Principal component scores: how do the dimensions of the $\mathbf{W}_{\bullet k}$ relate to its population analogue $W_k$?

Question

I'm currently studying principal component analysis, and I'd like some clarification with regards to the notation used for population and sample principal component scores/vectors. The population principal component scores/vectors are defined as follows:

For $\mathbf{X} \sim (\mathbf{\mu}, \Sigma)$ and $\Sigma = \Gamma \Lambda \Gamma^T$ with $r$ the rank of $\Sigma$, write $\mathbf{X}_{\text{cent}} = \mathbf{X} - \mathbf{\mu}$ for the centered vector.

The principal component score $W_k$ corresponding to the index $k \le r$ is the projection of $\mathbf{X}_{\text{cent}}$ in the direction of $\mathbf{\eta}_k$: $$W_k = \mathbf{\eta}^T_k \mathbf{X}_{\text{cent}} = \mathbf{\eta}^T_k ( \mathbf{X} - \mathbf{\mu})$$ And for $\kappa \le r$, the $\kappa$-dimensional principal component vector $\mathbf{W}^{(\kappa)}$ is $$\mathbf{W}^{(\kappa)} = \begin{bmatrix} W_1 \\ \vdots \\ W_\kappa \end{bmatrix} = \Gamma^T_\kappa \mathbf{X}_{\text{cent}} = [\mathbf{\eta}_1 \dots \mathbf{\eta}_\kappa]^T \mathbf{X}_\text{cent}$$ $\mathbf{W}^{(\kappa)} = [ W_1 \dots W_\kappa]^T$ is the $\kappa$-dimensional principal component vector.

The sample principal component scores/vectors are defined as follows:

For data $\mathbb{X} \sim \text{Sam}(\overline{\mathbf{X}}, S)$ with $S = \hat{\Gamma} \hat{\Lambda} \hat{\Gamma}^T$, the $k$th sample principal component score $\mathbf{W}_{\bullet k}$ is the row vector $$\mathbf{W}_{\bullet k} = \mathbf{\hat{\eta}}_k^T \mathbb{X}_{\text{cent}} = \mathbf{\hat{\eta}}_k^T (\mathbb{X} - \overline{\mathbf{X}})$$ And for $\kappa \le r$ the principal component data $\mathbb{W}^{(\kappa)}$ consist of the first $\kappa$ principal component vectors $\mathbf{W}_{\bullet k}$ with $k \le \kappa$: $$\mathbb{W}^{(\kappa)} = \begin{bmatrix} \mathbf{W}_{\bullet 1} \\ \vdots \\ \mathbf{W}_{\bullet k} \end{bmatrix} = \mathbf{\hat{\Gamma}}^T_\kappa \mathbb{X}_{\text{cent}}$$

The introduction of the $\mathbf{W}_{\bullet k}$ confused me. Whereas the $W_k$ have a single dimension (they seem to just be scalars), the $\mathbf{W}_{\bullet k}$ seem to have two dimensions (including the dot $\bullet$, which I assume means something about across all rows). How do the dimensions of the $\mathbf{W}_{\bullet k}$ relate to its population analogue $W_k$?

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EDIT:

With regards to Javier TG's answer, if $\mathbf{X}$ has dimensions $d \times 1$, $\mathbb{X}$ has dimensions $d \times n$, and $\mathbb{W}^{(\kappa)}$ has dimensions $k \times n$, then $\mathbf{W}_{\bullet k}$ would have dimensions $1 \times n$, no? So the dot $\bullet$ could mean that it is over all rows, and therefore $\bullet k$ means that it is over all rows but only column $k$. I mean, it even said that "... $\mathbb{W}^{(\kappa)}$ consist of the first $\kappa$ principal component vectors $\mathbf{W}_{\bullet k}$ ...", so $\mathbf{W}_{\bullet k}$ can't be a scalar, right?

Answer 1

The difference of the second half of the question w.r.t. the first half, is that now we are using estimations of the population parameters in order to get the principal component scores/ vectors $(\mu \approx \overline{\mathbf{X}}, \, \Sigma \approx S)$. By doing this, principal directions will also be estimated as the eigenvectors of $S$. This is denoted by using hat notation $\eta_k^T \to \hat{\eta}_k^T$.

On the other hand, $\mathbb{X}$ has $d\times n$ dimensions, as it's said in the question, where $n$ is the number of samples (each column is a sample). Given this, $\mathbf{W}_{\bullet k} = \hat{\eta}_k^T \mathbb{X}_{\text{cent}}$ will have $1\times n$ dimensions (where each column is the projection of each centered sample, given by $\mathbb{X}_{\text{cent}}=(\mathbb{X}-\bar{\mathbf{X}})$, in the estimated principal direction $\hat{\eta}_k$).

Given this, in relation to the notation of $\mathbf{W}_{\bullet k}$, we can conclude that $k$ is used to express which estimated principal direction is being used to project the samples contained in $\mathbb{X}_{\text{cent}}$. With respect to the bullet, $\bullet$, I agree. It seems to express that all the samples of $\mathbb{X}$ are being used.