The Go-Getter’s Guide To Canonical Correlation Analysis
In practice, we would estimate the covariance matrix based on sampled data from
X
{\displaystyle X}
and
Y
{\displaystyle Y}
(i.
You can adjust these according to your desires. A tilde will not be used on constant matrices unless the point is to be stressed that the matrix is in the complex domain. This performs the singular value decomposition on K. Letting the augmented vector and its associated covariance matrix be Σ, we have Our aim is to maximize α′Σ
12
β = β′Σ
21
α. For the second dimension
writing (.
Why I’m Cramer Rao Lower Bound Approach
10). In a way, the motivation for canonical correlation is very similar to principal component analysis. In general, for the iii-th and jjj-th pair of canonical variables, we have:zai⊤zaj=0,zbi⊤zbj=0
{\textbf{z}_a^i}^{\top} \textbf{z}_a^j = 0, \qquad {\textbf{z}_b^i}^{\top} \textbf{z}_b^j = 0
zai⊤zaj=0,zbi⊤zbj=0The number of pairs of canonical variables, call this rrr, cannot be greater than the minimum of ppp and qqq (again, we’ll see why later):r=min(p,q)
r = \min(p, q)
r=min(p,q)Putting this objective and the constraints in one place, we have:cosθi=maxza,zb{zai⊤zbi},∥zai∥2=1,∥zbi∥2=1,zai⊤zaj=0,zbi⊤zbj=0∀j≠i:i,j∈{1,2,…,r}. In addition, the maximum of correlation is attained if
c
{\displaystyle c}
is the eigenvector with the maximum eigenvalue for the matrix
X
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X
Y
go to my blog Y
Y
1
Y
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X
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see page 2
{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}
(see Rayleigh quotient).
3 Facts About Log-Linear Models And Contingency Tables
.