research article quality evaluation based on multivariate

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 639652 10 pageshttpdxdoiorg1011552013639652

Research ArticleQuality Evaluation Based on Multivariate Statistical Methods

Shen Yin1 Xiangping Zhu1 and Hamid Reza Karimi2

1 College of Engineering Bohai University Liaoning 121013 China2Department of Engineering Faculty of Engineering and Science of the University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Xiangping Zhu xiangpingzhu2010gmailcom

Received 22 October 2013 Accepted 6 November 2013

Academic Editor Xiaojie Su

Copyright copy 2013 Shen Yin et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Quality prediction models are constructed based on multivariate statistical methods including ordinary least squares regression(OLSR) principal component regression (PCR) partial least squares regression (PLSR) and modified partial least squaresregression (MPLSR) The prediction model constructed by MPLSR achieves superior results compared with the other threemethods from both aspects of fitting efficiency and prediction ability Based on it further research is dedicated to selecting keyvariables to directly predict the product quality with satisfactory performance The prediction models presented are more efficientthan tradition ones and can be useful to support human experts in the evaluation and classification of the product quality Theeffectiveness of the quality prediction models is finally illustrated and verified based on the practical data set of the red wine

1 Introduction

An accurate evaluation of the wine quality is of significancefor vintners to perform wine classification and target mar-keting However since influenced by numerous factors suchas grape varieties yeast strains wine making technologiesand human experiences [1] evaluating wine quality is themain challenge for both the food industry and wine sciencecommunity Traditionally wine quality is given by humanexperts or obtained by analyzing chemical compounds inthe wine [2 3] Nevertheless besides the issues like timeconsuming and complexity thesemethods sometimes cannotmeet the requirements when winemakers want to knowthe quality before the wine has been vinified for examplepredicting wine quality at the time of selecting grapes

Amongst the various influence factors of the wine qualitygrape is the most basic and important factor for makinghigh quality wine and some grape physicochemical indexeshave a strong relation to the wine quality Since the sugarin grapes is the raw material for yeast to produce alcoholits content plays a crucial role in the fermentation processand almost determines the alcoholic level of the wine [4]In addition to the sugar with studying three kinds of grapes(Cabernet Sauvignon and Merlot) which consume the samematuration Fang et al have found that the flavonoid varieties

in red wines are greatly affected by the varieties of grapes[5] Although there exists malate synthesizing by yeast viafumarate or oxaloacetate pathways the amount of that can benegligible compared with the malate originating from grapesand the high malate levels in grapes follow the high malatelevels in wines [6] All of these research results have revealedthe important relationship between the grape physicochemi-cal indexes and the wine quality and the necessity to evaluatethe wine quality from such grape indexes

In this paper based on a data set of wine qualitiesand grape physicochemical indexes wine quality predictionmodels are constructed with multivariate statistical methodsIn order to obtain an efficient wine quality prediction modela comparison amongst the models established by ordinaryleast squares regression principal component regressionpartial least squares regression and a modified partial leastsquare regression is made and the best model is selectedout The calibration set includes 50 grape physicochemicalindexes Dealing with so much data is a time consumingand complex task With the correlation analysis it has beenfound that there exists multicollinearity problems in somegrape physicochemical indexes Under the framework ofwine quality predictionmodel a suggestion regarding the useof fewer grape physicochemical indexes to predict the winequality under the promise of prediction accuracy is proposed

2 Mathematical Problems in Engineering

Compared with the majority of the methods reported forwine quality analysis the method discussed in this paperprovides a simpler and more convenient way to predictthe wine quality Besides what is most remarkable of theproposedmethod is the increasing possibility for winemakersto predict the wine quality before the complete wine makingprocess and make appropriate decisions in advance such asthe grape selection wine classification and target marketing

Although investigations on evaluating wine quality onlybased on grape physicochemical indexes are rare in thewine science community the relationship between grapesand metabolites in wine is available in many literatureresources Since the multiple biochemical process occurswith the grape ripening the grape harvest time is influentialin the wine composition For example the yeast-derivedmetabolites including volatile esters dimethyl sulfide glyc-erol and mannoproteins will increase with harvest date [7]In addition the climate also has a profound influence ongrape and finally influences the wine metabolites Among themost important climate change-related effects are advancedharvest times and temperatures increased grape sugar con-centrations that lead to high wine alcohol level lower levelsof acids and the modification of varietal aroma compounds[8 9] Moreover with the help of 1H NMR spectroscopicanalysis and multivariate statistics methods such as PCAand OPLS-DA Son et al further proved that the grapevarieties influence the wine quality significantly and theyalso succeeded in separating the wines vinified from fourdifferent grape varieties [10] Besides it is known from otherliterature that the genetic factors are also in a relation withgrape and wine quality [11] Given background of wines andgrapes as well as the relevant influential factors achieved bythis literature this paper focuses on studying the relationshipbetween wine quality and grape physicochemical indexes byapplying multivariate statistical methods

The rest of this paper is structured as follows Section 2makes a brief description of the multivariate statistical meth-ods including ordinary least squares regression principalcomponent regression partial least squares regression and amodified partial least square regression Section 3 discussesthe red wine making process with relevant illustrations Thewine predictionmodels are established based onmultivariatestatistical methods in Section 4 and a comparison amongthese models is provided to select the best model Further-more the suggestion about using fewer grape physicochemi-cal indexes to predict the wine quality is proposed as well inthis section Finally the conclusions are presented in the lastsection

2 Modeling Algorithms

Multivariate statistical analysis is a powerful tool for solvingthewine analysis problemswhich often involve large amountsof data and has been wildly applied in many relevant studiessuch as analyzing the elements in wines by PLS regression[12] investigating the relationship betweenwine compositionand vintage via principal component analysis (PCA) and par-tial least squares (PLS) [13 14] and utilizing the visible-near

infrared spectroscopy and chemometrics to classify Rieslingwines from different countries with the help of PCA [15]In this section besides describing three classical multivariatestatistics methods a modified partial least squares regression(MPLSR) method is also introduced which has a simplermath description compared with standard PLS regression[16]

21 Ordinary Least Squares Regression (OLSR) Ordinaryleast squares regression (OLSR) is a standard approach toprovide the approximate solution for the overdeterminedsystems through minimizing the sum of the errors createdin the results of every single equation Generally OLSR oftenappears in the situation where only one dimension responsevariable is involved However OLSR is also capable of solvingthe problems of more than one dimension response variablesand the algorithm can be briefly introduced as follows

Step 1 Collect 119899 samples of the measurable variables andresponse variables and normalize them to zero mean andunit variance denoted as 119883 = [119909

1sdot sdot sdot 119909119898] isin 119877

119899times119898 119884 =

[1199101sdot sdot sdot 119910119901] isin 119877119899times119901The following steps will be performed only

when119883119879119883 is a full rank matrix

Step 2 Minimize the sum of errors 119890119895

120573119895= argmin 119890119879

119895119890119895= argmin (119910

119895minus 119883120573119895)

119879

(119910119895minus 119883120573119895)

119895 = 1 sdot sdot sdot 119901

(1)

According to the invertibility of 119883119879119883 the 120573119895can also be

calculated by

120573119895= (119883119879119883)

minus1

119883119879119910119895 (2)

Step 3 Use the 119890119895and 120573

119895to form 119864 isin 119877

119899times119901 and 119872 isin 119877119898times119901

respectively and the final model can be expressed as

119884 = 119883119872 + 119864 (3)

22 Principal Component Regression (PCR) In practice theindependent variables may be highly collinear This phe-nomenon is the so-called multicollinearity and it is knownthat such collinearity problems can sometimes lead to seriousstability problems when the OLSR is applied [17] Alternativemethods to OLSR are proposed in order to overcome thisproblem The most popular methods are the principal com-ponent regression (PCR) and partial least squares regression(PLSR) The collinearity problems are efficiently solved inPCR algorithm by introducing principal components (PCs)which preserve the significant information in the originaldata set and the algorithm can be briefly formulated asfollows

Step 1 Perform normalization on the gathered measurablevariables and the responses variables presented as 119883 =

[1199091sdot sdot sdot 119909119898] isin 119877

119899times119898 and 119884 = [1199101sdot sdot sdot 119910119901] isin 119877

119899times119901where 119909

119894and 119910

119894have the zero mean and the unit variance

Mathematical Problems in Engineering 3

Step 2 Implement singular value decomposition (SVD) onthe covariance matrix of independent variable set

1

119899 minus 1

119883119879119883 = 119875Λ119875

119879

Λ = diag (1205821sdot sdot sdot 120582119898) 120582

1gt sdot sdot sdot gt 120582

119898gt 0

(4)

in which 119875 is the so-called loading matrix of covariancematrix

Step 3 Use appropriate criteria [18] to determine the numberof principle components and calculate the score matrix 119879

119879 = 119883119875pc isin 119877119899times119897

(5)

where

119875pc sub 119875 = [119875pc119875res] isin 119877119898times119898

119875pc = [1199011sdot sdot sdot 119901119897] isin 119877119898times119897

119875res = [119901119897+1

sdot sdot sdot 119901119898] isin 119877119898times(119898minus119897)

119897 the number of the principal components

(6)

Step 4 Preform OLS regression between the score matrix119879 and the dependent matrix 119884 and obtain the final model

120573119894= (119879119879119879)

minus1

119879119879119910119894 119894 = 1 sdot sdot sdot 119901

119861 = [1205731sdot sdot sdot 120573119901] isin 119877119898times119901

119884 = 119879119861 + 119864 = 119883119872 + 119864 119872 = 119875pc119861

(7)

23 Partial Least Squares Regression (PLSR) Different fromthe PCR which only considers the outer relation of 119883 blockpartial least squares regression (PLSR) takes both outer rela-tions (119883 and119884 block individually) and inner relation (linkingboth blocks) into account and thus the latent variables (LVs)of119883 and 119884 have a strong relation which ensures the ability ofPLSR model to make prediction from measurable variables[19] Owing to this property PLSR is widely used in manyfields such as fault detection [20] neuroimaging [21 22]and wine analysis [13] Many improvements have been madein PLS since 1982 when Herman Wold proposed the PLSalgorithm [16 23ndash25] In this section we would like to onlyconsider the standard PLSR method as follows

Step 1 Normalize the collected data sets of 119883 and 119884expressed as 119883 = [119909


119899times119898 and 119884 = [1199101sdot sdot sdot 119910119901] isin

119877119899times119901 where 119909

119894and 119910

119894are vectors with zero mean and unit

variance

Step 2 Iteratively calculate the following equations 120574 times

(119901119896 119902119896) = arg max

119901119896=1119902119896=1

119901119879

119896119883119879119884119902119896 119896 = 1 sdot sdot sdot 120574

119906119896= 119883119896119901119896 V

119896= 119884119896119902119896

119883119896= 119906119896119888119879

119896+ 119883119896+1

119884119896= 119906119896119903119879

119896+ 119884119896+1

119888119896=

119883119879

119896119906119896

1003817100381710038171003817119906119896

1003817100381710038171003817

2 119903

119896=

119884119879

119896119906119896

1003817100381710038171003817119906119896

1003817100381710038171003817

2

119883119896+1

= 119883119896minus 119906119896119888119879

119896 119884

119896+1= 119884119896minus 119906119896119903119879

119896

(8)

where the 119901119896 119902119896and 119906

119896 V119896are the loading vector and score

vector of 119883 and 119884 respectively and the 120574 is the numberof latent variables (LVs) which is usually determined by thecross-validation criteria [26]

Step 3 Store 119901119896 119906119896 119888119896 and 119903

119896into 119875 119880 119862 and 119877 and the

result of standard PLSR algorithm can be expressed as

119880 = 119883119875 119883 = 119880119862119879+ 119864

119884 = 119880119877119879+ 119865 = 119883119872 + 119865 119872 = 119875119877

119879

(9)

24Modified Partial Least Squares Regression (MPLSR) BothPCR and PLSR project the original data set into principalcomponents space or latent variables space and residue spaceand the lower dimension of principal components spaceor latent variables space overcomes the multicollinearityproblem which is a hamper for the OLSR However thesedimension reductionmethods alsomake PCR and PLSR havethe risk that useful information will lose in selected PCs orLVs Although OLSR builds the model maintaining all ofinformation from the origin data set it always comes to a haltwhen the data set exists the problem of multicollinearity orthe number of samples smaller than the number of variableswhich are two common phenomenons in practice In orderto solve these problems Yin et al proposed the MPLSRwhich has been validated on the industrial benchmark ofTennessee Eastman process for fault detection and a goodresult has been obtained [16] MPLSR also has a simpler mathdescription compared with PCR or PLSR and the algorithmcan be expressed as follows

Step 1 Gather all the 119899 samples of measurable variables andresponse variables and stack them into 119883 and 119884 respec-tively Normalize them to zero mean and unit variancedenoted as 119883 = [119909



119877119899times119901

Step 2 Calculate the regression coefficient matrix119872

1

119899

119884119879119883 asymp 119872

119879119883119879119883

119899

119872 = (119883119879119883)

minus1

119883119879119884 if rank (119883119879119883) = 119898

119872 = (119883119879119883)

dagger

119883119879119884 if rank (119883119879119883) lt 119898

(10)

in which the (119883119879119883)dagger is the pseudoinverse of119883119879119883


(a) Vendage (b) Making the must (c) Fermentation

(f) Bottling (e) Aging (d) Pressing and setting

Figure 1 Flow sheet of winemaking

Step 3 Obtain the final model of MPLSR

119884 = 119883119872 + 119864119910 (11)

where 119864119910is the residue part of 119884 which is uncorrelated with

119883

3 Winemaking Process

All wines are practically made in a common process fromgrapes harvesting to bottling For red wines the winemakingprocess can be roughly divided into six steps

Firstly grape harvesting is performed to supply the rawmaterial for winemaking For the grape compositions whichmay positively or negatively influence wine chemistry andsensory properties or vary with the maturity of grapes theharvest dates should be carefully considered Traditionallythe levels of grape sugar acids and PH are used to determinethe harvest dates However Jackson and Lombard have pro-posed that suchmeasures alone are not sufficient to accuratelypredict wine composition because many key grape-derivedcompounds in wine do not track with sugar accumulation[27] In many cases the optimal time for picking lasts onlya very few days Grape harvesting or vendage can be mademechanically or manually Although mechanical harvester isa cost-effective choice there are some disadvantages whichwill affect the wine quality [28] Thus in order to guaranteethe wine quality some regions still harvest grapes manuallyAs soon as grapes have been harvested separating stems fromgrapes and crushing grapes are necessarily performed in thewinery [29 30] Fermentation is the most significant stageof vinification and the main reaction during this stage isconverting sugars to ethanol with the assistance of yeast Thechemical reaction can be described as

C6H12O6

yeast997888997888997888rarr 2C

2H5OH + 2CO

2(12)

The natural yeast which is already presented on grapesmay give unpredictable results depending on the exact typesof yeast thus the cultured yeast is always added to themust in

order to ensure a successful fermentationThe amount of timeconsumed by a wine ferment process varies depending on thetype of grape and the methods adopted by winemakers Gen-erally 10 to 30 days will be consumed during the fermentationwhich takes place in large vats [31] After the fermentation themust (now wine) is drawn off to press away the skins whichare the so-called cap floats contributing to the color andflavour of red wine In order to acquire a relatively clear winestatical settlement is necessary to separate the sediments anddead yeast cells from wine Since the vintage can significantlyimpact the flavour and quality of wine [32] most winemakersprefer to let wines undergo a period of aging in oak orredwood barrels to improve the wine quality and the coolunderground cellars are perfect for maturing wines Duringthe period of aging winemakers will check the woodenbarrels at regular intervals Sometimes moving wine fromone barrel to another or from barrel to stainless steel tankis necessary and nitrogen gas is used to protect wine fromexposing to large amounts of oxygen [33 34] Dependingon the type and quality of the wine desired the aging ofred wine may last from several months to several yearsHowever not all of wines should go through the period ofagingWhile somewineswill improve quality with age otherscan and should be drunk immediately because the tanninwhich gives the wine its flavor will precipitate out Finally thewine is prepared for bottling For getting a good presentationof wines especially for the commercial wines filtrationbefore bottling is often used to make the wine bright andclear by removing suspended particles The red wine makingprocess described above can be roughly presented in theflowsheet of Figure 1 (the pictures used in this flowsheet canbe downloaded from httpwwwwhitmanedu)

4 Results and Discussion

In this section multivariate statistical methods are utilizedto establish the models of the relationship between thegrape physicochemical indexes and the wine quality Thefitting efficiency and predicting ability of these methods


Table 1 The wine quality of one sample given by human experts

Human expert 1 2 3 4 5 6 7 8 9 10 Mean pointsPresentation (15 points) 10 12 10 7 10 12 12 10 11 10 104Fragrance (30 points) 18 24 24 18 24 22 22 18 27 20 217Mouth feel (44 points) 24 33 37 29 28 26 22 26 34 29 288Overall feeling (11 points) 8 9 10 8 8 7 8 8 9 8 83Sum points (100 points) 60 78 81 62 70 67 64 62 81 67 692

are compared by corresponding figures or indexes Afteranalyzing the influence of grape physicochemical indexes onthe wine quality a suggestion is proposed to reduce the usageof grape physicochemical indexes in efficient wine qualityprediction

41 Data Preprocessing Thedata set (mathematicalmodelingofficial site httpwwwmcmeducn) involved in modelingconsists of the information of red wine qualities and 50 grapephysicochemical indexes For some grape physicochemicalindexes two or three measurements are preformed and themean values are taken as the results of such indexes

Corresponding to grape samples wine samples have beenvinified and collected simultaneously to avoid the differencesin wine quality caused by different vintages which influencethe compositions in wine significantly [13] Ten humanexperts have been involved in the evaluation of wine qualities(100 points in total) from four aspects presentation (15points) fragrance (30 points) mouth feel (44 points) andoverall feeling (11 points) The total points are calculated bysumming the points of four aspects given by the humanexperts and the ultimate wine quality of each sample obtainedfrom averaging the 10 total points Taking one of the samplesfor example the detailed information can be found in Table 1

It is known that the data set used to construct the modelscannot be applied to verify the performance of the modelsBased on such guideline the samples are divided into twoparts which are named calibration set and verification setrespectively The calibration set formed by 27 samples isused to construct the wine quality prediction models and theverification set formed by 10 samples is used to verify thepredicting ability of the models

42 Modeling and Comparison In order to simplify theproblem some assumptions are made as follows

(1) For all of wines involved in this paper the vinificationprocesses are identical including the materials addedduring the wine making process

(2) All of wines have been vinified by the winemakerswho possessed the same vinification experience

(3) The wine qualities have been evaluated by humanexperts as soon as the wines have been vinified

All of the above assumptions are made to ensure that thedifferences of wine qualities are only caused by differences ofthe grapes This paper merely aims to study the relationshipbetween the grape physicochemical indexes and the winequality

As aforementioned OLSR is not satisfactory to build theregression equation when the calibration set suffers the prob-lem of multicollinearity or the number of samples is smallerthan the number of variables By applying correlativityanalysis on the calibration set strong correlations have beenfound in some grape physicochemical indexes especiallythe reducing sugar and glucose whose correlation coefficientreaches 0982 The correlation coefficients which are greaterthan 095 are presented in Table 3 Besides the number ofsamples consumed by the grape physicochemical indexes isalso smaller than the number of its measure variables Thusit is obvious that OLSR is incapable of establishing the modelbased on the grape physicochemical indexes andwine qualityby using the calibration set

Apart fromOLSR all the discussedmultivariate statisticalmethods like PCR PLSR and MPLSR have been utilized toconstruct the relevant models Two commonly used indexesthat is the root mean squared error of calibration (RMSEC)and root mean error of prediction (RMSEP) are mainlyconsidered here for evaluating the fitting efficiency andprediction ability of different methods RMSEC and RMSEPcan be described as follows

RMSEC =radicsum119873

119894=1(119910119894minus 119910cal119894)

2

119873

RMSEP =radicsum119872

119894=1(119910119894minus 119910pred119894)

2

119872

(13)

where 119910119894is the measured value for the 119894th sample 119910cal119894 and

119910pred119894 are the method fitted and model predicted responsevalues for the 119894th sample respectively 119873 and 119872 are thenumber of calibration samples and verification samplesrespectively

The CPV2 (cumulative percent variance) [18] with select-ing 95 as CPV is firstly applied to selected the number ofPCs and 16 PCs are obtained for PCR 4 LVs are selected forPLSR by utilizing cross-validation [26 35] With maintainingnearly all the information from the origin data set MPLSRis not necessary to select the PCs or LVs that are the maindifficulties for PCR and PLSR

Figures 2(a) 2(b) 2(c) 2(d) 2(e) and 2(f) show thefitting efficiency of different models obtained by PCR PLSRand MPLSR For only preserving the significant variabilityinformation in the grape physicochemical indexes whenchoosing PCs the fitting efficiency of PCR is the worst amongthese three methods Different from PCR PLSR selects LVs


100

90

80

70

60

50

40

40 60 80 100

PCR fitted wine quality

Actu

al w

ine q

ualit

y

(a)

0 10 20 30

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90

Wine samples

Win

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lity

(b)

50 60 70 80 90 100

50

60

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PCR predicted wine quality

(g)

Actu

al w

ine q

ualit

y

50 60 70 80 90 100

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PLSR fitted wine quality

Actu

al w

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ualit

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(c)

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(d)

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PLSR predicted wine quality

Actu

al w

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ualit

y

(h)

MPLSR fitted wine quality50 60 70 80 90 100

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Actu

al w

ine q

ualit

y

(e)

0 10 20 30

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Actual wine qualityFitted wine quality

(f)

MPLSR predicted wine quality50 60 70 80 90 100

50

60

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80

90

100Ac

tual

win

e qua

lity

(i)

Figure 2 (a)ndash(f) denote fitted quality versus actual quality (g)ndash(i) denote predicted quality versus actual quality (a) (b) (g) are plotted byPCR (c) (d) (h) are plotted by PLSR (e) (f) (i) are plotted by MPLSR

under the consideration of both the relation among physic-ochemical indexes and the relationship between the grapephysicochemical indexes and the wine quality As it is shownin Figure 2 the model constructed by PLSR is obviouslysuperior to the PCRmodelThe best fitting efficiency belongsto the MPLSR model in which the fitted wine quality equalsthe actual quality ForMPLSRmodel there exists no problemfor selecting the PCs or LVs which is often the source oferrors between the fitted values and actual values due to onlykeeping the PCs or LVs and ignoring the residues based onsome criterion [18 26] With the RMSEC values that denotethe fitting ability of different models Table 2 can also showthat the fitting efficiency of the MPLSR model is the bestamong the three models and PCR model is the worst

Table 2 RMSEC and RMSEP values

PCR PLSR MPLSRRMSEC 453 282 0RMSEP 426 232 131

The goal of regression models is to predict the winequality from new measured grape physicochemical indexesIn order to compare the prediction ability of thesemodels theRMSEP values of each model are calculated Correspondingto the maximum RMSEP value of the PCR model andthe minimum RMSEP value of the MPLSR model thePCR model has the worst prediction ability while the best


Table 3 Correlation coefficients of some grape physicochemical indexes

Total content of amino acids Resveratrol Flavonol Reducing sugar Reducing sugarProline CIS resveratrol Quercetin Fructose Glucose

Correlation coefficient 0978 0978 0968 0967 0982

predicting ability belongs to MPLSR model which is similarto the results of fitting efficiency of these three modelsFigures 2(g) and 2(h) give visual results The blue points inthe figures should locate on the red line if the model canprecisely predict the wine quality and based on this ruleit can be seen that the best predicting ability of the MPLSRmodel can be also derived

According to Figure 2 and Table 2 it is obvious thatdifferent multivariate statistical methods will significantlyinfluence the result of the predicting models Due to theproblems of multicollinearity and the number of samplessmaller than the number of variables in the calibrationset the OLSR is absolutely unusable for establishing theprediction model With the minimal values of RMSEC andRMSEP MPLSR obtains a satisfying result for both the fittingefficiency and predicting ability

43 Selecting the Key Grape Physicochemical Indexes It isknown that 50 grape physicochemical indexes are includedin the calibration set and sufficient information has been pro-vided by these indexes However it is a time consuming andcomplex work to measure so many grape physicochemicalindexes From the regression equations it has been foundthat some grape physicochemical indexes contribute little tothe wine quality Besides Table 3 also shows that some grapephysicochemical indexes are redundant From these pointsit is necessary to analyze the contribution of every grapephysicochemical index and select the optimal grape physic-ochemical indexes for efficiently predicting the wine quality

In order to analyze the contribution of the every grapephysicochemical index to the wine quality a contributionratio (CR) is introduced in and it can be formulated as follows

119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898

119861 = [1205731 1205732sdot sdot sdot 120573119898] isin 1198771times119898

119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)

where 119861 is the coefficient matrix of regression equation and1205730is a constant The benefit of this definition for CR is that

the sum of all CR119894(119894 = 1 sdot sdot sdot 119898) equals 1

As the best prediction model the MPLSR model isapplied to analyze the CR of every grape physicochemicalindex and select the optimal indexes Utilizing one of thesamples the CR of every grape physicochemical index canbe obtained as shown in Table 4 The CR with a negativevalue means that the index has a negative contribution tothe wine quality From the table it is obvious that somegrape physicochemical indexes influence the wine qualitysignificantly especially alanine protein total sugar solublesolids and dry matter content while some indexes have little

0

0

20 30 40 50

50

10

Grape physicochemical indexes

minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids

Figure 3 The CRs of 50 grape physicochemical indexes

influence or even no influence on the wine quality such astrans resveratrol glucoside CIS resveratrol glucoside andisorhamnetin whose CRs equal 0 An illustrated expressionabout the CRs is shown in Figure 3

To get rid of these useless indexes which contribute littleto the wine quality or have a strong correlation with othergrape physicochemical indexes a CR threshold is applied andthe grape physicochemical index will be ignored if its CRis lower than the CR threshold Comparing the regressionresults with different CR thresholds 1 is selected as thethreshold and 13 indexes are ignored which are marked inTable 5 with asterisk (lowast) Based on the new calibration setwhich includes 37 grape physicochemical indexes a newcalibrationmodel is established by using theMPLSRmethodThe CRs of the selected grape physicochemical indexes in thenew model are presented in Figure 4 As can be seen fromthe figure alanine protein total sugar and soluble solidsstill have a great influence on the wine quality Moreoverfructose PH and titratable acid also contribute to thewine quality significantly However the dry matter contentwhich influences the wine quality greatly in the old modelconstructed by 50 grape physicochemical indexes contributesto the wine quality in the new model slightly

A comparison between the newmodel and the old modelis made in Figure 5 From Figures 5(a) 5(b) 5(d) and 5(e) itis obvious that the fitting efficiency of the newmodel is nearlyequal to the old model and a same result can also be derivedfrom Table 5 with the equal RMSEC values of both modelsThe verification set is utilized to test the predicting ability ofthe newmodel As can be seen fromTable 5 although 13 grapephysicochemical indexes have been ignored the RMSEPvalue of the new model does not significantly deviate fromthat of the model built by all the 50 grape physicochemical


Table 4 The CR of 50 grape physicochemical indexes

Index CRTotal amino acids 249Aspartic acid minus213

Threonine 1918Serine minus149

Glutamic acid 1025Prolinelowast 089Glycine 1400Alanine minus3834

Cystine minus240

Valine minus1373

Methionine minus2525

Isoleucine 1728Leucine 2047Tyrosine minus268

Phenylalanine 156Lysine minus516Histidine 877Arginine minus229

Protein 4529VClowast minus078Anthocyaninlowast 023Tartaric acid minus697Malic acid 741Citric acid minus190POAdagger 634Browning degree minus697

DPPH free radical 654Total phenolic minus1577

Tannin 471Grape flavonoidlowast minus014

Resveratrollowast minus005

Trans-RGdaggerlowast 0CIS-RGdaggerlowast 012Transresveratrol 221CIS resveratrollowast 0Flavonollowast 038MCdaggerlowast 074Quercetinlowast 029Kaempferollowast minus002

Isorhamnetinlowast 0Total sugar minus4956

Reducing sugar 1067Fructose 908Glucose 1072Soluble solids minus4656

PH minus368

Titratable acid 1132Solid acid ratio minus2514

Dry matter content 3992Ear weight 143lowast13 grepe physicochemical indexes which are ignoreddaggerPOA polyphenol oxidase activity trans-RG transresveratrol glucosideCIS-RG CIS resveratrol glucoside MC myricetin cannabiscetin

Table 5 The values of RMSEC and RMSEP

Old model New modelRMSEC 0 0RMSEP 131 168

0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent

Grape physicochemical indexesC

ontr

ibut

ion

ratio

(CR)

Figure 4 The CRs of 37 grape physicochemical indexes in the newmodel

indexes A slight difference also can be found as well bycomparing Figures 5(c) and 5(f)

Whether from the results shown in Figure 2 or fromthe indexes of RMSEC and RMSEP presented in Table 2the model constructed by MPLSR has a relative satisfyingperformance for predicting the wine quality based on thegrape physicochemical indexes With the contribution ratioanalysis it is interesting to note that the contributions ofsome grape physicochemical indexes that is trans resveratrolglucoside CIS resveratrol glucoside and isorhamnetin canbe negligible while some grape physicochemical indexesthat is protein soluble solids and total sugar have a signif-icant influence on the final wine quality Although 13 grapephysicochemical indexes have been ignored the new modelconstructed based on the remaining indexes with MPLSRmethod still presents a good wine quality predicting ability

5 Conclusion

In this paper four multivariate statistical methods that isOLSR PCR PLSR and MPLSR are firstly reviewed Basedon these methods and the real data obtained in practicewine quality prediction models have been constructed Withthe superior fitting efficiency and better predicting abilityrepresented by RMSEC and RMSEP respectively the modelbuilt by MPLSR outperforms the other three models Severalgrape physicochemical indexes such as protein solublesolids and total sugar are found to have significant contribu-tions to the final wine quality while others are insignificantThrough ignoring the insignificant grape physicochemicalindexes the model constructed by key indexes could presenta satisfactory wine quality predicting ability The efficiency



50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)

Figure 5 A comparison between the model with 50 grape physicochemical indexes and the model with 37 grape physicochemical indexes(a) (b) (c) are plotted by old model constructed with 50 grape physicochemical indexes And (d) (e) (f) are plotted by the new modelconstructed with 37 grape physicochemical indexes

of the MPLSR model is essential to be validated on a largerdata set Moreover robust wine quality predictionmodels aremeaningful to be proposed in the future work

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (no 61304102) and the Natural ScienceFoundation of Liaoning Province China (no 2013020002)

References

[1] S Charters and S Pettigrew ldquoThe dimensions of wine qualityrdquoFood Quality and Preference vol 18 no 7 pp 997ndash1007 2007

[2] P Cortez A Cerdeira F Almeida T Matos and J Reis ldquoMod-eling wine preferences by data mining from physicochemicalpropertiesrdquoDecision Support Systems vol 47 no 4 pp 547ndash5532009

[3] S Buratti S Benedetti M Scampicchio and E C PangerodldquoCharacterization and classification of Italian Barbera winesby using an electronic nose and an amperometric electronictonguerdquo Analytica Chimica Acta vol 525 no 1 pp 133ndash1392004

[4] J Herrera A Guesalaga and E Agosin ldquoShortwave-nearinfrared spectroscopy for non-destructive determination ofmaturity of wine grapesrdquoMeasurement Science and Technologyvol 14 no 5 pp 689ndash697 2003

[5] F Fang J-M Li P Zhang et al ldquoEffects of grape varietyharvest date fermentation vessel and wine ageing on flavonoidconcentration in redwinesrdquoFoodResearch International vol 41no 1 pp 53ndash60 2008

[6] P R Gayon D Dubourdieu B Doneche and A LonvaudEds Handbook of Enology the Microbiology of Wine andVinifications John Wiley amp Sons New York NY USA 2006

[7] K Bindon C Varela J Kennedy H Holt and M HerderichldquoRelationships between harvest time and wine compositionin vitis vinifera l cv cabernet sauvignon 1 grape and winechemistryrdquo Food Chemistry vol 138 no 1 pp 1696ndash1705 2013

[8] R Mira de Orduna ldquoClimate change associated effects on grapeand wine quality and productionrdquo Food Research Internationalvol 43 no 7 pp 1844ndash1855 2010

[9] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension vehicle systems using


t-s fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[10] H-S Son G-S Hwang H-J Ahn W-M Park C-H Leeand Y-S Hong ldquoCharacterization of wines from grape varietiesthrough multivariate statistical analysis of 1H NMR spectro-scopic datardquo Food Research International vol 42 no 10 pp1483ndash1491 2009

[11] G E Pereira J-P Gaudillere C van Leeuwen et al ldquo1HNMR and chemometrics to characterize mature grape berriesin four wine-growing areas in Bordeaux Francerdquo Journal ofAgricultural and Food Chemistry vol 53 no 16 pp 6382ndash63892005

[12] D A Guillen M Palma R Natera R Romero and C G Bar-roso ldquoDetermination of the age of Sherry wines by regressiontechniques using routine parameters and phenolic and volatilecompoundsrdquo Journal of Agricultural and Food Chemistry vol53 no 7 pp 2412ndash2417 2005

[13] S A Bellomarino R M Parker X A Conlan N W Barnettand M J Adams ldquoPartial least squares and principal com-ponents analysis of wine vintage by high performance liquidchromatography with chemiluminescence detectionrdquoAnalyticaChimica Acta vol 678 no 1 pp 34ndash38 2010

[14] X Shi Y Lv Z Fei and J Liang ldquoA multivariable statisticalprocess monitoring method based on multiscale analysis andprincipal curvesrdquo International Journal of Innovative Comput-ing Information and Control vol 9 no 4 pp 1781ndash1800 2013

[15] L Liu D Cozzolino W U Cynkar et al ldquoPreliminary studyon the application of visible-near infrared spectroscopy andchemometrics to classify Riesling wines from different coun-triesrdquo Food Chemistry vol 106 no 2 pp 781ndash786 2008

[16] S Yin S X Ding P Zhang A Hagahni and A Naik ldquoStudyon modifications of pls approach for process monitoringrdquo inProceedings of the 18th IFACWorld Congress

[17] S Weisberg Applied Linear Regression vol 528 John Wiley ampSons New York NY USA 2005

[18] S Valle W Li and S J Qin ldquoSelection of the number ofprincipal components the variance of the reconstruction errorcriterion with a comparison to other methodsrdquo Industrial andEngineering Chemistry Research vol 38 no 11 pp 4389ndash44011999

[19] P Geladi and B R Kowalski ldquoPartial least-squares regression atutorialrdquo Analytica Chimica Acta vol 185 pp 1ndash17 1986

[20] S yin Data-driven design of fault diagnosis systems [PhDthesis] Universitat Duisburg-Essen Fakultat fut Ingenieurwis-senschaften Ingenieurwissenschaften-CampusDuisburgAbtei-lung Elektrotechink und Informationstechnik Automatisierungund komplexe Systeme 2012

[21] A Krishnan L J Williams A R McIntosh and H AbdildquoPartial Least Squares (PLS) methods for neuroimaging atutorial and reviewrdquo NeuroImage vol 56 no 2 pp 455ndash4752011

[22] X Shi Y Lv Z Fei and J Liang ldquoFokrul alom mazarbhuiyaandmuhammad abulaish clustering periodic frequent patternsusing fuzzy statistical parametersrdquo International Journal ofInnovative Computing Information and Control vol 8 no 3 pp2275ndash2283 2012

[23] V E Vinzi Handbook of Partial Least Squares ConceptsMethods and Applications Springer New York NY USA 2010

[24] H Wold ldquoSoft modelling the basic design and some exten-sionsrdquo Systems Under Indirect Observation vol 2 pp 589ndash5911982

[25] H Li X Jing and H R Karimi ldquoOutput-feedback basedh-infinity control for active suspension systems with controldelayrdquo IEEE Transactions on Industrial Electronics vol 61 no1 pp 436ndash446 2014

[26] S Wold M Sjostrom and L Eriksson ldquoPLS-regression a basictool of chemometricsrdquoChemometrics and Intelligent LaboratorySystems vol 58 no 2 pp 109ndash130 2001

[27] D I Jackson and P B Lombard ldquoEnvironmental and manage-ment practices affecting grape composition and wine quality-areviewrdquoAmerican Journal of Enology andViticulture vol 44 no4 pp 409ndash430 1993

[28] Q Zhou P Shi J Lu and S Xu ldquoAdaptive output-feedbackfuzzy tracking control for a class of nonlinear systemsrdquo IEEETransactions on Fuzzy Systems vol 19 no 5 pp 972ndash982 2011

[29] Y Fukayama D Tanaka and T Kataoka ldquoSeparation of indi-vidual instrument sounds in monaural music signals by apply-ing statistical least-squares criterionrdquo International Journal ofInnovative Computing Information and Control vol 8 no 3 pp2275ndash2283 2012

[30] X Xie S Yin H Gao and K Okyay ldquoAsymptotic stabilityand stabilisation of uncertain delta operator systems with time-varying delaysrdquo IET Control Theory amp Applications vol 7 no 8pp 1071ndash1078 2013

[31] D Li Z Wang and H Gao ldquoDistributed h-infinity filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[32] I S HornseyThe Chemistry and Biology of Winemaking RoyalSociety of Chemistry 2007

[33] Q Zhou P Shi S Xu and H Li ldquoObserver-based adaptiveneural network control for nonlinear stochastic systems withtime-delayrdquo IEEE Transactions on Neural Networks and Learn-ing Systems vol 24 no 1 pp 71ndash80 2013

[34] C J Tsai T Chang Y Yang MWu and Y Li ldquoAn approach foruser authentication on non-keyboard devices usingmouse clickcharacteristics and statistical-based classificationrdquo InternationalJournal of Innovative Computing Information and Control vol8 no 11 pp 7875ndash7886 2012

[35] X Su Z Li Y Feng and L Wu ldquoNew global exponential sta-bility criteria for interval-delayed neural networksrdquo Proceedingsof the Institution of Mechanical Engineers I vol 225 no 1 pp125ndash136 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Compared with the majority of the methods reported forwine quality analysis the method discussed in this paperprovides a simpler and more convenient way to predictthe wine quality Besides what is most remarkable of theproposedmethod is the increasing possibility for winemakersto predict the wine quality before the complete wine makingprocess and make appropriate decisions in advance such asthe grape selection wine classification and target marketing

Although investigations on evaluating wine quality onlybased on grape physicochemical indexes are rare in thewine science community the relationship between grapesand metabolites in wine is available in many literatureresources Since the multiple biochemical process occurswith the grape ripening the grape harvest time is influentialin the wine composition For example the yeast-derivedmetabolites including volatile esters dimethyl sulfide glyc-erol and mannoproteins will increase with harvest date [7]In addition the climate also has a profound influence ongrape and finally influences the wine metabolites Among themost important climate change-related effects are advancedharvest times and temperatures increased grape sugar con-centrations that lead to high wine alcohol level lower levelsof acids and the modification of varietal aroma compounds[8 9] Moreover with the help of 1H NMR spectroscopicanalysis and multivariate statistics methods such as PCAand OPLS-DA Son et al further proved that the grapevarieties influence the wine quality significantly and theyalso succeeded in separating the wines vinified from fourdifferent grape varieties [10] Besides it is known from otherliterature that the genetic factors are also in a relation withgrape and wine quality [11] Given background of wines andgrapes as well as the relevant influential factors achieved bythis literature this paper focuses on studying the relationshipbetween wine quality and grape physicochemical indexes byapplying multivariate statistical methods

The rest of this paper is structured as follows Section 2makes a brief description of the multivariate statistical meth-ods including ordinary least squares regression principalcomponent regression partial least squares regression and amodified partial least square regression Section 3 discussesthe red wine making process with relevant illustrations Thewine predictionmodels are established based onmultivariatestatistical methods in Section 4 and a comparison amongthese models is provided to select the best model Further-more the suggestion about using fewer grape physicochemi-cal indexes to predict the wine quality is proposed as well inthis section Finally the conclusions are presented in the lastsection

2 Modeling Algorithms

Multivariate statistical analysis is a powerful tool for solvingthewine analysis problemswhich often involve large amountsof data and has been wildly applied in many relevant studiessuch as analyzing the elements in wines by PLS regression[12] investigating the relationship betweenwine compositionand vintage via principal component analysis (PCA) and par-tial least squares (PLS) [13 14] and utilizing the visible-near

infrared spectroscopy and chemometrics to classify Rieslingwines from different countries with the help of PCA [15]In this section besides describing three classical multivariatestatistics methods a modified partial least squares regression(MPLSR) method is also introduced which has a simplermath description compared with standard PLS regression[16]

21 Ordinary Least Squares Regression (OLSR) Ordinaryleast squares regression (OLSR) is a standard approach toprovide the approximate solution for the overdeterminedsystems through minimizing the sum of the errors createdin the results of every single equation Generally OLSR oftenappears in the situation where only one dimension responsevariable is involved However OLSR is also capable of solvingthe problems of more than one dimension response variablesand the algorithm can be briefly introduced as follows

Step 1 Collect 119899 samples of the measurable variables andresponse variables and normalize them to zero mean andunit variance denoted as 119883 = [119909


119899times119898 119884 =

[1199101sdot sdot sdot 119910119901] isin 119877119899times119901The following steps will be performed only

when119883119879119883 is a full rank matrix

Step 2 Minimize the sum of errors 119890119895

120573119895= argmin 119890119879

119895119890119895= argmin (119910

119895minus 119883120573119895)

119879

(119910119895minus 119883120573119895)

119895 = 1 sdot sdot sdot 119901

(1)

According to the invertibility of 119883119879119883 the 120573119895can also be

calculated by

120573119895= (119883119879119883)

minus1

119883119879119910119895 (2)

Step 3 Use the 119890119895and 120573

119895to form 119864 isin 119877

119899times119901 and 119872 isin 119877119898times119901

respectively and the final model can be expressed as

119884 = 119883119872 + 119864 (3)

22 Principal Component Regression (PCR) In practice theindependent variables may be highly collinear This phe-nomenon is the so-called multicollinearity and it is knownthat such collinearity problems can sometimes lead to seriousstability problems when the OLSR is applied [17] Alternativemethods to OLSR are proposed in order to overcome thisproblem The most popular methods are the principal com-ponent regression (PCR) and partial least squares regression(PLSR) The collinearity problems are efficiently solved inPCR algorithm by introducing principal components (PCs)which preserve the significant information in the originaldata set and the algorithm can be briefly formulated asfollows

Step 1 Perform normalization on the gathered measurablevariables and the responses variables presented as 119883 =

[1199091sdot sdot sdot 119909119898] isin 119877

119899times119898 and 119884 = [1199101sdot sdot sdot 119910119901] isin 119877

119899times119901where 119909

119894and 119910

119894have the zero mean and the unit variance



1

119899 minus 1

119883119879119883 = 119875Λ119875

119879



119898gt 0

(4)



119879 = 119883119875pc isin 119877119899times119897

(5)

where



119875res = [119901119897+1



(6)


120573119894= (119879119879119879)

minus1

119879119879119910119894 119894 = 1 sdot sdot sdot 119901


119884 = 119879119861 + 119864 = 119883119872 + 119864 119872 = 119875pc119861

(7)





119877119899times119901 where 119909

119894and 119910


variance


(119901119896 119902119896) = arg max

119901119896=1119902119896=1

119901119879

119896119883119879119884119902119896 119896 = 1 sdot sdot sdot 120574

119906119896= 119883119896119901119896 V

119896= 119884119896119902119896

119883119896= 119906119896119888119879

119896+ 119883119896+1

119884119896= 119906119896119903119879

119896+ 119884119896+1

119888119896=

119883119879

119896119906119896

1003817100381710038171003817119906119896

1003817100381710038171003817

2 119903

119896=

119884119879

119896119906119896

1003817100381710038171003817119906119896

1003817100381710038171003817

2

119883119896+1

= 119883119896minus 119906119896119888119879

119896 119884

119896+1= 119884119896minus 119906119896119903119879

119896

(8)

where the 119901119896 119902119896and 119906



Step 3 Store 119901119896 119906119896 119888119896 and 119903

119896into 119875 119880 119862 and 119877 and the


119880 = 119883119875 119883 = 119880119862119879+ 119864

119884 = 119880119877119879+ 119865 = 119883119872 + 119865 119872 = 119875119877

119879

(9)





119877119899times119901


1

119899

119884119879119883 asymp 119872

119879119883119879119883

119899

119872 = (119883119879119883)

minus1

119883119879119884 if rank (119883119879119883) = 119898

119872 = (119883119879119883)

dagger

119883119879119884 if rank (119883119879119883) lt 119898

(10)







119884 = 119883119872 + 119864119910 (11)


119883




C6H12O6

yeast997888997888997888rarr 2C

2H5OH + 2CO

2(12)




















119894=1(119910119894minus 119910cal119894)

2

119873


119894=1(119910119894minus 119910pred119894)

2

119872

(13)






100

90

80

70

60

50

40

40 60 80 100


Actu

al w

ine q

ualit

y

(a)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(b)

50 60 70 80 90 100

50

60

70

80

90

100


(g)

Actu

al w

ine q

ualit

y

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(c)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(d)

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(h)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(e)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity


(f)


50

60

70

80

90

100Ac

tual

win

e qua

lity

(i)














119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898


119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)




0

0

20 30 40 50

50

10


minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids










Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

119899 minus 1

119883119879119883 = 119875Λ119875

119879



119898gt 0

(4)



119879 = 119883119875pc isin 119877119899times119897

(5)

where



119875res = [119901119897+1



(6)


120573119894= (119879119879119879)

minus1

119879119879119910119894 119894 = 1 sdot sdot sdot 119901


119884 = 119879119861 + 119864 = 119883119872 + 119864 119872 = 119875pc119861

(7)





119877119899times119901 where 119909

119894and 119910


variance


(119901119896 119902119896) = arg max

119901119896=1119902119896=1

119901119879

119896119883119879119884119902119896 119896 = 1 sdot sdot sdot 120574

119906119896= 119883119896119901119896 V

119896= 119884119896119902119896

119883119896= 119906119896119888119879

119896+ 119883119896+1

119884119896= 119906119896119903119879

119896+ 119884119896+1

119888119896=

119883119879

119896119906119896

1003817100381710038171003817119906119896

1003817100381710038171003817

2 119903

119896=

119884119879

119896119906119896

1003817100381710038171003817119906119896

1003817100381710038171003817

2

119883119896+1

= 119883119896minus 119906119896119888119879

119896 119884

119896+1= 119884119896minus 119906119896119903119879

119896

(8)

where the 119901119896 119902119896and 119906



Step 3 Store 119901119896 119906119896 119888119896 and 119903

119896into 119875 119880 119862 and 119877 and the


119880 = 119883119875 119883 = 119880119862119879+ 119864

119884 = 119880119877119879+ 119865 = 119883119872 + 119865 119872 = 119875119877

119879

(9)





119877119899times119901


1

119899

119884119879119883 asymp 119872

119879119883119879119883

119899

119872 = (119883119879119883)

minus1

119883119879119884 if rank (119883119879119883) = 119898

119872 = (119883119879119883)

dagger

119883119879119884 if rank (119883119879119883) lt 119898

(10)







119884 = 119883119872 + 119864119910 (11)


119883




C6H12O6

yeast997888997888997888rarr 2C

2H5OH + 2CO

2(12)




















119894=1(119910119894minus 119910cal119894)

2

119873


119894=1(119910119894minus 119910pred119894)

2

119872

(13)






100

90

80

70

60

50

40

40 60 80 100


Actu

al w

ine q

ualit

y

(a)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(b)

50 60 70 80 90 100

50

60

70

80

90

100


(g)

Actu

al w

ine q

ualit

y

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(c)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(d)

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(h)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(e)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity


(f)


50

60

70

80

90

100Ac

tual

win

e qua

lity

(i)














119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898


119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)




0

0

20 30 40 50

50

10


minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids










Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119884 = 119883119872 + 119864119910 (11)


119883




C6H12O6

yeast997888997888997888rarr 2C

2H5OH + 2CO

2(12)




















119894=1(119910119894minus 119910cal119894)

2

119873


119894=1(119910119894minus 119910pred119894)

2

119872

(13)






100

90

80

70

60

50

40

40 60 80 100


Actu

al w

ine q

ualit

y

(a)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(b)

50 60 70 80 90 100

50

60

70

80

90

100


(g)

Actu

al w

ine q

ualit

y

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(c)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(d)

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(h)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(e)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity


(f)


50

60

70

80

90

100Ac

tual

win

e qua

lity

(i)














119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898


119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)




0

0

20 30 40 50

50

10


minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids










Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























119894=1(119910119894minus 119910cal119894)

2

119873


119894=1(119910119894minus 119910pred119894)

2

119872

(13)






100

90

80

70

60

50

40

40 60 80 100


Actu

al w

ine q

ualit

y

(a)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(b)

50 60 70 80 90 100

50

60

70

80

90

100


(g)

Actu

al w

ine q

ualit

y

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(c)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(d)

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(h)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(e)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity


(f)


50

60

70

80

90

100Ac

tual

win

e qua

lity

(i)














119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898


119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)




0

0

20 30 40 50

50

10


minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids










Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

90

80

70

60

50

40

40 60 80 100


Actu

al w

ine q

ualit

y

(a)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(b)

50 60 70 80 90 100

50

60

70

80

90

100


(g)

Actu

al w

ine q

ualit

y

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(c)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity

(d)

50 60 70 80 90 100

50

60

70

80

90

100


Actu

al w

ine q

ualit

y

(h)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(e)

0 10 20 30

50

60

70

80

90

Wine samples

Win

e qua

lity


(f)


50

60

70

80

90

100Ac

tual

win

e qua

lity

(i)














119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898


119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)




0

0

20 30 40 50

50

10


minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids










Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119910 minus 1205730= 119861119883 CR

119894=

120573119894119909119894

119910 minus 1205730

119894 = 1 sdot sdot sdot 119898


119883 = [1199091 1199092sdot sdot sdot 119909119898]119879

isin 1198771times119898

(14)




0

0

20 30 40 50

50

10


minus50

Con

trib

utio

n ra

tio (C

R)

Alanine

Protein

Total sugarSoluble

Dry matter content

solids










Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Cystine minus240

Valine minus1373











PH minus368





0 20 30 40 50

1

08

06

04

02

0

minus02

minus04

10

Alanine

Protein

Total sugar

Soluble solids

Fructose

PHTitratableacid

Dry mattercontent


ontr

ibut

ion

ratio

(CR)




5 Conclusion




50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(a)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity

(b)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(c)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y (a)

(d)

90

85

80

75

70

65

60

55

50

0 5 10 15 20 25 30

Wine samples

Win

e qua

lity


(e)


50

60

70

80

90

100

Actu

al w

ine q

ualit

y

(f)





Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article quality evaluation based on multivariate

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