research article functional principal components...

Research ArticleFunctional Principal Components Analysis ofShanghai Stock Exchange 50 Index

Zhiliang Wang Yalin Sun and Peng Li

College of Mathematics and Informatics North China University of Water Conservancy and Hydroelectric PowerZhengzhou 450000 China

Correspondence should be addressed to Zhiliang Wang wzlncwueducn

Received 28 February 2014 Revised 23 May 2014 Accepted 4 July 2014 Published 22 July 2014

Academic Editor Seenith Sivasundaram

Copyright copy 2014 Zhiliang Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The main purpose of this paper is to explore the principle components of Shanghai stock exchange 50 index by means offunctional principal component analysis (FPCA) Functional data analysis (FDA) deals with random variables (or process) withrealizations in the smooth functional space One of the most popular FDA techniques is functional principal component analysiswhich was introduced for the statistical analysis of a set of financial time series from an explorative point of view FPCA is thefunctional analogue of the well-known dimension reduction technique in the multivariate statistical analysis searching for lineartransformations of the random vector with themaximal variance In this paper we studied themonthly return volatility of Shanghaistock exchange 50 index (SSE50) Using FPCA to reduce dimension to a finite level we extracted the most significant componentsof the data and some relevant statistical features of such related datasets The calculated results show that regarding the samples asrandom functions is rational Compared with the ordinary principle component analysis FPCA can solve the problem of differentdimensions in the samples And FPCA is a convenient approach to extract the main variance factors

1 Introduction

In the present study of data analysis we have learned thedata we research is either cross-sectional data or panel dataIn the practical research however we often meet with suchdata which has functional characteristics Functional datais multivariate data with an ordering on the dimensions[1] The data seem to deserve the label ldquofunctionalrdquo sincethey so clearly reflect the smooth curves that we assumegenerated them The typical dataset of this sort consistsof time series and cross-sectional data such as the timeseries of stock price and some datasets even may take oncurves or images Advances in data collection and storagehave tremendously increased the presence of such functionaldata whose graphical representations are curves imagesor shapes The theoretical and practical developments infunctional data analysis aremainly from the last four decadesdue to the rapid development of computer recording andstoring facilities As a new area of statistics functionaldata analysis extends existing methodologies and theories

from the fields of data analysis generalized linear modelsmultivariate data analysis nonparametric statistics andmanyothers Recently there were several impressive attempts toanalyze functional dataset such as Ramsay et al [2ndash5] whoproposed some new concepts and methods in the field ofFDA

FPCA is the functional analogue of the well-knowndimension reduction technique in the multivariate statisticalanalysis and is useful in determining the common factorsor trends that are present in the dynamics of the underlyingrecovered functions

The advance of FPCA can be seen when Karhunen [7]and Loeve [8] independently developed a theory on theoptimal series expansion of a continuous stochastic processMotivated by a dataset of growth curve Rao [9] developedsome preliminary ideas on FPCA and proposed statisticaltests for the equality of average growth curves over a period oftime Much later Dauxois et al [10] introduced a functionalexposition of PCA with applications to statistical inferenceSeveral other notable developments have arisen out of the

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 365204 7 pageshttpdxdoiorg1011552014365204

2 Discrete Dynamics in Nature and Society

Table 1 The differences in notation between PCA and FPCA [6]

PCA FPCAVariables 119883 = [119909

1 1199092 119909

119899] 119909119894= [1199091119894 119909

119901119894]1015840

119894 = 1 119899 119891 (119905) = [1198911(119905) 1198912(119905) 119891

119899

(119905)] 119905 isin [1199091 119909119901]

Data Vectors isin 119877119875 Curves isin 1198712[1199091 119909119901]

Covariance Matrix 119881 = Cov(119883) isin 119877119875 times 119877119875Operator 119881 bounded between 119909

1and 119909

119901

120601119896(119905) isin 119871

2[1199091 119909119901] int119909119901

1199091

119881120585119896(119905) 119889119905 = 120582

119896120585119896(119905)

119881 1198712[1199091 119909119901] rarr 119871

2[1199091 119909119901]

Eigen structure Vector Φ119896isin 119877

119881Φ119896= 120582119896Φ119896 for 1 le 119896 le min(119899 119901)

Function 120601119896(119905) isin 119871

2[1199091 119909119901]

int119909119901

1199091

119881120601119896(119905) 119889119905 = 120582

119896120601119896(119905) for 1 le 119896 le 119899

Components Random variables in 119877119875 Random variables in 1198712[1199091 119909119901]

systematic research of the functional data analysis groupnamed the Toulouse School of Functional Data Analysis [11]

In recent years Hall and Hosseini-Nasab [12 13] showedhow the properties of functional principal component anal-ysis can be elucidated through stochastic expansions andrelated results Yao et al [14] proposed a FPCA procedurevia a conditional expectation method which is aimed atestimating functional principal component scores for sparselongitudinal data Hall and Vial [15] have investigated theproperties of FPCA and have given some insights intomethodology and convergence rates Di et al [16] introducedmultilevel FPCA which is designed to extract the intra- andintersubject geometric components of multilevel functionaldata Based on FPCA Hyndman and Shang [17] proposedgraphical tools for visualizing functional data and detectingfunctional outliers

Due to the theoretical and practical developments FPCAhas been successfully applied to many practical problemssuch as the analysis of cornea curvature in the human eye[18] the analysis of electronic commerce [19] the analysisof growth curve [20] the analysis of income density [21]the analysis of implied volatility surface in finance [22] theanalysis of longitudinal primary biliary liver cirrhosis [23]and the analysis of spectroscopy data [24] FurthermoreHyndman and Shahid Ullah [25] proposed a smoothed androbust FPCAandused it to forecast age-specificmortality andfertility rates

The objective of this paper is to study the monthlyvolatility of return of Shanghai 50 index which consists of50 stocks Treating stock price series as random functionin a space spanned by finite dimensional functional baseswe intensively explore methods of functional data analysisespecially functional principal component analysis

In the area of finance some impressive papers withthe functional data analysis are found such as Ramsay andRamsey [26] Muller and Ulrich [27] andMiao [28] But fewrepublications are found with research on the increasinglyflourishing Chinese financial market This paper will fill theblank both in theory and in application

Our study can be described as an exploratory dataapproach

Data collection 997888rarr Data Analysis 997888rarr ConclusionsThis paper is organized as follows In Section 2 we describethe functional principal component analysis (FPCA) which

plays a significant role in the development of functionaldata analysis It is also an essential ingredient of functionalprincipal component regression (FPCR) Section 3 will illus-trate the empirical study with the application of the theoryin Section 2 Some further discussion and a conclusion arepresented in Section 4

2 Methodology

As mentioned before an important tool in the functionaldata analysis toolbox is FPCA that is functional principalcomponent analysis The main idea of FPCA is just likemultivariate principal component analysis (PCA) but itsprincipal component weights or harmonics are functions oftime They carry the main features of the functional dataobject and can be interpreted separately

The differences in notation between PCA and FPCA aresummarized in Table 1

The basic assumption of FDA is that data generatingprocess can be described as a smooth function FPCAfinds the set of orthogonal principal component function bymaximizing the variance along each component

The first functional principal component 1206011(119905) is defined

by

1206011(119905) = arg max

Φ2= 1

1

119873

119873

sum

119894=1

(int119871

120601(119905)119891119894(119905) 119889119905)

2

(1)

subject to10038171003817100381710038171206011

1003817100381710038171003817 = 1 (2)

The 119896th functional principal component 120601119896(119905) can be found

analogously subject to the additional constraint

int119871

120601119895(119905) 120601119896(119905) 119889119905 = 0 forall119895 lt 119896 (3)

The sample covariance function of 119891(119909) = [1198911(119909) 1198912(119909)

119891119899(119909)] 119909 isin [119909

1 119909119901] is given by

119881 (119904 119905) =1

119873

119873

sum

119894=1

119891119894(119904) 119891119894(119905) (4)

where function 119891119894(119905) has usually been first centered

Discrete Dynamics in Nature and Society 3

Covariance operator 119881 extends the concept of a samplecovariance matrix to functional data it is easy to show that119881is a positive compact symmetric linear operator It is obviousthat

(119881120601119870) (119905) = 120582

119896120601119870(119905) 120582

1ge 1205822ge sdot sdot sdot ge 0 (5)

Detailed calculation procedure is provided below

Step 1 The data we need in this paper is collected throughsome public resources such as WIND database

Step 2 The data we get may be dirty so data preprocessingis necessary Then the raw data are collected cleaned andorganized

Step 3 The data are next converted to functional formThrough this step the raw data for observation 119894 are usedto define a function 119891

119894that can be evaluated at all values of

119905 over interval [1199091 119909119901] In order to do this a basis must be

specified A nonparametric method is used to estimate 119891119894(119905)

for 119905 isin [1199091 119909119901] 119894 = 1 119899

Then we express each function as a linear combinationof basic functions and approximate each function by a finitenumber of basis functions 120593

119896 Consider

119891119894(119905) asymp

119870

sum

119896=1

120573119894119896120593119896 119894 = 1 119899 (6)

Some popular basis functions such as polynomial basisfunctions Bernstein polynomial basis functions Fourierbasis functions and wavelet basis function and B-spline areused to estimate the functions B-spline is our first choicebecause of its goodness of fitting nonperiodic data in ourstudy

Step 4 The functionmay also need to be registered or alignedin order to show some important features Vertical amplitudevariation and horizontal variation can be separated by thisstep In our study this step is not used due to our datacharacteristics

Step 5 Next a variety of preliminary displays and sum-mary statistics are developed For example first and secondderivative curves estimated from these data using techniquesdiscussed before are displayed and we can elude that somecurves have larger variation while other curves are with lessimpressed variation

Step 6 Then exploratory analyses such as FPCA can becarried out

The first principal component can be found by solving

1198811206011(119905) = 120582

11206011(119905)

100381710038171003817100381712060111003817100381710038171003817 = 1 (7)

Step 7 The 119896th functional principal component is a solutionof

119881120601119896(119905) = 120582

119896120601119896(119905)

10038171003817100381710038171206011198961003817100381710038171003817 = 1 (8)

0 2 4 6 8 10 12minus60

minus40

minus20

0

20

40

60

Figure 1 The monthly rate of return of 50 stocks

subject to

int119871

120601119895(119905) 120601119896(119905) = 0 forall119895 lt 119896 (9)

Step 8 Accumulative percentage of explained variance iscalculated and some discussion and economic explanationabout the functional principal component are providedfinally

3 Application

We now represent the monthly rate of return of 50 stocks inFigure 1 which constitute the SSE50 index

As we can see almost nothing can be seen in this form ofplot So some work must be taken to study the data

Then the datasets are converted to functional formwhichmeans functions that can be evaluated at all values of Tover some intervalThe 50 functions are displayed in Figure 2with the estimated mean function in bold There are featuresin this data too subtle to see in this type of plot

An impression is that some curves are high (with goodinvestment return) and that some curves are low (with not sogood investment return)

We therefore conclude that some of the variation fromcurve to curve can be explained at the level of certainderivatives The fact that derivatives are of interest is furtherreason to think of the records as functions rather than vectorsof observations in discrete time

Next we will give the fitted curve of the 50 curves we havegot With the stock code 600019 we can see that the fittedresult is pretty good as illustrated in Figure 3

Figures 4 and 5 display the first and second derivativecurves estimated from these data using techniques discussedbefore We can elude that some curves have larger variationwhile other curves are with less impressed variation

Now in Figure 6 we illustrate the variance-covariancestructure of return rate The peak point at the middle of thediagonal represents the largest variance in October

At last the principal component functions are repre-sented in Figures 7ndash11 as perturbations of the mean function


2 4 6 8 10 12minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Arguments

Func

tions

Figure 2The functional form of 50 stocks Note the black bold lineis the mean function

2 4 6 8 10 12minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Arguments

Func

tions

Case 2 RMS residual = 00025067

Figure 3 The functional form of the stock code 600019

2 4 6 8 10 12

0

50

Arguments

Func

tions

minus150

minus100

minus50

Figure 4 The first derivative curves

2 4 6 8 10 12

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Arguments

Func

tions

Figure 5 The second derivative curves

0

5

10

15

0

5

10

15

minus100

minus50

0

50

100

150

Figure 6 The variance-covariance structure of return rate

Table 2 Accumulative percentage of explained variance

PC The percentage of explained variancePC1 3022PC2 2207PC3 1991PC4 1102PC5 754Total 9076

by adding and subtracting a multiple of each principalcomponent function

Table 2 includes accumulative percentage of explainedvariance

We can see that the first principal component function(Figure 7) which accounts for 3022 percent of the variationhas always had an obvious positive effect on the mean func-tion between February and April 2011 In fact the conceptof high-speed rail provoked the Chinese financial marketvigorously during that period Therefore we can reasonably


4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles

PCA function 1 (percentage of variability 30)

1 2 3 4 5 6 7 8 9 1110 12

Figure 7 The first principal component function

3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12

Figure 8 The second principal component function

believe that the first principal component represents thespeculation boom

The second principal component function (Figure 8)which accounts for 2207 percent of the variance is seen topick up the influence of tighten monetary policy to controlthe excessive price rises especially the price of real estate

With the stock price getting lower and lower more andmore investors believe the current price is worth taking riskwhich forms some power of buying driving price brieflyreboundedThe third principal component (Figure 9) whichaccounts for 1991 percent of the variation is believed to bethe representative of the influence

With the excess drop in price a growing number ofblue chips are underestimated and the investment value willpromote a price returnThe effect is summarized to the fourthprincipal component (Figure 10) which accounts for 1102percent of the variation

3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

1 2 3 4 5 6 7 8 9 1110 12

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles


Figure 9 The third principal component function

3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12

Figure 10 The fourth principal component function

Thefifthprincipal component (Figure 11) which accountsfor only 75 percent of the variation having little effect on themean function will not be discussed in this paper

4 Conclusion and Further Discussion

FPCA attempts to find the dominant modes of variationaround an overall trend function and is thus a key techniquein functional data analysis As we described before moderndata analysis has benefited and will continue to benefitgreatly from the development of functional data analysisIn this paper we mainly illustrate the functional principalcomponents analysis by the research on monthly returnrate of stocks constituting Shanghai 50 index We extractedthe main variance factors over time by extracting principalcomponent regarding the samples as random functionswhich has strong theoretical and practical value A functionalfeature of the proposed approach that distinguishes it from


3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12

Figure 11 The fifth principal component function

established methods for spot volatility analysis is that it isgeared towards the analysis of observations drawn from allrealizations of the volatility process rather than observationsfrom a single realization As we described before the firstprincipal component function has always had an obviouspositive effect on the mean function and thus could besummarized as the representation of the speculation boomThe second principal component function on the other handhaving outstanding negative effect on the mean functionbetween May to July 2011 is seen to pick up the influence offiscal austerity to curb the fast rising prices

In addition the proposed FPCA method is easy to pro-gram and implement By smoothing the underlying functionsor curves the principal components we need are extractedeasily The fast computational speed of our method makes itfeasible to be applied in empirical studies with a large numberof observations

Besides the proposed method in this paper several othermethods such as curves classification nonparametric analy-sis and functional depth analysis can be utilized to analyzefunctional dataThesemethods will be considered in the nextstudy Moreover in order to emphasize the interest of doingthe functional approach and compare the correspondingresults we will treat the curves as high dimensional standardvectors in the future work

Conflict of Interests

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

References

[1] P Hall H Muller and J Wang ldquoProperties of principal com-ponent methods for functional and longitudinal data analysisrdquoAnnals of Statistics vol 34 no 3 pp 1493ndash1517 2006

[2] J O Ramsay and C J Dalzell ldquoSome tools for functionaldata analysis (with discussion)rdquo Journal of the Royal StatisticalSociety B vol 53 no 3 pp 539ndash572 1991

[3] J O Ramsay and B W Silverman Applied Functional DataAnalysis Methods and Case Studies Springer New York NYUSA 2002

[4] J O Ramsay and B W Silverman Functional Data AnalysisSpringer New York NY USA 2nd edition 2005

[5] J O Ramsay G Hooker and S Graves Functional DataAnalysis with R and MATLAB Springer New York NY USA2009

[6] H L Shang A Survey of Functional Principal ComponentAnalysis (Working Paper0611) Department of Econometricsand Business Statistics Monash University 2011

[7] KKarhunen ldquoZur Spektraltheorie stochastischer Prozesserdquo vol1946 no 34 7 pages 1946

[8] M Loeve ldquoFonctions aleatoires a decomposition orthogonalexponentiellerdquo La Revue Scientique vol 84 pp 159ndash162 1946

[9] C R Rao ldquoSome statistical methods for comparison of growthcurvesrdquo Biometrics vol 14 no 1 pp 1ndash17 1958

[10] J Dauxois and A Pousse Les analyses factorielles en calcul desprobabilities et en statistique essai drsquoetude synthetique [PhD the-sis] lrsquoUniversite Paul-Sabatier de Toulouse Toulouse France1976

[11] J Dauxois A Pousse and Y Romain ldquoAsymptotic theoryfor the principal component analysis of a vector randomfunction some applications to statistical inferencerdquo Journal ofMultivariate Analysis vol 12 no 1 pp 136ndash154 1982

[12] P Hall and M Hosseini-Nasab ldquoOn properties of functionalprincipal components analysisrdquo Journal of the Royal StatisticalSociety B Statistical Methodology vol 68 no 1 pp 109ndash1262006

[13] P Hall and M Hosseini-Nasab ldquoTheory for high-order boundsin functional principal components analysisrdquo MathematicalProceedings of the Cambridge Philosophical Society vol 146 no1 pp 225ndash256 2009

[14] F Yao H Muller and J Wang ldquoFunctional data analysis forsparse longitudinal datardquo Journal of the American StatisticalAssociation vol 100 no 470 pp 577ndash590 2005

[15] P Hall and C Vial ldquoAssessing the finite dimensionality offunctional datardquo Journal of the Royal Statistical Society B vol68 no 4 pp 689ndash705 2006

[16] C Di C M Crainiceanu B S Caffo and N M Punjabi ldquoMul-tilevel functional principal component analysisrdquo The Annals ofApplied Statistics vol 3 no 1 pp 458ndash488 2009

[17] R J Hyndman and H L Shang ldquoRainbow plots bagplotsand boxplots for functional datardquo Journal of Computational andGraphical Statistics vol 19 no 1 pp 29ndash45 2010

[18] N Locantore J S Marron D G Simpson N Tripoli J TZhang and K L Cohen ldquoRobust principal component analysisfor functional datardquo Test vol 8 no 1 pp 1ndash73 1999

[19] S Wang W Jank and G Shmueli ldquoExplaining and forecastingonline auction prices and their dynamics using functional dataanalysisrdquo Journal of Business amp Economic Statistics vol 26 no2 pp 144ndash160 2008

[20] J M Chiou and P L Li ldquoFunctional clustering and identifyingsubstructures of longitudinal datardquo Journal of the Royal Statisti-cal Society B Statistical Methodology vol 69 no 4 pp 679ndash6992007

[21] A Kneip and K J Utikal ldquoInference for density familiesusing functional principal component analysisrdquo Journal of the


American Statistical Association vol 96 no 454 pp 519ndash5322001

[22] F Cont and J Fonseca ldquoThe dynamics of implied volatilitysurfacesrdquo Quantitative Finance vol 2 no 1 pp 45ndash60 2002

[23] F Yao H G Muller and J L Wang ldquoFunctional linearregression analysis for longitudinal datardquo Annals of Statisticsvol 33 no 6 pp 2873ndash2903 2005

[24] F Yao and H Muller ldquoFunctional quadratic regressionrdquo Bio-metrika vol 97 no 1 pp 49ndash64 2010

[25] R J Hyndman and M Shahid Ullah ldquoRobust forecastingof mortality and fertility rates a functional data approachrdquoComputational Statistics and Data Analysis vol 51 no 10 pp4942ndash4956 2007

[26] J O Ramsay and J B Ramsey ldquoFunctional data analysisof the dynamics of the monthly index of nondurable goodsproductionrdquo Journal of Econometrics vol 107 no 1-2 pp 327ndash344 2002

[27] H G Muller and M S Ulrich ldquoFunctional data analysis forvolatilityrdquo Journal of Economics Literature ClassificationCodes C14 C51 C52 G12 G17 2011

[28] H Miao ldquoPotential applications of function data analysisin high-frequency financial researchrdquo Journal of Business ampFinancial Affairs vol 2 no 1 article e125 2013

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Stochastic AnalysisInternational Journal of


Table 1 The differences in notation between PCA and FPCA [6]

PCA FPCAVariables 119883 = [119909

1 1199092 119909

119899] 119909119894= [1199091119894 119909

119901119894]1015840

119894 = 1 119899 119891 (119905) = [1198911(119905) 1198912(119905) 119891

119899

(119905)] 119905 isin [1199091 119909119901]

Data Vectors isin 119877119875 Curves isin 1198712[1199091 119909119901]

Covariance Matrix 119881 = Cov(119883) isin 119877119875 times 119877119875Operator 119881 bounded between 119909

1and 119909

119901

120601119896(119905) isin 119871

2[1199091 119909119901] int119909119901

1199091

119881120585119896(119905) 119889119905 = 120582

119896120585119896(119905)

119881 1198712[1199091 119909119901] rarr 119871

2[1199091 119909119901]

Eigen structure Vector Φ119896isin 119877

119881Φ119896= 120582119896Φ119896 for 1 le 119896 le min(119899 119901)

Function 120601119896(119905) isin 119871

2[1199091 119909119901]

int119909119901

1199091

119881120601119896(119905) 119889119905 = 120582

119896120601119896(119905) for 1 le 119896 le 119899

Components Random variables in 119877119875 Random variables in 1198712[1199091 119909119901]

systematic research of the functional data analysis groupnamed the Toulouse School of Functional Data Analysis [11]

In recent years Hall and Hosseini-Nasab [12 13] showedhow the properties of functional principal component anal-ysis can be elucidated through stochastic expansions andrelated results Yao et al [14] proposed a FPCA procedurevia a conditional expectation method which is aimed atestimating functional principal component scores for sparselongitudinal data Hall and Vial [15] have investigated theproperties of FPCA and have given some insights intomethodology and convergence rates Di et al [16] introducedmultilevel FPCA which is designed to extract the intra- andintersubject geometric components of multilevel functionaldata Based on FPCA Hyndman and Shang [17] proposedgraphical tools for visualizing functional data and detectingfunctional outliers

Due to the theoretical and practical developments FPCAhas been successfully applied to many practical problemssuch as the analysis of cornea curvature in the human eye[18] the analysis of electronic commerce [19] the analysisof growth curve [20] the analysis of income density [21]the analysis of implied volatility surface in finance [22] theanalysis of longitudinal primary biliary liver cirrhosis [23]and the analysis of spectroscopy data [24] FurthermoreHyndman and Shahid Ullah [25] proposed a smoothed androbust FPCAandused it to forecast age-specificmortality andfertility rates

The objective of this paper is to study the monthlyvolatility of return of Shanghai 50 index which consists of50 stocks Treating stock price series as random functionin a space spanned by finite dimensional functional baseswe intensively explore methods of functional data analysisespecially functional principal component analysis

In the area of finance some impressive papers withthe functional data analysis are found such as Ramsay andRamsey [26] Muller and Ulrich [27] andMiao [28] But fewrepublications are found with research on the increasinglyflourishing Chinese financial market This paper will fill theblank both in theory and in application

Our study can be described as an exploratory dataapproach

Data collection 997888rarr Data Analysis 997888rarr ConclusionsThis paper is organized as follows In Section 2 we describethe functional principal component analysis (FPCA) which

plays a significant role in the development of functionaldata analysis It is also an essential ingredient of functionalprincipal component regression (FPCR) Section 3 will illus-trate the empirical study with the application of the theoryin Section 2 Some further discussion and a conclusion arepresented in Section 4

2 Methodology

As mentioned before an important tool in the functionaldata analysis toolbox is FPCA that is functional principalcomponent analysis The main idea of FPCA is just likemultivariate principal component analysis (PCA) but itsprincipal component weights or harmonics are functions oftime They carry the main features of the functional dataobject and can be interpreted separately

The differences in notation between PCA and FPCA aresummarized in Table 1

The basic assumption of FDA is that data generatingprocess can be described as a smooth function FPCAfinds the set of orthogonal principal component function bymaximizing the variance along each component

The first functional principal component 1206011(119905) is defined

by

1206011(119905) = arg max

Φ2= 1

1

119873

119873

sum

119894=1

(int119871

120601(119905)119891119894(119905) 119889119905)

2

(1)

subject to10038171003817100381710038171206011

1003817100381710038171003817 = 1 (2)

The 119896th functional principal component 120601119896(119905) can be found

analogously subject to the additional constraint

int119871

120601119895(119905) 120601119896(119905) 119889119905 = 0 forall119895 lt 119896 (3)

The sample covariance function of 119891(119909) = [1198911(119909) 1198912(119909)

119891119899(119909)] 119909 isin [119909

1 119909119901] is given by

119881 (119904 119905) =1

119873

119873

sum

119894=1

119891119894(119904) 119891119894(119905) (4)

where function 119891119894(119905) has usually been first centered



(119881120601119870) (119905) = 120582

119896120601119870(119905) 120582









for 119905 isin [1199091 119909119901] 119894 = 1 119899


119896 Consider

119891119894(119905) asymp

119870

sum

119896=1

120573119894119896120593119896 119894 = 1 119899 (6)






1198811206011(119905) = 120582

11206011(119905)

100381710038171003817100381712060111003817100381710038171003817 = 1 (7)


119881120601119896(119905) = 120582

119896120601119896(119905)

10038171003817100381710038171206011198961003817100381710038171003817 = 1 (8)

0 2 4 6 8 10 12minus60

minus40

minus20

0

20

40

60


subject to

int119871

120601119895(119905) 120601119896(119905) = 0 forall119895 lt 119896 (9)


3 Application











2 4 6 8 10 12minus50

minus40

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tions


2 4 6 8 10 12minus50

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minus10

0

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Arguments

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tions



2 4 6 8 10 12

0

50

Arguments

Func

tions

minus150

minus100

minus50


2 4 6 8 10 12

minus400

minus300

minus200

minus100

0

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0

5

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0

5

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minus50

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4

3

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1

0

minus1

minus2

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minus4

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or ce

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aria

bles


1 2 3 4 5 6 7 8 9 1110 12


3

2

1

0

minus1

minus2

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Arguments

Har

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or ce

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1 2 3 4 5 6 7 8 9 1110 12






3

2

1

0

minus1

minus2

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1 2 3 4 5 6 7 8 9 1110 12

Arguments

Har

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or ce

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3

2

1

0

minus1

minus2

minus3

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Arguments

Har

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or ce

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1 2 3 4 5 6 7 8 9 1110 12






3

2

1

0

minus1

minus2

minus3

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Arguments

Har

mon

ics f

or ce

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bles


1 2 3 4 5 6 7 8 9 1110 12







References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119881120601119870) (119905) = 120582

119896120601119870(119905) 120582









for 119905 isin [1199091 119909119901] 119894 = 1 119899


119896 Consider

119891119894(119905) asymp

119870

sum

119896=1

120573119894119896120593119896 119894 = 1 119899 (6)






1198811206011(119905) = 120582

11206011(119905)

100381710038171003817100381712060111003817100381710038171003817 = 1 (7)


119881120601119896(119905) = 120582

119896120601119896(119905)

10038171003817100381710038171206011198961003817100381710038171003817 = 1 (8)

0 2 4 6 8 10 12minus60

minus40

minus20

0

20

40

60


subject to

int119871

120601119895(119905) 120601119896(119905) = 0 forall119895 lt 119896 (9)


3 Application











2 4 6 8 10 12minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Arguments

Func

tions


2 4 6 8 10 12minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Arguments

Func

tions



2 4 6 8 10 12

0

50

Arguments

Func

tions

minus150

minus100

minus50


2 4 6 8 10 12

minus400

minus300

minus200

minus100

0

100

200

300

400

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Arguments

Func

tions


0

5

10

15

0

5

10

15

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minus50

0

50

100

150








4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

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Arguments

Har

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ics f

or ce

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aria

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1 2 3 4 5 6 7 8 9 1110 12


3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

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aria

bles


1 2 3 4 5 6 7 8 9 1110 12






3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

1 2 3 4 5 6 7 8 9 1110 12

Arguments

Har

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or ce

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3

2

1

0

minus1

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Arguments

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1 2 3 4 5 6 7 8 9 1110 12






3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12







References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 4 6 8 10 12minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Arguments

Func

tions


2 4 6 8 10 12minus50

minus40

minus30

minus20

minus10

0

10

20

30

40

50

Arguments

Func

tions



2 4 6 8 10 12

0

50

Arguments

Func

tions

minus150

minus100

minus50


2 4 6 8 10 12

minus400

minus300

minus200

minus100

0

100

200

300

400

500

Arguments

Func

tions


0

5

10

15

0

5

10

15

minus100

minus50

0

50

100

150








4

3

2

1

0

minus1

minus2

minus3

minus4

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Arguments

Har

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or ce

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1 2 3 4 5 6 7 8 9 1110 12


3

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1

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minus2

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Arguments

Har

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or ce

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1 2 3 4 5 6 7 8 9 1110 12






3

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1 2 3 4 5 6 7 8 9 1110 12

Arguments

Har

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or ce

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3

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Arguments

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1 2 3 4 5 6 7 8 9 1110 12






3

2

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minus5

minus6

Arguments

Har

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ics f

or ce

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ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12







References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4

3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

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1 2 3 4 5 6 7 8 9 1110 12


3

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1

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Arguments

Har

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or ce

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1 2 3 4 5 6 7 8 9 1110 12






3

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1 2 3 4 5 6 7 8 9 1110 12

Arguments

Har

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3

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Arguments

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1 2 3 4 5 6 7 8 9 1110 12






3

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minus2

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minus5

minus6

Arguments

Har

mon

ics f

or ce

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ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12







References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3

2

1

0

minus1

minus2

minus3

minus4

minus5

minus6

Arguments

Har

mon

ics f

or ce

nter

ed v

aria

bles


1 2 3 4 5 6 7 8 9 1110 12







References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article functional principal components...

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