# Korélasi kanonik

Luncat ka: pituduh, sungsi

Dina statistik, analisa canonical correlation, dimimitian ku Harold Hotelling, nyaeta turus keur nyieun matrik cross-covariance.

## Harti

Ditangtukeun dua kolom vektor $X = (x_1, \dots, x_n)'$ jeung $Y = (y_1, \dots, y_m)'$ variabel acak momen kadua, salah sahiji hartina cross-covarian $\Sigma _{12} = \operatorname{cov}(X, Y)$ nu bakal jadi matriks $n \times m$ numana asupan $(i, j)$ nyaeta kovarian $\operatorname{cov}(x_i, y_j)$.

Analisa korelasi kanonik nyobaan $a$ sarta $b$ saperti dina variabele acak $a' X$ jeung $b' Y$ ngamaksimalkeun korélasi $\rho = \operatorname{cor}(a' X, b' Y)$. Variabel acak $U = a' X$ jeung $V = b' Y$ ngarupakeun pasangan munggaran variabel kanonik. Saterusna vektor ngamaksimalkeun subyek korelasi nu sarua keur negeskeun yen hal ieu teu pakait jeung pasangan munggaran variabel kanonik; hasilna nyaeta pasangan kadua variabel kanonik. Ieu prosedur terus lumangsung salila $\min\{m,n\}$ kali.

## Komputasi

### Bukti

Anggap $\Sigma _{11} = \operatorname{cov}(X, X)$ jeung $\Sigma _{22} = \operatorname{cov}(Y, Y)$. Parameter nu dimaksimalkeun nyaeta

$\rho = \frac{a' \Sigma _{12} b}{\sqrt{a' \Sigma _{11} a} \sqrt{b' \Sigma _{22} b}}.$

Lengkah kahiji nyaeta ngahartikeun parobahan basis jeung hartina

$c = \Sigma _{11} ^{1/2} a,$
$d = \Sigma _{22} ^{1/2} b.$

$\rho = \frac{c' \Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1/2} d}{\sqrt{c' c} \sqrt{d' d}}.$

$c' \Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1/2} d \leq \left(c' \Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1/2} \Sigma _{22} ^{-1/2} \Sigma _{21} \Sigma _{11} ^{-1/2} c \right)^{1/2} \left(d' d \right)^{1/2},$
$\rho \leq \frac{\left(c' \Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1/2} \Sigma _{22} ^{-1/2} \Sigma _{21} \Sigma _{11} ^{-1/2} c \right)^{1/2}}{\left(c' c \right)^{1/2}}.$

Hal ieu sarua lamun vektor $d$ jeung $\Sigma _{22} ^{-1/2} \Sigma _{21} \Sigma _{11} ^{-1/2} c$ kolinier. Tambahanna, korelasi maksimum kahontal lamun $c$ nyaeta vektor eigen ku nilai maksimal vektor eigen keur matrik $\Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1} \Sigma _{21} \Sigma _{11} ^{-1/2}$ (tempo Rayleigh quotient). Pasangan saterusna bakal kapanggih ku cara nurunkeun gedena nilai eigen. Sacara ortogonal dijamin ku matrik korelasi nu simetri.

### Solusi

Solusina nyaeta:

• $c$ nyaeta vektoreigen $\Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1} \Sigma _{21} \Sigma _{11} ^{-1/2}$
• $d$ nyaeta sabanding jeung $\Sigma _{22} ^{-1/2} \Sigma _{21} \Sigma _{11} ^{-1/2} c$

Papasanganna nyaeta:

• $d$ nyaeta vektor eigen $\Sigma _{22} ^{-1/2} \Sigma _{21} \Sigma _{11} ^{-1} \Sigma _{12} \Sigma _{22} ^{-1/2}$
• $c$ nyaeta sabanding jeung $\Sigma _{11} ^{-1/2} \Sigma _{12} \Sigma _{22} ^{-1/2} d$

Variabel kanonik dihartikeun ku:

$U = c' \Sigma _{11} ^{-1/2} X = a' X$
$V = d' \Sigma _{22} ^{-1/2} Y = b' Y$

## Uji hipotesa

Unggal baris bisa diuji signifikan-na ku cara metoda nu bakal dijentrekeun. Lamun $p$ observasi mandiri dina sampel sarta $\widehat{\rho}_i$ nyaeta korelasi estimasi $i = 1,\dots, \min\{m,n\}$. Keur baris ka-$i$, uji statistik nyaeta:

$\chi ^2 = - \left( p - 1 - \frac{1}{2}(m + n + 1)\right) \ln \prod _ {j = i} ^p (1 - \widehat{\rho}_j^2),$

nu deukeut kana sebaran chi-square numana $(m - i + 1)(n - i + 1)$ tingkat kabebasan keur $p$ nu gede.[1]

## Pamakean praktis

Tipe husus keur koralsi kanonik dina psikologi nyaeta nyokot dua runtuyan variabel tur nempo sabaraha ilahar diantara dua uji. Contona, anjeun nyokot dua uji personal multidimensi nu geus aya saperti [MMPI]] jeung NEO. Ku nempo kumaha faktor MMPI pakait jeung faktor NEO, anjeun bakal meunang hal nu jentre kumaha dimensi nu ilahar antara dua uji sarta sakumaha beda varian nu dibagikeun. Contona, anjeun bisa manggihkeun yen versi leuwih atawa neuroticis diitung keur materi nu ngabagikeun varian antara dua uji.

## Rujukan jeung tumbu kaluar

1. Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis, Academic Press.