is described Section 3 contains the methods used tobuild the linear and nonlinear predictive models includingordinary least squares regression regularized methods thatis ridge LASSO and elastic net regressions nonlinearleast squares regression and artificial neural networks asmultilayer perceptron and radial basis function network InSection 3 the criterion used to evaluate the performance ofthemodels calculated usingmean square error in the processof cross-validation is also defined Section 4 describes theprocedures used for building models and their evaluation in119877 language and STATISTICA software The obtained resultsare analyzed and discussed in Section 5 Finally in Section 6the performed work is concluded

2 Material

The predictive models were built using the training dataof athletes practising race walking The analysis involved

a group of colts and juniors from Poland Among thecompetitors were the finalists in the Polish Junior IndoorChampionships and the Polish Junior Championships Thedata of race walkers was recorded during the 2011-2012 seasonin the form of trainingmeans and training loadsThe trainingmean is the type of work performed while the training load isthe amount ofwork at a particular intensity done by an athleteduring exercise [1] In the material which has been collected11 means of training were distinguished The material wasdrawn from the annual training cycle for the following fourphases transition general preparation special preparationand starting phase The training data has the form of sums oftraining loads completed in onemonth of the chosen trainingphaseThematerial included 122 training patternsmade by 21race walkers

Control of the training process in race walking requiresdifferent tests of physical fitness at every training levelBecause this research concerns the competitors in colt andjunior categories thus in order to determine a unifiedcriterion of the level of training a result for 3000m racewalking was used The choice of the distance of 3000m isvalid because this is the indoor walking competition

The description of the variables under consideration andtheir basic statistics are presented in Table 1 The variablesare as follows arithmetic mean of 119909 minimum value 119909minmaximum value 119909max standard deviation SD and coefficientof variation 119881 = SD119909 sdot 100 The qualitative variablesare 1198831 1198832 1198833 1198834 which take their values from the set0 1Theother variables that is1198835 11988318 are quantitativevariables If the value at inputs 1198831 1198832 1198833 is 0 it meansthat the transitional period is considered Setting the value1 on one of the inputs 1198831 1198832 1198833 it means the trainingperiod is selected The variable 1198834 represents the gender ofthe competitor where the value 0 denotes a female while thevalue 1 denotes amale and the age is represented by1198835 Basicsomatic features of race walkers such as weight and height arepresented in the form of BMI (1198836) expressed by the formula

BMI = 119872

1198672 [kgm2] (1)

where 119872 is the body weight [kg] and 119867 is the body height[m] The variable 1198837 denotes the current result over 3 km inseconds Training loads are characterized by the followingvariables running exercises (1198838) walking with differentlevels of intensity (1198839 11988310 11988311) exercises forming differ-ent types of endurance (11988312 11988313 11988314) exercises formingtechniques (11988315) exercises forming muscle strength (11988316)exercises forming general fitness (11988317) and warming upexercises (11988318)

An example of data used for building the model has theform

x5 = [0 1 0 0

23 2209 800 32 400 112 20 16 324 48 8

280 640 400]

1199105= 800

(2)










189 14 24 30 156


193 164 221 17 87


9626 795 1210 877 91


309 0 56 106 344

1198839


2246 57 440 961 428

11988310


532 0 120 346 651

11988311


79 0 30 94 1197


89 0 24 5 560


83 0 324 86 1032


129 0 56 161 1250

11988315


44 0 12 42 960


902 0 360 1048 1163


5220 120 720 1099 210


3173 150 420 725 228


3 Methods


X1

Xp

YY = f(X1 Xp)





= 1199080 +11988311199081 + sdot sdot sdot +119883119901119908119901

= 1199080 +

119901

sum

119895=1119883119895119908119895

(3)


= 11988311199081 + sdot sdot sdot +119883119901119908119901

=

119901

sum

119895=1119883119895119908119895

(4)


119894=

[1199091198941 119909119894119901] 119894 = 1 119899 119909




119910119894= 11990911989411199081 + sdot sdot sdot + 119909

119894119901119908119901

=

119901

sum

119895=1119909119894119895119908119895= x119894w

(5)


of

X =

[[[[[[[

[

11990911 11990912 sdot sdot sdot 1199091119901

11990921 11990922 sdot sdot sdot 1199092119901

d

1199091198991 1199091198992 sdot sdot sdot 119909

119899119901

]]]]]]]

]

(6)


y = Xw (7)

where y = [1199101 119910119899]119879



119894


119890119894= 119910119894minus119910119894= 119910119894minus

119901

sum

119895=1119909119894119895119908119895= 119910119894minus x119894w (8)



where e = [1198901 119890119899]119879 and y = [1199101 119910119899]



w = argminw

(119869 (w sdot)) (10)




119869 (w) = e119879e

= (y minusXw)119879

(y minusXw)

=1003817100381710038171003817y minusXw1003817100381710038171003817

22

(11)


w = (X119879X)minus1X119879y (12)



119869 (w 120582) = e119879e+120582w119879w

= (y minusXw)119879

(y minusXw) + 120582w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w

22

(13)



w = (X119879X+120582I)minus1X119879y (14)



119869 (w 120582) = e119879e+120582z119879w

= (y minusXw)119879

(y minusXw) + 120582z119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w1

(15)

where z = [1199111 119911119901]119879 119911119895= sgn(119908




119869 (w 1205821 1205822) = e119879e+1205821z119879w +1205822w

119879w

= (y minusXw)119879

(y minusXw) + 1205821z119879w

+1205822w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +1205821 w1 +1205822 w

22

(16)




119910119894= 119891 (x

119894w) (17)


119869 (w) =

119899

sum

119894=11198902119894=

119899

sum

119894=1(119910119894minus119910119894)2

=

119899

sum

119894=1(119910119894minus119891 (x

119894w))

2

(18)





119886119896=

119901

sum

119895=1119909119895119908(1)119896119895

(19)

where 119896 = 1 119898 and 119908(1)119896119895




119911119896= ℎ (119886

119896) (20)


119886 =

119898

sum

119896=1119911119896119908(2)119896

(21)

where119908(2)119896


in the form of

119910 = 120590 (119886) (22)



119910119894=

119901

sum

119895=1120601119895(119909119894119895)119908119895= 120593 (x

119894)w (23)



119895 In the RBF


119895

120593 (x119894 c) = 120593 (

1003817100381710038171003817x119894 minus c1003817100381710038171003817) (24)


119895





MSECV =1119899

119899

sum

119894=1(119910119894minus119910minus119894)2

RMSECV = radicMSECV

(25)





MSET =1119899

119899

sum

119894=1(119910119894minus119910119894)2

RMSET = radicMSET

(26)
















119895are








119891 (x119894w k) =

119901

sum

119895=1119909119894119895119908119895+

119901

sum

119895=11199092119894119895V119895 (27)







V4 = 0)






0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]


0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]





1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in












189 14 24 30 156


193 164 221 17 87


9626 795 1210 877 91


309 0 56 106 344

1198839


2246 57 440 961 428

11988310


532 0 120 346 651

11988311


79 0 30 94 1197


89 0 24 5 560


83 0 324 86 1032


129 0 56 161 1250

11988315


44 0 12 42 960


902 0 360 1048 1163


5220 120 720 1099 210


3173 150 420 725 228


3 Methods


X1

Xp

YY = f(X1 Xp)





= 1199080 +11988311199081 + sdot sdot sdot +119883119901119908119901

= 1199080 +

119901

sum

119895=1119883119895119908119895

(3)


= 11988311199081 + sdot sdot sdot +119883119901119908119901

=

119901

sum

119895=1119883119895119908119895

(4)


119894=

[1199091198941 119909119894119901] 119894 = 1 119899 119909




119910119894= 11990911989411199081 + sdot sdot sdot + 119909

119894119901119908119901

=

119901

sum

119895=1119909119894119895119908119895= x119894w

(5)


of

X =

[[[[[[[

[

11990911 11990912 sdot sdot sdot 1199091119901

11990921 11990922 sdot sdot sdot 1199092119901

d

1199091198991 1199091198992 sdot sdot sdot 119909

119899119901

]]]]]]]

]

(6)


y = Xw (7)

where y = [1199101 119910119899]119879



119894


119890119894= 119910119894minus119910119894= 119910119894minus

119901

sum

119895=1119909119894119895119908119895= 119910119894minus x119894w (8)



where e = [1198901 119890119899]119879 and y = [1199101 119910119899]



w = argminw

(119869 (w sdot)) (10)




119869 (w) = e119879e

= (y minusXw)119879

(y minusXw)

=1003817100381710038171003817y minusXw1003817100381710038171003817

22

(11)


w = (X119879X)minus1X119879y (12)



119869 (w 120582) = e119879e+120582w119879w

= (y minusXw)119879

(y minusXw) + 120582w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w

22

(13)



w = (X119879X+120582I)minus1X119879y (14)



119869 (w 120582) = e119879e+120582z119879w

= (y minusXw)119879

(y minusXw) + 120582z119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w1

(15)

where z = [1199111 119911119901]119879 119911119895= sgn(119908




119869 (w 1205821 1205822) = e119879e+1205821z119879w +1205822w

119879w

= (y minusXw)119879

(y minusXw) + 1205821z119879w

+1205822w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +1205821 w1 +1205822 w

22

(16)




119910119894= 119891 (x

119894w) (17)


119869 (w) =

119899

sum

119894=11198902119894=

119899

sum

119894=1(119910119894minus119910119894)2

=

119899

sum

119894=1(119910119894minus119891 (x

119894w))

2

(18)





119886119896=

119901

sum

119895=1119909119895119908(1)119896119895

(19)

where 119896 = 1 119898 and 119908(1)119896119895




119911119896= ℎ (119886

119896) (20)


119886 =

119898

sum

119896=1119911119896119908(2)119896

(21)

where119908(2)119896


in the form of

119910 = 120590 (119886) (22)



119910119894=

119901

sum

119895=1120601119895(119909119894119895)119908119895= 120593 (x

119894)w (23)



119895 In the RBF


119895

120593 (x119894 c) = 120593 (

1003817100381710038171003817x119894 minus c1003817100381710038171003817) (24)


119895





MSECV =1119899

119899

sum

119894=1(119910119894minus119910minus119894)2

RMSECV = radicMSECV

(25)





MSET =1119899

119899

sum

119894=1(119910119894minus119910119894)2

RMSET = radicMSET

(26)
















119895are








119891 (x119894w k) =

119901

sum

119895=1119909119894119895119908119895+

119901

sum

119895=11199092119894119895V119895 (27)







V4 = 0)






0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]


0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]





1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119894


119890119894= 119910119894minus119910119894= 119910119894minus

119901

sum

119895=1119909119894119895119908119895= 119910119894minus x119894w (8)



where e = [1198901 119890119899]119879 and y = [1199101 119910119899]



w = argminw

(119869 (w sdot)) (10)




119869 (w) = e119879e

= (y minusXw)119879

(y minusXw)

=1003817100381710038171003817y minusXw1003817100381710038171003817

22

(11)


w = (X119879X)minus1X119879y (12)



119869 (w 120582) = e119879e+120582w119879w

= (y minusXw)119879

(y minusXw) + 120582w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w

22

(13)



w = (X119879X+120582I)minus1X119879y (14)



119869 (w 120582) = e119879e+120582z119879w

= (y minusXw)119879

(y minusXw) + 120582z119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +120582 w1

(15)

where z = [1199111 119911119901]119879 119911119895= sgn(119908




119869 (w 1205821 1205822) = e119879e+1205821z119879w +1205822w

119879w

= (y minusXw)119879

(y minusXw) + 1205821z119879w

+1205822w119879w

=1003817100381710038171003817y minusXw1003817100381710038171003817

22 +1205821 w1 +1205822 w

22

(16)




119910119894= 119891 (x

119894w) (17)


119869 (w) =

119899

sum

119894=11198902119894=

119899

sum

119894=1(119910119894minus119910119894)2

=

119899

sum

119894=1(119910119894minus119891 (x

119894w))

2

(18)





119886119896=

119901

sum

119895=1119909119895119908(1)119896119895

(19)

where 119896 = 1 119898 and 119908(1)119896119895




119911119896= ℎ (119886

119896) (20)


119886 =

119898

sum

119896=1119911119896119908(2)119896

(21)

where119908(2)119896


in the form of

119910 = 120590 (119886) (22)



119910119894=

119901

sum

119895=1120601119895(119909119894119895)119908119895= 120593 (x

119894)w (23)



119895 In the RBF


119895

120593 (x119894 c) = 120593 (

1003817100381710038171003817x119894 minus c1003817100381710038171003817) (24)


119895





MSECV =1119899

119899

sum

119894=1(119910119894minus119910minus119894)2

RMSECV = radicMSECV

(25)





MSET =1119899

119899

sum

119894=1(119910119894minus119910119894)2

RMSET = radicMSET

(26)
















119895are








119891 (x119894w k) =

119901

sum

119895=1119909119894119895119908119895+

119901

sum

119895=11199092119894119895V119895 (27)







V4 = 0)






0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]


0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]





1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119886119896=

119901

sum

119895=1119909119895119908(1)119896119895

(19)

where 119896 = 1 119898 and 119908(1)119896119895




119911119896= ℎ (119886

119896) (20)


119886 =

119898

sum

119896=1119911119896119908(2)119896

(21)

where119908(2)119896


in the form of

119910 = 120590 (119886) (22)



119910119894=

119901

sum

119895=1120601119895(119909119894119895)119908119895= 120593 (x

119894)w (23)



119895 In the RBF


119895

120593 (x119894 c) = 120593 (

1003817100381710038171003817x119894 minus c1003817100381710038171003817) (24)


119895





MSECV =1119899

119899

sum

119894=1(119910119894minus119910minus119894)2

RMSECV = radicMSECV

(25)





MSET =1119899

119899

sum

119894=1(119910119894minus119910119894)2

RMSET = radicMSET

(26)
















119895are








119891 (x119894w k) =

119901

sum

119895=1119909119894119895119908119895+

119901

sum

119895=11199092119894119895V119895 (27)







V4 = 0)






0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]


0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]





1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in












119895are








119891 (x119894w k) =

119901

sum

119895=1119909119894119895119908119895+

119901

sum

119895=11199092119894119895V119895 (27)







V4 = 0)






0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]


0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]





1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









V4 = 0)






0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]


0 2 4 6 8 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]





1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1 2 3 4 5 6 7 8 9 100

50

100

150


RMSECVRMSET

RMSE

CV[s

] RM

SET

[s]




6 Conclusions





Acknowledgment


References







































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in
























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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