Probabilistic PCA for morphological traits

Data

(source: Villéger et al, PLoS One, 2012, see also the mFD R package vignette)

\(n = 45\) fish species (from the Terminos Lagoon, Gulf of Mexico); \(p = 16\) morphological traits :

• logM : log(mass)
• Ogsf : Oral gape surface
• OgSh : Oral gape shape
• OgPo : Oral gape position
• GrLg : Gill raker length
• GtLg : Gut length
• EySz : Eye size
• EyPo : Eye position
• BdSh : Body transversal shape
• BdSf : Body transversal surface
• PfPo : Pectoral fin position
• PfSh : Aspect ratio of the pectoral fin
• CpHt : Caudal peduncle throttling
• CfSh : Aspect ratio of the caudal fin
• FsRt : Fins surface ratio
• FsSf : Fins surface to body size ratio

\[ Y = [Y_{ij}]: n \times p \text{ data matrix} \]

Scaled traits \[ \widetilde{Y}_{ij} = \frac{Y_{ij} - \overline{Y}_j}{\widehat{\sigma}_j}. \]

Principal component analysis

Singular value decomposition of \(\widetilde{Y}\) = eigenvalue decomposition of the empirical correlation matrix of \(Y\)

Probabilistic PCA

\[\begin{align*} \label{eq:factorModel} \{Z_i\}_i & \text{ iid:} & Z_i & \sim \mathcal{N}(0, I_q) \\ \{Y_i\}_i & \text{ indep.} \mid Z: & (Y_i \mid Z_i) & \sim \mathcal{N}(A^\intercal Z_i, \sigma^2 I_p) \end{align*}\]

## lambda =

##  [1] 5.20511221 2.37961176 2.02898869 1.65345227 1.46016972 0.77295469
##  [7] 0.68568306 0.52512096 0.35907309 0.30554429 0.26000287 0.11819477
## [13] 0.08601438 0.06937744 0.05119802 0.03950177

## lambda.hat =

##  [1] 5.20511221 2.37961176 2.02898869 1.65345227 1.46016972 0.77295469
##  [7] 0.68568306 0.52512096 0.35907309 0.30554429 0.26000287 0.07285728
## [13] 0.07285728 0.07285728 0.07285728 0.07285728

Estimated covariance matrix

\[ \widehat{A}^\intercal \widehat{A} + \widehat{\sigma}^2 I_p \]