Multivariate AnalysisMultivariate Analysis

IrlandiaIrlandia GinanjarGinanjar

JurusanJurusan StatistikaStatistikaJurusanJurusan StatistikaStatistika

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AnalisisAnalisis MultivariatMultivariat

AnalisisAnalisis multivariatmultivariat adalahadalah suatusuatu studistudi tentangtentangb bb b i b li b l dd d dd dbeberapabeberapa variabelvariabel randomrandom dependentdependent secarasecarasimultansimultan..

AnalisisAnalisis iniini merupakanmerupakan pengembanganpengembangan daridarianalisisanalisis univariatunivariat..

UntukUntuk mendapatkanmendapatkan hasilhasil analisisanalisis yangyang tepattepatakanakan diperlukandiperlukan asumsiasumsi--asumsiasumsi distribusionaldistribusional..pp

frameworkframework matematismatematis relatifrelatif lebihlebih complexcomplex jikajikadibandingkandibandingkan dengandengan analisisanalisis univariatunivariatdibandingkandibandingkan dengandengan analisisanalisis univariatunivariat..

AnalisisAnalisis iniini telahtelah digunakandigunakan seringsering digunakandigunakan dididuniadunia nyatanyataduniadunia nyatanyata..

BeberapaBeberapa DistribusiDistribusi MultivariatMultivariatBeberapaBeberapa DistribusiDistribusi MultivariatMultivariat

DistribusiDistribusi NormalNormal MultivariatMultivariat

PengembanganPengembangan daridari DistribusiDistribusi NormalNormal

Di ib iDi ib i Wi hWi hDistribusiDistribusi WishartWishart

PengembanganPengembangan daridari DistribusiDistribusi ChiChi––SquareSquareg gg g qq

StatistikStatistik dandan distribusidistribusi Hotelling’sHotelling’s TT22

bb d id i i iki ik dd di ib idi ib iPenembanganPenembangan daridari statistikstatistik dandan distribusidistribusi

Student’sStudent’s––tt kuadratkuadrat

StatistikStatistik Willk’sWillk’s LambdaLambda

PengembanganPengembangan daridari rasiorasio daridari duadua statistikstatistik ChiChiPengembanganPengembangan daridari rasiorasio daridari duadua statistikstatistik ChiChi––

SquareSquare

BeberapaBeberapa UkuranUkuran MultivariatMultivariatBeberapaBeberapa UkuranUkuran MultivariatMultivariatVektorVektor RataRata--ratarata

KoleksiKoleksi ratarata--ratarata daridari variabelvariabel--variabelvariabel yangyangdikajidikaji

MatriksMatriks KovariansKovariansKoleksiKoleksi VariansVarians dandan KovariansKovarians daridari variabelvariabel--KoleksiKoleksi VariansVarians dandan KovariansKovarians daridari variabelvariabelvariabelvariabel yangyang dikajidikaji

MatriksMatriks KorelasiKorelasiMatriksMatriks KorelasiKorelasiKoleksiKoleksi koefisienkoefisien korelasikorelasi daridari variabelvariabel--variabelvariabelyangyang dikajidikajiyangyang dikajidikaji

TheThe GeneralizedGeneralized VarianceVarianceii d id i ikik iiDeterminanDeterminan daridari MatriksMatriks VariansVarians

BeberapaBeberapa TesTes SignifikansiSignifikansiM l i iM l i iMultivariatMultivariat

TesTes SignifikansiSignifikansi vektorvektor ratarata--ratarata tunggaltunggal

TesTes kesamaankesamaan daridari duadua vektorvektor ratarata--ratarataTesTes kesamaankesamaan daridari duadua vektorvektor ratarata ratarata

TesTes kesamaankesamaan daridari beberapabeberapa vektorvektor ratarata--ratarata

TesTes SignifikansiSignifikansi matriksmatriks kovarianskovarians tunggaltunggal

TesTes kesamaankesamaan daridari duadua matriksmatriks kovarianskovariansTesTes kesamaankesamaan daridari duadua matriksmatriks kovarianskovarians

TesTes kesamaankesamaan daridari beberapabeberapa matriksmatriks kovarianskovarians

TesTes independensiindependensi daridari beberapabeberapa setset variatvariat

TT i d d ii d d i d id i i ti tTesTes independensiindependensi daridari variatvariat

BeberapaBeberapa TeknikTeknik MultivariatMultivariatBeberapaBeberapa TeknikTeknik MultivariatMultivariat

TheThe Hotelling’sHotelling’s –– TT22 StatisticStatisticgg

TheThe MultivariateMultivariate AnalysisAnalysis ofof VarianceVariance andandCovarianceCovarianceCovarianceCovariance

TheThe MultivariateMultivariate ExperimentalExperimental DesignsDesigns

ThTh M lti i tM lti i t P filP fil A l iA l iTheThe MultivariateMultivariate ProfileProfile AnalysisAnalysis

TheThe MultivariateMultivariate RegressionRegression AnalysisAnalysis

TheThe GeneralizedGeneralized MultivariateMultivariate AnalysisAnalysis ofof VarianceVariance

TheThe PrincipalPrincipal ComponentComponent AnalysisAnalysisTheThe PrincipalPrincipal ComponentComponent AnalysisAnalysis

TheThe FactorFactor AnalysisAnalysis

BeberapaBeberapa teknikteknik Multivariate Multivariate h ih iTechniquesTechniques

TheThe CanonicalCanonical CorrelationCorrelation AnalysisAnalysisyy

TheThe DiscriminatoryDiscriminatory AnalysisAnalysis

TheThe ClusterCluster AnalysisAnalysisTheThe ClusterCluster AnalysisAnalysis

TheThe MultidimensionalMultidimensional ScalingScaling

TheThe CorrespondenceCorrespondence AnalysisAnalysis

TheThe ClassificationClassification TreesTrees

TheThe PathPath AnalysisAnalysis

TheThe StructuralStructural EquationsEquations ModelsModelsTheThe StructuralStructural EquationsEquations ModelsModels

TheThe SeeminglySeemingly UnrelatedUnrelated RegressionRegression ModelsModels

Jenis Data DasarJenis Data Dasar

TerdapatTerdapat duadua jenisjenis data data dasardasar ::

N M i (Q li i )N M i (Q li i )1.1. Non Metric (Qualitative)Non Metric (Qualitative)–– DataData nonnon metrikmetrik bisabisa berupaberupa aatributtribut,, atauatau sifatsifat kategorikkategorik yangyang

menunjukkanmenunjukkan atauatau menggambarkanmenggambarkan suatusuatu oobyekbyek..jj gggg yy–– VariabelVariabel yangyang diukurdiukur menggunakanmenggunakan skalaskala nominalnominal dandan ordinalordinal

umumnyaumumnya merupakanmerupakan variabelvariabel nonnon metricmetric

22 MetricMetric (Quantitative)(Quantitative)22.. MetricMetric (Quantitative)(Quantitative)–– PengukuranPengukuran dilakukandilakukan sehinggasehingga suatusuatu oobyekbyek dapatdapat diketahuidiketahui

perbedaannyaperbedaannya dalamdalam jumlahjumlah atauatau derajatderajat..–– VariabelVariabel yangyang diukurdiukur mengunakanmengunakan skalaskala IntervalInterval dandan RatioRatio

merupakanmerupakan variabelvariabel metricmetric

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Klasifikasi metode data analitisKlasifikasi metode data analitis

l ifik il ifik i dd dd li ili i dd dib idib iKlasifikasiKlasifikasi metodemetode datadata analitisanalitis dapatdapat dibagidibagimenjadimenjadi ::

1.1. DependenceDependence MethodMethodDapatDapat didefinisikandidefinisikan sebagaisebagai suatusuatu metodemetode didi manamana suatusuatu•• DapatDapat didefinisikandidefinisikan sebagaisebagai suatusuatu metodemetode didi manamana suatusuatuvariabelvariabel atauatau kumpulankumpulan variabelvariabel yangyang diketahuidiketahui sebagaisebagaivariabelvariabel dependendependen diprediksidiprediksi atauatau dijelaskandijelaskan oleholeh variabelvariabel--

i b li b l l il i di bdi b b ib i i b li b l i d di d dvariabelvariabel yangyang lainlain yangyang disebutdisebut sebagaisebagai variabelvariabel independenindependen..

2.2. InterdependenceInterdependence MethodMethodAd l hAd l h s ts t t dt d didi tid ktid k dd s ts t tt s k l ks k l k•• AdalahAdalah suatusuatu metodemetode dimanadimana tidaktidak adaada satusatu atauatau sekelompoksekelompokvariabelvariabel yangyang didefinisikandidefinisikan sebagaisebagai independenindependen ataupunataupunvariabelvariabel dependendependen ..

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Classification of Multivariate TechniquesClassification of Multivariate TechniquesClassification of Multivariate TechniquesClassification of Multivariate Techniques((Dependence Method)Dependence Method)

Dependent Variable (s)Dependent Variable (s)One More than One

Independent Metric Non Metric Metric Non MetrikVariale(s)

OneMetric • Simple • Discriminan analysis • Canonical • Multiple group

Non Metric

Regression

• t-test

• Logistic regression

• Discrete Discriminan Analysis

Correlation

• Manova

discriminananalysis (MDA)

Analysis

More than One

Metric • Multiple • Discriminan analysis • Canonical • Multiple groupMetric

Non Metric

Multipleregression

• Anova

Discriminan analysis• Logistic regression

• Discrete Discriminan A l i

Canonical Correlation

• Manova

Multiple group discriminan analysis (MDA)

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Analysis• Conjoint Analysis

Klasifikasi of Multivariate TechniquesKlasifikasi of Multivariate Techniques((Interdependence Method)Interdependence Method)((Interdependence Method)Interdependence Method)

Number of variableType of Data

Metric Non Metric

Two • Simple Corelation • Two way contingencyTwo Simple Corelation table

M lti

More than two• Principal Componen

Analysis

• Multiway Contingency table

• Loglinear model• Factor Analysis • Corespondence

Analysis

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S U f l M t iS U f l M t iSome Useful MatricesSome Useful Matrices

Diagonal Matrices

Identity MatricesIdentity Matrices

Symmetric Matrices

Idempotent Matrices

Orthogonal MatricesOrthogonal Matrices

Diagonal MatricesDiagonal Matrices

A diagonal matrix is a square matrix that has values on the diagonal with all goff-diagonal entities being zero.

⎡ ⎤⎢ ⎥⎢ ⎥

11

22

a 0 0 00 a 0 0⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦

22

33

44

0 a 0 00 0 a 00 0 0 a

Identity MatricesIdentity Matrices

A di l i hA diagonal matrix where • the diagonal elements all equal 1.g q• All other elements equal 0.

IA = AI = AIA = AI = A

⎡ ⎤⎢ ⎥= ⎢ ⎥

1 0 0 00 1 0 0I = ⎢ ⎥⎢ ⎥⎣ ⎦0 0 1 00 0 0 1

I

Idempotent MatricesIdempotent Matrices

Any matrix A such that A2 = A is said to be of idempotentbe of idempotent.

Orthogonal MatricesOrthogonal MatricesAny square matrix A with rows that areAny square matrix A with rows that are mutually perpendicular and have unit lengths is said to be orthogonal, i.e.,

A’A I A 1 A’A’A = I or A-1 = A’.

Eigenvalues and EigenvectorsEigenvalues and EigenvectorsFor a square matrix A the scalars λFor a square matrix A, the scalars, λ, satisfying the polynomial equation |A λI| 0 ll d th i l f|A - λI| = 0 are called the eigenvalues of A.

A square matrix A is said to have eigenvectors e such that for every λ theeigenvectors e such that for every λ the equation Ae = λe.

Usually e is normailized, e’e = 1.

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