\[ \begin{equation} \begin{bmatrix}y \\ x \end{bmatrix} \sim \mathcal{N}\left( \begin{bmatrix} \mu_y \\ \mu_x \end{bmatrix}, \underset{(m + p)\times(m + p)}{\begin{bmatrix} \Sigma_{yy} & \Sigma_{yx}\\ \Sigma_{xy} & \Sigma_{xx} \end{bmatrix}} \right) \end{equation} \] There are \(\frac{1}{2}(p + m)(p + m + 1)\) unknowns to identify this model.

equivalently,

\[ \begin{equation} \mathbf{y} = \beta_0 + \boldsymbol{\beta}^t\mathbf{x} + \boldsymbol{\varepsilon} \end{equation} \]

also, we can express, \(\boldsymbol{\beta} = \Sigma_{xx}^{-1}\Sigma_{xy}\)

\[\begin{bmatrix}w \\ z \end{bmatrix} \sim \mathcal{N}\left( \begin{bmatrix} \mu_w \\ \mu_z \end{bmatrix}, \underset{(m + p)\times(m + p)}{\begin{bmatrix} \Sigma_{ww} & \Sigma_{wz}\\ \Sigma_{zw} & \Sigma_{zz} \end{bmatrix}} \right)\]

\(\frac{1}{2}(p + m)(p + m + 1)\) unknowns can be reduced and parameterized.

A subspace in predictor space spanned by subset of predictor components is relevant for the response. Also, only a subspace of response space spanned by subset of response components is informative.

Education

• Teaching Statistics
• Creating Examples

design <- crossing(
  gamma = c(0.1, 1.2),
  relpos = c("1:5", "5:9")) %>%
  mutate(relpos = map(
    relpos, ~eval(parse(text = .x))),
    sim_obj = map2(gamma, relpos,
      ~simrel(
        n      = 500,
        p      = 10, m = 3,
        q      = 10,
        relpos = list(.y),
        ypos   = list(1:3),
        gamma  = .x,
        eta    = 0, R2 = 0.8,
        type   = "multivariate"
      )
    ))

# A tibble: 4 x 3
  gamma relpos sim_obj     
  <dbl> <chr>  <list>      
1   0.1 1:5    <S3: simrel>
2   0.1 5:9    <S3: simrel>
3   1.2 1:5    <S3: simrel>
4   1.2 5:9    <S3: simrel>

Estimator under comparison

• Principal Component Regression (PCR)
• Partial Least Squares (PLS)