  Important but rare and subtle processes driving galaxy morphology and
star-formation may be missed by traditional spiral, elliptical, irregular or
S\'ersic bulge/disk classifications. To overcome this limitation, we use a
principal component analysis of non-parametric morphological indicators
(concentration, asymmetry, Gini coefficient, $M_{20}$, multi-mode, intensity
and deviation) measured at rest-frame $B$-band (corresponding to HST/WFC3 F125W
at 1.4 $< z <$ 2) to trace the natural distribution of massive ($>10^{10}
M_{\odot}$) galaxy morphologies. Principal component analysis (PCA) quantifies
the correlations between these morphological indicators and determines the
relative importance of each. The first three principal components (PCs) capture
$\sim$75 per cent of the variance inherent to our sample. We interpret the
first principal component (PC) as bulge strength, the second PC as dominated by
concentration and the third PC as dominated by asymmetry. Both PC1 and PC2
correlate with the visual appearance of a central bulge and predict galaxy
quiescence. PC1 is a better predictor of quenching than stellar mass, as as
good as other structural indicators (S\'ersic-n or compactness). We divide the
PCA results into groups using an agglomerative hierarchical clustering method.
Unlike S\'ersic, this classification scheme separates compact galaxies from
larger, smooth proto-elliptical systems, and star-forming disk-dominated clumpy
galaxies from star-forming bulge-dominated asymmetric galaxies. Distinguishing
between these galaxy structural types in a quantitative manner is an important
step towards understanding the connections between morphology, galaxy assembly
and star-formation.
