  Principal Geodesic Analysis (PGA) is applied to a climate time series. First,
we transform each multidimensional sequence into the path signature. Since the
signature lives in a curved space, usual principal component analysis (PCA) is
not applicable. Instead, we treat the signature space as a geodesic manifold.
By replacing the notion of straight lines with that of geodesics, the domain of
PCA can be extended to the curved space. Then, the first principal component is
derived as the geodesic that minimizes the unexplained variance in the data. As
an application, we computed the leading modes for one-year NINO SST time
series, which are divided into segments that represent annual variations of
monthly averages. It is interesting that some months in a year reveal
characteristic undulations that could indicate the early signs of upcoming El
Nin\~{o} events.
